Using the Time-Weighted Residual Method in Forced Vibration Analysis of Timoshenko Beam under Moving Load

Document Type : Research Article

Authors

1 Department of Civil Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran

2 Department of Civil Engineering, Isfahan University of Technology, Isfahan, Iran

3 Department of Civil Engineering, Isfahan University of Technology, Isfahan , Iran

Abstract

In this study, formulation of a recently proposed time-weighted residual method has been developed for the vibration analysis of Timoshenko beams under moving loads. Employing pre-integration relations as well as equilibrium equations is the main idea of this method. In the first step of the proposed method, the time interval is subdivided into a number of sub-intervals and then the acceleration field in each time step is defined as the combination of an unknown function and a series of exponential basis functions with constant coefficients. Finally, the solution of the problem is computed by the time-weighted residual method along with exact satisfaction of the initial and the boundary conditions at the two ends of the beam. Storing the information of solution at each time step on the exponential coefficients is the most important advantage of this method so that the solution is progressed in time without the need to discretize beams and only by using an appropriate recursive relation to update the exponential coefficients. In order to investigate the accuracy and efficiency of the proposed method, the results of solving three sample problems of constant and accelerated moving load on the beams with different boundary conditions, are compared with the results of the finite element method. This comparison illustrates the speed and accuracy of the proposed method in estimating the internal shear forces and bending moments rather than those obtained by the finite element method.

Keywords

Main Subjects

References

[1]  B. Movahedian, B. Boroomand, S. Mansouri, A robust time‐space formulation for large‐scale scalar wave problems using exponential basis functions, International Journal for Numerical Methods in Engineering, 114(7) (2018) 719-748.
[2]  J L. Frýba, Vibration of solids and structures under moving loads, Springer Science & Business Media, 2013.
[3]  A. Florence, Traveling force on a Timoshenko beam, Journal of Applied Mechanics, 32(2) (1965) 351-358.
 [4] C.R. Steele, The Timoshenko beam with a moving load, Journal of Applied Mechanics, 35(3) (1968) 481-488.
[5]  S. Mackertich, Moving load on a Timoshenko beam, The Journal of the Acoustical Society of America, 88(2) (1990) 1175-1178.
 [6]H. Lee, Dynamic response of a rotating Timoshenko shaft subject to axial forces and moving loads, Journal of Sound and Vibration, 181(1) (1995) 169-177.
 [7] R.-T. Wang, Vibration of multi-span Timoshenko beams to a moving force, Journal of sound and vibration, 207(5) (1997) 731-742.
[8]  S.E. Azam, M. Mofid, R.A. Khoraskani, Dynamic response of Timoshenko beam under moving mass, Scientia Iranica, 20(1) (2013) 50-56.
[9]  D. Roshandel, M. Mofid, A. Ghannadiasl, Modal analysis of the dynamic response of Timoshenko beam under moving mass, Scientia Iranica. Transaction A, Civil Engineering, 22(2) (2015) 331.
[10]  M. Kargarnovin, D. Younesian, Dynamics of Timoshenko beams on Pasternak foundation under moving load, Mechanics research communications, 31(6) (2004) 713-723.
[11]  M. Shafiei, N. Khaji, Analytical solutions for free and forced vibrations of a multiple cracked Timoshenko beam subject to a concentrated moving load, Acta Mechanica, 221(1-2) (2011) 79.
[12]  J.Q. Jiang, W.-Q. Chen, Y.-H. Pao, Reverberation-ray analysis of continuous Timoshenko beams subject to moving loads, Journal of Vibration and Control, 18(6) (2012) 774-784.
[13]  L. Ding, H.-P. Zhu, L. Wu, Effects of axial load and structural damping on wave propagation in periodic Timoshenko beams on elastic foundations under moving loads, Physics Letters A, 380(31-32) (2016) 2335-2341.
[14]  Y.-H. Lin, Vibration analysis of Timoshenko beams traversed by moving loads, Journal of Marine Science and Technology, 2(4) (1994) 25-35.
 [15] P. Lou, G.-l. Dai, Q.-y. Zeng, Dynamic analysis of a Timoshenko beam subjected to moving concentrated forces using the finite element method, Shock and Vibration, 14(6) (2007) 459-468.
[16]  V. Sarvestan, H.R. Mirdamadi, M. Ghayour, Vibration analysis of cracked Timoshenko beam under moving load with constant velocity and acceleration by spectral finite element method, International Journal of Mechanical Sciences, 122 (2017) 318-330.
 [17]  S.S. Kourehli, S. Ghadimi, R. Ghadimi, Crack identification in Timoshenko beam under moving mass using RELM, Steel and Composite Structures, 28(3) (2018) 279-288.
[18]  S.S. Kourehli, S. Ghadimi, R. Ghadimi, Vibration analysis and identification of breathing cracks in beams subjected to single or multiple moving mass using online sequential extreme learning machine, Inverse Problems in Science and Engineering, 27(8) (2019) 1057-1080.
 [19]  X. Zhang, H. Liang, M. Zhao, Fundamental solution and its validation by numerical inverse Laplace transformation and FEM for a damped Timoshenko beam subjected to impact and moving loads, Journal of Vibration and Control, 25(3) (2019) 593-611.
Amirkabir Journal of Civil Engineering
Volume 53, Issue 4
July 2021
Pages 1339-1354

Files

Share

How to cite

Statistics