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Research Papers

Coupling Ritz Method and Triangular Quadrature Rule for Moving Mass Problem

[+] Author and Article Information
S. A. Eftekhari, A. A. Jafari

 Department of Mechanical Engineering, K. N. Toosi University, P. O. Box 19395-1999, 19697 Tehran, Iran

J. Appl. Mech 79(2), 021018 (Feb 24, 2012) (14 pages) doi:10.1115/1.4005577 History: Received December 27, 2010; Revised October 02, 2011; Posted February 01, 2012; Published February 13, 2012; Online February 24, 2012

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.

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Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Simply supported beam with a moving mass

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Figure 2

Spectral radius of TQR amplification matrix with the Type I and Type II sample time points for different values of m (α=1)

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Figure 3

Spectral radius of TQR amplification matrix with Types I and II sample time points for different values of Ns and α (m = 4)

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Figure 4

Algorithmic damping ratio of the TQR with the Type I and Type II sample time points for different values of m (α=1)

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Figure 5

Algorithmic damping ratio of the TQR with Types I and II sample time points for different values of Ns and α (m = 4)

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Figure 6

Relative period error of the TQR with the Type I and Type II sample time points for different values of m (α=1)

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Figure 7

Relative period error of the TQR with Types I and II sample time points for different values of Ns and α (m = 4)

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Figure 8

Comparison between the TQR solutions and those of the Newmark and Houbolt schemes at different time levels and moving load velocities for a fixed time step size

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Figure 9

Comparison between the TQR solutions and those of the Newmark and Houbolt schemes at different time levels and moving load velocities for a fixed time step size (numerical results for clamped-clamped beam/bar)

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Figure 10

Comparison between the TQR solutions and those of the Newmark and Houbolt schemes at different time levels and moving load velocities for a fixed time step size (numerical results for clamped-free beam/bar)

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Figure 11

Comparison between the TQR solutions and those of the Newmark and Houbolt schemes at different time levels and moving load velocities for a fixed time step size (numerical results for clamped-clamped beam/bar)

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Figure 12

Comparison between the TQR solutions and those of the Newmark and Houbolt schemes at different time levels and moving load velocities for a fixed time step size (numerical results for clamped-free beam/bar)

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