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Research Papers

# Dynamic Analysis of Laminated Composite Coated Beams Carrying Multiple Accelerating Oscillators Using a Coupled Finite Element-Differential Quadrature Method

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

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

M. Farid

Department of Mechanical Engineering, Shiraz University, Shiraz, Iran

M. Khani

Department of Civil Engineering, Shiraz University, Shiraz, Iran

J. Appl. Mech 76(6), 061001 (Jul 21, 2009) (13 pages) doi:10.1115/1.3114969 History: Received December 27, 2006; Revised November 18, 2008; Published July 21, 2009

## Abstract

In this paper, a numerical algorithm using a coupled finite element-differential quadrature (DQ) method is proposed for the dynamic analysis of laminated composite coated beams subjected to a stream of accelerating oscillators. The finite element method with cubic Hermitian interpolation functions is used to discretize the spatial domain. The DQ method is then employed to discretize the time domain. The resulting set of algebraic equations can be solved by either direct methods or iterative methods. It is revealed that the DQ method stands out in numerical accuracy, as well as in computational efficiency, over the well-known standard finite difference schemes, such as the Newmark, Wilson $θ$, Houbolt, and central difference methods, for the cases considered. Furthermore, in the numerical examples, the effects of various parameters having something to do with the title problem, such as lamina thickness, orientation of the coats, arrival time intervals, velocities, and accelerations of the oscillators on the dynamic behavior of the system, are investigated. The technique presented in this investigation is general and can be easily applied to any time-dependent problem.

## Figures

Figure 11

Central displacement of a simply supported beam subjected to a moving point load (v=0.75 m/s, n=6, and m=21)

Figure 12

Central displacement of a simply supported beam subjected to a moving point load (v=0.75 m/s and n=6)

Figure 13

Central displacement of a simply supported beam subjected to a moving point load (v=2.5 m/s, n=4, and m=11)

Figure 1

Convergence of solutions with respect to the number of sample time points for v=0.1 m/s(n=4)

Figure 2

Convergence of solutions with respect to the number of sample time points for v=0.2 m/s(n=4)

Figure 3

Convergence of solutions with respect to the number of sample time points for v=0.75 m/s(n=4)

Figure 4

Convergence of solutions with respect to the number of finite elements for v=0.1 m/s(m=115)

Figure 5

Convergence of solutions with respect to the number of finite elements for v=0.2 m/s(m=60)

Figure 6

Convergence of solutions with respect to the number of finite elements for v=0.75 m/s(m=21)

Figure 7

Central displacement of a simply supported beam subjected to a moving point load (v=0.1 m/s, n=4, and m=115)

Figure 8

Central displacement of a simply supported beam subjected to a moving point load (v=0.1 m/s and n=4)

Figure 9

Central displacement of a simply supported beam subjected to a moving point load (v=0.2 m/s, n=4, and m=60)

Figure 10

Central displacement of a simply supported beam subjected to a moving point load (v=0.2 m/s, and n=4)

Figure 14

Central displacement of a simply supported beam subjected to a moving point load (v=2.5 m/s and n=4)

Figure 15

Central displacement of a simply supported beam subjected to a moving point load (v=5 m/s, n=4, and m=8)

Figure 16

Central displacement of a simply supported beam subjected to a moving point load (v=5 m/s and n=4)

Figure 17

Convergence of solutions with respect to the number of sample time points for v=0.05 m/s(n=8)

Figure 24

Convergence of solutions with respect to the number of sample time points for v=7 m/s(n=4)

Figure 25

Convergence of solutions with respect to the number of time elements for v=0.45 m/s(n=4)

Figure 26

Convergence of solutions with respect to the number of time elements for v=1.5 m/s(n=4)

Figure 27

Convergence of solutions with respect to the number of sample time points for v=6 m/s(n=6)

Figure 28

Convergence of solutions with respect to the number of finite elements for v=6 m/s(m=25)

Figure 29

Central displacement of a clamped-clamped beam subjected to a moving oscillator (v=6 m/s, n=6, and m=37)

Figure 30

Central displacement of a clamped-clamped beam subjected to a moving oscillator (v=6 m/s and n=6)

Figure 31

Central displacement of a simply supported beam subjected to a moving oscillator (v=6 m/s, n=6, and m=53)

Figure 32

Central displacement of a simply supported beam subjected to a moving oscillator (v=6 m/s and n=6)

Figure 18

Convergence of solutions with respect to the number of sample time points for v=0.25 m/s(n=6)

Figure 19

Convergence of solutions with respect to the number of sample time points for v=0.5 m/s(n=4)

Figure 20

Convergence of solutions with respect to the number of sample time points for v=1 m/s(n=6)

Figure 21

Effect of δ-value on accuracy and stability of numerical results (v=2 m/s and n=4)

Figure 22

Effect of δ-value on accuracy and stability of numerical results (v=4 m/s and n=6)

Figure 23

Convergence of solutions with respect to the number of sample time points for v=3 m/s(n=4)

Figure 33

Central displacement of a simply supported beam subjected to a moving oscillator (v=1.5 m/s, n=6, and m=51)

Figure 34

Central displacement of a simply supported beam subjected to a moving oscillator (v=1.5 m/s and n=6)

Figure 35

Laminated composite coated beam cross section

Figure 36

Effects of parameters h/H and θ on the dynamic magnification factor, v=30 m/s; θ=0 (dotted line), θ=45 (dashed line), and θ=90 (solid line)

Figure 37

Effects of parameters h/H and θ on the dynamic magnification factor, v=30 m/s; h/H=0 (dotted line), h/H=0.5 (dashed line), and h/H=0.85 (solid line)

Figure 38

Influence of h/H and velocity of moving oscillator on the dynamic magnification factor, θ=0; h/H=0 (dotted line), h/H=0.5 (dashed line), and h/H=1 (solid line)

Figure 39

Influence of θ and velocity of moving oscillator on the dynamic magnification factor, h/H=0; θ=0 (dotted line), θ=45 (dashed line), and θ=90 (solid line)

Figure 41

Maximum central deflections of the beam due to two oscillators for v2≤v1, v1=30 m/s; v2=10 m/s (dotted line), v2=20 m/s (dashed line), and v2=30 m/s (solid line)

Figure 43

Maximum central deflections of the beam due to two oscillators for a2≤a1, a1=20 m/s2; a2=0 m/s (dotted line), a2=10 m/s2 (dashed line), and a2=20 m/s2 (solid line) (v1=v2=30 m/s)

Figure 46

Maximum central deflections of the beam due to two oscillators for |a2|≥|a1|, a1=−20 m/s2; a2=−20 m/s2 (dotted line), a2=−30 m/s2 (dashed line), and a2=−40 m/s2 (solid line) (v1=v2=50 m/s)

Figure 40

Variation in dynamic magnification factors versus the equivalent bending stiffness of the beam; v=15 m/s (dotted line), v=30 m/s (dashed line), and v=45 m/s (solid line)

Figure 42

Maximum central deflections of the beam due to two oscillators for v2≥v1, v1=30 m/s; v2=30 m/s (dotted line), v2=40 m/s (dashed line), and v2=50 m/s (solid line)

Figure 44

Maximum central deflections of the beam due to two oscillators for a2≥a1, a1=20 m/s2; a2=20 m/s2 (dotted line), a2=30 m/s2 (dashed line), and a2=40 m/s2 (solid line) (v1=v2=30 m/s)

Figure 45

Maximum central deflections of the beam due to two oscillators for |a2|≤|a1|, a1=−20 m/s2; a2=0 (dotted line), a2=−10 m/s2 (dashed line), and a2=−20 m/s2 (solid line) (v1=v2=50 m/s)

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