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Research Papers

A Frequency Domain Method for Calculating the Failure Probability of Nonlinear Systems With Random Uncertainty

[+] Author and Article Information
Haitao Liao

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

Wenwang Wu

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

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received October 13, 2017; final manuscript received January 24, 2018; published online March 30, 2018. Assoc. Editor: Maurizio Porfiri.

J. Vib. Acoust 140(4), 041019 (Mar 30, 2018) (9 pages) Paper No: VIB-17-1456; doi: 10.1115/1.4039405 History: Received October 13, 2017; Revised January 24, 2018

A hybrid approach is proposed to evaluate the probability of unacceptable performance with respect to uncertain parameters. The evaluation of structural reliability and the solution of maximum vibration response are performed simultaneously. A constrained optimization problem is deduced for which several techniques have been developed to obtain the reliability index. The nonlinear equality constraints of the optimization problem are constructed based on the harmonic balance equations, the optimality condition of the maximum vibration response with respect to the vibration frequency and the limit state failure function. With the nonlinear equality constraints imposed on the harmonic balance equations and the derivative of the maximum vibration response with respect to the vibration frequency, the inner loop for solving the maximum vibration response is avoided. The sensitivity gradients are derived by virtue of the adjoint method. The original optimization formulation is then solved by means of the sequential quadratic programming method (SQP) method. Finally, the developed approach has been verified by comparison with reference values from Monte Carlo simulation (MCS). Numerical results reveal that the proposed method is capable of predicting the failure probability of nonlinear structures with random uncertainty.

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References

Haldar, A. , and Mahadevan, S. , 2000, Probability, Reliability, and Statistical Methods in Engineering Design, Wiley, New York.
Rackwitz, R. , 2001, “ Reliability Analysis—A Review and Some Perspectives,” Struct. Saf., 23(4), pp. 365–395. [CrossRef]
Hohenbichler, M. , and Rackwitz, R. , 1982, “ First-Order Concepts in System Reliability,” Struct. Saf., 1(3), pp. 177–188. [CrossRef]
Hohenbichler, M. , Gollwitzer, S. , Kruse, W. , and Rackwitz, R. , 1987, “ New Light on First- and Second-Order Reliability Methods,” Struct. Saf., 4(4), pp. 267–284. [CrossRef]
Der, K. A. , Lin, H. Z. , and Hwang, S. J. , 1987, “ Second-Order Reliability Approximations,” J. Eng. Mech., 113(8), pp. 1208–1225. [CrossRef]
Kiureghian, A. D. , and Stefano, M. D. , 1991, “ Efficient Algorithm for Second-Order Reliability Analysis,” J. Eng. Mech., 117(12), pp. 2904–2923. [CrossRef]
Enevoldsen, I. , and Sørensen, J. D. , 1994, “ Reliability-Based Optimization in Structural Engineering,” Struct. Saf., 15(3), pp. 169–196. [CrossRef]
Tu, J. , Choi, K. K. , and Park, Y. H. , 1999, “ A New Study on Reliability-Based Design Optimization,” ASME J. Mech. Des., 121(4), pp. 557–564. [CrossRef]
Hasofer, A. M. , and Lind, N. C. , 1974, “ Exact and Invariant Second-Moment Code Format,” J. Eng. Mech. Div., 100(1), pp. 111–121.
Rackwitz, R. , and Flessler, B. , 1978, “ Structural Reliability Under Combined Random Load Sequences,” Comput. Struct., 9(5), pp. 489–494. [CrossRef]
Keshtegar, B. , 2016, “ Stability Iterative Method for Structural Reliability Analysis Using a Chaotic Conjugate Map,” Nonlinear Dyn., 84(4), pp. 2161–2174. [CrossRef]
Keshtegar, B. , and Meng, Z. , 2017, “ A Hybrid Relaxed First-Order Reliability Method for Efficient Structural Reliability Analysis,” Struct. Saf., 66, pp. 84–93. [CrossRef]
Bucher, C. , and Most, T. , 2008, “ A Comparison of Approximate Response Functions in Structural Reliability Analysis,” Probab. Eng. Mech., 23(2–3), pp. 154–163. [CrossRef]
Luo, X. , Li, X. , Zhou, J. , and Cheng, T. , 2012, “ A Kriging-Based Hybrid Optimization Algorithm for Slope Reliability Analysis,” Struct. Saf., 34(1), pp. 401–406. [CrossRef]
Kroetz, H. M. , Tessari, R. K. , and Beck, A. T. , 2017, “ Performance of Global Metamodeling Techniques in Solution of Structural Reliability Problems,” Adv. Eng. Software, 114, pp. 394–404. [CrossRef]
Hall, K. C. , Thomas, J. P. , and Clark, W. S. , 2002, “ Computation of Unsteady Nonlinear Flows in Cascades Using a Harmonic Balance Technique,” AIAA J., 40(5), pp. 879–886. [CrossRef]
Guillot, L. , Vigué, P. , Vergez, C. , and Cochelin, B. , 2017, “ Continuation of Quasi-Periodic Solutions With Two-Frequency Harmonic Balance Method,” J. Sound Vib., 394, pp. 434–450. [CrossRef]
Cochelin, B. , and Vergez, C. , 2009, “ A High Order Purely Frequency-Based Harmonic Balance Formulation for Continuation of Periodic Solutions,” J. Sound Vib., 324(1–2), pp. 243–262. [CrossRef]
Dai, H. , Yue, X. , Yuan, J. , and Atluri, S. N. , 2014, “ A Time Domain Collocation Method for Studying the Aeroelasticity of a Two Dimensional Airfoil With a Structural Nonlinearity,” J. Comput. Phys., 270, pp. 214–237. [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]
Grolet, A. , and Thouverez, F. , 2012, “ Free and Forced Vibration Analysis of a Nonlinear System With Cyclic Symmetry:Application to a Simplified Model,” J. Sound Vib., 331(12), pp. 2911–2928. [CrossRef]
Grolet, A. , and Thouverez, F. , 2015, “ Computing Multiple Periodic Solutions of Nonlinear Vibration Problems Using the Harmonic Balance Method and Groebner Bases,” Mech. Syst. Signal Process., 52–53, pp. 529–547. [CrossRef]
Liao, H. , 2015, “ Piecewise Constrained Optimization Harmonic Balance Method for Predicting the Limit Cycle Oscillations of an Airfoil With Various Nonlinear Structures,” J. Fluids Struct., 55, pp. 324–346. [CrossRef]
Coudeyras, N. , Sinou, J. J. , and Nacivet, S. , 2009, “ A New Treatment for Predicting the Self-Excited Vibrations of Nonlinear Systems With Frictional Interfaces: The Constrained Harmonic Balance Method, With Application to Disc Brake Squeal,” J. Sound Vib., 319(3–5), pp. 1175–1199. [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]
Liao, H. , 2014, “ Nonlinear Dynamics of Duffing Oscillator With Time Delayed Term,” CMES: Comput. Model. Eng. Sci., 103(3), pp. 155–187.
Cameron, T. M. , and Griffin, J. H. , 1989, “ An Alternating Frequency/Time Domain Method for Calculating the Steady-State Response of Nonlinear Dynamic Systems,” ASME J. Appl. Mech., 56(1), pp. 149–154. [CrossRef]
Blommaert, M. , Dekeyser, W. , Baelmans, M. , Gauger, N. R. , and Reiter, D. , 2017, “ A Practical Globalization of One-Shot Optimization for Optimal Design of Tokamak Divertors,” J. Comput. Phys., 328, pp. 399–412. [CrossRef]
Liao, H. , 2016, “ Efficient Sensitivity Analysis Method for Chaotic Dynamical Systems,” J. Comput. Phys., 313, pp. 57–75. [CrossRef]
Maute, K. , Nikbay, M. , and Farhat, C. , 2003, “ Sensitivity Analysis and Design Optimization of Three‐Dimensional Non‐Linear Aeroelastic Systems by the Adjoint Method,” Int. J. Numer. Methods Eng., 56(6), pp. 911–933. [CrossRef]
Wright, S. , and Nocedal, J. , 2006, Numerical Optimization, 2nd ed., Springer, New York.
Wright, S. , and Nocedal, J. , 2006, Sequential Quadratic Programming, Springer, New York, pp. 529–562.

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Schematic illustration of a two-dimensional LSF

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