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Research Papers

A New Mixed Finite Element–Differential Quadrature Formulation for Forced Vibration of Beams Carrying Moving Loads

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

Department of Mechanical Engineering, K. N. Toosi University, 19395-1999 Tehran, Iran

J. Appl. Mech 78(1), 011020 (Oct 26, 2010) (16 pages) doi:10.1115/1.4002037 History: Received December 29, 2008; Revised June 21, 2010; Posted June 23, 2010; Published October 26, 2010; Online October 26, 2010

In this paper, a new version of mixed finite element–differential quadrature formulation is presented for solving time-dependent problems. The governing partial differential equation of motion of the structure is first reduced to a set of ordinary differential equations (ODEs) in time using the finite element method. The resulting system of ODEs is then satisfied at any discrete time point apart and changed to a set of algebraic equations by the application of differential quadrature method (DQM) for time derivative discretization. The resulting set of algebraic equations can be solved by either direct methods (such as the Gaussian elimination method) or iterative methods (such as the Gauss–Seidel method). The mixed formulation enjoys the strong geometry flexibility of the finite element method and the high accuracy, low computational efforts, and efficiency of the DQM. The application of the formulation is then shown by solving some moving load class of problems (i.e., moving force, moving mass, and moving oscillator problems). The stability property and computational efficiency of the scheme are also 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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Figures

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Figure 3

Effect of δ-value on the stability of the DQ time integration scheme for even and odd numbers of sampling time points

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Figure 2

Stability of the DQ time integration scheme for the solution of first-order ODEs for different types of sampling time points

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Figure 1

Simply supported beam with a moving oscillator

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Figure 6

Convergence of solutions with respect to the number of sampling time points for moving load problem of a simply supported beam for different values of Tf/T(n=2, nT=1). Note that at a given sampling point, the time interval for all schemes is equal (i.e., ΔtDQ=ΔtWilson=ΔtNewmark=ΔtHoubolt).

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Figure 7

Convergence of solutions with respect to the number of sampling time points for moving load problem of a simply supported beam for different values of Tf/T and n(nT=1)

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Figure 8

Effect of δ-point on the accuracy of DQ solutions (n=2, nT=1)

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Figure 4

Stability of the DQ time integration scheme for the solution of second-order ODEs for different types of sampling time points

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Figure 5

Effect of δ-value on the stability of the DQ time integration scheme for the solution of second-order ODEs

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Figure 9

Convergence of solutions with respect to the number of time points for moving load problem with clamped supports (n=8, nT=1)

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Figure 10

Midspan deflection and maximum deflection of the simply supported beam due to two oscillators traveling with a velocity of 4 m/s for ΔT=0 (solid line), ΔT=1 s (dashed line), and ΔT=1.5 s (dotted line)

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Figure 11

Central displacements of the simply supported beam for different moving load models (v=4 m/s, ΔT=1.5 s)

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