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Research Papers

An Analytical Study on Forced Vibration of Beams Carrying a Number of Two Degrees-of-Freedom Spring–Damper–Mass Subsystems

[+] Author and Article Information
Jun Chen, Dawei Dong, Bing Yan, Chunrong Hua

Department of Mechanical Engineering,
Southwest Jiaotong University,
Chengdu 610031, China

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received March 14, 2016; final manuscript received July 6, 2016; published online August 19, 2016. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 138(6), 061011 (Aug 19, 2016) (15 pages) Paper No: VIB-16-1119; doi: 10.1115/1.4034252 History: Received March 14, 2016; Revised July 06, 2016

This paper presents an analytical solution for the forced vibration of beams carrying a number of two degrees-of-freedom (DOF) spring–damper–mass (SDM) systems. The beam is divided into a series of distinct sub-beams at the spring connection points and the point of force action. The 2DOF SDM systems are replaced with a set of effective springs with complex stiffness, and the compatibility of the placement and the force at the common interface of two adjacent sub-beams is systematically organized. Then, the boundary conditions are enforced, and the governing matrix equation is formulated. Next, the closed-form expression for the frequency response function (FRF) was determined analytically. The presented method can simultaneously consider arbitrary boundary conditions and any number of 2DOF SDM systems. Furthermore, regardless of the number of subsystems, none of the associated matrices is larger than 4 × 4, which provides a significant computational advantage. To validate the accuracy and reliability of the proposed method, some results are compared with the corresponding results obtained using the conventional finite element method (FEM); good agreement is observed between the results of the two approaches. Finally, the effects of the system parameters on the vibration transmission in the beam and subsystem vibration reduction are studied.

Copyright © 2016 by ASME
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Figures

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Fig. 1

An elastically constrained beam carrying a number of 2DOF SDM systems

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Fig. 2

(a) Force equilibrium and displacement compatibility of the sub-beam system and (b) the 2DOF spring-system replaced by four complex effective stiffnesses

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Fig. 3

Equilibrium conditions of the mth beam loading the excitation force

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Fig. 4

A cantilever beam with three single-DOF SDM systems (E = 7 × 1010 N/m2, I = 6.4 × 10−6 m4, m = ρ × A = 0.675 kg, L = 1 m, F0 = 10 l1 = 0.1 m, l2 = 0.5 m, l3 = 0.9 m, mi = 0.1 kg, ki = 0.1 N/m, ci = 0.1 N·S/m, and i = 1, 2, 3) [28]

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Fig. 5

Comparison of the frequency response curves obtained using different methods: (a) the beam system illustrated in Fig.4, (b) the beam system illustrated in Fig. 6 with clamped-free boundary condition, and (c) the beam system illustrated in Fig. 6 with elastically constrained boundary condition (FR with the presented method, FR with FEM, and FR with ANCM in Ref. [28])

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Fig. 6

An elastically constrained beam carrying three 2DOF SDM systems

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Fig. 7

Frequency response at P1, P2, P3, and P4, as illustrated in Fig. 4(a) force is applied at P1, (b) force is applied at P2, and (c) force is applied at P3 (FR at P1 with the presented method,· FR at P1 with FEM, FR at P2 with the presented method, FR at P3 with FEM, FR at P3 with the presented method, FR at P3 with FEM, FR at P4 with the presented method, and FR at P4 with FEM)

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Fig. 8

Effect of the number of subsystems on the frequency response at the right end of the beam (N = 0, N = 1, N = 3, and N = 5)

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Fig. 9

Effect of the stiffness of the subsystem on the frequency response at the right end of the beam (kij = 3333.33/2 kN/m, kij = 3333.33 kN/m, kij = 3333.33 × 2 kN/m, and kij = 3333.33 × 4 kN/m)

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Fig. 10

Effect of the mass of the subsystem on the frequency response at the right end of the beam (mi = 266.67/2 kg, mi = 266.67 kg, mi = 266.67 × 2 kg, and mi = 266.67 × 4 kg)

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Fig. 11

Effect of the moment of inertia of the subsystem on the frequency response at the right end of the beam (Ji = 20/2 kg·m2, Ji = 20 kg·m2, Ji = 20 × 2 kg·m2, and Ji = 20 × 4 kg·m2)

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Fig. 12

Effect of the damping of the subsystem on the frequency response at the right end of the beam (ci = 66.67/2 N·s/m, ci = 66.67 N·s/m, ci = 66.67 × 2 N·s/m, and ci = 66.67 × 4 N·s/m)

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Fig. 13

An elastically constrained beam carrying a 2DOF SDM system

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Fig. 14

Effect of the position of the subsystem on its frequency response (d = 0 m, d = 0.3 m, d = 0.6 m, and d = 0.9 m)

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Fig. 15

Effect of the stiffness of the subsystem on its frequency response (kij = 10,000/10 kN/m, kij = 10,000 kN/m, kij = 10,000 × 10 kN/m, and kij = 10,000 × 100 kN/m)

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Fig. 16

Effect of the damping of the subsystem on its frequency response (ci = 200/10 N·s/m, ci = 200 N·s/m, ci = 200 × 10 N·s/m, and ci = 200 × 100 N·s/m)

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Fig. 17

Effect of the elastic support stiffness on the frequency response of the subsystem (Ki = 15,000/10 kN/m, Ki = 15,000 kN/m, Ki = 15,000 × 10 kN/m, and Ki = 15,000 × 100 kN/m)

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Fig. 18

Effect of the elastic support damping on the frequency response of the subsystem (Ci = 300/10 N·s/m, Ci = 300 N·s/m, Ci = 300 × 10 N·s/m, and Ci = 300 × 100 N·s/m)

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