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Recent advances and comparisons between harmonic balance and Volterra-based nonlinear frequency response analysis methods

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Abstract

Harmonic balance and Volterra-based analysis methods are well known, but the capabilities of these methods have been limited by significant issues of complexity which either constrain their application to relatively simple cases, or limit the accuracy of analysis in more complex cases. This study briefly summarizes recent results which effectively extend the capabilities of both harmonic balance and Volterra-based analysis by making complex analyses much more feasible. The new capabilities and performance of the two approaches are then evaluated and compared using benchmark case studies of a Duffing oscillator and a nonlinear automotive damper. The results offer new insights and lead to different conclusions on the relative merits of harmonic balance versus Volterra-based analysis relative to prior studies and similar benchmark analyses.

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Center for Nonlinear Dynamics and Control, Villanova University, 800 Lancaster Ave, Villanova, PA, 19085, USA

    J. C. Peyton Jones & K. S. A. Yaser

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  1. J. C. Peyton Jones
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  2. K. S. A. Yaser
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Correspondence to J. C. Peyton Jones.

Appendix

Appendix

Expressions for \(H_{5}(\cdot ),{\ldots },H_{7}(\cdot )\) for nonlinear damping case:

$$\begin{aligned}&H_5 \left( {j\omega _1 ,\ldots ,j\omega _5 } \right) = - H_1 \left( {j(\omega _1 +\omega _2 +\omega _3 +\omega _4 +\omega _5 )} \right) \\&\quad \times \left( {\begin{array}{l} d_2 \left( {\begin{array}{l} 2\left( {j\omega _1 +j\omega _2 } \right) \left( {j\omega _3 +j\omega _4 +j\omega _5 } \right) H_2 \left( {j\omega _1 +j\omega _2 } \right) H_3 \left( {j\omega _3 ,j\omega _4 ,j\omega _5 } \right) \\ +2j\omega _1 \left( {j\omega _2 +j\omega _3 +j\omega _4 +j\omega _5 } \right) H_1 \left( {j\omega _1 } \right) H_4 \left( {j\omega _2 ,j\omega _3 ,j\omega _4 ,j\omega _5 } \right) \\ \end{array}} \right) \\ +\,d_3 \left( {\begin{array}{l} 3j\omega _1 \left( {j\omega _2 +j\omega _3 } \right) \left( {j\omega _4 +j\omega _5 } \right) H_1 \left( {j\omega _1 } \right) H_2 \left( {j\omega _2 ,j\omega _3 } \right) H_2 \left( {j\omega _4 ,j\omega _5 } \right) \\ +3j\omega _1 j\omega _2 \left( {j\omega _3 +j\omega _4 +j\omega _5 } \right) H_1 \left( {j\omega _1 } \right) H_1 \left( {j\omega _2 } \right) H_3 \left( {j\omega _3 ,j\omega _4 ,j\omega _5 } \right) \\ \end{array}} \right) \\ \end{array}} \right) \\&H_6 \left( {j\omega _1 ,\ldots ,j\omega _6 } \right) = - H_1 \left( {j(\omega _1 +\omega _2 +\omega _3 +\omega _4 +\omega _5 +\omega _6 )} \right) \\&\quad \times \left( {\begin{array}{l} d_2 \left( {\begin{array}{l} 2j\omega _1 \left( {j\omega _2 +j\omega _3 +j\omega _4 +j\omega _5 +j\omega _6 } \right) H_1 \left( {j\omega _1 } \right) H_5 \left( {j\omega _2 ,j\omega _3 ,j\omega _4 ,j\omega _5 ,j\omega _6 } \right) \\ +2\left( {j\omega _1 +j\omega _2 } \right) \left( {j\omega _3 +j\omega _4 +j\omega _5 +j\omega _6 } \right) H_2 \left( {j\omega _1 ,j\omega _2 } \right) H_4 \left( {j\omega _3 ,j\omega _4 ,j\omega _5 ,j\omega _6 } \right) \\ \left( {j\omega _1 +j\omega _2 +j\omega _3 } \right) \left( {j\omega _4 +j\omega _5 +j\omega _6 } \right) H_3 \left( {j\omega _1 ,j\omega _2 ,j\omega _3 } \right) H_3 \left( {j\omega _4 ,j\omega _5 ,j\omega _6 } \right) \\ \end{array}} \right) \\ +\,d_3 \left( {\begin{array}{l} 3j\omega _1 j\omega _2 \left( {j\omega _3 +j\omega _4 +j\omega _5 +j\omega _6 } \right) H_1 \left( {j\omega _1 } \right) H_1 \left( {j\omega _2 } \right) H_4 \left( {j\omega _3 ,j\omega _4 ,j\omega _5 ,j\omega _6 } \right) \\ +\left( {j\omega _1 +j\omega _2 } \right) \left( {j\omega _3 +j\omega _4 } \right) \left( {j\omega _5 +j\omega _6 } \right) H_2 \left( {j\omega _1 ,j\omega _2 } \right) H_2 \left( {j\omega _3 ,j\omega _4 } \right) H_2 \left( {j\omega _5 ,j\omega _6 } \right) \\ +3j\omega _1 \left( {j\omega _2 +j\omega _3 } \right) \left( {j\omega _4 +j\omega _5 +j\omega _6 } \right) H_1 \left( {j\omega _1 } \right) H_2 \left( {j\omega _2 ,j\omega _3 } \right) H_3 \left( {j\omega _4 ,j\omega _5 ,j\omega _6 } \right) \\ \end{array}} \right) \\ \end{array}} \right) \\&H_7 \left( {j\omega _1 ,\ldots ,j\omega _7 } \right) =-H_1 \left( {j(\omega _1 +\cdots +\omega _7 )} \right) \\&\quad \times \left( {\begin{array}{l} d_2 \left( {\begin{array}{l} 2j\omega _1 \left( {j\omega _2 +j\omega _3 +j\omega _4 +j\omega _5 +j\omega _6 +j\omega _7 } \right) H_1 \left( {j\omega _1 } \right) H_6 \left( {j\omega _2 ,j\omega _3 ,j\omega _4 ,j\omega _5 ,j\omega _6 ,j\omega _7 } \right) \\ +2\left( {j\omega _1 +j\omega _2 } \right) \left( {j\omega _3 +j\omega _4 +j\omega _5 +j\omega _6 +j\omega _7 } \right) H_2 \left( {j\omega _1 ,j\omega _2 } \right) H_5 \left( {j\omega _3 ,j\omega _4 ,j\omega _5 ,j\omega _6 ,j\omega _7 } \right) \\ +2\left( {j\omega _1 +j\omega _2 +j\omega _3 } \right) \left( {j\omega _4 +j\omega _5 +j\omega _6 +j\omega _7 } \right) H_3 \left( {j\omega _1 ,j\omega _2 ,j\omega _3 } \right) H_4 \left( {j\omega _4 ,j\omega _5 ,j\omega _6 ,j\omega _7 } \right) \\ \end{array}} \right) \\ +\,d_3 \left( {\begin{array}{l} 6j\omega _1 \left( {j\omega _2 +j\omega _3 } \right) \left( {j\omega _4 +j\omega _5 +j\omega _6 +j\omega _7 } \right) H_1 \left( {j\omega _1 } \right) H_2 \left( {j\omega _2 ,j\omega _3 } \right) H_4 \left( {j\omega _4 ,j\omega _5 ,j\omega _6 ,j\omega _7 } \right) \\ +3\left( {j\omega _1 +j\omega _2 } \right) \left( {j\omega _3 +j\omega _4 } \right) \left( {j\omega _5 +j\omega _6 +j\omega _7 } \right) H_2 \left( {j\omega _1 ,j\omega _2 } \right) H_2 \left( {j\omega _3 ,j\omega _4 } \right) H_3 \left( {j\omega _5 ,j\omega _6 ,j\omega _7 } \right) \\ +3j\omega _1 \left( {j\omega _2 +j\omega _3 +j\omega _4 } \right) \left( {j\omega _5 +j\omega _6 +j\omega _7 } \right) H_1 \left( {j\omega _1 } \right) H_3 \left( {j\omega _2 ,j\omega _3 ,j\omega _4 } \right) H_3 \left( {j\omega _5 ,j\omega _6 ,j\omega _7 } \right) \\ \end{array}} \right) \\ \end{array}} \right) \\ \end{aligned}$$

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Peyton Jones, J.C., Yaser, K.S.A. Recent advances and comparisons between harmonic balance and Volterra-based nonlinear frequency response analysis methods. Nonlinear Dyn 91, 131–145 (2018). https://doi.org/10.1007/s11071-017-3860-z

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