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Research Papers

The Reduced Space Shooting Method for Calculating the Peak Periodic Solutions of Nonlinear Systems

[+] Author and Article Information
Haitao Liao

Institute of Advanced Structure Technology,
Beijing Institute of Technology,
Beijing 100081, China
e-mail: ht0819@163.com

Wenwang Wu

Institute of Advanced Structure Technology,
Beijing Institute of Technology,
Beijing 100081, China

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 20, 2017; final manuscript received February 18, 2018; published online April 12, 2018. Assoc. Editor: Brian Feeny.

J. Comput. Nonlinear Dynam 13(6), 061001 (Apr 12, 2018) (9 pages) Paper No: CND-17-1173; doi: 10.1115/1.4039682 History: Received April 20, 2017; Revised February 18, 2018

A hybrid approach which combines the reduced sequential quadratic programing (SQP) method with the shooting method is proposed to search the worst resonance response of nonlinear systems. The shooting method is first employed to construct the nonlinear equality constraints for the constrained optimization problem. Then, the complex optimization problem is simplified and solved numerically by the reduced SQP method. By virtue of the coordinate basis decomposition scheme which exploits the gradients of nonlinear equality constraints, the nonlinear equality constraints are eliminated, resulting in a simple optimization problem subject to bound constraints. Moreover, the second-order correction (SOC) technique is adopted to overcome Maratos effect. The novelty of the approach described lies in the capability to efficiently handle nonlinear equality constraints. The effectiveness of the proposed algorithm is demonstrated by two benchmark examples seen in the literature.

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Figures

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Fig. 4

The profile of the NLTVA coupled to a Duffing oscillator

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Fig. 1

Amplitude–frequency response curves of the Duffing equation

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Fig. 2

Evolution of the optimization algorithm

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Fig. 3

Numerical optimization results of the Duffing oscillator obtained by the proposed method

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Fig. 6

The frequency response curves obtained by using the continuation method for F = 0.19

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Fig. 7

Evolution of the optimization iterations for F = 0.15

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Fig. 8

Evolution of the optimization iterations for F = 0.19

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Fig. 9

The optimal solutions for the resonance peaks obtained by the present method: (a) F = 0.15 and (b) F = 0.15

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Fig. 5

The frequency response curves obtained by using the continuation method for F = 0.15: (a) the lower amplitude curves and (b) the isolated solution branches

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