Analytical and numerical investigation of beam-spring systems with varying stiffness: a comparison of consistent and lumped mass matrices considerations

Mohammed Alkinidri; Rab Nawaz; Hani Alahmadi; Mohammed Alkinidri; Rab Nawaz; Hani Alahmadi

doi:10.3934/math.20241016

AIMS Mathematics

2024, Volume 9, Issue 8: 20887-20904. doi: 10.3934/math.20241016

Previous Article Next Article

Research article

Analytical and numerical investigation of beam-spring systems with varying stiffness: a comparison of consistent and lumped mass matrices considerations

1.
Department of Mathematics, College of Science and Arts, King Abdulaziz University, Rabigh, Saudi Arabia
2.
Department of Mathematics, Comsats University Islamabad, Pakistan
3.
Center for Applied Mathematics and Bioinformatics, Gulf University for Science and Technology, Hawally 32093, Kuwait
4.
Mathematics Department, College of Science, Jouf University, P. O. Box 2014, Sakaka, Saudi Arabia

Received: 07 March 2024 Revised: 22 May 2024 Accepted: 29 May 2024 Published: 28 June 2024
MSC : 35P15, 74K10, 74S05

This study examined the vibration behavior of a beam with linear spring attachments using finite element analysis. It aims to determine the natural frequency with both consistent/coupled mass and lumped mass matrices. The natural frequencies and corresponding mode shapes were correctly determined which formed the basis of any further noise vibration and severity calculations and impact or crash analysis. In order to obtain eigenfrequencies subject to the attached spring, the characteristic equation was obtained by eigenfunctions expansion whose roots were extracted using the root-finding technique. The finite element method by coupled and lumped mass matrices was then used to determine complete mode shapes against various eigenfrequencies. The mode shapes were then analyzed subject to supports with varying stiffness thereby comparing the analytical and numerical results in case of consistent and lumped masses matrices so as to demonstrate how the present analysis could prove more valuable in mathematical and engineering contexts. Utilizing a consistent mass matrix significantly enhanced accuracy compared to a lumped mass matrix, thereby validating the preference for the former, even with a limited number of beam elements. The results indicated that substantial deflection occurred at the beam's endpoints, supporting the dynamic behavior of the spring-beam system.
- beam,
- eigenfrequency,
- CM matrix,
- LM matrix,
- springs,
- variable stiffness,
- mode shape,
- FEM
Citation: Mohammed Alkinidri, Rab Nawaz, Hani Alahmadi. Analytical and numerical investigation of beam-spring systems with varying stiffness: a comparison of consistent and lumped mass matrices considerations[J]. AIMS Mathematics, 2024, 9(8): 20887-20904. doi: 10.3934/math.20241016

Related Papers:

Abstract

This study examined the vibration behavior of a beam with linear spring attachments using finite element analysis. It aims to determine the natural frequency with both consistent/coupled mass and lumped mass matrices. The natural frequencies and corresponding mode shapes were correctly determined which formed the basis of any further noise vibration and severity calculations and impact or crash analysis. In order to obtain eigenfrequencies subject to the attached spring, the characteristic equation was obtained by eigenfunctions expansion whose roots were extracted using the root-finding technique. The finite element method by coupled and lumped mass matrices was then used to determine complete mode shapes against various eigenfrequencies. The mode shapes were then analyzed subject to supports with varying stiffness thereby comparing the analytical and numerical results in case of consistent and lumped masses matrices so as to demonstrate how the present analysis could prove more valuable in mathematical and engineering contexts. Utilizing a consistent mass matrix significantly enhanced accuracy compared to a lumped mass matrix, thereby validating the preference for the former, even with a limited number of beam elements. The results indicated that substantial deflection occurred at the beam's endpoints, supporting the dynamic behavior of the spring-beam system.

References

[1]	R. E. D. Bishop, D. C. Johnson. The mechanics of vibration, Cambridge University Press, 1960.
[2]	W. Zhang, S. Zhang, J. Wei, Y. Huang, Flexural behavior of SFRC-NC composite beams: an experimental and numerical analytical study, Structures, 60 (2024), 105823. https://doi.org/10.1016/j.istruc.2023.105823 doi: 10.1016/j.istruc.2023.105823
[3]	P. Zhang, P. Schiavone, H. Qing, Dynamic stability analysis of porous functionally graded beams under hygro-thermal loading using nonlocal strain gradient integral model, Appl. Math. Mech., 44 (2023), 2071–2092. https://doi.org/10.1007/s10483-023-3059-9 doi: 10.1007/s10483-023-3059-9
[4]	A. Khanfer, L. Bougoffa, On the nonlinear system of fourth-order beam equations with integral boundary conditions, AIMS Math., 6 (2021), 11467–11481. https://doi.org/10.3934/math.2021664 doi: 10.3934/math.2021664
[5]	M. G$\ddot{u}$rg$\ddot{u}$ze, On the vibrations of restrained beams and rods with heavy masses, J. Sound Vib., 100 (1985), 588–589. https://doi.org/10.1016/S0022-460X(85)80009-2 doi: 10.1016/S0022-460X(85)80009-2
[6]	T. Liu, P. Feng, Y. Bai, S. Bai, J. Yang, F. Zhao, Flexural performance of curved-pultruded GFRP arch beams subjected to varying boundary conditions, Eng. Struct., 308 (2024), 117962. https://doi.org/10.1016/j.engstruct.2024.117962 doi: 10.1016/j.engstruct.2024.117962
[7]	H. R. $\ddot{O}$z, Calculation of the natural frequencies of a beam-mass system using the finite element method, Math. Comput. Appl., 5 (2000), 67–76. https://doi.org/10.3390/mca5020067 doi: 10.3390/mca5020067
[8]	E. $\ddot{O}$zkaya, Linear transverse vibrations of a simply supported beam carrying concentrated masses, Math. Comput. Appl., 6 (2001), 147–152. https://doi.org/10.3390/mca6020147 doi: 10.3390/mca6020147
[9]	R. O. Grossi, B. Arenas, A variational approach to the vibration of tapered beams with elastically restrained ends, J. Sound Vib., 195 (1996), 507–511. https://doi.org/10.1006/jsvi.1996.0439 doi: 10.1006/jsvi.1996.0439
[10]	R. C. Smith, K. L. Bowers, J. Lund, A fully Sinc-Galerkin method for Euler-Bernoulli beam models, Numer. Methods Partial Differ. Equations, 8 (1992), 171–202. https://doi.org/10.1002/num.1690080207 doi: 10.1002/num.1690080207
[11]	O. Moaaz, A. E. Abouelregal, F. Alsharari, Lateral vibration of an axially moving thermoelastic nanobeam subjected to an external transverse excitation, AIMS Math., 8 (2023), 2272–2295. https://doi.org/10.3934/math.2023118 doi: 10.3934/math.2023118
[12]	M. Baccouch, The local discontinuous Galerkin method for the fourth-order Euler–Bernoulli partial differential equation in one space dimension. Part I: superconvergence error analysis, J. Sci. Comput., 59 (2014), 795–840. https://doi.org/10.1007/s10915-013-9782-0 doi: 10.1007/s10915-013-9782-0
[13]	J. Xie, Z. Zhang, Efficient high-order physical property-preserving difference methods for nonlinear fourth-order wave equation with damping, Comput. Math. Appl., 142 (2023), 64–83. https://doi.org/10.1016/j.camwa.2023.04.012 doi: 10.1016/j.camwa.2023.04.012
[14]	D. Shi, L. Wang, X. Liao, New estimates of mixed finite element method for fourth-order wave equation, Math. Methods Appl. Sci., 40 (2023), 4448–4461. https://doi.org/10.1002/mma.4316 doi: 10.1002/mma.4316
[15]	Y. Liu, C. S. Gurram, The use of He's variational iteration method for obtaining the free vibration of an Euler-Bernoulli beam, Math. Comput. Modell., 50 (2009), 1545–1552. https://doi.org/10.1016/j.mcm.2009.09.005 doi: 10.1016/j.mcm.2009.09.005
[16]	M. N. Hamdan, L. A. Latif, On the numerical convergence of discretization methods for the free vibrations of beams with attached inertia elements, J. Sound Vib., 169 (1994), 527–545. https://doi.org/10.1006/jsvi.1994.1032 doi: 10.1006/jsvi.1994.1032
[17]	M. Jafari, H. Djojodihardjo, K. A. Ahmad, Vibration analysis of a cantilevered beam with spring loading at the tip as a generic elastic structure, Appl. Mech. Mater., 629 (2014), 407–413. https://doi.org/10.4028/www.scientific.net/AMM.629.407 doi: 10.4028/www.scientific.net/AMM.629.407
[18]	G. Kanwal, R. Nawaz, N. Ahmed, Analyzing the effect of rotary inertia and elastic constraints on a beam supported by a wrinkle elastic foundation: a numerical investigation, Buildings, 13 (2023), 1457. https://doi.org/10.3390/buildings13061457 doi: 10.3390/buildings13061457
[19]	P. K. Banerjee, R. Butterfield, Boundary element method in engineering science, McGraw-Hill Education, 1981.
[20]	O. C. Zeinkeinwicz, Finite element method, Butterworth Heineman, 2005.
[21]	M. Petyt, Introduction to finite element vibration analysis, 2 Eds., Cambridge Univrsiy Press, 2015. https://doi.org/10.1017/CBO9780511761195
[22]	L. Euler, De motu vibratorio laminarum elasticarum, ubi plures novae vibrationum species hactenus non pertractatae evolvuntur, Novi Commentarii Academiae Scientiarum Petropolitanae, 1773.
[23]	A. W. Leissa, M. S. Qatu, Vibrations of continuous systems, McGraw-Hill Education, 2011.
[24]	S. M. Han, H. Benaroya, T. Wei, Dynamics of transversely vibrating beams using four engineering theories, J. Sound Vib., 225 (1999), 935–988. https://doi.org/10.1006/jsvi.1999.2257 doi: 10.1006/jsvi.1999.2257
[25]	T. Nawaz, M. Afzal, R. Nawaz, The scattering analysis of trifurcated waveguide involving structural discontinuities, Adv. Mech. Eng., 11 (2019), 282. https://doi.org/10.1177/1687814019829282 doi: 10.1177/1687814019829282
[26]	A. J. Ferreira, N. Fantuzzi, MATLAB codes for finite element analysis, Springer, 2009. https://doi.org/10.1007/978-3-030-47952-7
[27]	E. Kreyszig, Advanced engineering mathematics, John Wiley & Sons, Inc., 2009.
[28]	A. Bosten, V. Denoël, A. Cosimo, J. Linn, O. Brüls, A beam contact benchmark with analytic solution, ZAMM J. Appl. Math. Mech., 103 (2023), e202200151. https://doi.org/10.1002/zamm.202200151 doi: 10.1002/zamm.202200151
[29]	M. Bobková, L. Pospíšil, Numerical solution of bending of the beam with given friction, Mathematics, 9 (2021), 898. https://doi.org/10.3390/math9080898 doi: 10.3390/math9080898
[30]	L. He, A. J. Valocchi, C. A. Duarte, A transient global-local generalized FEM for parabolic and hyperbolic PDEs with multi-space/time scales, J. Comput. Phys., 488 (2023), 112179. https://doi.org/10.1016/j.jcp.2023.112179 doi: 10.1016/j.jcp.2023.112179
[31]	L. He, A. J. Valocchi, C. A. Duarte, An adaptive global-local generalized FEM for multiscale advection-diffusion problems, Comput. Methods Appl. Mech. Eng., 418 (2024), 116548. https://doi.org/10.1016/j.cma.2023.116548 doi: 10.1016/j.cma.2023.116548
[32]	A. Yaseen, R. Nawaz, Acoustic radiation through a flexible shell in a bifurcated circular waveguide, Math. Meth. Appl. Sci., 46 (2023), 6262–6278. https://doi.org/10.1002/mma.8902 doi: 10.1002/mma.8902
[33]	M. Alkinidri, S. Hussain, R. Nawaz, Analysis of noise attenuation through soft vibrating barriers: an analytical investigation, AIMS Math., 8 (2023), 18066–18087. https://doi.org/10.3934/math.2023918 doi: 10.3934/math.2023918

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)