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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Nayfeh, A. H. , and Mook, D. T. , 2008, Nonlinear Oscillations, Wiley, New York.
Akhavan, H. , and Ribeiro, P. , 2017, “Geometrically Non-Linear Periodic Forced Vibrations of Imperfect Laminates With Curved Fibres by the Shooting Method,” Comp. Part B: Eng., 109, pp. 286–296. [CrossRef]
Sinou, J. J. , Didier, J. , and Faverjon, B. , 2015, “Stochastic Non-Linear Response of a Flexible Rotor With Local Non-Linearities,” Int. J. Non-Linear Mech., 74, pp. 92–99. [CrossRef]
Gong, G. , and Dunne, J. F. , 2011, “Efficient Exceedance Probability Computation for Randomly Uncertain Nonlinear Structures With Periodic Loading,” J. Sound Vib., 330(10), pp. 2354–2368. [CrossRef]
Dou, S. , and Jensen, J. S. , 2015, “Optimization of Nonlinear Structural Resonance Using the Incremental Harmonic Balance Method,” J. Sound Vib., 334, pp. 239–254. [CrossRef]
Xiong, H. , Kong, X. , Li, H. , and Yang, Z. , 2017, “Vibration Analysis of Nonlinear Systems With the Bilinear Hysteretic Oscillator by Using Incremental Harmonic Balance Method,” Commun. Nonlinear Sci. Numer. Simul., 42, pp. 437–450. [CrossRef]
Attili, B. S. , and Syam, M. I. , 2008, “Efficient Shooting Method for Solving Two Point Boundary Value Problems,” Chaos, Solitons Fractals, 35(5), pp. 895–903. [CrossRef]
Ardeh, H. A. , and Allen, M. S. , 2016, “Multiharmonic Multiple-Point Collocation: A Method for Finding Periodic Orbits of Strongly Nonlinear Oscillators,” ASME J. Comput. Nonlinear Dyn., 11(4), p. 041006. [CrossRef]
Wang, X. F. , and Zhu, W. D. , 2015, “A Modified Incremental Harmonic Balance Method Based on the Fast Fourier Transform and Broyden's Method,” Nonlinear Dyn., 81(1–2), pp. 981–989. [CrossRef]
Hoang, T. , Duhamel, D. , Foret, G. , Yin, H. P. , and Argoul, P. , 2017, “Frequency Dependent Iteration Method for Forced Nonlinear Oscillators,” Appl. Math. Modell., 42, pp. 441–448. [CrossRef]
Javed, U. , Abdelkefi, A. , and Akhtar, I. , 2016, “An Improved Stability Characterization for Aeroelastic Energy Harvesting Applications,” Commun. Nonlinear Sci. Numer. Simul., 36, pp. 252–265. [CrossRef]
Peeters, M. , Viguié, R. , Sérandour, G. , Kerschen, G. , and Golinval, J. C. , 2009, “Nonlinear Normal Modes—Part II: Toward a Practical Computation Using Numerical Continuation Techniques,” Mech. Syst. Signal Process, 23(1), pp. 195–216. [CrossRef]
Wang, F. , 2015, “Bifurcations of Nonlinear Normal Modes Via the Configuration Domain and the Time Domain Shooting Methods,” Commun. Nonlinear Sci. Numer. Simul., 20(2), pp. 614–628. [CrossRef]
Lee, K. H. , Han, H. S. , and Park, S. , 2017, “Bifurcation Analysis of Coupled Lateral/Torsional Vibrations of Rotor Systems,” J. Sound Vib., 386, pp. 372–389. [CrossRef]
Kim, S. , and Palazzolo, A. B. , 2017, “Shooting With Deflation Algorithm-Based Nonlinear Response and Neimark-Sacker Bifurcation and Chaos in Floating Ring Bearing Systems,” ASME J. Comput. Nonlinear Dyn., 12(3), p. 031003. [CrossRef]
Habib, G. , Detroux, T. , Viguié, R. , and Kerschen, G. , 2015, “Nonlinear Generalization of Den Hartog' s Equal-Peak Method,” Mech. Syst. Signal Process, 52–53, pp. 17–28. [CrossRef]
Detroux, T. , Habib, G. , Masset, L. , and Kerschen, G. , 2015, “Performance, Robustness and Sensitivity Analysis of the Nonlinear Tuned Vibration Absorber,” Mech. Syst. Signal Process, 60–61, pp. 799–809. [CrossRef]
Förster, A. , and Krack, M. , 2016, “An Efficient Method for Approximating Resonance Curves of Weakly-Damped Nonlinear Mechanical Systems,” Comput. Struct., 169, pp. 81–90. [CrossRef]
Liao, H. , and Sun, W. , 2013, “A New Method for Predicting the Maximum Vibration Amplitude of Periodic Solution of Non-Linear System,” Nonlinear Dyn., 71(3), pp. 569–582. [CrossRef]
Liao, H. , 2015, “Optimization Analysis of Duffing Oscillator With Fractional Derivatives,” Nonlinear Dyn., 79(2), pp. 1311–1328. [CrossRef]
Dednam, W. , and Botha, A. E. , 2015, “Optimized Shooting Method for Finding Periodic Orbits of Nonlinear Dynamical Systems,” Eng. Comput., 31(4), pp. 749–762. [CrossRef]
Nocedal, J. , and Wright, S. J. , 2006, Numerical Optimization, 2nd ed., Springer, New York.
Nocedal, J. , and Wright, S. J. , 2006, Sequential Quadratic Programming, Springer, New York.
Byrd, R. H. , and Nocedal, J. , 1990, “An Analysis of Reduced Hessian Methods for Constrained Optimization,” Math. Prog., 49(1–3), pp. 285–323. [CrossRef]
Schmid, C. , 1994, “Reduced Hessian Successive Quadratic Programming for Large-Scale Process Optimization,” Ph.D. thesis, Carnegie Mellon University, Pittsburgh, PA.
Biegler, L. T. , Nocedal, J. , Schmid, C. , and Ternet, D. , 2000, “Numerical Experience With a Reduced Hessian Method for Large Scale Constrained Optimization,” Comput. Opt. Appl., 15(1), pp. 45–67. [CrossRef]
Biegler, L. T. , Nocedal, J. , and Schmid, C. , 1995, “A Reduced Hessian Method for Large-Scale Constrained Optimization,” SIAM J. Opt., 5(2), pp. 314–347. [CrossRef]
Bonis, I. , and Theodoropoulos, C. , 2012, “Model Reduction-Based Optimization Using Large-Scale Steady-State Simulators,” Chem. Eng. Sci, 69(1), pp. 69–80. [CrossRef]
Zhang, S. Y. , and Deng, Z. C. , 2006, “Group Preserving Schemes for Nonlinear Dynamic System Based on RKMK Methods,” Appl. Math. Comput., 175(1), pp. 497–507.
Lazarus, A. , and Thomas, O. , 2010, “A Harmonic-Based Method for Computing the Stability of Periodic Solutions of Dynamical Systems,” C. R. Méc., 338(9), pp. 510–517. [CrossRef]
Boggs, P. T. , and Tolle, J. W. , 1995, “Sequential Quadratic Programming,” Acta Numer., 4, pp. 1–51. [CrossRef]
Ugray, Z. , Lasdon, L. , Plummer, J. , Glover, F. , Kelly, J. , and Marti, R. , 2007, “Scatter Search and Local NLP Solvers: A Multistart Framework for Global Optimization,” INFORMS J. Comput., 19(3), pp. 328–340. [CrossRef]


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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. 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. 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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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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