Document Zbl 1507.74163

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(2022)</a>. </div> <div class="abstract">Summary: The simplest frequency formulation for nonlinear oscillators is introduced and proved, and a modification is suggested. A fractal vibration in a porous medium is introduced, and its low-frequency property is elucidated by the frequency formulation. It reveals that the inertia force in a fractal space is equivalent to a couple of the inertia force and the damping force for the traditional differential model.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7537362">Cited in <strong>6</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A74H45" title="MSC2020">74H45</a> </td> <td class="space"> Vibrations in dynamical problems in solid mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A74K05" title="MSC2020">74K05</a> </td> <td class="space"> Strings </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A74S40" title="MSC2020">74S40</a> </td> <td class="space"> Applications of fractional calculus in solid mechanics </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3ADuffing+equation">Duffing equation</a>; <a href="/?q=ut%3Aperiodic+solution">periodic solution</a>; <a href="/?q=ut%3Atwo-scale+fractional+derivative">two-scale fractional derivative</a>; <a href="/?q=ut%3Afractional+calculus">fractional calculus</a>; <a href="/?q=ut%3Astring+vibration">string vibration</a>; <a href="/?q=ut%3Aporous+medium">porous medium</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1507.74163" data-ciurl="/ci/07537362" data-biburl="/bibtex/07537362.bib" data-amsurl="/amsrefs/07537362.bib" data-xmlurl="/xml/07537362.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07537362.pdf" title="Zbl 1507.74163 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1142/S0218348X22500463" aria-label="DOI for “A modified frequency-amplitude formulation for fractal vibration systems”" title="10.1142/S0218348X22500463">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Ji, Q. P., Wang, J., Lu, L. X. and Ge, C. F., Li-He’s modified homotopy perturbation method coupled with the energy method for the dropping shock response of a tangent nonlinear packaging system, J. Low Freq. Noise Vibr. Act. Control40 (2021) 675-682.</td> </tr><tr> <td>[2]</td> <td class="space">Song, H. Y., A thermodynamic model for a packing dynamical system, Therm. Sci.24 (2020) 2331-2335.</td> </tr><tr> <td>[3]</td> <td class="space">Mei, Y., Liu, Y. Q. and He, J. H., On the mountain-river-desert relation, Therm. Sci.25(6) (2021) 4817-4822.</td> </tr><tr> <td>[4]</td> <td class="space">He, J. H. and Mostapha, D. R., Insight into the significance of Hall current and Joule heating on the dynamics of Darcy-Forchheimer peristaltic flow of Rabinowitsch fluid, J. Math.2021 (2021) 3638807, https://doi.org/10.1155/2021/3638807. · <a href="/1477.76108" class="nowrap">Zbl 1477.76108</a></td> </tr><tr> <td>[5]</td> <td class="space">He, J. H. and Abd-Elazem, N. Y., Insights into partial slips and temperature jumps of a nanofluid flow over a stretched or shrinking surface, Energies14 (2021) 6691.</td> </tr><tr> <td>[6]</td> <td class="space">He, J. H., Moatimid, G. M. and Mostapha, D. R., Nonlinear instability of two streaming-superposed magnetic Reiner-Rivlin fluids by He-Laplace method, J. Electroanal. Chem.895 (2021) 115388.</td> </tr><tr> <td>[7]</td> <td class="space">Jin, X., Liu, M., Pan, F., Li, Y. and Fan, J., Low frequency of a deforming capillary vibration, part 1: Mathematical model, J. Low Freq. Noise Vibr. Act. Control38 (2019) 1676-1680.</td> </tr><tr> <td>[8]</td> <td class="space">He, J. and Jin, X., A short review on analytical methods for the capillary oscillator in a nanoscale deformable tube, Math. Methods Appl. Sci. (2020), https://doi.org/10.1002/mma.6321.</td> </tr><tr> <td>[9]</td> <td class="space">He, C. H., Liu, C., He, J. H., Shirazi, A. H. and Mohammad-Sedighi, H., Passive atmospheric water harvesting utilizing an ancient Chinese ink slab and its possible applications in modern architecture, Facta Univ., Mech. Eng.19 (2021) 229-239, https://doi.org/10.22190/FUME201203001H.</td> </tr><tr> <td>[10]</td> <td class="space">He, J.-H. and El-Dib, Y. O., Homotopy perturbation method for Fangzhu oscillator, J. Math. Chem.58 (2020) 2245-2253. · <a href="/1470.34050" class="nowrap">Zbl 1470.34050</a></td> </tr><tr> <td>[11]</td> <td class="space">He, J.-H. and El-Dib, Y. O., Homotopy perturbation method with three expansions for Helmholtz-Fangzhu oscillator, Int. J. Mod. Phys. B35 (2021) 2150244, https://doi.org/10.1142/S0217979221502441. · <a href="/1494.34075" class="nowrap">Zbl 1494.34075</a></td> </tr><tr> <td>[12]</td> <td class="space">He, J. H., Qie, N., He, C. H. and Gepreel, K.. Fast identification of the pull-in voltage of a nano/micro-electromechanical system, J. Low Freq. Noise Vibr. Act. Control (2022), https://doi.org/10.1177/14613484211068252.</td> </tr><tr> <td>[13]</td> <td class="space">Yao, X. and He, J. H., On fabrication of nanoscale non-smooth fibers with high geometric potential and nanoparticle’s non-linear vibration, Therm. Sci.24 (2020) 2491-2497.</td> </tr><tr> <td>[14]</td> <td class="space">Palacios-Pineda, L. M., Elias-Zuniga, A., Martinez-Romero, O. and Olvera-Trejo, D., Investigation of the fractal response of a nonlinear packaging system, Fractals (2021), https://doi.org/10.1142/S0218348X22500074. · <a href="/1487.34095" class="nowrap">Zbl 1487.34095</a></td> </tr><tr> <td>[15]</td> <td class="space">Elías-Zúñiga, A., Romero-Martínez, O., Olvera-Trejo, D. and Palacios-Pineda, L. M., Fractal equation of motion of a non-Gaussian polymer chain: investigating its dynamic fractal response using an ancient Chinese algorithm, J. Math. Chem. (2022), https://doi.org/10.1007/s10910-021-01310-x. · <a href="/1487.81157" class="nowrap">Zbl 1487.81157</a></td> </tr><tr> <td>[16]</td> <td class="space">Elías-Zúñiga, A., Martínez-Romero, O., Olvera-Trejo, D. and Palacios-Pineda, L. M., An efficient ancient Chinese algorithm to investigate the dynamics response of a fractal microgravity forced oscillator, Fractals29 (2021) 2150144. · <a href="/1487.34095" class="nowrap">Zbl 1487.34095</a></td> </tr><tr> <td>[17]</td> <td class="space">He, J. H., Amer, T. S., Elnaggar, S. and Galal, A. A., Periodic property and instability of a rotating pendulum system, Axioms10 (2021) 191.</td> </tr><tr> <td>[18]</td> <td class="space">Anjum, N., He, J. H., Ain, Q. T. and Tian, D., Li-He’s modified homotopy perturbation method for doubly-clamped electrically actuated microbeams-based microelectromechanical system, Facta Univ., Mech. Eng.19(4) (2021) 601-612, https://doi.org/10.22190/FUME210112025A.</td> </tr><tr> <td>[19]</td> <td class="space">He, J.-H., El-Dib, Y. O. and Mady, A. A., Homotopy perturbation method for the fractal Toda oscillator, Fractal Fract.5 (2021) 93, https://doi.org/10.3390/fractalfract5030093.</td> </tr><tr> <td>[20]</td> <td class="space">He, C. H., Tian, D., Moatimid, G. M., Salman, H. F. and Zekry, M. H., Hybrid Rayleigh-van der Pol-Duffing oscillator (HRVD): Stability analysis and controller, J. Low Freq. Noise Vibr. Act. Control (2021), https://doi.org/10.1177/14613484211026407.</td> </tr><tr> <td>[21]</td> <td class="space">Wang, K. J., On new abundant exact traveling wave solutions to the local fractional Gardner equation defined on Cantor sets, Math. Methods Appl. Sci. (2021), https://doi.org/10.1002/mma.7897. · <a href="/1529.35127" class="nowrap">Zbl 1529.35127</a></td> </tr><tr> <td>[22]</td> <td class="space">Wang, K. J., Generalized variational principle and periodic wave solution to the modified equal width-Burgers equation in nonlinear dispersion media, Phys. Lett. A419 (2021) 127723, https://doi.org/10.1016/j.physleta.2021.127723. · <a href="/07413640" class="nowrap">Zbl 07413640</a></td> </tr><tr> <td>[23]</td> <td class="space">Wang, K. J. and Zhang, P. L., Investigation of the periodic solution of the time-space fractional Sasa-Satsuma equation arising in the monomode optical fibers, Europhys. Lett. (2021), https://doi.org/10.1209/0295-5075/ac2a62.</td> </tr><tr> <td>[24]</td> <td class="space">Wang, K. J. and Wang, G. D., Study on the nonlinear vibration of embedded carbon nanotube via the Hamiltonian-based method, J. Low Freq. Noise Vibr. Act. Control (2021), https://doi.org/10.1177/14613484211032757.</td> </tr><tr> <td>[25]</td> <td class="space">Han, C., Wang, Y. L. and Li, Z. Y., Numerical solutions of space fractional variable-coefficient KdV-modified KdV equation by Fourier spectral method, Fractals (2021), https://doi.org/10.1142/S0218348X21502467. · <a href="/1506.35199" class="nowrap">Zbl 1506.35199</a></td> </tr><tr> <td>[26]</td> <td class="space">Dan, D. D., Zhang, W., Wang, Y. L. and Ban, T. T., Using piecewise reproducing kernel method and Legendre polynomial for solving a class of the time variable fractional order advection-reaction-diffusion equation, Therm. Sci.25 (2021) 1261-1268.</td> </tr><tr> <td>[27]</td> <td class="space">He, J. H., The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators, J. Low Freq. Noise Vibr. Act. Control38 (2019) 1252-1260, https://doi.org/10.1177/1461348419844145.</td> </tr><tr> <td>[28]</td> <td class="space">He, J.-H., The simplest approach to nonlinear oscillators, Results Phys.15 (2019) 102546.</td> </tr><tr> <td>[29]</td> <td class="space">Qie, N., Hou, W. F. and He, J. H., The fastest insight into the large amplitude vibration of a string, Rep. Mech. Eng.2 (2020) 1-5, https://doi.org/10.31181/rme200102001q.</td> </tr><tr> <td>[30]</td> <td class="space">Liu, C. X., A short remark on He’s frequency formulation, J. Low Freq. Noise Vibr. Act. Control40 (2021) 672-674.</td> </tr><tr> <td>[31]</td> <td class="space">Feng, G. Q., He’s frequency formula to fractal undamped Duffing equation, J. Low Freq. Noise Vibr. Act. Control (2021), https://doi.org/10.1177/1461348421992608.</td> </tr><tr> <td>[32]</td> <td class="space">Wu, Y. and Liu, Y. P., Residual calculation in He’s frequency-amplitude formulation, J. Low Freq. Noise Vibr. Act. Control40 (2021) 1040-1047.</td> </tr><tr> <td>[33]</td> <td class="space">Wang, K. L., A new fractal model for the soliton motion in a microgravity space, Int. J. Numer. Methods Heat Fluid Flow (2020), https://doi.org/10.1108/HFF-05-2020-0247.</td> </tr><tr> <td>[34]</td> <td class="space">Wang, K. L., He’s frequency formulation for fractal nonlinear oscillator arising in a microgravity space, Numer. Methods Partial Differ. Equ.37 (2021) 1374-1384. · <a href="/07776019" class="nowrap">Zbl 07776019</a></td> </tr><tr> <td>[35]</td> <td class="space">Wang, K. L. and Wei, C. F., A powerful and simple frequency formula to nonlinear fractal oscillators, J. Low Freq. Noise Vibr. Act. Control (2020), https://doi.org/10.1177/1461348420947832.</td> </tr><tr> <td>[36]</td> <td class="space">Wang, K. L., A new fractal transform frequency formulation for fractal nonlinear oscillator, Fractals29 (2021) 2150062. · <a href="/1492.34014" class="nowrap">Zbl 1492.34014</a></td> </tr><tr> <td>[37]</td> <td class="space">He, J. H., Skrzypacz, P. S., Zhang, Y. N. and Pang, J., Approximate periodic solutions to microelectromechanical system oscillator subject to magnetostatic excitation, Math. Methods Appl. Sci. (2020), https://doi.org/10.1002/mma.7018.</td> </tr><tr> <td>[38]</td> <td class="space">Zuo, Y.-T., A gecko-like fractal receptor of a three-dimensional printing technology: A fractal oscillator, J. Math. Chem.59 (2021) 735-744. · <a href="/1466.92309" class="nowrap">Zbl 1466.92309</a></td> </tr><tr> <td>[39]</td> <td class="space">Elias-Zuniga, A., Palacios-Pineda, L. M., Jimenez-Cedeno, I. H., Martinez-Romero, O. and Olvera-Trejo, D., Analytical solution of the fractal cubi-quintic Duffing equation, Fractals29 (2021) 2150080. · <a href="/1489.34002" class="nowrap">Zbl 1489.34002</a></td> </tr><tr> <td>[40]</td> <td class="space">Elias-Zuniga, A.et al., Equivalent power-form representation of the fractal Toda oscillator, Fractals29 (2021) 2150034. · <a href="/1481.78022" class="nowrap">Zbl 1481.78022</a></td> </tr><tr> <td>[41]</td> <td class="space">Elias-Zuniga, A.et al., He’s frequency-amplitude formulation for nonlinear oscillators using Jacobi elliptic functions, J. Low Freq. Noise Vibr. Act. Control39 (2020) 1261-1223.</td> </tr><tr> <td>[42]</td> <td class="space">Elias-Zuniga, A.et al., A fractal model for current generation in porous electrodes, J. Electroanal. Chem.880 (2021) 114883.</td> </tr><tr> <td>[43]</td> <td class="space">Elías-Zúñiga, A.et al., Enhanced He’s frequency-amplitude formulation for nonlinear oscillators, Results Phys.19 (2020) 103626, https://doi.org/10.1016/j.rinp.2020.103626.</td> </tr><tr> <td>[44]</td> <td class="space">Wang, K. J. and Wang, G. D., Gamma function method for the nonlinear cubic-quintic Duffing oscillators, J. Low Freq. Noise Vibr. Act. Control (2021), https://doi.org/10.1177/14613484211044613.</td> </tr><tr> <td>[45]</td> <td class="space">He, C. H., Wang, J. H. and Yao, S. W., A complement to period/frequency estimation of a nonlinear oscillator, J. Low Freq. Noise Vibr. Act. Control38 (2019) 992-995.</td> </tr><tr> <td>[46]</td> <td class="space">He, J.-H. and García, A., The simplest amplitude-period formula for non-conservative oscillators, Rep. Mech. Eng.2 (2021) 143-148.</td> </tr><tr> <td>[47]</td> <td class="space">Xiao, B. Q.et al., A fractal model for predicting oxygen effective diffusivity of a porous media with rough surface under dry and wet conditions, Fractals29 (2021) 2150076. · <a href="/1500.76079" class="nowrap">Zbl 1500.76079</a></td> </tr><tr> <td>[48]</td> <td class="space">Liu, G. N.et al., A study on gas drainage considering coupling process of fracture-pore microstructure and coal deformation, Fractals29 (2021) 2150065. · <a href="/1498.76090" class="nowrap">Zbl 1498.76090</a></td> </tr><tr> <td>[49]</td> <td class="space">Su, H. B.et al., A fractal Monte Carlo approach to model oil and water two-phase seepage in low-permeability reservoirs with rough surface, Fractals29 (2021) 2150003. · <a href="/1500.76088" class="nowrap">Zbl 1500.76088</a></td> </tr><tr> <td>[50]</td> <td class="space">Liu, Y. H.et al., Fractal analysis of digit rock cores, Fractals28 (2020) 2050144.</td> </tr><tr> <td>[51]</td> <td class="space">Liu, G. N.et al., Study on evolution of fractal dimension for fractured coal seam under multi-field coupling, Fractals28 (2020) 2050072.</td> </tr><tr> <td>[52]</td> <td class="space">He, J.-H., Kou, S.-J., He, C.-H., Zhang, Z.-W. and Gepreel, K. A., Fractal oscillation and its frequency-amplitude property, Fractals29 (2021) 2150105. · <a href="/1489.34014" class="nowrap">Zbl 1489.34014</a></td> </tr><tr> <td>[53]</td> <td class="space">He, J. H. and Ain, Q. T., New promises and future challenges of fractal calculus: from two-scale thermodynamics to fractal variational principle, Therm. Sci.24 (2020) 659-681.</td> </tr><tr> <td>[54]</td> <td class="space">He, J. H., Fractal calculus and its geometrical explanation, Results Phys.10 (2018) 272-276.</td> </tr><tr> <td>[55]</td> <td class="space">He, J. H., On the fractal variational principle for the Telegraph equation, Fractals29 (2021) 2150022. · <a href="/1482.35005" class="nowrap">Zbl 1482.35005</a></td> </tr><tr> <td>[56]</td> <td class="space">Ling, W. W. and Wu, P. X., A fractal variational theory of the Broer-Kaup system in shallow water waves, Therm. Sci.25 (2021) 2051-2056.</td> </tr><tr> <td>[57]</td> <td class="space">Ma, X. S. and Zhang, L. N., He’s fractal calculus and its application to fractal Korteweg-de-Vries equation, Therm. Sci.25 (2021) 2149-2154.</td> </tr><tr> <td>[58]</td> <td class="space">Wei, C. F., Two-scale transform for 2-D fractal heat equation in a fractal space, Therm. Sci.25 (2021) 2339-2345.</td> </tr><tr> <td>[59]</td> <td class="space">He, C. H., Liu, C., He, J. H. and Gepreel, K. A., Low frequency property of a fractal vibration model for a concrete beam, Fractals29 (2021) 2150117, https://doi.org/10.1142/S0218348X21501176C. · <a href="/1482.74083" class="nowrap">Zbl 1482.74083</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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