In this paper, a mixed method that combines the Ritz method and the triangular quadrature rule (TQR) is presented for solving time-dependent problems. In this study, the Ritz method is first used to discretize the spatial partial derivatives. The TQR is then employed to analogize the temporal derivatives. The resulting algebraic formulation is a triangular matrix equation, which reduces to the solution of a system of algebraic equations of the size of the problem for each time step. This requires less computational effort compared to the differential quadrature method (DQM) where a larger system of the size of the problem should be solved within each time element. The mixed formulation combines the simplicity of the Ritz method and accuracy and computational efficiency of the TQR. The stability property and computational efficiency of the scheme are discussed in detail. Numerical results show that the proposed mixed methodology can be used as an efficient tool for handling the time-dependent problems.
References
1.
Katz
, R.
, Lee
, C. W.
, Ulsoy
, A.
, and Scott
, R. A.
, 1987, “Dynamic Stability and Response of a Beam Subjected to a Deflection Dependent Moving Load
,” ASME J. Vibr. Acoust., Stress, Reliab. Des.
, 109
, pp. 361
–365
.2.
Fryba
, L.
, 1972, Vibration of Solids and Structures Under Moving Loads
, Noordhoff International
, Groningen, the Netherlands
.3.
Nelson
, H. D.
, and Conover
, R. A.
, 1971, “Dynamic Stability of a Beam Carrying Moving Masses
,” ASME J. Appl. Mech.
, 38
, pp. 1003
–1006
.4.
Hutton
, D. V.
, and Counts
, J.
, 1974, “Deflections of a Beam Carrying a Moving Mass
,” ASME J. Appl. Mech.
, 41
, pp. 803
–804
.5.
Benedetti
, G. A.
, 1974, “Dynamic Stability of a Beam Loaded by a Sequence of Moving Mass Particles
,” ASME J. Appl. Mech.
, 41
, pp. 1069
–1071
.6.
Saigal
, S.
, 1986, “Dynamic Behavior of Beam Structures Carrying Moving Masses
,” ASME J. Appl. Mech.
, 53
, pp. 222
–224
.7.
Duffy
, D. G.
, 1990, “The Response of an Infinite Railroad Track to a Moving, Vibrating Mass
,” ASME J. Appl. Mech.
, 57
, pp. 66
–73
.8.
Esmailzadeh
, E.
, and Ghorashi
, M.
, 1995, “Vibration Analysis of Beams Traversed by Uniform Partially Distributed Moving Masses
,” J. Sound Vib.
, 184
, pp. 9
–17
.9.
Rao
, G. V.
, 2000, “Linear Dynamics of an Elastic Beam Under Moving Loads
,” ASME J. Vibr. Acoust.
, 122
, pp. 281
–289
.10.
Lee
, K.-Y.
, and Renshaw
, A. A.
, 2000, “Solution of the Moving Mass Problem Using Complex Eigenfunction Expansions
,” ASME J. Appl. Mech.
, 67
, pp. 823
–827
.11.
Nikkhoo
, A.
, Rofooei
, F. R.
, and Shadnam
, M. R.
, 2007, “Dynamic Behavior and Modal Control of Beams Under Moving Mass
,” J. Sound Vib.
, 306
, pp. 712
–724
.12.
Dehestani
, M.
, Mofid
, M.
, and Vafai
, A.
, 2009, “Investigation of Critical Influential Speed for Moving Mass Problems on Beams
,” Appl. Math. Model.
, 33
(10
), pp. 3885
–3895
.13.
Kiani
, K.
, Nikkhoo
, A.
, and Mehri
, B.
, 2009, “Parametric Analyses of Multispan Viscoelastic Shear Deformable Beams Under Excitation of a Moving Mass
,” ASME J. Vibr. Acoust.
, 131
, p. 051009
.14.
Mofid
, M.
, Tehranchi
, A.
, and Ostadhossein
, A.
, 2010, “On the Viscoelastic Beam Subjected to Moving Mass
,” Adv. Eng. Software
, 41
(2
), pp. 240
–247
.15.
Khalili
, S. M. R.
, Jafari
, A. A.
, and Eftekhari
, S. A.
, 2010, “A Mixed Ritz-DQ Method for Forced Vibration of Functionally Graded Beams Carrying Moving Loads
,” Compos. Struct.
, 92
(10
), pp. 2497
–2511
.16.
Oni
, S. T.
, and Omolofe
, B.
, 2011, “Dynamic Response of Prestressed Rayleigh Beam Resting on Elastic Foundation and Subjected to Masses Traveling at Varying Velocity
,” ASME J. Vibr. Acoust.
, 133
, p. 041005
.17.
Wu
, J -J.
, 2005, “Dynamic Analysis of an Inclined Beam Due to Moving Loads
,” J. Sound Vibr.
, 288
, pp. 107
–131
.18.
Fung
, T. C.
, 1998, “Complex Time-Step Newmark Methods With Controllable Numerical Dissipation
,” Int. J. Numer. Methods Eng.
, 41
, pp. 65
–93
.19.
Kujawski
, J.
, and Gallagher
, R. H.
, 1989, “A Generalized Least-Squares Family of Algorithms for Transient Dynamic Analysis
,” Earthquake Eng. Struct. Dyn.
, 18
, pp. 539
–550
.20.
Gellert
, M.
, 1978, “A New Algorithm for Integration of Dynamic Systems
,” Comput. Struct.
, 9
, pp. 401
–408
.21.
Mancuso
, M.
, and Ubertini
, F.
, 2001, ” Collocation Methods With Controllable Dissipation for Linear Dynamics
,” Comput. Methods Appl. Mech. Eng.
, 190
, pp. 3607
–3621
.22.
Tanaka
, M.
, and Chen
, W.
, 2001, “Coupling Dual Reciprocity BEM and Differential Quadrature Method for Time-Dependent Diffusion Problems
,” Appl. Math. Model.
, 25
, pp. 257
–268
.23.
Fung
, T. C.
, 2002, “Stability and Accuracy of Differential Quadrature Method in Solving Dynamic Problems
,” Comput. Methods Appl. Mech. Eng.
, 191
, pp. 1311
–1331
.24.
Chen
, W.
, and Tanaka
, M.
, 2002, “A Study on Time Schemes for DRBEM Analysis of Elastic Impact Wave
,” Comput. Mech.
, 28
, pp. 331
–338
.25.
Eftekhari
, S. A.
, Farid
, M.
, and Khani
, M.
, 2009, “Dynamic Analysis of Laminated Composite Coated Beams Carrying Multiple Accelerating Oscillators Using a Coupled Finite Element-Differential Quadrature Method
,” ASME J. Appl. Mech.
, 76
, p. 061001
.26.
Eftekhari
, S. A.
, and Khani
, M.
, 2010, “A Coupled Finite Element-Differential Quadrature Element Method and its Accuracy for Moving Load Problem
,” Appl. Math. Model.
, 34
, pp. 228
–237
.27.
Jafari
, A. A.
, and Eftekhari
, S. A.
, 2011, “A New Mixed Finite Element-Differential Quadrature Formulation for Forced Vibration of Beams Carrying Moving Loads
,” ASME J. Appl. Mech.
, 78
, p. 011020
.28.
Mancuso
, M.
, Ubertini
, F.
, and Momanyi
, F. X.
, 2000, “Time Continuous Galerkin Methods for Linear Heat Conduction Problems
,” Comput. Methods Appl. Mech. Eng.
, 189
, pp. 91
–106
.29.
Li
, X. D.
, and Wiberg
, N.-E.
, 1996, “Structural Dynamic Analysis by a Time-Discontinuous Galerkin Finite Element Method
,” Int. J. Numer. Methods Eng.
, 39
, pp. 2131
–2152
.30.
Hughes
, T. J. R.
, 1987, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis
, Prentic-Hall
, Englewood Cliffs, NJ
.31.
Zienkiewicz
, O. C.
, and Taylor
, R. L.
, 2000, The Finite Element Method
, 5th ed., Butterworth-Heinemann
, Oxford
.32.
Bathe
, K. J.
, and Wilson
, E. L.
, 1973, “Stability and Accuracy Analysis of Direct Integration Methods
,” Earthquake Eng. Struct. Dyn.
, 1
, pp. 283
–291
.33.
Park
, K. C.
, 1975, “An Improved Stiffy Stable Method for Direct Integration of Nonlinear Structural Dynamic Equations
,” ASME J. Appl. Mech.
, 42
, pp. 464
–470
.34.
Hughes
, T. J. R.
, 1976, “Stability, Convergence and Growth and Decay of the Average Acceleration Method in Nonlinear Structural Dynamics
,” Comput. Struct.
, 6
, pp. 313
–324
.35.
Hilber
, H. M.
, Hughes
, T. J. R.
, and Taylor
, R. L.
, 1977, “Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics
,” Earthquake Eng. Struct. Dyn.
, 5
, pp. 283
–292
.36.
Hilber
, H. M.
, and Hughes
, T. J. R.
, 1978, “Collocation, Dissipation and ‘Overshoot’ for Time Integration Schemes in Structural Dynamics
,” Earthquake Eng. Struct. Dyn.
, 6
, pp. 99
–118
.37.
Wood
, W. L.
, Bossak
, M.
, and Zeinkiewicz
, O. C.
, 1981, “An Alpha Modification of Newmark’s Method
,” Int. J. Numer. Methods Eng.
, 15
, pp. 1562
–1566
.38.
Chung
, J.
, and Hulbert
, G. M.
, 1993, “A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method
,” ASME J. Appl. Mech.
, 60
, pp. 371
–375
.39.
Zhou
, X.
, and Tamma
, K. K.
, 2004, “Design, Analysis, and Synthesis of Generalized Single Step Single Solve and Optimal Algorithms for Structural Dynamics
,” Int. J. Numer. Methods Eng.
, 59
, pp. 597
–668
.40.
Zhou
, X.
, Tamma
, K. K.
, and Sha
, D.
, 2005, “Design Spaces, Measures and Metrics for Evaluating Quality of Time Operators and Consequences Leading to Improved Algorithms by Design-Illustration to Structural Dynamics
,” Int. J. Numer. Methods Eng.
, 64
, pp. 1841
–1870
.41.
Bathe
, K. J.
, and Wilson
, E. L.
, 1976, Numerical Methods in Finite Element Analysis
, Prentic-Hall
, Englewood Cliffs, NJ
.42.
Meirovitch
, L.
, 1967, Analytical Methods in Vibrations
, Macmillan
, New York
.43.
Rao
, S. S.
, 2007, Vibration of Continuous Systems
, John Wiley and Sons
, New York
.Copyright © 2012
by American Society of Mechanical Engineers
You do not currently have access to this content.
