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Nonlinear dynamics of viscoelastic flexible structural systems by finite element method

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Abstract

This article presents a nonlinear displacement based finite elements model to study and analyze the nonlinear dynamic response of flexible double wishbone structural vehicle suspension system considering damping effect which was not previously discussed elsewhere. Due to large deflection and moderate rotation encountered during passing over road bumps, the kinematic nonlinearity is included through von Kármán strain component. Elastic undamped as well as viscous and viscoelastic damping mechanism are considered and compared. Considering the viscoelastic damping mechanism, the viscoelastic damping mechanism is modeled based on the integral constitutive form, which is recast into an incremental form suitable for finite element implementation. Additionally, the revolute joint element is adopted to incorporate the joint flexibility in the double wishbone system. The plane frame element is adopted to model the suspension links by using Timoshenko beam theory. The developed nonlinear finite element equations of motion are solved through the incremental iterative Newmark technique. The developed procedure is verified by comparing the obtained results with analytical solution and excellent agreement is observed. The applicability of the developed procedure is demonstrated by conducting parametric studies to show the effects of the road irregularities profiles, the vehicle speed, and the material damping coefficients on the nonlinear vibrations response of the double wishbone suspension systems. The obtained results are supportive in the design and manufacturing processes of these structural systems.

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References

  1. Rahman AAA, Mahmoud FF (2016) Analysis of nanocontact problems of layered viscoelastic solids with surface energy effects under different loading patterns. Acta Mech 227(2):527–548. https://doi.org/10.1007/s00707-015-1473-5

    Article  MathSciNet  MATH  Google Scholar 

  2. Abdelrahman AA, El-Shafei AG, Mahmoud FF (2016) Influence of surface energy on the nanoindentation response of elastically-layered viscoelastic materials. Int J Mech Mater Des 12(2):193–209. https://doi.org/10.1007/s10999-015-9301-6

    Article  Google Scholar 

  3. Abdelrahman AA, Eltaher MA, Kabeel AM, Abdraboh AM, Hendy AA (2019) Free and forced analysis of perforated beams. Steel Compos Struct 31(5):489–502. https://doi.org/10.12989/scs.2019.31.5.489

    Article  Google Scholar 

  4. Abdelrahman AA, Nabawy AE, Abdelhaleem AM, Alieldin SS (2019) Dynamic finite element analysis of flexible double wishbone suspension systems with different damping mechanisms. Eur J Comput Mech 5:573–604. https://doi.org/10.13052/ejcm2642-2085.2862

    Article  MathSciNet  Google Scholar 

  5. Abdelrahman AA, El-Shafei AG (2020) Modeling and analysis of the transient response of viscoelastic solids. Waves Random Complex Media. https://doi.org/10.1080/17455030.2020.1714790

    Article  Google Scholar 

  6. Akbaş ŞD (2014) Wave propagation analysis of edge cracked circular beams under impact force. PLoS ONE 9(6):e100496. https://doi.org/10.1371/journal.pone.0100496

    Article  Google Scholar 

  7. Akbaş ŞD (2015) Wave propagation of a functionally graded beam in thermal environments. Steel Compos Struct 19(6):1421–1447. https://doi.org/10.12989/scs.2015.19.6.1421

    Article  Google Scholar 

  8. Akbaş ŞD (2016) Forced vibration analysis of viscoelastic nanobeams embedded in an elastic medium. Smart Struct Syst 18(6):1125–1143. https://doi.org/10.12989/sss.2016.18.6.1125

    Article  Google Scholar 

  9. Akbas SD (2020) Dynamic responses of laminated beams under a moving load in thermal environment. Steel Compos Struct 35(6):729–737. https://doi.org/10.12989/scs.2020.35.6.729

    Article  MathSciNet  Google Scholar 

  10. Akbas SD (2020) Modal analysis of viscoelastic nanorods under an axially harmonic load. Adv Nano Res 8(4):277–282. https://doi.org/10.12989/anr.2020.8.4.277

    Article  Google Scholar 

  11. Akbas SD, Fagheeh YA, Assie AS, Eltaher MA (2020) Dynamic analysis of visco-elastic functionally graded porous thick beams under pulse load. Eng Comput. https://doi.org/10.1007/s00366-020-01070-3

    Article  Google Scholar 

  12. Alotta G, Barrera O, Cocks A, Di Paola M (2018) The finite element implementation of 3D fractional viscoelastic constitutive models. Finite Elem Anal Des 146:28–41. https://doi.org/10.1016/j.finel.2018.04.003

    Article  MathSciNet  Google Scholar 

  13. Assie AE, Eltaher MA, Mahmoud FF (2010) The response of viscoelastic-frictionless bodies under normal impact. Int J Mech Sci 52(3):446–454. https://doi.org/10.1016/j.ijmecsci.2009.11.005

    Article  Google Scholar 

  14. Assie AE, Eltaher MA, Mahmoud FF (2010) Modeling of viscoelastic contact-impact problems. Appl Math Model 34(9):2336–2352. https://doi.org/10.1016/j.apm.2009.11.001

    Article  MathSciNet  MATH  Google Scholar 

  15. Assie AE, Eltaher MA, Mahmoud FF (2011) Behavior of a viscoelastic composite plates under transient load. J Mech Sci Technol 25(5):1129. https://doi.org/10.1007/s12206-011-0302-6

    Article  Google Scholar 

  16. Awati PS, Judulkar LM (2014) Modal and stress analysis of lower wishbone arm along with topology. Int J Appl Innov Eng Manag 3(5):296–302

    Google Scholar 

  17. Azevêdo A, Vasconcelos A, dos Santos S (2016) Dynamic analysis of elastically supported Timoshenko beam. Revista Interdisciplinar de Pesquisa em Engenharia 2(34):71–85

    Google Scholar 

  18. Bathe KJ (2006) Finite element procedures. Klaus-Jurgen Bathe, Englewood Cliffs

    MATH  Google Scholar 

  19. Bratland M, Haugen B, Rølvåg T (2014) Modal analysis of active flexible multibody systems containing PID controllers with non-collocated sensors and actuators. Finite Elem Anal Des 91:16–29. https://doi.org/10.1016/j.finel.2014.06.011

    Article  Google Scholar 

  20. Brigham JC, Aquino W (2009) Inverse viscoelastic material characterization using pod reduced-order modeling in acoustic–structure interaction. Comput Methods Appl Mech Eng 198(9–12):893–903. https://doi.org/10.1016/j.cma.2008.10.018

    Article  MATH  Google Scholar 

  21. Capsoni A, Viganò GM, Bani-Hani K (2013) On damping effects in Timoshenko beams. Int J Mech Sci 73:27–39. https://doi.org/10.1016/j.ijmecsci.2013.04.001

    Article  Google Scholar 

  22. Carrera E, Giunta G, Petrolo M (2011) Beam structures: classical and advanced theories. Wiley, Hoboken

    Book  Google Scholar 

  23. Chen LQ, Yang XD (2005) Stability in parametric resonance of axially moving viscoelastic beams with time-dependent speed. J Sound Vib 284(3–5):879–891. https://doi.org/10.1016/j.jsv.2004.07.024

    Article  Google Scholar 

  24. Chen Z, Lim CW (2018) Static-dynamic relationship for flexural free vibration of extensible beams. Int J Struct Stab Dyn 18(09):1871010. https://doi.org/10.1142/S0219455418710104

    Article  MathSciNet  Google Scholar 

  25. Cook RD (2007) Concepts and applications of finite element. Wiley

  26. Chopra AK (2012) Dynamics of structures. Pearson Education, Upper Saddle River, pp 409–429

    Google Scholar 

  27. Diaz MI, Aquino W, Bonnet M (2015) A modified error in constitutive equation approach for frequency-domain viscoelasticity imaging using interior data. Comput Methods Appl Mech Eng 296:129–149. https://doi.org/10.1016/j.cma.2015.07.025

    Article  MathSciNet  MATH  Google Scholar 

  28. Ding H, Zhang GC, Chen LQ, Yang SP (2012) Forced vibrations of supercritically transporting viscoelastic beams. J Vib Acoust. https://doi.org/10.1115/1.4006184

    Article  Google Scholar 

  29. Ebrahimi F, Hosseini SHS (2020) Nonlinear dynamics and stability of viscoelastic nanoplates considering residual surface stress and surface elasticity effects: a parametric excitation analysis. Eng Comput. https://doi.org/10.1007/s00366-019-00906-x

    Article  Google Scholar 

  30. Fahimi S, Baghani M, Zakerzadeh MR, Eskandari A (2018) Developing a visco-hyperelastic material model for 3D finite deformation of elastomers. Finite Elem Anal Des 140:1–10. https://doi.org/10.1016/j.finel.2017.10.009

    Article  MathSciNet  Google Scholar 

  31. Feng H, Cui X, Li G (2012) Static and dynamic analysis of Timoshenko beam using nodal integration technique. Int J Appl Mech 4(04):1250045. https://doi.org/10.1142/S1758825112500457

    Article  Google Scholar 

  32. Findley WN, Davis FA (2013) Creep and relaxation of nonlinear viscoelastic materials. Courier Corporation, New York

    Google Scholar 

  33. Froio D, Rizzi E (2016) Analytical solution for the elastic bending of beams lying on a variable Winkler support. Acta Mech 227(4):1157–1179. https://doi.org/10.1007/s00707-015-1508-y

    Article  MathSciNet  MATH  Google Scholar 

  34. Gadade B, Todkar RG (2015) Design, analysis of A-type front lower suspension arm in commercial vehicle. Int Res J Eng Technol 2(7):759–766

    Google Scholar 

  35. Ghayesh MH, Amabili M (2012) Nonlinear dynamics of axially moving viscoelastic beams over the buckled state. Comput Struct 112:406–421. https://doi.org/10.1016/j.compstruc.2012.09.005

    Article  Google Scholar 

  36. Gholipour A, Ghayesh MH, Hussain S (2020) A continuum viscoelastic model of Timoshenko NSGT nanobeams. Eng Comput. https://doi.org/10.1007/s00366-020-01017-8

    Article  Google Scholar 

  37. Grandhi RV (1990) Optimum design of space structures with active and passive damping. Eng Comput 6(3):177–183

    Article  Google Scholar 

  38. Güler D (2006) Dynamic analysis of double wishbone suspension. Master’s Thesis, İzmir Institute of Technology

  39. Hamed MA, Abo-bakr RM, Mohamed SA, Eltaher MA (2020) Influence of axial load function and optimization on static stability of sandwich functionally graded beams with porous core. Eng Comput. https://doi.org/10.1007/s00366-020-01023-w

    Article  Google Scholar 

  40. Hamed MA, Mohamed NA, Eltaher MA (2020) Stability buckling and bending of nanobeams including cutouts. Eng Comput. https://doi.org/10.1007/s00366-020-01063-2

    Article  Google Scholar 

  41. Hammerand DC (1999) Geometrically-linear and nonlinear analysis of linear viscoelastic composites using the finite element method. Doctoral dissertation, Virginia Tech. http://hdl.handle.net/10919/28893

  42. Hassaan GA (2014) Car dynamics using quarter model and passive suspension, part I: effect of suspension damping and car speed. Int J Comput Tech 1(2):1–9

    Google Scholar 

  43. Hatami H, Hosseini M (2019) Elastic-plastic analysis of bending moment–axial force interaction in metallic beam of T-section. J Appl Comput Mech 5(1):162–173. https://doi.org/10.22055/JACM.2018.25857.1298

    Article  Google Scholar 

  44. Hemin MM, Rahman MM, Omar RM (2011) Dynamic analysis of vehicle arm based on finite element approach. J Adv Sci Eng Res 1(1):124–137

    Google Scholar 

  45. Hieu D, Hai NQ (2019) Free vibration analysis of quintic nonlinear beams using equivalent linearization method with a weighted averaging. J Appl Comput Mech 5(1):46–57. https://doi.org/10.22055/JACM.2018.24919.1217

    Article  Google Scholar 

  46. Hilton HH (2009) Viscoelastic Timoshenko beam theory. Mech Time-Depend Mat 13(1):1–10

    Article  Google Scholar 

  47. Ijagbemi CO, Oladapo BI, Campbell HM, Ijagbemi CO (2016) Design and simulation of fatigue analysis for a vehicle suspension system (VSS) and its effect on global warming. Procedia Eng 159:124–132. https://doi.org/10.1016/j.proeng.2016.08.135

    Article  Google Scholar 

  48. Inman DJ (2014) Engineering vibration. Pearson Education, Englewood Cliffs

    Google Scholar 

  49. Jena RM, Chakraverty S, Jena SK (2019) Dynamic response analysis of fractionally damped beams subjected to external loads using homotopy analysis method. J Appl Comput Mech 5(2):355–366. https://doi.org/10.22055/JACM.2019.27592.1419

    Article  Google Scholar 

  50. Joshi, A., & Singh, K. K. (2019). Structural analysis and comparison of multiaxle vehicle suspension system. Available at SSRN 3394038

  51. Kahoul H, Belhour S, Bellaouar A, Dron JP (2019) Fatigue life prediction of upper arm suspension using strain life approach. J Eng Des Technol. https://doi.org/10.1108/JEDT-03-2018-0047

    Article  Google Scholar 

  52. Karimiasl M, Ebrahimi F, Mahesh V (2019) Postbuckling analysis of piezoelectric multiscale sandwich composite doubly curved porous shallow shells via Homotopy Perturbation Method. Eng Comput. https://doi.org/10.1007/s00366-019-00841-x

    Article  Google Scholar 

  53. Keramat A, Shirazi KH (2014) Finite element based dynamic analysis of viscoelastic solids using the approximation of volterra integrals. Finite Elem Anal Des 86:89–100. https://doi.org/10.1016/j.finel.2014.03.010

    Article  MathSciNet  Google Scholar 

  54. Kiasat MS, Zamani HA, Aghdam MM (2014) On the transient response of viscoelastic beams and plates on viscoelastic medium. Int J Mech Sci 83:133–145. https://doi.org/10.1016/j.ijmecsci.2014.03.007

    Article  Google Scholar 

  55. Kim W, Lee JH (2018) An improved explicit time integration method for linear and nonlinear structural dynamics. Comput Struct 206:42–53. https://doi.org/10.1016/j.compstruc.2018.06.005

    Article  Google Scholar 

  56. Kohnke P (2013) ANSYS theory reference manual. Ansys Inc., Canonsburg

    Google Scholar 

  57. Korchagin V, Dolbow J, Stepp D (2007) A theory of amorphous viscoelastic solids undergoing finite deformations with application to hydrogels. Int J Solids Struct 44(11–12):3973–3997. https://doi.org/10.1016/j.ijsolstr.2006.11.002

    Article  MATH  Google Scholar 

  58. Lakes R, Lakes RS (2009) Viscoelastic materials. Cambridge University Press, Cambridge

    Book  Google Scholar 

  59. Latifi M, Kharazi M, Ovesy HR (2017) Effect of integral viscoelastic core on the nonlinear dynamic behaviour of composite sandwich beams with rectangular cross sections. Int J Mech Sci 123:141–150. https://doi.org/10.1016/j.ijmecsci.2017.02.007

    Article  Google Scholar 

  60. Lee D, Woo Y, Lee S, Han C (2008) Design consideration of the nonlinear specifications in the automotive body. Finite Elem Anal Des 44(14):851–861. https://doi.org/10.1016/j.finel.2008.06.002

    Article  Google Scholar 

  61. Li B, Wang S, Wu X, Wang B (2014) Dynamic response of continuous beams with discrete viscoelastic supports under sinusoidal loading. Int J Mech Sci 86:76–82. https://doi.org/10.1016/j.ijmecsci.2014.02.005

    Article  Google Scholar 

  62. Li YX, Hu ZJ, Sun LZ (2016) Dynamical behavior of a double-beam system interconnected by a viscoelastic layer. Int J Mech Sci 105:291–303. https://doi.org/10.1016/j.ijmecsci.2015.11.023

    Article  Google Scholar 

  63. Liu P, Huda MN, Tang Z, Sun L (2020) A self-propelled robotic system with a visco-elastic joint: dynamics and motion analysis. Eng Comput 36(2):655–669. https://doi.org/10.1007/s00366-019-00722-3

    Article  Google Scholar 

  64. Luo W, Xia Y, Weng S (2015) Vibration of Timoshenko beam on hysteretically damped elastic foundation subjected to moving load. Sci China Phys Mech Astron 58(8):84601. https://doi.org/10.1007/s11433-015-5664-9

    Article  Google Scholar 

  65. Mahmoud FF, El-Shafei AG, Attia MA, Rahman AA (2013) Analysis of quasistatic frictional contact problems in nonlinear viscoelasticity with large deformations. Int J Mech Sci 66:109–119. https://doi.org/10.1016/j.ijmecsci.2012.11.001

    Article  Google Scholar 

  66. Martin O (2016) A modified variational iteration method for the analysis of viscoelastic beams. Appl Math Model 40(17–18):7988–7995. https://doi.org/10.1016/j.apm.2016.04.011

    Article  MathSciNet  MATH  Google Scholar 

  67. Mohamed N, Mohamed SA, Eltaher MA (2020) Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model. Eng Comput. https://doi.org/10.1007/s00366-020-00976-2

    Article  Google Scholar 

  68. Moory-Shirbani M, Sedighi HM, Ouakad HM, Najar F (2018) Experimental and mathematical analysis of a piezoelectrically actuated multilayered imperfect microbeam subjected to applied electric potential. Compos Struct 184:950–960. https://doi.org/10.1016/j.compstruct.2017.10.062

    Article  Google Scholar 

  69. Morin B, Legay A, Deü JF (2018) Reduced order models for dynamic behavior of elastomer damping devices. Finite Elem Anal Des 143:66–75. https://doi.org/10.1016/j.finel.2018.02.001

    Article  Google Scholar 

  70. Nabawy AE, Abdelrahman AA, Abdalla WS, Abdelhaleem AM, Alieldin SS (2019) Analysis of the dynamic behavior of the double wishbone suspension system. Int J Appl Mech. https://doi.org/10.1142/S1758825119500443

    Article  Google Scholar 

  71. Nakamura N (2017) Time history response analysis using extended Rayleigh damping model. Procedia Eng 199:1472–1477. https://doi.org/10.1016/j.proeng.2017.09.408

    Article  Google Scholar 

  72. Namadchi AH, Jandaghi E, Alamatian J (2020) A new model-dependent time integration scheme with effective numerical damping for dynamic analysis. Eng Comput. https://doi.org/10.1007/s00366-020-00960-w

    Article  Google Scholar 

  73. Odabaşı V, Maglio S, Martini A, Sorrentino S (2019) Static stress analysis of suspension systems for a solar-powered car. FME Trans 47(1):70–75. https://doi.org/10.5937/fmet1901070O

    Article  Google Scholar 

  74. Oñate E (2013) Structural analysis with the finite element method. Linear statics: volume 2: beams, plates and shells. Springer Science & Business Media, New York

    MATH  Google Scholar 

  75. Pascon JP, Coda HB (2017) Finite deformation analysis of visco-hyperelastic materials via solid tetrahedral finite elements. Finite Elem Anal Des 133:25–41. https://doi.org/10.1016/j.finel.2017.05.007

    Article  MathSciNet  Google Scholar 

  76. Payette GS, Reddy JN (2010) Nonlinear quasi-static finite element formulations for viscoelastic Euler–Bernoulli and Timoshenko beams. Int J Numer Methods Biomed Eng 26(12):1736–1755. https://doi.org/10.1002/cnm.1262

    Article  MathSciNet  MATH  Google Scholar 

  77. Payette GS, Reddy JN (2013) A nonlinear finite element framework for viscoelastic beams based on the high-order Reddy beam theory. J Eng Mater Technol. https://doi.org/10.1115/1.4023185

    Article  Google Scholar 

  78. Persson P, Flodén O, Pedersen B (2020) Predicting vibroacoustic performance of thin-walled lightweight structures during conceptual design. Finite Elem Anal Des 169:103342. https://doi.org/10.1016/j.finel.2019.103342

    Article  MathSciNet  Google Scholar 

  79. Ramesh UA, Siddhartha D, Sethu RA, Ramanan N, Karthi V (2017) Modeling and analysis of lower wishbone for independent suspension system for commercial vehicles. J Chem Pharm Sci 2:166–168

    Google Scholar 

  80. Reddy, J. N. (2013). An introduction to continuum mechanics. Cambridge university press.‏

  81. Rezaiee-Pajand M, Karimi-Rad M (2015) More accurate and stable time integration scheme. Eng Comput 31(4):791–812. https://doi.org/10.1007/s00366-014-0390-x

    Article  Google Scholar 

  82. Sankar SL, Kumar GA, Suresh AV (2018) Design and analysis of wishbones in double wishbone suspension system. Int J Veh Struct Syst 10(4):260–262. https://doi.org/10.4273/ijvss.10.4.06

    Article  Google Scholar 

  83. Santamaría, P. A. C., Sierra, A., & Estrada, O. A. G. (2018). Stress analysis of a suspension control arm

  84. Sedighi HM, Shirazi KH, Noghrehabadi AR, Yildirim AHMET (2012) Asymptotic investigation of buckled beam nonlinear vibration. Iran J Sci Technol Trans Mechan Eng 36(M2):107–116

    Google Scholar 

  85. Sedighi HM, Shirazi KH, Attarzadeh MA (2013) A study on the quintic nonlinear beam vibrations using asymptotic approximate approaches. Acta Astronaut 91:245–250. https://doi.org/10.1016/j.actaastro.2013.06.018

    Article  Google Scholar 

  86. Sedighi HM (2014) Size-dependent dynamic pull-in instability of vibrating electrically actuated microbeams based on the strain gradient elasticity theory. Acta Astronaut 95:111–123. https://doi.org/10.1016/j.actaastro.2013.10.020

    Article  Google Scholar 

  87. Sedighi H, Daneshmand F (2015) Nonlinear transversely vibrating beams by the homotopy perturbation method with an auxiliary term. J Appl Comput Mech 1(1):1–9. https://doi.org/10.22055/jacm.2014.10545

    Article  Google Scholar 

  88. Shariati A, Hosseini SHS, Bayrami SS, Ebrahimi F, Toghroli A (2020) Effect of residual surface stress on parametrically excited nonlinear dynamics and instability of viscoelastic piezoelectric nanoelectromechanical resonators. Eng Comput. https://doi.org/10.1007/s00366-019-00916-9

    Article  Google Scholar 

  89. Singh KV (2016) Eigenvalue and eigenvector computation for discrete and continuous structures composed of viscoelastic materials. Int J Mech Sci 110:127–137. https://doi.org/10.1016/j.ijmecsci.2016.03.009

    Article  Google Scholar 

  90. Wideberg J, Bordons C, Luque P, Mántaras DA, Marcos D, Kanchwala H (2014) Development and experimental validation of a dynamic model for electric vehicle with in hub motors. Procedia-Soc Behav Sci 160:84–91. https://doi.org/10.1016/j.sbspro.2014.12.119

    Article  Google Scholar 

  91. Wojciech S, Adamiec-Wójcik I (1993) Nonlinear vibrations of spatial viscoelastic beams. Acta Mech 98(1–4):15–25. https://doi.org/10.1007/BF01174290

    Article  MATH  Google Scholar 

  92. Won SG, Bae SH, Cho JR, Bae SR, Jeong WB (2013) Three-layered damped beam element for forced vibration analysis of symmetric sandwich structures with a viscoelastic core. Finite Elem Anal Des 68:39–51. https://doi.org/10.1016/j.finel.2013.01.004

    Article  MathSciNet  MATH  Google Scholar 

  93. Xiaodong Y, Li-Qun C (2006) Non-linear forced vibration of axially moving viscoelastic beams. Acta Mech Solida Sin 19(4):365–373. https://doi.org/10.1007/s10338-006-0643-3

    Article  Google Scholar 

  94. Yu H, Zhao C, Zheng H (2017) A higher-order variable cross-section viscoelastic beam element via ANCF for kinematic and dynamic analyses of two-link flexible manipulators. Int J Appl Mech 9(08):1750116. https://doi.org/10.1142/S1758825117501162

    Article  Google Scholar 

  95. Zghal S, Bouazizi ML, Bouhaddi N, Nasri R (2015) Model reduction methods for viscoelastic sandwich structures in frequency and time domains. Finite Elem Anal Des 93:12–29. https://doi.org/10.1016/j.finel.2014.08.003

    Article  Google Scholar 

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Authors and Affiliations

  1. Mechanical Design and Production Department, Faculty of Engineering, Zagazig University, P.O. Box 44519, Zagazig, Egypt

    A. A. Abdelrahman, A. E. Nabawy, A. M. Abdelhaleem, S. S. Alieldin & M. A. Eltaher

  2. Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, Saudi Arabia

    M. A. Eltaher

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  1. A. A. Abdelrahman
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Abdelrahman, A.A., Nabawy, A.E., Abdelhaleem, A.M. et al. Nonlinear dynamics of viscoelastic flexible structural systems by finite element method. Engineering with Computers 38 (Suppl 1), 169–190 (2022). https://doi.org/10.1007/s00366-020-01141-5

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