Research Papers

Vibration of Flexible Structures Under Nonlinear Boundary Conditions

[+] Author and Article Information
Xiao-Ye Mao

Shanghai Institute of Applied
Mathematics and Mechanics,
Shanghai University,
149 Yan Chang Road,
Shanghai 200072, China
e-mail: maoxiaoye1987920@aliyun.com

Hu Ding

Shanghai Institute of Applied
Mathematics and Mechanics,
Shanghai University,
149 Yan Chang Road,
Shanghai 200072, China
e-mail: dinghu3@shu.edu.cn

Li-Qun Chen

Shanghai Key Laboratory of Mechanics
in Energy Engineering,
Department of Mechanics,
Shanghai Institute of Applied
Mathematics and Mechanics,
Shanghai University,
149 Yan Chang Road,
Shanghai 200072, China
e-mail: lqchen@staff.shu.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 9, 2017; final manuscript received September 6, 2017; published online September 21, 2017. Editor: Yonggang Huang.

J. Appl. Mech 84(11), 111006 (Sep 21, 2017) (11 pages) Paper No: JAM-17-1434; doi: 10.1115/1.4037883 History: Received August 09, 2017; Revised September 06, 2017

The nonlinear response of a flexible structure, subjected to generally supported conditions with nonlinearities, is investigated for the first time. An analytical procedure is proposed first. Moreover, a simulation technique usually employed in static analysis is developed for confirmation. Generally, ordinary perturbation methods could analyze dynamics of flexible structures with linear boundary conditions. As nonlinear boundaries are taken into account, they are out of operation for the modal shape that is hardly to be obtained, which is the key to the analysis. In order to overcome this, nonlinear boundary conditions are rescaled and the technique of modal revision is employed. Consequently, each governing equation with different time-scales could be analyzed exactly according to corresponding rescaled boundary conditions. The total response of any point at the flexible structure will be composed by harmonic responses yielded by the analytical method. Furthermore, the differential quadrature element method (DQEM), a numerical simulation technique could satisfy boundary conditions strictly, is introduced to certify analytical results. The comparison shows a reasonable agreement between these two methods. In fact, the accuracy of the analytical method for nonlinear boundaries could be explained in theory. Based on the certification, boundary nonlinearities are discussed in detail analytically and found to play an important role in responses. Because of the important role played by the nonlinear factors in the vibration and control of the flexible structure, this paper will open the vibration analysis and numerical study of the flexible structure with nonlinear constraints.

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Fig. 1

Model of a nonlinearly supported flexible structure

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Fig. 3

Normalized shape of U21 and U22

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Fig. 4

Normalized shape for corrected solutions with respect to boundary damping: (a) normalized shape of U23 and (b) normalized shape of U24

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Fig. 5

Time-domain response of the middle point with Fv = 5 N/m and Ω = 327 rad/s: (a) response in 2 s and (b) steady-state response

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Fig. 6

Forced vibration for the first mode: (a) amplitude–frequency curve and (b) phase–frequency curve

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Fig. 7

Comparison between response with and without boundary nonlinearity: (a) amplitude–frequency curve and (b) phase–frequency curve

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Fig. 8

Response for p2: (a) amplitude–frequency curve at the middle of U21, (b) amplitude–frequency curve at the middle of U22, (c) amplitude–frequency curve at the middle of U23, and (d) amplitude–frequency curve at the middle of U24

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Fig. 9

Total responses for some nodal points on the flexible structure by analytical method and DQEM: (a) total response at the middle point and (b) total response on the left boundary



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