A comprehensive investigation of dual-damped Euler-Bernoulli beam under moving mass on Pasternak foundation via spectral and central difference methods

Adebola Samuel Adeoye; Ezekiel Olaoluwa Omole; Sule Adekunle Jimoh; Victoria Iyadunni Ayodele; Taiwo Aanu Ogunlusi

doi:10.46481/jnsps.2026.3087

Authors

Adebola Samuel Adeoye
Department of Mathematical Sciences, Achievers University, Owo, Ondo State, Nigeria
Ezekiel Olaoluwa Omole
[email protected]

Department of Mathematics, Federal University of Technology and Environmental Sciences, Iyin-Ekiti, Ekiti State, Nigeria
Sule Adekunle Jimoh
Department of Mathematical Sciences, Achievers University, Owo, Ondo State, Nigeria
Victoria Iyadunni Ayodele
Department of Computer Science and Mathematics, Nigeria Police Academy, Wudil, Kano State, Nigeria
Taiwo Aanu Ogunlusi
Department of Mathematics, Federal University of Oye-Ekiti, Ekiti State, Nigeria

Keywords:

Euler-Bernoulli beam, Chebyshev collocation, Central difference, Damping coefficients, Pasternak foundation

Abstract

This study investigates the dynamic behavior of a dual-damped Euler-Bernoulli beam carrying a moving mass and supported on a Pasternak foundation, with the aim of improving prediction and control of vibration in structural systems. The beam model incorporates both internal (material) and external (viscous) damping and is discretized spatially using the Chebyshev collocation method (high-order spectral accuracy). Time integration is performed with an explicit central-difference scheme. Convergence and stability of the high-order spatial discretization with explicit time stepping are assessed, and extensive parametric studies are carried out over inertia, foundation shear stiffness, and damping coefficients. Numerical results validated across simulation cases show clear repeatable trends: increases in mass per unit length, foundation shear stiffness, and viscoelastic (internal and external) damping all substantially reduce peak deflection amplitudes and mid-span displacement. The moving-mass model predicts lower critical speeds and larger dynamic amplification better than equivalent moving-force approximations. The combined dual-damped, high-order numerical scheme provides an accurate efficient tool for analyzing beam dynamics under moving loads. Results highlight practical design levers (inertia, shear stiffness, damping) that engineers can tune to mitigate vibration in civil, structural, and biomechanical applications, and identify parameter ranges most susceptible to resonance.

Dimensions

REFERENCES

[1] E.Winkler, Die Lehre von der Elastizität und Festigkeit, Dominicus, Prague, 1867, 225. https://archive.org/embed/bub_gb_25E5AAAAcAAJ

[2] P. L. Pasternak, On a New Method of Analysis of an Elastic Foundation by Means of Two Foundation Constants (in Russian), Gosudarstvennoe Izdatel’stvo Literaturi po Stroitel’stvu i Arkhitekture, Moscow, 1954, 21. https://api.semanticscholar.org/CorpusID:124773833.

[3] A.D. Kerr, ``Elastic and viscoelastic foundation models'', Journal of Applied Mechanics (ASME) 31 (1964) 491. https://doi.org/10.1115/1.3629667.

[4] A. D. Kerr, ``A study of a new foundation model'', Acta Mechanica 1 (1965) 135. https://doi.org/10.1007/BF01174308.

[5] M. Di Paola, F. Marino & M. Zingales, ``A generalized model of elastic foundation based on long-range interactions: Integral and fractional formulation'', International Journal of Solids and Structures 46 (2009) 3124. https://doi.org/10.1016/j.ijsolstr.2009.03.024.

[6] R. Lewandowski,P. Litewka ,M. Łasecka-Plura & Z. M. Pawlak, ``Dynamics of structures, frames, and plates with viscoelastic dampers or layers: A literature review'', Buildings 13 (2023) 2223. https://doi.org/10.3390/buildings13092223.

[7] J. M. Gere & S. P. Timoshenko, Mechanics of Materials, 4th ed., PWS Publishing, Boston, 1997, 312. https://books.google.com.ng/books?id=BKJRAAAAMAAJ.

[8] A. S. Adeoye & T. O. Awodola, ``Dynamic behavior of moving distributed masses of orthotropic rectangular plates with clamped--clamped boundary conditions resting on a constant elastic bi-parametric foundation'', International Journal of Chemistry, Mathematics and Physics 4 (2020) 71. https://doi.org/10.22161/ijcmp.4.4.2.

[9] D. Younesian, M. R. Eslami & M. Atashipour, ``Elastic and viscoelastic foundations: A review on linear and nonlinear modeling'', Archive of Applied Mechanics 89 (2019) 229. https://doi.org/10.1007/s11071-019-04977-9.

[10] I. Esen, ``Dynamic response of a functionally graded Timoshenko beam on two-parameter elastic foundations due to a variable velocity moving mass'', International Journal of Mechanical Sciences 153 (2019) 21. https://doi.org/10.1016/j.ijmecsci.2019.01.033.

[11] D. J. Inman, Engineering Vibration, 4th ed., Pearson Education, Upper Saddle River, NJ, 2014, 215. https://doi.org/10.1201/9781003402695.

[12] J. N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, 2nd ed., John Wiley & Sons, New York, 2002, pp. 60--150. https://openlibrary.org/works/OL1994170W/Energy_principles_and_variational_methods_in_applied_mechanics.

[13] G. Karami & M. Garnich, ``Effective moduli and failure considerations for composites with periodic fiber waviness,'' Composite Structures 67 (2005) 461. https://doi.org/10.1016/j.compstruct.2004.01.014.

[14] F. Naeim, Dynamics of Structures—Theory and Applications to Earthquake Engineering, Third EditionAnil K.Chopra2007. Pearson Prentice Hall, Upper Saddle River, New Jersey, Pp. 912, Earthquake Spectra, 23 (2007) 491. https://doi.org/10.1193/1.2720354.

[15] R. S. Lakes, Viscoelastic Materials, Cambridge University Press, Cambridge, 2009, 1. https://doi.org/10.1017/CBO9780511626722.

[16] R. L. Bagley & P. J. Torvik, ``A theoretical basis for the application of fractional calculus to viscoelasticity'', Journal of Rheology 27 (1983) 201. https://doi.org/10.1122/1.549724.

[17] N. M. Newmark, ``A method of computation for structural dynamics'', Journal of the Engineering Mechanics Division (ASCE) 85 (1959) 67. http://dx.doi.org/10.1061/jsdeag.0000555.

[18] H. M. Hilber, T. J. R. Hughes & R. L. Taylor, ``Improved numerical dissipation for time integration algorithms in structural dynamics'', Earthquake Engineering & Structural Dynamics 5 (1977) 283. https://doi.org/10.1002/eqe.4290050306.

[19] J. Chung & G. M. Hulbert, ``A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-$alpha$ method'', Journal of Applied Mechanics (ASME) 60 (1993) 371. https://doi.org/10.1115/1.2900803.

[20] O. C. Zienkiewicz, R. L. Taylor & J. Z. Zhu, The finite element method: its basis and fundamentals,in Computer Methods in Applied Mechanics and Engineering, 195 (2006) 6584. https://doi.org/10.1016/b978-1-85617-633-0.00020-4.

[21] L. Frýba, Vibration of Solids and Structures under Moving Loads, Noordhoff International Publishing, Groningen (1972); reprinted by Springer, Dordrecht, 1999, pp. 60--120. https://books.google.lu/books?id=PV7aAAAAIAAJ&hl=de&num=20&source=gbs_book_other_versions_r&cad=4

[22] Y.-B. Yang & C. W. Lin, ``Vehicle–bridge interaction dynamics and potential applications'', Journal of Sound and Vibration 284 (2005) 205. https://doi.org/10.1016/j.jsv.2004.06.032.

[23] M. H. Ghayesh, H. Farokhi, Y. Zhang & A. Gholipour, ``Nonlinear coupled moving-load excited dynamics of beam–mass structures'', Archives of Civil and Mechanical Engineering, 20 (2020) 45. https://doi.org/10.1007/s43452-020-00040-2.

[24] S. S. Law & X.-Q. Zhu, Moving Loads--Dynamic Analysis and Identification Techniques, CRC Press, Boca Raton, FL (2011). ISBN: 978-0-203-84142-6. http://dx.doi.org/10.1201/b10561.

[25] Awodola, T. O., & Adeoye, A. S., ``Vibration of Orthotropic Rectangular Plates Under the Action of Moving Distributed Masses and Resting on a Variable Elastic Pasternak Foundation with Clamped End Conditions'', International Journal of Advanced Engineering Research and Science 8 (2021) 026. http://dx.doi.org/10.22161/ijaers.86.4.

[26] M. Pirmoradian, M. Keshmiri & H. Karimpour, ``On the parametric excitation of a Timoshenko beam due to intermittent passage of moving masses: instability and resonance analysis'', Acta Mechanica 226 (2015) 1241. https://doi.org/10.1007/s00707-014-1240-z.

[27] A. S. Adeoye & T. O. Awodola, ``Dynamic response to moving distributed masses of pre-stressed uniform Rayleigh beam resting on variable elastic Pasternak foundation'', Edelweiss Applied Science and Technology 2 (2018) 1. https://doi.org/10.33805/2576.8484.106.

[28] İ. Esen, ``Dynamic Response of a Beam Due to an Accelerating Moving Mass Using Moving Finite Element Approximation'', Mathematical and Computational Applications 16 (2011) 171. https://doi.org/10.3390/mca16010171.

[29] X. Liu, Y. Wang & X. Ren, ``Optimal vibration control of moving-mass beam systems with uncertainty'', Journal of Low Frequency Noise, Vibration and Active Control 39 (2020) 803. https://doi.org/10.1177/1461348419844150.

[30] S. Zhao, X. Tang & Y. Du, ``Machine learning prediction and evaluation for structural damage comfort of suspension footbridge'', Buildings 14 (2024) 1344. https://doi.org/10.3390/buildings14051344.

[31] L. N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia (2000) 1. https://doi.org/0.1137/1.9780898719598.

[32] C. Canuto, M. Y. Hussaini, A. Quarteroni & T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer, Berlin Heidelberg, 2006, pp. 60--120. https://doi.org/10.1007/978-3-540-30726-6.

[33] J. Shen, T. Tang & L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, Berlin Heidelberg, 2011, pp. 60--150. https://doi.org/10.1007/978-3-540-71041-7.

[34] J. A. C. Weideman & S. C. Reddy, ``A MATLAB differentiation matrix suite'', ACM Transactions on Mathematical Software 26 (2000) 465. https://doi.org/10.1145/365723.365727.

[35] J. P. Berrut & L. N. Trefethen, ``Barycentric Lagrange interpolation'', SIAM Review, 46 (2004) 501. https://doi.org/10.1137/S0036144502417715.

[36] L. N. Trefethen, ``Is Gauss quadrature better than Clenshaw--Curtis?'', SIAM Review 50 (2008) 67. https://doi.org/10.1137/060659831.

[37] L. N. Trefethen & D. Bau, Numerical Linear Algebra, SIAM, Philadelphia, 1997, pp. 90--150. https://doi.org/10.1137/1.9780898719574.

[38] N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, 2002, pp. 50-150. https://doi.org/10.1137/1.9780898718027.

[39] B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge University Press, Cambridge, 1996, pp. 30--80. https://doi.org/10.1017/CBO9780511626357.

[40] J. P. Boyd, Chebyshev & Fourier Spectral Methods, Lecture Notes in Engineering, Springer, Berlin, 1989, pp. 50--120. https://doi.org/10.1007/978-3-642-83876-7.

[41] J. Chung & G. M. Hulbert, ``A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-$alpha$ method'', Journal of Applied Mechanics (ASME) 60 (1993) 371. https://doi.org/10.1115/1.2900803.

[42] K.-J. Bathe, Finite Element Procedures, Prentice Hall, Upper Saddle River, NJ, 1996, pp. 100--200 . ISBN: 978-0133014587. https://www.scribd.com/document/266613452/Finite-Element-Procedures-K-J-Bathe-1996

[43] K. C. Park & P. G. Underwood, ``A variable-step central difference method for structural dynamics analysis --- Part 1. theoretical aspects'', Computer Methods in Applied Mechanics and Engineering 22 (1980) 241. https://doi.org/10.1016/0045-7825(80)90087-0.

[44] P. G. Underwood & K. C. Park, ``A variable-step central difference method for structural dynamics analysis --- Part 2. implementation and performance evaluation'', Computer Methods in Applied Mechanics and Engineering 23 (1980) 259. https://doi.org/10.1016/0045-7825(80)90009-2.

[45] G. M. Hulbert & J. Chung, ``Explicit time integration algorithms for structural dynamics with optimal numerical dissipation'', Computer Methods in Applied Mechanics and Engineering, 137 (1996) 175. https://doi.org/10.1016/S0045-7825(96)01036-5.

[46] L. Chen, C. J. Jia & M. F. Lei, ``An improved model of the Pasternak foundation beam umbrella arch considering the generalized shear force'', Journal of Central South University 32 (2025) 1503. https://doi.org/10.1007/s11771-024-5777-2.

[47] J. S. Carvajal-Muñoz, C. A. Vega-Posada & J. C. Saldarriaga-Molina, ``Simplified Dynamic Analysis of Beam-Column Elements with Generalized End-Boundary Conditions on Non-Homogeneous Pasternak Elastic Foundation'', in Geo-Congress 2022, ASCE (2022) 216. https://doi.org/10.1061/9780784484043.021.

[48] S. W. Gong & K. Y. Lam, ``Analysis of layered composite beam to underwater shock including structural damping and stiffness effects'', Shock and Vibration 9(2002) 283. https://doi.org/10.1155/2002/574056.

[49] G. Karami & M. Garnich, ``Effective moduli and failure considerations for composites with periodic fiber waviness'', Composite Structures 67 (2005) 461. https://doi.org/10.1016/j.compstruct.2004.02.005.

[50] E. O. Omole, E. O. Adeyefa, V. I. Ayodele, A. Shokri & Y. Wang,``Ninth-order multistep collocation formulas for solving models of partial differential equations arising in fluid dynamics: design and implementation strategies'', Axioms 12 (2023) 891. https://doi.org/10.3390/axioms12090891.

[51] O. T. Olotu, E. O. Omole, R. O. Folaranmi, B. M. Yisa, P. O. Adeniran & J. A. Gbadeyan, ``Natural vibration analysis of axially graded tapered Rayleigh beams on a quadratic bi-parametric elastic foundation'', Nigerian Journal of Technology Development 22 (2025) 79. https://doi.org/10.63746/njtd.v22i1.3341.

[52] J. S. Adekunle & A. Alimi, ``Dynamic response of non-prismatic Bernoulli--Euler beam with exponentially varying thickness resting on variable elastic foundation'', Asian Research Journal of Mathematics 6 (2017) 1. https://doi.org/10.9734/ARJOM/2017/35785.

[53] L. Frýba, Vibration of Solids and Structures under Moving Loads, Springer Netherlands, Dordrecht, 1972, pp. 50--120. https://doi.org/10.1007/978-94-011-9685-7.