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class="vector-toc-numb">2</span> <span>Definitions</span> </div> </a> <ul id="toc-Definitions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rolling_resistance_coefficient" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rolling_resistance_coefficient"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Rolling resistance coefficient</span> </div> </a> <ul id="toc-Rolling_resistance_coefficient-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Measurement" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Measurement"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Measurement</span> </div> </a> <ul id="toc-Measurement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Physical_formulae" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Physical_formulae"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Physical formulae</span> </div> </a> <ul id="toc-Physical_formulae-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rolling_resistance_coefficient_examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rolling_resistance_coefficient_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Rolling resistance coefficient examples</span> </div> </a> <ul id="toc-Rolling_resistance_coefficient_examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dependence_on_diameter" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dependence_on_diameter"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Dependence on diameter</span> </div> </a> <button aria-controls="toc-Dependence_on_diameter-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Dependence on diameter subsection</span> </button> <ul id="toc-Dependence_on_diameter-sublist" class="vector-toc-list"> <li id="toc-Stagecoaches_and_railroads" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stagecoaches_and_railroads"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Stagecoaches and railroads</span> </div> </a> <ul id="toc-Stagecoaches_and_railroads-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pneumatic_tires" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pneumatic_tires"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Pneumatic tires</span> </div> </a> <ul id="toc-Pneumatic_tires-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dependence_on_applied_torque" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dependence_on_applied_torque"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Dependence on applied torque</span> </div> </a> <button aria-controls="toc-Dependence_on_applied_torque-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Dependence on applied torque subsection</span> </button> <ul id="toc-Dependence_on_applied_torque-sublist" class="vector-toc-list"> <li id="toc-All_wheels" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#All_wheels"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>All wheels</span> </div> </a> <ul id="toc-All_wheels-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Railroad_steel_wheels" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Railroad_steel_wheels"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Railroad steel wheels</span> </div> </a> <ul id="toc-Railroad_steel_wheels-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pneumatic_tires_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pneumatic_tires_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Pneumatic tires</span> </div> </a> <ul id="toc-Pneumatic_tires_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dependence_on_wheel_load" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dependence_on_wheel_load"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Dependence on wheel load</span> </div> </a> <button aria-controls="toc-Dependence_on_wheel_load-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Dependence on wheel load subsection</span> </button> <ul id="toc-Dependence_on_wheel_load-sublist" class="vector-toc-list"> <li id="toc-Railroad_steel_wheels_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Railroad_steel_wheels_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Railroad steel wheels</span> </div> </a> <ul id="toc-Railroad_steel_wheels_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pneumatic_tires_3" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pneumatic_tires_3"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Pneumatic tires</span> </div> </a> <ul id="toc-Pneumatic_tires_3-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dependence_on_curvature_of_roadway" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Dependence_on_curvature_of_roadway"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Dependence on curvature of roadway</span> </div> </a> <button aria-controls="toc-Dependence_on_curvature_of_roadway-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Dependence on curvature of roadway subsection</span> </button> <ul id="toc-Dependence_on_curvature_of_roadway-sublist" class="vector-toc-list"> <li id="toc-General" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>General</span> </div> </a> <ul id="toc-General-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Sound" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sound"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Sound</span> </div> </a> <ul id="toc-Sound-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Factors_that_contribute_in_tires" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Factors_that_contribute_in_tires"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Factors that contribute in tires</span> </div> </a> <ul id="toc-Factors_that_contribute_in_tires-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Railroads:_Components_of_rolling_resistance" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Railroads:_Components_of_rolling_resistance"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Railroads: Components of rolling resistance</span> </div> </a> <ul id="toc-Railroads:_Components_of_rolling_resistance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Comparing_rolling_resistance_of_highway_vehicles_and_trains" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Comparing_rolling_resistance_of_highway_vehicles_and_trains"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Comparing rolling resistance of highway vehicles and trains</span> </div> </a> <ul id="toc-Comparing_rolling_resistance_of_highway_vehicles_and_trains-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">16</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">17</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" 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class="mw-page-title-main">Rolling resistance</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B5%D0%BD%D0%B5_%D0%BF%D1%80%D0%B8_%D1%82%D1%8A%D1%80%D0%BA%D0%B0%D0%BB%D1%8F%D0%BD%D0%B5" title="Триене при търкаляне – Bulgarian" lang="bg" hreflang="bg" data-title="Триене при търкаляне" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D1%83%D1%81%D1%83_%D1%81%C4%95%D1%80%D1%82%C4%95%D0%BD%C4%95%D0%B2%C4%95" title="Кусу сĕртĕнĕвĕ – Chuvash" lang="cv" hreflang="cv" data-title="Кусу сĕртĕнĕвĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Valiv%C3%BD_odpor" title="Valivý odpor – Czech" lang="cs" hreflang="cs" data-title="Valivý odpor" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Rollwiderstand" title="Rollwiderstand – German" lang="de" hreflang="de" data-title="Rollwiderstand" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Veeretakistus" title="Veeretakistus – Estonian" lang="et" hreflang="et" data-title="Veeretakistus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Resistencia_a_la_rodadura" title="Resistencia a la rodadura – Spanish" lang="es" hreflang="es" data-title="Resistencia a la rodadura" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Rulrezisto" title="Rulrezisto – Esperanto" lang="eo" hreflang="eo" data-title="Rulrezisto" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%82%D8%A7%D9%88%D9%85%D8%AA_%D8%BA%D9%84%D8%AA%D8%B4%DB%8C" title="مقاومت غلتشی – Persian" lang="fa" hreflang="fa" data-title="مقاومت غلتشی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/R%C3%A9sistance_au_roulement" title="Résistance au roulement – French" lang="fr" hreflang="fr" data-title="Résistance au roulement" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A7%88%EC%B0%B0%EB%A0%A5#구름_마찰력" title="마찰력 – Korean" lang="ko" hreflang="ko" data-title="마찰력" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A5%87%E0%A4%B2%E0%A5%8D%E0%A4%B2%E0%A4%A8_%E0%A4%98%E0%A4%B0%E0%A5%8D%E0%A4%B7%E0%A4%A3" title="बेल्लन घर्षण – Hindi" lang="hi" hreflang="hi" data-title="बेल्लन घर्षण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Attrito#Attrito_volvente" title="Attrito – Italian" lang="it" hreflang="it" data-title="Attrito" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/G%C3%B6rd%C3%BCl%C5%91_ellen%C3%A1ll%C3%A1s" title="Gördülő ellenállás – Hungarian" lang="hu" hreflang="hu" data-title="Gördülő ellenállás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Rolweerstand" title="Rolweerstand – Dutch" lang="nl" hreflang="nl" data-title="Rolweerstand" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%BB%A2%E3%81%8C%E3%82%8A%E6%8A%B5%E6%8A%97" title="転がり抵抗 – Japanese" lang="ja" hreflang="ja" data-title="転がり抵抗" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Tarcie_toczne" title="Tarcie toczne – Polish" lang="pl" hreflang="pl" data-title="Tarcie toczne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D1%80%D0%B5%D0%BD%D0%B8%D0%B5_%D0%BA%D0%B0%D1%87%D0%B5%D0%BD%D0%B8%D1%8F" title="Трение качения – Russian" lang="ru" hreflang="ru" data-title="Трение качения" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Vierint%C3%A4vastus" title="Vierintävastus – Finnish" lang="fi" hreflang="fi" data-title="Vierintävastus" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Rullmotst%C3%A5nd" title="Rullmotstånd – Swedish" lang="sv" hreflang="sv" data-title="Rullmotstånd" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%89%E0%AE%B0%E0%AF%81%E0%AE%B3%E0%AF%8D_%E0%AE%8E%E0%AE%A4%E0%AE%BF%E0%AE%B0%E0%AF%8D%E0%AE%A4%E0%AE%BF%E0%AE%B1%E0%AE%A9%E0%AF%8D" title="உருள் எதிர்திறன் – Tamil" lang="ta" hreflang="ta" data-title="உருள் எதிர்திறன்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D1%80%D1%82%D1%8F_%D0%BA%D0%BE%D1%87%D0%B5%D0%BD%D0%BD%D1%8F" title="Тертя кочення – Ukrainian" lang="uk" hreflang="uk" data-title="Тертя кочення" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ma_s%C3%A1t_l%C4%83n" title="Ma sát lăn – Vietnamese" lang="vi" hreflang="vi" data-title="Ma sát lăn" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Force resisting the motion when a body rolls on a surface</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Rolling_Resistance_2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Rolling_Resistance_2.png/220px-Rolling_Resistance_2.png" decoding="async" width="220" height="294" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/1e/Rolling_Resistance_2.png 1.5x" data-file-width="246" data-file-height="329" /></a><figcaption>Figure 1  Hard wheel rolling on and deforming a soft surface, resulting in the <a href="/wiki/Reaction_(physics)" title="Reaction (physics)">reaction force</a> <i><b>R</b></i> from the surface having a component that opposes the motion. (<i><b>W</b></i> is some vertical load on the axle, <i><b>F</b></i> is some towing force applied to the axle, <i><b>r</b></i> is the wheel radius, and both friction with the ground and friction at the axle are assumed to be negligible and so are not shown. The wheel is rolling to the left at constant speed.) Note that <i><b>R</b></i> is the resultant force from non-uniform pressure at the wheel-roadbed contact surface. This pressure is greater towards the front of the wheel due to hysteresis.</figcaption></figure> <p><b>Rolling resistance</b>, sometimes called <b>rolling friction</b> or <b>rolling drag</b>, is the force resisting the <a href="/wiki/Motion_(physics)" class="mw-redirect" title="Motion (physics)">motion</a> when a body (such as a <a href="/wiki/Ball" title="Ball">ball</a>, <a href="/wiki/Tire" title="Tire">tire</a>, or <a href="/wiki/Wheel" title="Wheel">wheel</a>) <a href="/wiki/Rolling" title="Rolling">rolls</a> on a surface. It is mainly caused by <a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">non-elastic</a> effects; that is, not all the energy needed for deformation (or movement) of the wheel, roadbed, etc., is recovered when the pressure is removed. Two forms of this are <a href="/wiki/Hysteresis" title="Hysteresis">hysteresis</a> losses (see <a class="mw-selflink-fragment" href="#Primary_cause">below</a>), and permanent <a href="/wiki/Plastic_deformation" class="mw-redirect" title="Plastic deformation">(plastic) deformation</a> of the object or the surface (e.g. soil). Note that the <a href="/wiki/Frictional_contact_mechanics" title="Frictional contact mechanics">slippage</a> between the wheel and the surface also results in energy dissipation. Although some researchers have included this term in rolling resistance, some suggest that this dissipation term should be treated separately from rolling resistance because it is due to the applied torque to the wheel and the resultant slip between the wheel and ground, which is called <b>slip loss</b> or <b>slip resistance.</b><sup id="cite_ref-SAE_MOBILUS_1-0" class="reference"><a href="#cite_note-SAE_MOBILUS-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In addition, only the so-called slip resistance involves <a href="/wiki/Friction" title="Friction">friction</a>, therefore the name "rolling friction" is to an extent a misnomer. </p><p>Analogous with <a href="/wiki/Sliding_friction" class="mw-redirect" title="Sliding friction">sliding friction</a>, rolling resistance is often expressed as a coefficient times the normal force. This coefficient of rolling resistance is generally much smaller than the coefficient of sliding friction.<sup id="cite_ref-Peck_2-0" class="reference"><a href="#cite_note-Peck-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Any coasting wheeled <a href="/wiki/Vehicle" title="Vehicle">vehicle</a> will gradually slow down due to rolling resistance including that of the bearings, but a <a href="/wiki/Train" title="Train">train</a> car with steel wheels running on steel <a href="/wiki/Rail_tracks" class="mw-redirect" title="Rail tracks">rails</a> will roll farther than a <a href="/wiki/Bus" title="Bus">bus</a> of the same mass with rubber tires running on <a href="/wiki/Asphalt_concrete" title="Asphalt concrete">tarmac/asphalt</a>. Factors that contribute to rolling resistance are the (amount of) deformation of the wheels, the deformation of the roadbed surface, and movement below the surface. Additional contributing factors include <a href="#Dependence_on_diameter">wheel diameter</a>,<sup id="cite_ref-Hibbeler_3-0" class="reference"><a href="#cite_note-Hibbeler-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="#Dependence_on_wheel_load">load on wheel</a>, surface adhesion, sliding, and relative micro-sliding between the surfaces of contact. The losses due to <a href="/wiki/Hysteresis" title="Hysteresis">hysteresis</a> also depend strongly on the material properties of the wheel or tire and the surface. For example, a <a href="/wiki/Tire" title="Tire">rubber tire</a> will have higher rolling resistance on a paved road than a <a href="/wiki/Steel" title="Steel">steel</a> <a href="/wiki/Wheelset_(rail_transport)" title="Wheelset (rail transport)">railroad wheel</a> on a steel rail. Also, <a href="/wiki/Sand" title="Sand">sand</a> on the ground will give more rolling resistance than <a href="/wiki/Concrete" title="Concrete">concrete</a>. Soil rolling resistance factor is not dependent on speed.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2025)">citation needed</span></a></i>]</sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Primary_cause">Primary cause</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=1" title="Edit section: Primary cause"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Pressure_distribution_for_viscoelastic_rolling_cylinders.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Pressure_distribution_for_viscoelastic_rolling_cylinders.png/220px-Pressure_distribution_for_viscoelastic_rolling_cylinders.png" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Pressure_distribution_for_viscoelastic_rolling_cylinders.png/330px-Pressure_distribution_for_viscoelastic_rolling_cylinders.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Pressure_distribution_for_viscoelastic_rolling_cylinders.png/440px-Pressure_distribution_for_viscoelastic_rolling_cylinders.png 2x" data-file-width="876" data-file-height="438" /></a><figcaption>Asymmetrical pressure distribution between rolling cylinders due to viscoelastic material behavior (rolling to the right).<sup id="cite_ref-CONTACT_4-0" class="reference"><a href="#cite_note-CONTACT-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </figcaption></figure> <p>The primary cause of pneumatic tire rolling resistance is <a href="/wiki/Hysteresis" title="Hysteresis">hysteresis</a>:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <blockquote><p>A characteristic of a deformable material such that the energy of deformation is greater than the energy of recovery. The rubber compound in a tire exhibits hysteresis. As the tire rotates under the weight of the vehicle, it experiences repeated cycles of deformation and recovery, and it dissipates the hysteresis energy loss as heat. Hysteresis is the main cause of energy loss associated with rolling resistance and is attributed to the <a href="/wiki/Viscoelasticity" title="Viscoelasticity">viscoelastic characteristics</a> of the rubber. </p><dl><dd>— National Academy of Sciences<sup id="cite_ref-NAS286_6-0" class="reference"><a href="#cite_note-NAS286-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></dd></dl></blockquote> <p>This main principle is illustrated in the figure of the rolling cylinders. If two equal cylinders are pressed together then the contact surface is flat. In the absence of surface friction, contact stresses are normal (i.e. perpendicular) to the contact surface. Consider a particle that enters the contact area at the right side, travels through the contact patch and leaves at the left side. Initially its vertical deformation is increasing, which is resisted by the hysteresis effect. Therefore, an additional pressure is generated to avoid interpenetration of the two surfaces. Later its vertical deformation is decreasing. This is again resisted by the hysteresis effect. In this case this decreases the pressure that is needed to keep the two bodies separate. </p><p>The resulting pressure distribution is asymmetrical and is shifted to the right. The <a href="/wiki/Line_of_action" title="Line of action">line of action</a> of the (aggregate) <a href="/wiki/Force#Normal" title="Force">vertical force</a> no longer passes through the centers of the cylinders. This means that a <a href="/wiki/Moment_(physics)" title="Moment (physics)">moment</a> occurs that tends to retard the rolling motion. </p><p>Materials that have a large hysteresis effect, such as rubber, which bounce back slowly, exhibit more rolling resistance than materials with a small hysteresis effect that bounce back more quickly and more completely, such as steel or <a href="/wiki/Silica" class="mw-redirect" title="Silica">silica</a>. <a href="/wiki/Low_rolling_resistance_tires" class="mw-redirect" title="Low rolling resistance tires">Low rolling resistance tires</a> typically incorporate silica in place of carbon black in their tread compounds to reduce low-frequency hysteresis without compromising traction.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Note that railroads also have hysteresis in the roadbed structure.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=2" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the broad sense, specific "rolling resistance" (for vehicles) is the force per unit vehicle weight required to move the vehicle on level ground at a constant slow speed where aerodynamic drag (air resistance) is insignificant and also where there are no traction (motor) forces or brakes applied. In other words, the vehicle would be coasting if it were not for the force to maintain constant speed.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> This broad sense includes wheel bearing resistance, the energy dissipated by vibration and oscillation of both the roadbed and the vehicle, and sliding of the wheel on the roadbed surface (pavement or a rail). </p><p>But there is an even broader sense that would include energy wasted by wheel slippage due to the <a href="#Depends_on_applied_torque">torque applied from the engine</a>. This includes the increased power required due to the increased velocity of the wheels where the tangential velocity of the driving wheel(s) becomes greater than the vehicle speed due to slippage. Since <a href="/wiki/Power_(physics)#Mechanical_power" title="Power (physics)">power is equal to force times velocity</a> and the wheel velocity has increased, the power required has increased accordingly. </p><p>The pure "rolling resistance" for a train is that which happens due to deformation and possible minor sliding at the wheel-road contact.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> For a rubber tire, an analogous energy loss happens over the entire tire, but it is still called "rolling resistance". In the broad sense, "rolling resistance" includes wheel bearing resistance, energy loss by shaking both the roadbed (and the earth underneath) and the vehicle itself, and by sliding of the wheel, road/rail contact. Railroad textbooks seem to cover all these resistance forces but do not call their sum "rolling resistance" (broad sense) as is done in this article. They just sum up all the resistance forces (including aerodynamic drag) and call the sum basic train resistance (or the like).<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Since railroad rolling resistance in the broad sense may be a few times larger than just the pure rolling resistance<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> reported values may be in serious conflict since they may be based on different definitions of "rolling resistance". The train's engines must, of course, provide the energy to overcome this broad-sense rolling resistance. </p><p>For tires, rolling resistance is defined as the energy consumed by a tire per unit distance covered. It is also called rolling friction or rolling drag. It is one of the forces that act to oppose the motion of a driver. The main reason for this is that when the tires are in motion and touch the surface, the surface changes shape and causes deformation of the tire.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>For highway motor vehicles, there is some energy dissipated in shaking the roadway (and the earth beneath it), the shaking of the vehicle itself, and the sliding of the tires. But, other than the <a href="#Depends_on_applied_torque">additional power required due to torque</a> and wheel bearing friction, non-pure rolling resistance doesn't seem to have been investigated, possibly because the "pure" rolling resistance of a rubber tire is several times higher than the neglected resistances.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Rolling_resistance_coefficient">Rolling resistance coefficient</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=3" title="Edit section: Rolling resistance coefficient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The "rolling resistance coefficient" is defined by the following equation:<sup id="cite_ref-NAS286_6-1" class="reference"><a href="#cite_note-NAS286-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F=C_{rr}N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>F</mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F=C_{rr}N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40774ba595e9340365a0f3e0367af9a1fdfe872d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.861ex; height:2.509ex;" alt="{\displaystyle \ F=C_{rr}N}"></span> where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> is the rolling resistance force (shown as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> in figure 1),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{rr}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{rr}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.377ex; height:2.509ex;" alt="{\displaystyle C_{rr}}"></span> is the dimensionless <b>rolling resistance coefficient</b> or <b>coefficient of rolling friction</b> (<b>CRF</b>), and</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> is the <a href="/wiki/Normal_force" title="Normal force">normal force</a>, the force perpendicular to the surface on which the wheel is rolling.</li></ul> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{rr}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{rr}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.377ex; height:2.509ex;" alt="{\displaystyle C_{rr}}"></span> is the force needed to push (or tow) a wheeled vehicle forward (at constant speed on a level surface, or zero grade, with zero air resistance) per unit force of weight. It is assumed that all wheels are the same and bear identical weight. Thus: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ C_{rr}=0.01}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mn>0.01</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ C_{rr}=0.01}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/715f969199ffbdbf56c2a4858864537a03ca3a94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.19ex; height:2.509ex;" alt="{\displaystyle \ C_{rr}=0.01}"></span> means that it would only take 0.01 pounds to tow a vehicle weighing one pound. For a 1000-pound vehicle, it would take 1000 times more tow force, i.e. 10 pounds. One could say that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{rr}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{rr}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.377ex; height:2.509ex;" alt="{\displaystyle C_{rr}}"></span> is in lb(tow-force)/lb(vehicle weight). Since this lb/lb is force divided by force, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{rr}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{rr}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.377ex; height:2.509ex;" alt="{\displaystyle C_{rr}}"></span> is dimensionless. Multiply it by 100 and you get the percent (%) of the weight of the vehicle required to maintain slow steady speed. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{rr}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{rr}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.377ex; height:2.509ex;" alt="{\displaystyle C_{rr}}"></span> is often multiplied by 1000 to get the parts per thousand, which is the same as kilograms (kg force) per metric ton (tonne = 1000 kg ),<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> which is the same as pounds of resistance per 1000 pounds of load or Newtons/kilo-Newton, etc. For the US railroads, lb/ton has been traditionally used; this is just <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2000C_{rr}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2000</mn> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2000C_{rr}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eaa901806c06576187f00cadf2b7e3bd4cb775f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.027ex; height:2.509ex;" alt="{\displaystyle 2000C_{rr}}"></span>. Thus, they are all just measures of resistance per unit vehicle weight. While they are all "specific resistances", sometimes they are just called "resistance" although they are really a coefficient (ratio)or a multiple thereof. If using pounds or kilograms as force units, mass is equal to weight (in earth's gravity a kilogram a mass weighs a kilogram and exerts a kilogram of force) so one could claim that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{rr}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{rr}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.377ex; height:2.509ex;" alt="{\displaystyle C_{rr}}"></span> is also the force per unit mass in such units. The SI system would use N/tonne (N/T, N/t), which is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1000gC_{rr}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1000</mn> <mi>g</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1000gC_{rr}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d4d63ab69873f3ca7a3e7726be3a0fcbcea07fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.143ex; height:2.509ex;" alt="{\displaystyle 1000gC_{rr}}"></span> and is force per unit mass, where <i>g</i> is the acceleration of gravity in SI units (meters per second square).<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>The above shows resistance proportional to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{rr}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{rr}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.377ex; height:2.509ex;" alt="{\displaystyle C_{rr}}"></span> but does not explicitly show any variation with speed, <a href="#Dependence_on_wheel_load">loads</a>, <a href="#Dependence_on_applied_torque">torque</a>, surface roughness, <a href="#Dependence_on_diameter">diameter</a>, tire inflation/wear, etc., because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{rr}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{rr}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.377ex; height:2.509ex;" alt="{\displaystyle C_{rr}}"></span> itself varies with those factors. It might seem from the above definition of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{rr}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{rr}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.377ex; height:2.509ex;" alt="{\displaystyle C_{rr}}"></span> that the rolling resistance is directly proportional to vehicle weight but <a href="#Dependence_on_wheel_load">it is not</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Measurement">Measurement</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=4" title="Edit section: Measurement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are at least two popular models for calculating rolling resistance. </p> <ol><li>"Rolling resistance coefficient (RRC). The value of the rolling resistance force divided by the wheel load. The Society of Automotive Engineers (SAE) has developed test practices to measure the RRC of tires. These tests (<a href="/wiki/SAE_J1269" title="SAE J1269">SAE J1269</a> and <a href="/wiki/SAE_J2452" title="SAE J2452">SAE J2452</a>) are usually performed<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2008)">citation needed</span></a></i>]</sup> on new tires. When measured by using these standard test practices, most new passenger tires have reported RRCs ranging from 0.007 to 0.014."<sup id="cite_ref-NAS286_6-2" class="reference"><a href="#cite_note-NAS286-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> In the case of bicycle tires, values of 0.0025 to 0.005 are achieved.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> These coefficients are measured on rollers, with power meters on road surfaces, or with <a href="/w/index.php?title=Coast-down_test&action=edit&redlink=1" class="new" title="Coast-down test (page does not exist)">coast-down tests</a>. In the latter two cases, the effect of air resistance must be subtracted or the tests performed at very low speeds.</li> <li>The coefficient of rolling resistance <i>b</i>, which has the dimension of <a href="/wiki/Length" title="Length">length</a>, is approximately (due to the <a href="/wiki/Small-angle_approximation" title="Small-angle approximation">small-angle approximation</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(\theta )=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos(\theta )=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3089b053a8d412639ae7f4652582e4b08089da4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.272ex; height:2.843ex;" alt="{\displaystyle \cos(\theta )=1}"></span>) equal to the value of the rolling resistance force times the <a href="/wiki/Radius" title="Radius">radius</a> of the wheel divided by the wheel load.<sup id="cite_ref-Hibbeler_3-1" class="reference"><a href="#cite_note-Hibbeler-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li> <li><a href="/w/index.php?title=ISO_18164:2005&action=edit&redlink=1" class="new" title="ISO 18164:2005 (page does not exist)">ISO 18164:2005</a> is used to test rolling resistance in Europe.</li></ol> <p>The results of these tests can be hard for the general public to obtain as manufacturers prefer to publicize "comfort" and "performance". </p> <div class="mw-heading mw-heading2"><h2 id="Physical_formulae">Physical formulae</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=5" title="Edit section: Physical formulae"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The coefficient of rolling resistance for a slow rigid wheel on a perfectly elastic surface, not adjusted for velocity, can be calculated by <sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2018)">citation needed</span></a></i>]</sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{rr}={\sqrt {z/d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{rr}={\sqrt {z/d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57b3c71bdeb68ae2f1f02894c090b21ee0a10513" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.266ex; height:4.843ex;" alt="{\displaystyle C_{rr}={\sqrt {z/d}}}"></span> where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> is the sinkage depth</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> is the diameter of the rigid wheel</li></ul> <p>The empirical formula for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{rr}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{rr}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.377ex; height:2.509ex;" alt="{\displaystyle C_{rr}}"></span> for cast iron mine car wheels on steel rails is:<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{rr}=0.0048(18/D)^{\frac {1}{2}}(100/W)^{\frac {1}{4}}={\frac {0.0643988}{\sqrt[{4}]{WD^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mn>0.0048</mn> <mo stretchy="false">(</mo> <mn>18</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>D</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mo stretchy="false">(</mo> <mn>100</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>W</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>0.0643988</mn> <mroot> <mrow> <mi>W</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mroot> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{rr}=0.0048(18/D)^{\frac {1}{2}}(100/W)^{\frac {1}{4}}={\frac {0.0643988}{\sqrt[{4}]{WD^{2}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcdf0492d842147e528ef3ff457a3c7fb80b175d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:46.402ex; height:6.176ex;" alt="{\displaystyle C_{rr}=0.0048(18/D)^{\frac {1}{2}}(100/W)^{\frac {1}{4}}={\frac {0.0643988}{\sqrt[{4}]{WD^{2}}}}}"></span> where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> is the wheel diameter in inches</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> is the load on the wheel in pounds-force</li></ul> <p>As an alternative to using <a href="#Rolling_resistance_coefficient"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{rr}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{rr}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.377ex; height:2.509ex;" alt="{\displaystyle C_{rr}}"></span></a> one can use <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>, which is a different <b>rolling resistance coefficient</b> or <b>coefficient of rolling friction</b> with dimension of length. It is defined by the following formula:<sup id="cite_ref-Hibbeler_3-2" class="reference"><a href="#cite_note-Hibbeler-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F={\frac {Nb}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mi>b</mi> </mrow> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F={\frac {Nb}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f489c4b79d6e2d6b06536939ae05508303a93a55" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.737ex; height:5.343ex;" alt="{\displaystyle F={\frac {Nb}{r}}}"></span> where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> is the rolling resistance force (shown in figure 1),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is the wheel radius,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> is the <b>rolling resistance coefficient</b> or <b>coefficient of rolling friction</b> with dimension of length, and</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> is the normal force (equal to <i>W</i>, not <i>R</i>, as shown in figure 1).</li></ul> <p>The above equation, where resistance is inversely proportional to radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> seems to be based on the discredited "Coulomb's law" (Neither Coulomb's inverse square law nor Coulomb's law of friction)<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2020)">citation needed</span></a></i>]</sup>. See <a href="#Dependence_on_diameter">dependence on diameter</a>. Equating this equation with the force per the <a href="#Rolling_resistance_coefficient">rolling resistance coefficient</a>, and solving for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>, gives <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> = <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{rr}r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{rr}r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/532a925f07b8de83675b265c5d746f4c39b8442b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.426ex; height:2.509ex;" alt="{\displaystyle C_{rr}r}"></span>. Therefore, if a source gives rolling resistance coefficient (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{rr}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{rr}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.377ex; height:2.509ex;" alt="{\displaystyle C_{rr}}"></span>) as a dimensionless coefficient, it can be converted to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>, having units of length, by multiplying <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{rr}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{rr}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1ed040cc91de63640b8e8e6d0b003314126e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.377ex; height:2.509ex;" alt="{\displaystyle C_{rr}}"></span> by wheel radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Rolling_resistance_coefficient_examples">Rolling resistance coefficient examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=6" title="Edit section: Rolling resistance coefficient examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Table of rolling resistance coefficient examples: <a rel="nofollow" class="external autonumber" href="http://auto.howstuffworks.com/tire4.htm">[3]</a> </p> <table class="wikitable"> <tbody><tr align="center"> <th><i><b>C</b></i><sub><i>rr</i></sub></th> <th><i><b>b</b></i></th> <th><b>Description</b> </th></tr> <tr> <td>0.0003 <i>to</i> 0.0004<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></td> <td></td> <td><a href="#Definitions">"Pure rolling resistance"</a> Railroad steel wheel on steel rail </td></tr> <tr> <td>0.0010 <i>to</i> 0.0015<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </td> <td>0.1 mm<sup id="cite_ref-Hibbeler_3-3" class="reference"><a href="#cite_note-Hibbeler-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></td> <td>Hardened steel ball bearings on steel </td></tr> <tr> <td>0.0010 <i>to</i> 0.0025<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ReferenceA_23-0" class="reference"><a href="#cite_note-ReferenceA-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </td> <td></td> <td>Special <a href="/wiki/Michelin" title="Michelin">Michelin</a> <a href="/wiki/Solar_car" title="Solar car">solar car</a>/<a href="/wiki/Eco-marathon" class="mw-redirect" title="Eco-marathon">eco-marathon</a> tires </td></tr> <tr> <td>0.0010 <i>to</i> 0.0024<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup></td> <td>0.5 mm<sup id="cite_ref-Hibbeler_3-4" class="reference"><a href="#cite_note-Hibbeler-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></td> <td>Railroad steel wheel on steel rail. Passenger rail car about 0.0020<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>0.0019 <i>to</i> 0.0065<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></td> <td></td> <td>Mine car cast iron wheels on steel rail </td></tr> <tr> <td>0.0022 <i>to</i> 0.0050<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </td> <td></td> <td>Production bicycle tires at 120 psi (8.3 bar) and 50 km/h (31 mph), measured on rollers </td></tr> <tr> <td>0.0050</td> <td></td> <td>Dirty tram rails (standard) with straights and curves<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2008)">citation needed</span></a></i>]</sup> </td></tr> <tr> <td>0.0045 <i>to</i> 0.0080<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup></td> <td></td> <td>Large truck <a href="/wiki/Semi-trailer" title="Semi-trailer">(Semi)</a> tires </td></tr> <tr> <td>0.0055<sup id="cite_ref-ReferenceA_23-1" class="reference"><a href="#cite_note-ReferenceA-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></td> <td></td> <td>Typical BMX bicycle tires used for solar cars </td></tr> <tr> <td>0.0065<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup></td> <td></td> <td>EU passenger car tyre label fuel efficiency class A (upper limit) </td></tr> <tr> <td>0.0062 <i>to</i> 0.0150<sup id="cite_ref-greenseal2003_31-0" class="reference"><a href="#cite_note-greenseal2003-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup></td> <td></td> <td>Car tire measurements </td></tr> <tr> <td>0.0100 <i>to</i> 0.0150<sup id="cite_ref-Gillespie_{{ISBN|1-56091-199-9}}_p117_32-0" class="reference"><a href="#cite_note-Gillespie_{{ISBN|1-56091-199-9}}_p117-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup></td> <td></td> <td>Ordinary car tires on concrete </td></tr> <tr> <td>0.0385 <i>to</i> 0.0730<sup id="cite_ref-treatise_33-0" class="reference"><a href="#cite_note-treatise-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup></td> <td></td> <td>Stage coach (19th century) on dirt road. Soft snow on road for worst case. </td></tr> <tr> <td>0.3000<sup id="cite_ref-Gillespie_{{ISBN|1-56091-199-9}}_p117_32-1" class="reference"><a href="#cite_note-Gillespie_{{ISBN|1-56091-199-9}}_p117-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup></td> <td></td> <td>Ordinary car tires on sand </td></tr></tbody></table> <p>For example, in earth gravity, a car of 1000 kg on asphalt will need a force of around 100 <a href="/wiki/Newton_(unit)" title="Newton (unit)">newtons</a> for rolling (1000 kg × 9.81 m/s<sup>2</sup> × 0.01 = 98.1 N). </p> <div class="mw-heading mw-heading2"><h2 id="Dependence_on_diameter">Dependence on diameter</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=7" title="Edit section: Dependence on diameter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Stagecoaches_and_railroads">Stagecoaches and railroads</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=8" title="Edit section: Stagecoaches and railroads"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>According to Dupuit (1837), rolling resistance (of wheeled carriages with wooden wheels with iron tires) is approximately inversely proportional to the square root of wheel diameter.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> This rule has been experimentally verified for cast iron wheels (8″ - 24″ diameter) on steel rail<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> and for 19th century carriage wheels.<sup id="cite_ref-treatise_33-1" class="reference"><a href="#cite_note-treatise-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> But there are other tests on carriage wheels that do not agree.<sup id="cite_ref-treatise_33-2" class="reference"><a href="#cite_note-treatise-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> Theory of a cylinder rolling on an elastic roadway also gives this same rule<sup id="cite_ref-Williams_36-0" class="reference"><a href="#cite_note-Williams-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> These contradict earlier (1785) tests by Coulomb of rolling wooden cylinders where Coulomb reported that rolling resistance was inversely proportional to the diameter of the wheel (known as "Coulomb's law").<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> This disputed (or wrongly applied) -"Coulomb's law" is still found in handbooks, however. </p> <div class="mw-heading mw-heading3"><h3 id="Pneumatic_tires">Pneumatic tires</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=9" title="Edit section: Pneumatic tires"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For pneumatic tires on hard pavement, it is reported that the effect of diameter on rolling resistance is negligible (within a practical range of diameters).<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Dependence_on_applied_torque">Dependence on applied torque</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=10" title="Edit section: Dependence on applied torque"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The driving <a href="/wiki/Torque" title="Torque">torque</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> to overcome rolling resistance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01f9a77bad8a7937735a5d022af56bfe40b4819e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.738ex; height:2.509ex;" alt="{\displaystyle R_{r}}"></span> and maintain steady speed on level ground (with no air resistance) can be calculated by: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\frac {V_{s}}{\Omega }}R_{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mi mathvariant="normal">Ω</mi> </mfrac> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {V_{s}}{\Omega }}R_{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/902ab4a0d828f3a1007b5a6e3c61af9032ba9d3c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.667ex; height:5.343ex;" alt="{\displaystyle T={\frac {V_{s}}{\Omega }}R_{r}}"></span> where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57d5d0bb9d326d015df44456d7c0648f80c0f1f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.359ex; height:2.509ex;" alt="{\displaystyle V_{s}}"></span> is the linear speed of the body (at the axle), and</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span> its rotational speed.</li></ul> <p>It is noteworthy that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{s}/\Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{s}/\Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aefa3b4399bd5f216ba0e827f6b13dc20dac7838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.199ex; height:2.843ex;" alt="{\displaystyle V_{s}/\Omega }"></span> is usually not equal to the radius of the rolling body as a result of wheel slip.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> The slip between wheel and ground inevitably occurs whenever a driving or braking torque is applied to the wheel.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> Consequently, the linear speed of the vehicle differs from the wheel's circumferential speed. It is notable that slip does not occur in driven wheels, which are not subjected to driving torque, under different conditions except braking. Therefore, rolling resistance, namely hysteresis loss, is the main source of energy dissipation in driven wheels or axles, whereas in the drive wheels and axles slip resistance, namely loss due to wheel slip, plays the role as well as rolling resistance.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> Significance of rolling or slip resistance is largely dependent on the <a href="/wiki/Tractive_force" class="mw-redirect" title="Tractive force">tractive force</a>, coefficient of friction, normal load, etc.<sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="All_wheels">All wheels</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=11" title="Edit section: All wheels"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>"Applied torque" may either be driving torque applied by a motor (often through a <a href="/wiki/Transmission_(mechanics)" class="mw-redirect" title="Transmission (mechanics)">transmission</a>) or a braking torque applied by <a href="/wiki/Brake" title="Brake">brakes</a> (including <a href="/wiki/Regenerative_brake" class="mw-redirect" title="Regenerative brake">regenerative braking</a>). Such torques results in energy dissipation (above that due to the basic rolling resistance of a freely rolling, i.e. except slip resistance). This additional loss is in part due to the fact that there is some slipping of the wheel, and for pneumatic tires, there is more flexing of the sidewalls due to the torque. Slip is defined such that a 2% slip means that the circumferential speed of the driving wheel exceeds the speed of the vehicle by 2%. </p><p>A small percentage slip can result in a slip resistance which is much larger than the basic rolling resistance. For example, for pneumatic tires, a 5% slip can translate into a 200% increase in rolling resistance.<sup id="cite_ref-Roberts-torque_47-0" class="reference"><a href="#cite_note-Roberts-torque-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> This is partly because the <a href="/wiki/Tractive_force" class="mw-redirect" title="Tractive force">tractive force</a> applied during this slip is many times greater than the rolling resistance force and thus much more power per unit velocity is being applied (recall <a href="/wiki/Power_(physics)#Mechanical_power" title="Power (physics)">power</a> = force x velocity so that power per unit of velocity is just force). So just a small percentage increase in circumferential velocity due to slip can translate into a loss of traction power which may even exceed the power loss due to basic (ordinary) rolling resistance. For railroads, this effect may be even more pronounced due to the low rolling resistance of steel wheels. </p><p>It is shown that for a passenger car, when the <a href="/wiki/Tractive_force" class="mw-redirect" title="Tractive force">tractive force</a> is about 40% of the maximum traction, the slip resistance is almost equal to the basic rolling resistance (hysteresis loss). But in case of a <a href="/wiki/Tractive_force" class="mw-redirect" title="Tractive force">tractive force</a> equal to 70% of the maximum traction, slip resistance becomes 10 times larger than the basic rolling resistance.<sup id="cite_ref-SAE_MOBILUS_1-1" class="reference"><a href="#cite_note-SAE_MOBILUS-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Railroad_steel_wheels">Railroad steel wheels</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=12" title="Edit section: Railroad steel wheels"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In order to apply any <a href="/wiki/Traction_(engineering)" class="mw-redirect" title="Traction (engineering)">traction</a> to the wheels, some slippage of the wheel is required.<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> For trains climbing up a grade, this slip is normally 1.5% to 2.5%. </p><p>Slip (also known as <a href="/wiki/Rail_adhesion#Forces_on_wheels" class="mw-redirect" title="Rail adhesion">creep</a>) is normally roughly directly proportional to <a href="/wiki/Tractive_force" class="mw-redirect" title="Tractive force">tractive effort</a>. An exception is if the tractive effort is so high that the wheel is close to substantial slipping (more than just a few percent as discussed above), then slip rapidly increases with tractive effort and is no longer linear. With a little higher applied tractive effort the wheel spins out of control and the <a href="/wiki/Rail_adhesion" class="mw-redirect" title="Rail adhesion">adhesion</a> drops resulting in the wheel spinning even faster. This is the type of slipping that is observable by eye—the slip of say 2% for traction is only observed by instruments. Such rapid slip may result in excessive wear or damage. </p> <div class="mw-heading mw-heading3"><h3 id="Pneumatic_tires_2">Pneumatic tires</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=13" title="Edit section: Pneumatic tires"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Rolling resistance greatly increases with applied torque. At high torques, which apply a tangential force to the road of about half the weight of the vehicle, the rolling resistance may triple (a 200% increase).<sup id="cite_ref-Roberts-torque_47-1" class="reference"><a href="#cite_note-Roberts-torque-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> This is in part due to a slip of about 5%. The rolling resistance increase with applied torque is not linear, but increases at a faster rate as the torque becomes higher. </p> <div class="mw-heading mw-heading2"><h2 id="Dependence_on_wheel_load">Dependence on wheel load</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=14" title="Edit section: Dependence on wheel load"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Railroad_steel_wheels_2">Railroad steel wheels</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=15" title="Edit section: Railroad steel wheels"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The rolling resistance coefficient, Crr, significantly decreases as the weight of the rail car per wheel increases.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> For example, an empty freight car had about twice the Crr as a loaded car (Crr=0.002 vs. Crr=0.001). This same "economy of scale" shows up in testing of mine rail cars.<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> The theoretical Crr for a rigid wheel rolling on an elastic roadbed shows Crr inversely proportional to the square root of the load.<sup id="cite_ref-Williams_36-1" class="reference"><a href="#cite_note-Williams-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p><p>If Crr is itself dependent on wheel load per an inverse square-root rule, then for an increase in load of 2% only a 1% increase in rolling resistance occurs.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Pneumatic_tires_3">Pneumatic tires</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=16" title="Edit section: Pneumatic tires"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For pneumatic tires, the direction of change in Crr (rolling resistance coefficient) depends on whether or not tire inflation is increased with increasing load.<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> It is reported that, if inflation pressure is increased with load according to an (undefined) "schedule", then a 20% increase in load decreases Crr by 3%. But, if the inflation pressure is not changed, then a 20% increase in load results in a 4% increase in Crr. Of course, this will increase the rolling resistance by 20% due to the increase in load plus 1.2 x 4% due to the increase in Crr resulting in a 24.8% increase in rolling resistance.<sup id="cite_ref-53" class="reference"><a href="#cite_note-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Dependence_on_curvature_of_roadway">Dependence on curvature of roadway</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=17" title="Edit section: Dependence on curvature of roadway"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="General">General</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=18" title="Edit section: General"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When a vehicle (<a href="/wiki/Motor_vehicle" title="Motor vehicle">motor vehicle</a> or railroad <a href="/wiki/Train" title="Train">train</a>) goes around a curve, rolling resistance usually increases. If the curve is not <a href="/wiki/Banked_curve#Frictionless_banked_curve" class="mw-redirect" title="Banked curve">banked</a> so as to exactly counter the <a href="/wiki/Centrifugal_force" title="Centrifugal force">centrifugal force</a> with an equal and opposing <a href="/wiki/Centripetal_force" title="Centripetal force">centripetal force</a> due to the banking, then there will be a net unbalanced sideways force on the vehicle. This will result in increased rolling resistance. <a href="/wiki/Banked_curve" class="mw-redirect" title="Banked curve">Banking</a> is also known as <a href="/wiki/Cant_(road/rail)" class="mw-redirect" title="Cant (road/rail)">"superelevation" or "cant"</a> (not to be confused with <a href="/wiki/Cant_(road/rail)#Rail" class="mw-redirect" title="Cant (road/rail)">rail cant</a> of a <a href="/wiki/Track_(rail_transport)#Rail" class="mw-redirect" title="Track (rail transport)">rail</a>). For railroads, this is called <a href="/wiki/Curve_resistance" class="mw-redirect" title="Curve resistance">curve resistance</a> but for <a href="/wiki/Road" title="Road">roads</a> it has (at least once) been called rolling resistance due to <a href="/wiki/Cornering_force" title="Cornering force">cornering</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Sound">Sound</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=19" title="Edit section: Sound"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Rolling friction generates sound (vibrational) energy, as mechanical energy is converted to this form of energy due to the friction. One of the most common examples of rolling friction is the movement of <a href="/wiki/Motor_vehicle" title="Motor vehicle">motor vehicle</a> tires on a <a href="/wiki/Roadway" class="mw-redirect" title="Roadway">roadway</a>, a process which generates <a href="/wiki/Roadway_noise" title="Roadway noise">sound</a> as a by-product.<sup id="cite_ref-54" class="reference"><a href="#cite_note-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> The sound generated by automobile and truck tires as they roll (especially noticeable at highway speeds) is mostly due to the percussion of the tire treads, and compression (and subsequent decompression) of air temporarily captured within the treads.<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Factors_that_contribute_in_tires">Factors that contribute in tires</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=20" title="Edit section: Factors that contribute in tires"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Several factors affect the magnitude of rolling resistance a tire generates: </p> <ul><li>As mentioned in the introduction: wheel radius, forward speed, surface adhesion, and relative micro-sliding.</li> <li>Material - different fillers and polymers in tire composition can improve traction while reducing hysteresis. The replacement of some carbon black with higher-priced silica–silane is one common way of reducing rolling resistance.<sup id="cite_ref-NAS286_6-3" class="reference"><a href="#cite_note-NAS286-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The use of exotic materials including nano-clay has been shown to reduce rolling resistance in high performance rubber tires.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> Solvents may also be used to swell solid tires, decreasing the rolling resistance.<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup></li> <li>Dimensions - rolling resistance in tires is related to the flex of sidewalls and the contact area of the tire<sup id="cite_ref-schwalbe_58-0" class="reference"><a href="#cite_note-schwalbe-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> For example, at the same pressure, wider <a href="/wiki/Bicycle_tire" title="Bicycle tire">bicycle tires</a> flex less in the sidewalls as they roll and thus have lower rolling resistance (although higher air resistance).<sup id="cite_ref-schwalbe_58-1" class="reference"><a href="#cite_note-schwalbe-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup></li> <li>Extent of inflation - Lower pressure in tires results in more flexing of the sidewalls and higher rolling resistance.<sup id="cite_ref-schwalbe_58-2" class="reference"><a href="#cite_note-schwalbe-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> This energy conversion in the sidewalls increases resistance and can also lead to overheating and may have played a part in the infamous <a href="/wiki/Ford_Explorer" title="Ford Explorer">Ford Explorer</a> <a href="/wiki/Firestone_vs_Ford_Motor_Company_controversy" class="mw-redirect" title="Firestone vs Ford Motor Company controversy">rollover accidents</a>.</li> <li>Over inflating tires (such a bicycle tires) may not lower the overall rolling resistance as the tire may skip and hop over the road surface. Traction is sacrificed, and overall rolling friction may not be reduced as the wheel rotational speed changes and slippage increases.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2008)">citation needed</span></a></i>]</sup></li> <li>Sidewall deflection is not a direct measurement of rolling friction. A high quality tire with a high quality (and supple) casing will allow for more flex per energy loss than a cheap tire with a stiff sidewall.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (February 2008)">citation needed</span></a></i>]</sup> Again, on a bicycle, a quality tire with a supple casing will still roll easier than a cheap tire with a stiff casing. Similarly, as noted by Goodyear truck tires, a tire with a "fuel saving" casing will benefit the fuel economy through many tread lives (i.e. retreading), while a tire with a "fuel saving" tread design will only benefit until the tread wears down.</li> <li>In tires, <a href="/wiki/Tire_tread" title="Tire tread">tread</a> thickness and shape has much to do with rolling resistance. The thicker and more contoured the tread, the higher the rolling resistance<sup id="cite_ref-schwalbe_58-3" class="reference"><a href="#cite_note-schwalbe-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> Thus, the "fastest" bicycle tires have very little tread and heavy duty trucks get the best fuel economy as the tire tread wears out.</li> <li>Diameter effects seem to be negligible, provided the pavement is hard and the range of diameters is limited. See <a href="#Dependence_on_diameter">dependence on diameter</a>.</li> <li>Virtually all world speed records have been set on relatively narrow wheels,<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (December 2007)">citation needed</span></a></i>]</sup> probably because of their aerodynamic advantage at high speed, which is much less important at normal speeds.</li> <li>Temperature: with both solid and pneumatic tires, rolling resistance has been found to decrease as temperature increases (within a range of temperatures: i.e. there is an upper limit to this effect)<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> For a rise in temperature from 30 °C to 70 °C the rolling resistance decreased by 20-25%.<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> Racers heat their tires before racing, but this is primarily used to increase tire friction rather than to decrease rolling resistance.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Railroads:_Components_of_rolling_resistance">Railroads: Components of rolling resistance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=21" title="Edit section: Railroads: Components of rolling resistance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a broad sense rolling resistance can be defined as the sum of components<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup>): </p> <ol><li>Wheel bearing torque losses.</li> <li>Pure rolling resistance.</li> <li>Sliding of the wheel on the rail.</li> <li>Loss of energy to the roadbed (and earth).</li> <li>Loss of energy to oscillation of railway rolling stock.</li></ol> <p>Wheel bearing torque losses can be measured as a rolling resistance at the wheel rim, <a href="#Rolling_resistance_coefficient">Crr</a>. Railroads normally use roller bearings which are either <a href="/wiki/Rolling-element_bearing#Cylindrical_roller" title="Rolling-element bearing">cylindrical</a> (Russia)<sup id="cite_ref-cars-Russia_63-0" class="reference"><a href="#cite_note-cars-Russia-63"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> or <a href="/wiki/Rolling-element_bearing#Tapered_roller" title="Rolling-element bearing">tapered</a> (United States).<sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> The specific rolling resistance in bearings varies with both wheel loading and speed.<sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> Wheel bearing rolling resistance is lowest with high axle loads and intermediate speeds of 60–80 km/h with a Crr of 0.00013 (axle load of 21 tonnes). For empty freight cars with axle loads of 5.5 tonnes, Crr goes up to 0.00020 at 60 km/h but at a low speed of 20 km/h it increases to 0.00024 and at a high speed (for freight trains) of 120 km/h it is 0.00028. The Crr obtained above is added to the Crr of the other components to obtain the total Crr for the wheels. </p> <div class="mw-heading mw-heading2"><h2 id="Comparing_rolling_resistance_of_highway_vehicles_and_trains">Comparing rolling resistance of highway vehicles and trains</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=22" title="Edit section: Comparing rolling resistance of highway vehicles and trains"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The rolling resistance of steel wheels on steel rail of a train is far less than that of the rubber tires wheels of an automobile or truck. The weight of trains varies greatly; in some cases they may be much heavier per passenger or per net ton of freight than an automobile or truck, but in other cases they may be much lighter. </p><p>As an example of a very heavy passenger train, in 1975, Amtrak passenger trains weighed a little over 7 tonnes per passenger,<sup id="cite_ref-rr_stat_66-0" class="reference"><a href="#cite_note-rr_stat-66"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> which is much heavier than an average of a little over one ton per passenger for an automobile. This means that for an Amtrak passenger train in 1975, much of the energy savings of the lower rolling resistance was lost to its greater weight. </p><p>An example of a very light high-speed passenger train is the <a href="/wiki/N700_Series_Shinkansen" title="N700 Series Shinkansen">N700 Series Shinkansen</a>, which weighs 715 tonnes and carries 1323 passengers, resulting in a per-passenger weight of about half a tonne. This lighter weight per passenger, combined with the lower rolling resistance of steel wheels on steel rail means that an N700 Shinkansen is much more energy efficient than a typical automobile. </p><p>In the case of freight, CSX ran an advertisement campaign in 2013 claiming that their freight trains move "a ton of freight 436 miles on a gallon of fuel", whereas some sources claim trucks move a ton of freight about 130 miles per gallon of fuel, indicating trains are more efficient overall. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=23" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Coefficient_of_friction" class="mw-redirect" title="Coefficient of friction">Coefficient of friction</a></li> <li><a href="/wiki/Low-rolling_resistance_tires" class="mw-redirect" title="Low-rolling resistance tires">Low-rolling resistance tires</a></li> <li><a href="/wiki/Maglev_(transport)" class="mw-redirect" title="Maglev (transport)">Maglev</a> (<a href="/wiki/Magnetic" class="mw-redirect" title="Magnetic">Magnetic</a> <a href="/wiki/Levitation_(physics)" title="Levitation (physics)">Levitation</a>, the elimination of rolling and thus rolling resistance)</li> <li><a href="/wiki/Rolling_element_bearing" class="mw-redirect" title="Rolling element bearing">Rolling element bearing</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=24" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-SAE_MOBILUS-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-SAE_MOBILUS_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-SAE_MOBILUS_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSinaEsfahanianYazdiAzadi2018" class="citation journal cs1">Sina, Naser; Esfahanian, Vahid; Yazdi, Mohammad Reza Hairi; Azadi, Shahram (18 April 2018). <a rel="nofollow" class="external text" href="https://saemobilus.sae.org/content/06-11-02-0014/">"SAE MOBILUS"</a>. <i>SAE International Journal of Passenger Cars - Mechanical Systems</i>. <b>11</b> (2): <span class="nowrap">167–</span>176. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4271%2F06-11-02-0014">10.4271/06-11-02-0014</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-04-19</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SAE+International+Journal+of+Passenger+Cars+-+Mechanical+Systems&rft.atitle=SAE+MOBILUS&rft.volume=11&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E167-%3C%2Fspan%3E176&rft.date=2018-04-18&rft_id=info%3Adoi%2F10.4271%2F06-11-02-0014&rft.aulast=Sina&rft.aufirst=Naser&rft.au=Esfahanian%2C+Vahid&rft.au=Yazdi%2C+Mohammad+Reza+Hairi&rft.au=Azadi%2C+Shahram&rft_id=https%3A%2F%2Fsaemobilus.sae.org%2Fcontent%2F06-11-02-0014%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-Peck-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Peck_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeck1859" class="citation book cs1">Peck, William Guy (1859). <a rel="nofollow" class="external text" href="https://archive.org/details/elementsmechani02peckgoog"><i>Elements of Mechanics: For the Use of Colleges, Academies, and High Schools</i></a>. A.S. Barnes & Burr: New York. p. <a rel="nofollow" class="external text" href="https://archive.org/details/elementsmechani02peckgoog/page/n142">135</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2007-10-09</span></span>. <q>rolling friction less than sliding friction.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Mechanics%3A+For+the+Use+of+Colleges%2C+Academies%2C+and+High+Schools&rft.pages=135&rft.pub=A.S.+Barnes+%26+Burr%3A+New+York&rft.date=1859&rft.aulast=Peck&rft.aufirst=William+Guy&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felementsmechani02peckgoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-Hibbeler-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hibbeler_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hibbeler_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Hibbeler_3-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Hibbeler_3-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Hibbeler_3-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHibbeler2007" class="citation book cs1">Hibbeler, R.C. (2007). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/engineeringmecha00hibb_4"><i>Engineering Mechanics: Statics & Dynamics</i></a></span> (Eleventh ed.). Pearson, Prentice Hall. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/engineeringmecha00hibb_4/page/441">441</a>–442. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780132038096" title="Special:BookSources/9780132038096"><bdi>9780132038096</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Mechanics%3A+Statics+%26+Dynamics&rft.pages=441-442&rft.edition=Eleventh&rft.pub=Pearson%2C+Prentice+Hall&rft.date=2007&rft.isbn=9780132038096&rft.aulast=Hibbeler&rft.aufirst=R.C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fengineeringmecha00hibb_4&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-CONTACT-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-CONTACT_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.kalkersoftware.org/downloads/user-guide.pdf">"User guide for CONTACT, Rolling and sliding contact with friction. Technical report TR09-03 version v16.1. VORtech, 2016"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">2017-07-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=User+guide+for+CONTACT%2C+Rolling+and+sliding+contact+with+friction.+Technical+report+TR09-03+version+v16.1.+VORtech%2C+2016.&rft_id=http%3A%2F%2Fwww.kalkersoftware.org%2Fdownloads%2Fuser-guide.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://deepblue.lib.umich.edu/handle/2027.42/4274">A handbook for the rolling resistance of pneumatic tires Clark, Samuel Kelly; Dodge, Richard N. 1979</a></span> </li> <li id="cite_note-NAS286-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-NAS286_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-NAS286_6-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-NAS286_6-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-NAS286_6-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://onlinepubs.trb.org/onlinepubs/sr/sr286.pdf">"Tires and Passenger Vehicle Fuel Economy: Informing Consumers, Improving Performance -- Special Report 286. National Academy of Sciences, Transportation Research Board, 2006"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">2007-08-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Tires+and+Passenger+Vehicle+Fuel+Economy%3A+Informing+Consumers%2C+Improving+Performance+--+Special+Report+286.+National+Academy+of+Sciences%2C+Transportation+Research+Board%2C+2006&rft_id=http%3A%2F%2Fonlinepubs.trb.org%2Fonlinepubs%2Fsr%2Fsr286.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.tyrepriceadvisor.co.uk/en/top-tyre-brands/falken-tyres">Tyres-Online: The Benefits of Silica in Tyre Design</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130204065301/http://www.tyrepriceadvisor.co.uk/en/top-tyre-brands/falken-tyres">Archived</a> 2013-02-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Астахов, p.85</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">An example of such usage for railroads is <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100626045718/http://www.istc.illinois.edu/about/SeminarPresentations/20091118.pdf">here</a>.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Деев, p. 79. Hay, p. 68</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Астахов, Chapt. IV, p. 73+; Деев, Sect. 5.2 p. 78+; Hay, Chapt. 6 "Train Resistance" p. 67+</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Астахов, Fig. 4.14, p. 107</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160408115014/http://indiatransportportal.com/wp-content/uploads/2016/02/india-rolling-resistance-itp-feb-2016.pdf">"Rolling Resistance and Fuel Saving"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://indiatransportportal.com/wp-content/uploads/2016/02/india-rolling-resistance-itp-feb-2016.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2016-04-08.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Rolling+Resistance+and+Fuel+Saving&rft_id=http%3A%2F%2Findiatransportportal.com%2Fwp-content%2Fuploads%2F2016%2F02%2Findia-rolling-resistance-itp-feb-2016.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">If one were to assume that the resistance coefficients (Crr) for motor vehicles were the same as for trains, then for trains the neglected resistances taken together have a Crr of about 0.0004 (see Астахов, Fig. 4.14, p.107 at 20km/hr and assume a total Crr =0.0010 based on Fig. 3.8, p.50 (plain bearings) and adjust for roller bearings based on a delta Crr of 0.00035 as read from Figs. 4.2 and 4.4 on pp. 74, 76). Compare this Crr of 0.0004 to motor vehicle tire Crr's of at least 10 times higher per "Rolling resistance coefficient examples" in this article</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">kgf/tonne is used by Астахов throughout his book</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Деев uses N/T notation. See pp. 78-84.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWillett" class="citation web cs1">Willett, Kraig. <a rel="nofollow" class="external text" href="http://www.biketechreview.com/tires/AFM_tire_crr.htm">"Roller Data"</a>. <i>www.biketechreview.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2017-08-05</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.biketechreview.com&rft.atitle=Roller+Data&rft.aulast=Willett&rft.aufirst=Kraig&rft_id=http%3A%2F%2Fwww.biketechreview.com%2Ftires%2FAFM_tire_crr.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuiggiani2018" class="citation book cs1">Guiggiani, Massimo (5 May 2018). <i>The Science of Vehicle Dynamics</i>. Springer Cham. p. 22. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-73220-6" title="Special:BookSources/978-3-319-73220-6"><bdi>978-3-319-73220-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Science+of+Vehicle+Dynamics&rft.pages=22&rft.pub=Springer+Cham&rft.date=2018-05-05&rft.isbn=978-3-319-73220-6&rft.aulast=Guiggiani&rft.aufirst=Massimo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Hersey, equation (2), p. 83</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Астахов, p. 81.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.tribology-abc.com/abc/cof.htm">"Coefficients of Friction in Bearing"</a>. <i>Coefficients of Friction</i><span class="reference-accessdate">. Retrieved <span class="nowrap">7 February</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Coefficients+of+Friction&rft.atitle=Coefficients+of+Friction+in+Bearing&rft_id=http%3A%2F%2Fwww.tribology-abc.com%2Fabc%2Fcof.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.paccar.ethz.ch/technics/pac_I_vs_II.html">ETH - PAC-Car II vs PAC-Car I - Technics</a></span> </li> <li id="cite_note-ReferenceA-23"><span class="mw-cite-backlink">^ <a href="#cite_ref-ReferenceA_23-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ReferenceA_23-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Roche, Schinkel, Storey, Humphris & Guelden, "Speed of Light." <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7334-1527-X" title="Special:BookSources/0-7334-1527-X">0-7334-1527-X</a></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">Hay, Fig. 6-2 p.72(worst case shown of 0.0036 not used since it is likely erroneous)</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">Астахов, Figs. 3.8, 3.9, 3.11, pp. 50-55; Figs. 2.3, 2.4 pp. 35-36. (Worst case is 0.0024 for an axle load of 5.95 tonnes with obsolete plain (friction --not roller) bearings</span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text">Астахов, Fig. 2.1, p.22</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">Hersey, Table 6, p.267</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.biketechreview.com/tires/images/AFM_tire_testing_rev8.pdf">"Roller Data"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Roller+Data&rft_id=http%3A%2F%2Fwww.biketechreview.com%2Ftires%2Fimages%2FAFM_tire_testing_rev8.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.michelintruck.com/michelintruck/tires-retreads/xone/xOne-fuel-savings.jsp">Crr for large truck tires per Michelin</a></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">Annex 1, Part A. Category C1 tyres (passenger car) of <a rel="nofollow" class="external text" href="https://eur-lex.europa.eu/legal-content/EN/TXT/?uri=uriserv%3AOJ.L_.2020.177.01.0001.01.ENG&toc=OJ%3AL%3A2020%3A177%3ATOC">Regulation (EU) 2020/740 of the European Parliament and of the Council of 25 May 2020 on the labelling of tyres with respect to fuel efficiency and other parameters</a></span> </li> <li id="cite_note-greenseal2003-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-greenseal2003_31-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20101030230105/http://greenseal.org/resources/reports/CGR_tire_rollingresistance.pdf">Green Seal 2003 Report</a></span> </li> <li id="cite_note-Gillespie_{{ISBN|1-56091-199-9}}_p117-32"><span class="mw-cite-backlink">^ <a href="#cite_ref-Gillespie_{{ISBN|1-56091-199-9}}_p117_32-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Gillespie_{{ISBN|1-56091-199-9}}_p117_32-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Gillespie <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-56091-199-9" title="Special:BookSources/1-56091-199-9">1-56091-199-9</a> p117</span> </li> <li id="cite_note-treatise-33"><span class="mw-cite-backlink">^ <a href="#cite_ref-treatise_33-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-treatise_33-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-treatise_33-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Baker, Ira O., "Treatise on roads and pavements". New York, John Wiley, 1914. Stagecoach: Table 7, p. 28. Diameter: pp. 22-23. This book reports a few hundred values of rolling resistance for various animal-powered vehicles under various condition, mostly from 19th century data.</span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text">Hersey, subsection: "End of dark ages", p.261</span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text">Hersey, subsection: "Static rolling friction", p.266.</span> </li> <li id="cite_note-Williams-36"><span class="mw-cite-backlink">^ <a href="#cite_ref-Williams_36-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Williams_36-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Williams, 1994, Ch. "Rolling contacts", eq. 11.1, p. 409.</span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text">Hersey, subsection: "Coulomb on wooden cylinders", p. 260</span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text">U.S. National Bureau of Standards, Fig. 1.13</span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text">Some<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch"><span title="The material near this tag possibly uses too-vague attribution or weasel words. (May 2012)">who?</span></a></i>]</sup> think that smaller tire wheels, all else being equal, tend to have higher rolling resistance than larger wheels. In some laboratory tests, however, such as <a rel="nofollow" class="external text" href="http://www.legslarry.beerdrinkers.co.uk/tech/GS.htm">Greenspeed test results (accessdate = 2007-10-27)</a>, smaller wheels appeared to have similar or lower losses than large wheels, but these tests were done rolling the wheels against a small-diameter drum, which would theoretically remove the advantage of large-diameter wheels, thus making the tests irrelevant for resolving this issue. Another counter example to the claim of smaller wheels having higher rolling resistance can be found in the area of <a href="/wiki/Soap_Box_Derby" title="Soap Box Derby">ultimate speed</a> soap box derby racing. In this race, the speeds have increased as wheel diameters have decreased by up to 50%. This might suggest that rolling resistance may not be increasing significantly with smaller diameter within a practical range, if any other of the many variables involved have been controlled for. See <a href="/wiki/Talk:Rolling_resistance#Pneumatic_tires,_effect_of_diameter" title="Talk:Rolling resistance">talk page</a>.</span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZéhilGavin2013" class="citation journal cs1">Zéhil, Gérard-Philippe; Gavin, Henri P. (2013). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ijsolstr.2012.11.020">"Three-dimensional boundary element formulation of an incompressible viscoelastic layer of finite thickness applied to the rolling resistance of a rigid sphere"</a>. <i>International Journal of Solids and Structures</i>. <b>50</b> (6): <span class="nowrap">833–</span>842. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ijsolstr.2012.11.020">10.1016/j.ijsolstr.2012.11.020</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Solids+and+Structures&rft.atitle=Three-dimensional+boundary+element+formulation+of+an+incompressible+viscoelastic+layer+of+finite+thickness+applied+to+the+rolling+resistance+of+a+rigid+sphere&rft.volume=50&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E833-%3C%2Fspan%3E842&rft.date=2013&rft_id=info%3Adoi%2F10.1016%2Fj.ijsolstr.2012.11.020&rft.aulast=Z%C3%A9hil&rft.aufirst=G%C3%A9rard-Philippe&rft.au=Gavin%2C+Henri+P.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.ijsolstr.2012.11.020&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZéhilGavin2013" class="citation journal cs1">Zéhil, Gérard-Philippe; Gavin, Henri P. (2013). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ijsolstr.2012.11.021">"Simple algorithms for solving steady-state frictional rolling contact problems in two and three dimensions"</a>. <i>International Journal of Solids and Structures</i>. <b>50</b> (6): <span class="nowrap">843–</span>852. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ijsolstr.2012.11.021">10.1016/j.ijsolstr.2012.11.021</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Solids+and+Structures&rft.atitle=Simple+algorithms+for+solving+steady-state+frictional+rolling+contact+problems+in+two+and+three+dimensions&rft.volume=50&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E843-%3C%2Fspan%3E852&rft.date=2013&rft_id=info%3Adoi%2F10.1016%2Fj.ijsolstr.2012.11.021&rft.aulast=Z%C3%A9hil&rft.aufirst=G%C3%A9rard-Philippe&rft.au=Gavin%2C+Henri+P.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.ijsolstr.2012.11.021&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZéhilGavin2013" class="citation journal cs1">Zéhil, Gérard-Philippe; Gavin, Henri P. (2013). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ijsolstr.2012.09.025">"Simplified approaches to viscoelastic rolling resistance"</a>. <i>International Journal of Solids and Structures</i>. <b>50</b> (6): <span class="nowrap">853–</span>862. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ijsolstr.2012.09.025">10.1016/j.ijsolstr.2012.09.025</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Solids+and+Structures&rft.atitle=Simplified+approaches+to+viscoelastic+rolling+resistance&rft.volume=50&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E853-%3C%2Fspan%3E862&rft.date=2013&rft_id=info%3Adoi%2F10.1016%2Fj.ijsolstr.2012.09.025&rft.aulast=Z%C3%A9hil&rft.aufirst=G%C3%A9rard-Philippe&rft.au=Gavin%2C+Henri+P.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.ijsolstr.2012.09.025&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSinaEsfahanianYazdiAzadi2018" class="citation journal cs1">Sina, Naser; Esfahanian, Vahid; Yazdi, Mohammad Reza Hairi; Azadi, Shahram (18 April 2018). <a rel="nofollow" class="external text" href="https://saemobilus.sae.org/content/06-11-02-0014/">"SAE MOBILUS"</a>. <i>SAE International Journal of Passenger Cars - Mechanical Systems</i>. <b>11</b> (2): <span class="nowrap">167–</span>176. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4271%2F06-11-02-0014">10.4271/06-11-02-0014</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-04-19</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SAE+International+Journal+of+Passenger+Cars+-+Mechanical+Systems&rft.atitle=SAE+MOBILUS&rft.volume=11&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E167-%3C%2Fspan%3E176&rft.date=2018-04-18&rft_id=info%3Adoi%2F10.4271%2F06-11-02-0014&rft.aulast=Sina&rft.aufirst=Naser&rft.au=Esfahanian%2C+Vahid&rft.au=Yazdi%2C+Mohammad+Reza+Hairi&rft.au=Azadi%2C+Shahram&rft_id=https%3A%2F%2Fsaemobilus.sae.org%2Fcontent%2F06-11-02-0014%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSinaHairi_YazdiEsfahanian2020" class="citation journal cs1">Sina, Naser; Hairi Yazdi, Mohammad Reza; Esfahanian, Vahid (2020-03-01). "A novel method to improve vehicle energy efficiency: Minimization of tire power loss". <i>Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering</i>. <b>234</b> (4): <span class="nowrap">1153–</span>1166. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F0954407019861241">10.1177/0954407019861241</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0954-4070">0954-4070</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:199099736">199099736</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Institution+of+Mechanical+Engineers%2C+Part+D%3A+Journal+of+Automobile+Engineering&rft.atitle=A+novel+method+to+improve+vehicle+energy+efficiency%3A+Minimization+of+tire+power+loss&rft.volume=234&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1153-%3C%2Fspan%3E1166&rft.date=2020-03-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A199099736%23id-name%3DS2CID&rft.issn=0954-4070&rft_id=info%3Adoi%2F10.1177%2F0954407019861241&rft.aulast=Sina&rft.aufirst=Naser&rft.au=Hairi+Yazdi%2C+Mohammad+Reza&rft.au=Esfahanian%2C+Vahid&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-45"><span class="mw-cite-backlink"><b><a href="#cite_ref-45">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSinaNasiriKarkhaneh2015" class="citation journal cs1">Sina, Naser; Nasiri, Sayyad; Karkhaneh, Vahid (2015-11-01). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/abs/pii/S0306261915004493">"Effects of resistive loads and tire inflation pressure on tire power losses and CO2 emissions in real-world conditions"</a>. <i>Applied Energy</i>. <b>157</b>: <span class="nowrap">974–</span>983. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.apenergy.2015.04.010">10.1016/j.apenergy.2015.04.010</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0306-2619">0306-2619</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Applied+Energy&rft.atitle=Effects+of+resistive+loads+and+tire+inflation+pressure+on+tire+power+losses+and+CO2+emissions+in+real-world+conditions&rft.volume=157&rft.pages=%3Cspan+class%3D%22nowrap%22%3E974-%3C%2Fspan%3E983&rft.date=2015-11-01&rft_id=info%3Adoi%2F10.1016%2Fj.apenergy.2015.04.010&rft.issn=0306-2619&rft.aulast=Sina&rft.aufirst=Naser&rft.au=Nasiri%2C+Sayyad&rft.au=Karkhaneh%2C+Vahid&rft_id=https%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fabs%2Fpii%2FS0306261915004493&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-46"><span class="mw-cite-backlink"><b><a href="#cite_ref-46">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSinaEsfahanianYazdiAzadi2018" class="citation journal cs1">Sina, Naser; Esfahanian, Vahid; Yazdi, Mohammad Reza Hairi; Azadi, Shahram (18 April 2018). <a rel="nofollow" class="external text" href="https://saemobilus.sae.org/content/06-11-02-0014/">"SAE MOBILUS"</a>. <i>SAE International Journal of Passenger Cars - Mechanical Systems</i>. <b>11</b> (2): <span class="nowrap">167–</span>176. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4271%2F06-11-02-0014">10.4271/06-11-02-0014</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-04-19</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SAE+International+Journal+of+Passenger+Cars+-+Mechanical+Systems&rft.atitle=SAE+MOBILUS&rft.volume=11&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E167-%3C%2Fspan%3E176&rft.date=2018-04-18&rft_id=info%3Adoi%2F10.4271%2F06-11-02-0014&rft.aulast=Sina&rft.aufirst=Naser&rft.au=Esfahanian%2C+Vahid&rft.au=Yazdi%2C+Mohammad+Reza+Hairi&rft.au=Azadi%2C+Shahram&rft_id=https%3A%2F%2Fsaemobilus.sae.org%2Fcontent%2F06-11-02-0014%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-Roberts-torque-47"><span class="mw-cite-backlink">^ <a href="#cite_ref-Roberts-torque_47-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Roberts-torque_47-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Roberts, Fig. 17: "Effect of torque transmission on rolling resistance", p. 71</span> </li> <li id="cite_note-48"><span class="mw-cite-backlink"><b><a href="#cite_ref-48">^</a></b></span> <span class="reference-text">Деев, p.30 including eq. (2.7) and Fig. 2.3</span> </li> <li id="cite_note-49"><span class="mw-cite-backlink"><b><a href="#cite_ref-49">^</a></b></span> <span class="reference-text">Астахов, Figs. 3.8, 3.9, 3.11, pp. 50-55. Hay, Fig. 60-2, p. 72 shows the same phenomena but has higher values for Crr and not reported here since the railroads in 2011 <a rel="nofollow" class="external autonumber" href="https://web.archive.org/web/20100626045718/http://www.istc.illinois.edu/about/SeminarPresentations/20091118.pdf">[1]</a>. were claiming about the same value as Астахов</span> </li> <li id="cite_note-50"><span class="mw-cite-backlink"><b><a href="#cite_ref-50">^</a></b></span> <span class="reference-text">Hersey, Table 6., p. 267</span> </li> <li id="cite_note-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-51">^</a></b></span> <span class="reference-text">Per this assumption, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=kN^{0.5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>k</mi> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0.5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=kN^{0.5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c7b6c9e14cf4741f040fc8fa66c3ff409250a1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.507ex; height:2.676ex;" alt="{\displaystyle F=kN^{0.5}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> is the rolling resistance force and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> is the normal load force on the wheel due to vehicle weight, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> is a constant. It can be readily shown by differentiation of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> with respect to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> using <a href="/wiki/Differentiation_rules#The_polynomial_or_elementary_power_rule" title="Differentiation rules">this rule</a> that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\operatorname {d} N \over N}=2{\operatorname {d} F \over F}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>N</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi>F</mi> </mrow> <mi>F</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\operatorname {d} N \over N}=2{\operatorname {d} F \over F}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/061fd7615cb9ebc48e8d5b990030e8c935a0a1c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.097ex; height:5.343ex;" alt="{\displaystyle {\operatorname {d} N \over N}=2{\operatorname {d} F \over F}}"></span></span> </li> <li id="cite_note-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-52">^</a></b></span> <span class="reference-text">Roberts, pp. 60-61.</span> </li> <li id="cite_note-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-53">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external autonumber" href="https://www.matec-conferences.org/articles/matecconf/pdf/2017/22/matecconf_icmaa2017_12004.pdf">[2]</a> A Relationship between Tyre Pressure and Rolling Resistance Force under Different Vehicle Speed | Apiwat Suyabodha |Department of Automotive Engineering, Rangsit University, Lak-hok, Pathumthani, Thailand | 2017</span> </li> <li id="cite_note-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-54">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHogan1973" class="citation journal cs1">Hogan, C. Michael (1973). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00159677">"Analysis of highway noise"</a>. <i>Water, Air, and Soil Pollution</i>. <b>2</b> (3): <span class="nowrap">387–</span>392. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1973WASP....2..387H">1973WASP....2..387H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00159677">10.1007/BF00159677</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:109914430">109914430</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Water%2C+Air%2C+and+Soil+Pollution&rft.atitle=Analysis+of+highway+noise&rft.volume=2&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E387-%3C%2Fspan%3E392&rft.date=1973&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A109914430%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF00159677&rft_id=info%3Abibcode%2F1973WASP....2..387H&rft.aulast=Hogan&rft.aufirst=C.+Michael&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252FBF00159677&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-55">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStachowiakBatchelor2000" class="citation book cs1">Stachowiak, Gwidon W.; Batchelor, Andrew William (2000). <i>Engineering tribology</i> (2. ed.). Oxford: Butterworth-Heinemann. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7506-7304-4" title="Special:BookSources/978-0-7506-7304-4"><bdi>978-0-7506-7304-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+tribology&rft.place=Oxford&rft.edition=2.&rft.pub=Butterworth-Heinemann&rft.date=2000&rft.isbn=978-0-7506-7304-4&rft.aulast=Stachowiak&rft.aufirst=Gwidon+W.&rft.au=Batchelor%2C+Andrew+William&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-56">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="http://144.206.159.178/ft/200/607426/12614863.pdf">http://144.206.159.178/ft/200/607426/12614863.pdf</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged March 2018">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external free" href="http://www.rubberchemtechnol.org/resource/1/rctea4/v3/i1/p19_s1?isAuthorized=no">http://www.rubberchemtechnol.org/resource/1/rctea4/v3/i1/p19_s1?isAuthorized=no</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged March 2018">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup></span> </li> <li id="cite_note-schwalbe-58"><span class="mw-cite-backlink">^ <a href="#cite_ref-schwalbe_58-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-schwalbe_58-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-schwalbe_58-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-schwalbe_58-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.schwalbetires.com/tech_info/rolling_resistance">"Schwalbe Tires: Rolling Resistance"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Schwalbe+Tires%3A+Rolling+Resistance&rft_id=http%3A%2F%2Fwww.schwalbetires.com%2Ftech_info%2Frolling_resistance&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARolling+resistance" class="Z3988"></span></span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.recumbents.com/mars/pages/proj/tetz/other/Crr.html">The Recumbent Bicycle and Human Powered Vehicle Information Center</a></span> </li> <li id="cite_note-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</a></b></span> <span class="reference-text">U.S National Bureau of Standards p.? and Williams p.?</span> </li> <li id="cite_note-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</a></b></span> <span class="reference-text">Roberts, "Effect of temperature", p.59</span> </li> <li id="cite_note-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-62">^</a></b></span> <span class="reference-text">Астахов, p. 74, Although Астахов list these components, he doesn't give the sum a name.</span> </li> <li id="cite_note-cars-Russia-63"><span class="mw-cite-backlink"><b><a href="#cite_ref-cars-Russia_63-0">^</a></b></span> <span class="reference-text">Шадур. Л. А. (editor). Вагоны <span class="languageicon">(in Russian)</span>(Railway cars). Москва, Транспорт, 1980. pp. 122 and figs. VI.1 p. 123 VI.2 p. 125</span> </li> <li id="cite_note-64"><span class="mw-cite-backlink"><b><a href="#cite_ref-64">^</a></b></span> <span class="reference-text">Association of American Railroads, Mechanical Division "Car and Locomotive Encyclopedia", New York, Simmons-Boardman, 1974. Section 14: "Axle journals and bearings". Almost all of the ads in this section are for the tapered type of bearing.</span> </li> <li id="cite_note-65"><span class="mw-cite-backlink"><b><a href="#cite_ref-65">^</a></b></span> <span class="reference-text">Астахов, Fig 4.2, p. 76</span> </li> <li id="cite_note-rr_stat-66"><span class="mw-cite-backlink"><b><a href="#cite_ref-rr_stat_66-0">^</a></b></span> <span class="reference-text">Statistics of railroads of class I in the United States, Years 1965 to 1975: Statistical summary. Washington DC, Association of American Railroads, Economics and Finance Dept. See table for Amtrak, p.16. To get the tons per passenger divide ton-miles (including locomotives) by passenger-miles. To get tons-gross/tons-net, divide gross ton-mi (including locomotives) (in the "operating statistics" table by the revenue ton-miles (from the "Freight traffic" table)</span> </li> </ol></div> <ul><li>Астахов П.Н. <span class="languageicon">(in Russian)</span> "Сопротивление движению железнодорожного подвижного состава" (Resistance to motion of railway rolling stock) Труды ЦНИИ МПС (ISSN 0372-3305). Выпуск 311 (Vol. 311). - Москва: Транспорт, 1966. – 178 pp. <a rel="nofollow" class="external text" href="http://oskicat.berkeley.edu/record=b12003170~S1">perm. record at UC Berkeley</a> (In 2012, full text was on the Internet but the U.S. was blocked)</li> <li>Деев В.В., Ильин Г.А., Афонин Г.С. <span class="languageicon">(in Russian)</span> "Тяга поездов" (Traction of trains) Учебное пособие. - М.: Транспорт, 1987. - 264 pp.</li> <li>Hay, William W. "Railroad Engineering" New York, Wiley 1953</li> <li><a href="/wiki/Mayo_D._Hersey" title="Mayo D. Hersey">Hersey, Mayo D.</a>, "Rolling Friction" <i>Transactions of the ASME</i>, April 1969 pp. 260–275 and <i>Journal of Lubrication Technology</i>, January 1970, pp. 83–88 (one article split between two journals) Except for the "Historical Introduction" and a survey of the literature, it is mainly about laboratory testing of mine railroad cast iron wheels of diameters 8″ to 24 done in the 1920s (almost a half century delay between experiment and publication).</li> <li>Hoerner, Sighard F., "Fluid dynamic drag", published by the author, 1965. (Chapt. 12 is "Land-Borne Vehicles" and includes rolling resistance (trains, autos, trucks).)</li> <li>Roberts, G. B., "Power wastage in tires", International Rubber Conference, Washington, D.C. 1959.</li> <li>U.S National Bureau of Standards, "Mechanics of Pneumatic Tires", Monograph #132, 1969–1970.</li> <li>Williams, J. A. <i>Engineering tribology'. Oxford University Press, 1994.</i></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Rolling_resistance&action=edit&section=25" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output 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.navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Tires548" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Tires" title="Template:Tires"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Tires" title="Template talk:Tires"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Tires" title="Special:EditPage/Template:Tires"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Tires548" style="font-size:114%;margin:0 4em"><a href="/wiki/Tire" title="Tire">Tires</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Outline_of_tires#Types" title="Outline of tires">Types</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tubeless_tire" title="Tubeless tire">Tubeless tire</a></li> <li><a href="/wiki/Radial_tire" title="Radial tire">Radial tire</a></li> <li><a href="/wiki/Low_rolling_resistance_tire" title="Low rolling resistance tire">Low rolling resistance tire</a></li> <li><a href="/wiki/Run-flat_tire" title="Run-flat tire">Run-flat tire</a></li> <li><a href="/wiki/Michelin_PAX_System" title="Michelin PAX System">Michelin PAX System</a></li> <li><a href="/wiki/Airless_tire" title="Airless tire">Airless tire</a></li> <li><a href="/wiki/Tweel" title="Tweel">Tweel</a></li> <li><a href="/wiki/Rain_tyre" title="Rain tyre">Rain tyre</a></li> <li><a href="/wiki/Snow_tire" title="Snow tire">Snow tire</a></li> <li><a href="/wiki/All-terrain_tire" class="mw-redirect" title="All-terrain tire">All-terrain tire</a></li> <li><a href="/wiki/Bar_grip" title="Bar grip">Bar grip</a></li> <li><a href="/wiki/Off-road_tire" title="Off-road tire">Knobby tire</a></li> <li><a href="/wiki/Off-roading#Large_tires" title="Off-roading">Large tire</a></li> <li><a href="/wiki/Mud-terrain_tyre" class="mw-redirect" title="Mud-terrain tyre">Mud-terrain tire</a></li> <li><a href="/wiki/Paddle_tire" title="Paddle tire">Paddle tire</a></li> <li><a href="/wiki/Orange_oil_tires" title="Orange oil tires">Orange oil tires</a></li> <li><a href="/wiki/Whitewall_tire" title="Whitewall tire">Whitewall tire</a></li> <li><a href="/wiki/Tire#Aircraft" title="Tire">Aircraft tire</a></li> <li><a href="/wiki/Tundra_tire" title="Tundra tire">Tundra tire</a></li> <li><a href="/wiki/Bicycle_tire" title="Bicycle tire">Bicycle tire</a></li> <li><a href="/wiki/Tubular_tire" class="mw-redirect" title="Tubular tire">Tubular tire</a></li> <li><a href="/wiki/Lego_tire" class="mw-redirect" title="Lego tire">Lego tire</a></li> <li><a href="/wiki/Motorcycle_tyre" title="Motorcycle tyre">Motorcycle tyre</a></li> <li><a href="/wiki/Tire_code#Tractor_tires" title="Tire code">Tractor tire</a></li> <li><a href="/wiki/Racing_slick" title="Racing slick">Racing slick</a></li> <li><a href="/wiki/Formula_One_tyres" title="Formula One tyres">Formula One tyres</a></li> <li><a href="/wiki/Spare_tire" title="Spare tire">Spare tire</a></li> <li><a href="/wiki/Continental_tire" title="Continental tire">Continental tire</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Outline_of_tires#Components" title="Outline of tires">Components</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tire_bead" title="Tire bead">Bead</a></li> <li><a href="/wiki/Beadlock" title="Beadlock">Beadlock</a></li> <li><a href="/wiki/Tire_tread" title="Tire tread">Tread</a></li> <li><a href="/wiki/Siping_(rubber)" title="Siping (rubber)">Siping (rubber)</a></li> <li><a href="/wiki/Valve_stem" title="Valve stem">Valve stem</a> <ul><li><a href="/wiki/Dunlop_valve" title="Dunlop valve">Dunlop valve</a></li> <li><a href="/wiki/Presta_valve" title="Presta valve">Presta valve</a></li> <li><a href="/wiki/Schrader_valve" title="Schrader valve">Schrader valve</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Outline_of_tires#Attributes" title="Outline of tires">Attributes</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Camber_thrust" title="Camber thrust">Camber thrust</a></li> <li><a href="/wiki/Circle_of_forces" title="Circle of forces">Circle of forces</a></li> <li><a href="/wiki/Cold_inflation_pressure" title="Cold inflation pressure">Cold inflation pressure</a></li> <li><a href="/wiki/Contact_patch" title="Contact patch">Contact patch</a></li> <li><a href="/wiki/Cornering_force" title="Cornering force">Cornering force</a></li> <li><a href="/wiki/Ground_pressure" title="Ground pressure">Ground pressure</a></li> <li><a href="/wiki/Pacejka#The_Pacejka_.22Magic_Formula.22_tire_models" class="mw-redirect" title="Pacejka">Pacejka's Magic Formula</a></li> <li><a href="/wiki/Pneumatic_trail" title="Pneumatic trail">Pneumatic trail</a></li> <li><a href="/wiki/Relaxation_length" title="Relaxation length">Relaxation length</a></li> <li><a class="mw-selflink selflink">Rolling resistance</a></li> <li><a href="/wiki/Self_aligning_torque" title="Self aligning torque">Self aligning torque</a></li> <li><a href="/wiki/Slip_angle" title="Slip angle">Slip angle</a></li> <li><a href="/wiki/Steering_ratio#Alternate_definition" title="Steering ratio">Steering ratio</a></li> <li><a href="/wiki/Tire_balance" title="Tire balance">Tire balance</a></li> <li><a href="/wiki/Tire_load_sensitivity" title="Tire load sensitivity">Tire load sensitivity</a></li> <li><a href="/wiki/Tire_uniformity" title="Tire uniformity">Tire uniformity</a></li> <li><a href="/wiki/Lateral_Force_Variation" class="mw-redirect" title="Lateral Force Variation">Lateral Force Variation</a></li> <li><a href="/wiki/Radial_Force_Variation" class="mw-redirect" title="Radial Force Variation">Radial Force Variation</a></li> <li><a href="/wiki/Traction_(engineering)" class="mw-redirect" title="Traction (engineering)">Traction (engineering)</a></li> <li><a href="/wiki/Treadwear_rating" class="mw-redirect" title="Treadwear rating">Treadwear rating</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Outline_of_tires#Behaviors" title="Outline of tires">Behaviors</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aquaplaning" title="Aquaplaning">Aquaplaning</a></li> <li><a href="/wiki/Groove_wander" title="Groove wander">Groove wander</a></li> <li><a href="/wiki/Slip_(vehicle_dynamics)" title="Slip (vehicle dynamics)">Slip (vehicle dynamics)</a></li> <li><a href="/wiki/Tramlining" title="Tramlining">Tramlining</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Outline_of_tires#Maintenance" title="Outline of tires">Maintenance</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tire_maintenance" title="Tire maintenance">Tire maintenance</a></li> <li><a href="/wiki/Tire_rotation" title="Tire rotation">Tire rotation</a></li> <li><a href="/wiki/Bicycle_pump" title="Bicycle pump">Bicycle pump</a></li> <li><a href="/wiki/Central_Tire_Inflation_System" class="mw-redirect" title="Central Tire Inflation System">Central Tire Inflation System</a></li> <li><a href="/wiki/Tire_mousse" title="Tire mousse">Tire mousse</a></li> <li><a href="/wiki/Tire-pressure_monitoring_system" title="Tire-pressure monitoring system">Tire-pressure monitoring system</a></li> <li><a href="/wiki/Tire-pressure_gauge" title="Tire-pressure gauge">Tire-pressure gauge</a></li> <li><a href="/wiki/Direct_TPMS" title="Direct TPMS">Direct TPMS</a></li> <li><a href="/wiki/Bead_breaker" title="Bead breaker">Bead breaker</a></li> <li><a href="/wiki/Tire_changer" title="Tire changer">Tire changer</a></li> <li><a href="/wiki/Tire_iron" title="Tire iron">Tire iron</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Outline_of_tires#Life_cycle" title="Outline of tires">Life cycle</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tire_manufacturing" title="Tire manufacturing">Tire manufacturing</a></li> <li><a href="/wiki/List_of_tire_companies" title="List of tire companies">List of tire companies</a></li> <li><a href="/wiki/Retread" title="Retread">Retread</a></li> <li><a href="/wiki/Waste_tires" class="mw-redirect" title="Waste tires">Waste tires</a></li> <li><a href="/wiki/Tire_recycling" title="Tire recycling">Tire recycling</a></li> <li><a href="/wiki/Tire_fire" title="Tire fire">Tire fire</a></li> <li><a href="/wiki/Blowout_(tire)" title="Blowout (tire)">Blowout</a></li> <li><a href="/wiki/Flat_tire" title="Flat tire">Flat tire</a></li> <li><a href="/wiki/Ozone_cracking" title="Ozone cracking">Ozone cracking</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Outline_of_tires#Organizations" title="Outline of tires">Organizations</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/European_Tyre_and_Rim_Technical_Organisation" title="European Tyre and Rim Technical Organisation">European Tyre and Rim Technical Organisation</a></li> <li><a href="/wiki/Tire_Society" title="Tire Society">Tire Society</a></li> <li><a href="/wiki/Tire_Science_and_Technology" title="Tire Science and Technology">Tire Science and Technology</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Outline_of_tires#Identification" title="Outline of tires">Identification</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Tire_code" title="Tire code">Tire code</a></li> <li><a href="/wiki/Plus_sizing" title="Plus sizing">Plus sizing</a></li> <li><a href="/wiki/Tire_label" class="mw-redirect" title="Tire label">Tire label</a></li> <li><a href="/wiki/UTQG" class="mw-redirect" title="UTQG">Uniform Tire Quality Grading (UTQG)</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2" style="font-weight:bold;"><div> <ul><li><span class="noviewer" 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