This paper presents a computational study on the dynamic response of a clamped–clamped non-uniform elastic beam supported by a two-parameter foundation and subjected to a concentrated moving load. The analysis incorporates the inertial contribution of the moving load. Unlike uniform models, both the mass per unit length and the moment of inertia of the beam are assumed to vary along the span LLL. The governing equations are solved using the Generalized Galerkin approach combined with a modified Struble’s method. Analytical formulations and numerical simulations demonstrate that, for equal natural frequencies, the critical speed of a clamped–clamped non-uniform beam under a moving force exceeds that of the corresponding moving mass problem. Consequently, resonance is reached earlier in the moving mass case. In addition, it is observed that, at fixed foundation modulus and axial force, increasing the shear modulus decreases the system’s response amplitude. Similarly, higher axial force reduces vibration amplitude when foundation and shear moduli are held constant. Finally, deflection profiles diminish with increasing foundation modulus, although a stronger influence is observed when the shear modulus is varied.

Keywords: Non-Uniform Elastic beam, two parameter foundations, Concentrated Loads, Moving force, Moving mass, Clamped-Clamped boundary conditions

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1. Ogunbamike OK. Seismic analysis of simply supported damped Rayleigh beams on elastic foundation. Asian Res J Math. 2020;16(11):31–47.
2. Chen D, Yang J, Guo W, Liu Y, Gu C. Vibration study of a composite pipeline supported on elastic foundation using a transfer matrix method. J Vib Control. 2021;doi:10.1177/1077546320985370.
3. Tonzani GM, Elishakoff I. Three alternative versions of theory for a Timoshenko-Ehrenfest beam on a Winkler foundation. Math Mech. 2021;40(2):279–95.
4. Ike CC. Fourier sine transform method for the free vibration of Euler-Bernoulli resting on Winkler foundation. Int J Darshan Inst Eng Res Emerg Technol. 2018;7(1).
5. Jena SK, Chakraverty S, Tornabene F. Dynamic behaviour of nanobeam embedded in constant, linear, parabolic and sinusoidal type of Winkler elastic foundation using first-order nonlocal strain gradient model. Mater Res Express. 2019;6(8):1–23.
6. Jiya M, Shaba AI. Analysis of a uniform Bernoulli-Euler beam on Winkler foundation subjected harmonic moving load. J Appl Sci Environ Manage. 2018;22(3):368–72.
7. Ma J, Liu F, Nie M, Wang J. Nonlinear free vibration of a beam on Winkler foundation with consideration of soil mass motion of finite depth. Nonlinear Dyn. 2018;92:429–41.
8. Phadke HD, Jaiswal OR. Dynamic response of railway track resting on variable foundation using finite element method. Arab J Sci Eng. 2020;45:4823–41.
9. Roc MA. Finite element and numerical vibration analysis of a Timoshenko and Euler-Bernoulli beam traversed by a moving high speed train. J Braz Soc Mech Sci Eng. 2021;7. doi: 10.1504/IJHVS.083057.
10. Wang Y, Zhou A, Fu T, Zhang W. Transient response of a sandwich beam with functionally graded porous core traversed by a non-uniform distributed moving mass. Int J Mech Mater Des. 2020;16(3):519–40.
11. Jimoh SA. On modal-asymptotic analysis to prestressed thick beam on bi-parametric foundations subjected to moving loads. Achiev J Sci Res. 2021;3(2):28–46.
12. Baran B. Transfer matrix formulation for dynamic response of Timoshenko beam resting on two-parameter elastic foundation subjected to moving load. J Struct Eng Appl Mech. 2021;4(2):99–100.
13. Saurabh K. Natural frequency of beam with axial material graduation resting on two-parameter elastic foundation. Trends Sci. 2022;19(6):1–12.
14. Jimoh A, Ajoge EO. Dynamic behaviour of uniform Bernoulli-Euler beam resting on bi-parametric foundations and subjected to distributed moving load with damping effect. J Basic Appl Res Int. 2020;26(3):33–9.
15. Oni ST, Jimoh A. Dynamic response to moving concentrated loads of simply supported prestressed Bernoulli-Euler beam resting on bi-parametric subgrade. Int J Sci Eng Res. 2016;7(9):584–600.
16. Esen I. Dynamic response of a functionally graded Timoshenko beam on two-parameter foundations due to a variable velocity moving mass. Int J Mech Sci. 2019;153–154:21–35.
17. Wattanasakulpong N. Multi-moving loads induced vibration of FG sandwich beams resting on Pasternak elastic foundation. Walailak J Sci Technol. 2021;18:1–12.
18. Sigueira LO, Cortez RL, Hoefel SS. Vibration analysis of an axial loaded Euler-Bernoulli beam on two-parameter foundations. Proc 25th ABCM Int Congr Mech Eng. Uberlândia, Brazil; 2019. p.1–4.
19. Oni ST, Jimoh A. Dynamic response to moving concentrated loads of non-uniform simply supported prestressed Bernoulli-Euler beam resting on bi-parametric subgrades. Int J Sci Eng Res. 2016;7(3):754–70.
20. Jimoh A, Ajoge EO. Dynamic response of non-uniform Rayleigh beam subjected to harmonically varying moving load. J Appl Math Comput. 2018;2(8):345–56.
21. Jimoh A, Ajoge EO. Distributed moving load on non-uniform Bernoulli-Euler beam resting on bi-parametric foundations. Int J Appl Math Bioinform. 2020;9(2):41–52.
22. Jimoh A, Ajoge EO. Distributed moving load on non-uniform Bernoulli-Euler beam resting on bi-parametric foundations. Int J Appl Math Stat Sci. 2019;9(2):1–16.
23. Fryba L. Vibration of solids and structures under moving loads. Groningen: Noordhoff; 1972.

This paper presents a computational study on the dynamic response of a clamped–clamped non-uniform elastic beam supported by a two-parameter foundation and subjected to a concentrated moving load. The analysis incorporates the inertial contribution of the moving load. Unlike uniform models, both the mass per unit length and the moment of inertia of the beam are assumed to vary along the span LLL. The governing equations are solved using the Generalized Galerkin approach combined with a modified Struble’s method. Analytical formulations and numerical simulations demonstrate that, for equal natural frequencies, the critical speed of a clamped–clamped non-uniform beam under a moving force exceeds that of the corresponding moving mass problem. Consequently, resonance is reached earlier in the moving mass case. In addition, it is observed that, at fixed foundation modulus and axial force, increasing the shear modulus decreases the system’s response amplitude. Similarly, higher axial force reduces vibration amplitude when foundation and shear moduli are held constant. Finally, deflection profiles diminish with increasing foundation modulus, although a stronger influence is observed when the shear modulus is varied.

Keywords: Non-Uniform Elastic beam, two parameter foundations, Concentrated Loads, Moving force, Moving mass, Clamped-Clamped boundary conditions

🔒 This is a subscription article

Full text is available to subscribers and institutional members. Please choose an option below to access it.

1. Ogunbamike OK. Seismic analysis of simply supported damped Rayleigh beams on elastic foundation. Asian Res J Math. 2020;16(11):31–47.
2. Chen D, Yang J, Guo W, Liu Y, Gu C. Vibration study of a composite pipeline supported on elastic foundation using a transfer matrix method. J Vib Control. 2021;doi:10.1177/1077546320985370.
3. Tonzani GM, Elishakoff I. Three alternative versions of theory for a Timoshenko-Ehrenfest beam on a Winkler foundation. Math Mech. 2021;40(2):279–95.
4. Ike CC. Fourier sine transform method for the free vibration of Euler-Bernoulli resting on Winkler foundation. Int J Darshan Inst Eng Res Emerg Technol. 2018;7(1).
5. Jena SK, Chakraverty S, Tornabene F. Dynamic behaviour of nanobeam embedded in constant, linear, parabolic and sinusoidal type of Winkler elastic foundation using first-order nonlocal strain gradient model. Mater Res Express. 2019;6(8):1–23.
6. Jiya M, Shaba AI. Analysis of a uniform Bernoulli-Euler beam on Winkler foundation subjected harmonic moving load. J Appl Sci Environ Manage. 2018;22(3):368–72.
7. Ma J, Liu F, Nie M, Wang J. Nonlinear free vibration of a beam on Winkler foundation with consideration of soil mass motion of finite depth. Nonlinear Dyn. 2018;92:429–41.
8. Phadke HD, Jaiswal OR. Dynamic response of railway track resting on variable foundation using finite element method. Arab J Sci Eng. 2020;45:4823–41.
9. Roc MA. Finite element and numerical vibration analysis of a Timoshenko and Euler-Bernoulli beam traversed by a moving high speed train. J Braz Soc Mech Sci Eng. 2021;7. doi: 10.1504/IJHVS.083057.
10. Wang Y, Zhou A, Fu T, Zhang W. Transient response of a sandwich beam with functionally graded porous core traversed by a non-uniform distributed moving mass. Int J Mech Mater Des. 2020;16(3):519–40.
11. Jimoh SA. On modal-asymptotic analysis to prestressed thick beam on bi-parametric foundations subjected to moving loads. Achiev J Sci Res. 2021;3(2):28–46.
12. Baran B. Transfer matrix formulation for dynamic response of Timoshenko beam resting on two-parameter elastic foundation subjected to moving load. J Struct Eng Appl Mech. 2021;4(2):99–100.
13. Saurabh K. Natural frequency of beam with axial material graduation resting on two-parameter elastic foundation. Trends Sci. 2022;19(6):1–12.
14. Jimoh A, Ajoge EO. Dynamic behaviour of uniform Bernoulli-Euler beam resting on bi-parametric foundations and subjected to distributed moving load with damping effect. J Basic Appl Res Int. 2020;26(3):33–9.
15. Oni ST, Jimoh A. Dynamic response to moving concentrated loads of simply supported prestressed Bernoulli-Euler beam resting on bi-parametric subgrade. Int J Sci Eng Res. 2016;7(9):584–600.
16. Esen I. Dynamic response of a functionally graded Timoshenko beam on two-parameter foundations due to a variable velocity moving mass. Int J Mech Sci. 2019;153–154:21–35.
17. Wattanasakulpong N. Multi-moving loads induced vibration of FG sandwich beams resting on Pasternak elastic foundation. Walailak J Sci Technol. 2021;18:1–12.
18. Sigueira LO, Cortez RL, Hoefel SS. Vibration analysis of an axial loaded Euler-Bernoulli beam on two-parameter foundations. Proc 25th ABCM Int Congr Mech Eng. Uberlândia, Brazil; 2019. p.1–4.
19. Oni ST, Jimoh A. Dynamic response to moving concentrated loads of non-uniform simply supported prestressed Bernoulli-Euler beam resting on bi-parametric subgrades. Int J Sci Eng Res. 2016;7(3):754–70.
20. Jimoh A, Ajoge EO. Dynamic response of non-uniform Rayleigh beam subjected to harmonically varying moving load. J Appl Math Comput. 2018;2(8):345–56.
21. Jimoh A, Ajoge EO. Distributed moving load on non-uniform Bernoulli-Euler beam resting on bi-parametric foundations. Int J Appl Math Bioinform. 2020;9(2):41–52.
22. Jimoh A, Ajoge EO. Distributed moving load on non-uniform Bernoulli-Euler beam resting on bi-parametric foundations. Int J Appl Math Stat Sci. 2019;9(2):1–16.
23. Fryba L. Vibration of solids and structures under moving loads. Groningen: Noordhoff; 1972.