Abstract
The harmonic balance method (HBM) is one of the most widely used methods in solving nonlinear vibration problems, and its accuracy and computational efficiency largely depend on the number of the harmonics selected. The adaptive harmonic balance (AHB) method is an improved HBM method. This paper presents a modified AHB method with the asymptotic harmonic selection (AHS) procedure. This new harmonic selection procedure selects harmonics from the frequency spectra of nonlinear terms instead of estimating the contribution of each harmonic to the whole nonlinear response, by which the additional calculation is avoided. A modified continuation method is proposed to deal with the variable size of nonlinear algebraic equations at different values of path parameters, and then all solution branches of the amplitude-frequency response are obtained. Numerical experiments are carried out to verify the performance of the AHB-AHS method. Five typical nonlinear dynamic equations with different types of nonlinearities and excitations are chosen as the illustrative examples. Compared with the classical HBM and Runge-Kutta methods, the proposed AHB-AHS method is of higher accuracy and better convergence. The AHB-AHS method proposed in this paper has the potential to investigate the nonlinear vibrations of complex high-dimensional nonlinear systems.
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References
DAI, H. H., YUE, X. K., YUAN, J. P., and ATLURI, S. N. A time domain collocation method for studying the aeroelasticity of a two dimensional airfoil with a structural nonlinearity. Journal of Computational Physics, 270, 214–237 (2014)
JACOB, B. P. and EBECKEN, N. F. F. An optimized implementation of the Newmarki/Newton-Raphson algorithm for the time integration of non-linear problems. Communications in Numerical Methods in Engineering, 10, 983–992 (1994)
EPUREANU, B. I. and DOWELL, E. H. Localized basis function method for computing limit cycle oscillations. Nonlinear Dynamics, 31, 151–166 (2003)
GILMORE, R. and STEER, M. Nonlinear circuit analysis using the method of harmonic balance— a review of the art, part II: advanced concepts. International Journal of Microwave and Millimeter-Wave Computer-Aided Engineering, 1, 159–180 (1991)
GILMORE, R. and STEER, M. Nonlinear circuit analysis using the method of harmonic balance — a review of the art, part I: introductory concepts. International Journal of Microwave and Millimeter-Wave Computer-Aided Engineering, 1, 22–37 (1991)
CHEN, H. Z., HOU, L., CHEN, Y. S., and YANG, R. Dynamic characteristics of flexible rotor with squeeze film damper excited by two frequencies. Nonlinear Dynamics, 87, 2463–2481 (2016)
GOURARY, M., ULYANOV, S., ZHAROV, M., RUSAKOV, S., GULLAPALLI, K. K., and MULVANEY, B. J. A robust and efficient oscillator analysis technique using harmonic balance. Computer Methods in Applied Mechanics and Engineering, 181, 451–466 (2000)
DETROUX, T., RENSON, L., MASSET, L., and KERSCHEN, G. The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems. Computer Methods in Applied Mechanics and Engineering, 296, 18–38 (2015)
SHEN, Y. J., WEN, S. F., LI, X. H., YANG, S. P., and XING, H. J. Dynamical analysis of fractional-order nonlinear oscillator by incremental harmonic balance method. Nonlinear Dynamics, 85, 1457–1467 (2016)
MEES, A. I. The describing function matrix. IMA Journal of Applied Mathematics, 10, 49–67 (1972)
SERT, O. and CIGEROGLU, E. Adaptive harmonic balance methods-a comparison. Special Topics in Structural Dynamics, Springer, New York, 279–289 (2016)
JU, R., FAN, W., and ZHU, W. D. An efficient Galerkin averaging-incremental harmonic balance method based on the fast fourier transform and tensor contraction. Journal of Vibration and Acoustics-Transactions of the ASME, 142, 061011 (2020)
LAU, S. L. and CHEUNG, Y. K. Amplitude incremental variational principle for nonlinear vibration of elastic systems. Journal of Applied Mechanics-Transactions of the ASME, 48, 959–964 (1981)
JONES, J. C. P., YASER, K. S. A., and STEVENSON, J. Automatic computation and solution of generalized harmonic balance equations. Mechanical Systems and Signal Processing, 101, 309–319 (2018)
LAU, S. L., CHEUNG, Y. K., and WU, S. Y. A variable parameter incrementation method for dynamic instability of linear and nonlinear elastic systems. Journal of Applied Mechanics-Transactions of the ASME, 49, 849–853 (1982)
LU, W., GE, F., WU, X. D., and HONG, Y. S. Nonlinear dynamics of a submerged floating moored structure by incremental harmonic balance method with FFT. Marine Structures, 31, 63–81 (2013)
LEUNG, A. Y. T. and CHUI, S. K. Non-linear vibration of coupled duffing oscillators by an improved incremental harmonic balance method. Journal of Sound and Vibration, 181, 619–633 (1995)
HOU, L., CHEN, Y. S., FU, Y. Q., CHEN, H. Z., LU, Z. Y., and LIU, Z. S. Application of the HB-AFT method to the primary resonance analysis of a dual-rotor system. Nonlinear Dynamics, 88, 2531–2551 (2017)
HOU, L. and CHEN, Y. S. Analysis of 1/2 sub-harmonic resonance in a maneuvering rotor system. Science China Technological Sciences, 57, 203–209 (2013)
ZHANG, Z. Y., CHEN, Y. S., and LI, Z. G. Influencing factors of the dynamic hysteresis in varying compliance vibrations of a ball bearing. SCIENCE CHINA Technological Sciences, 58, 775–782 (2015)
SUN, C. Z., CHEN, Y. S., and HOU, L. Nonlinear dynamical behaviors of a complicated dual-rotor aero-engine with rub-impact. Archive of Applied Mechanics, 88, 1305–1324 (2018)
CHEN, H. Z., CHEN, Y. S., HOU, L., and LI, Z. G. Bifurcation analysis of rotor-squeeze film damper system with fluid inertia. Mechanism and Machine Theory, 81, 129–139 (2014)
TAGHIPOUR, J., HADDAD-KHODAPARAST, H., FRISWELL, M. I, SHAW, A. D., JALALI, H., and JAMIA, N. Harmonic-balance-based parameter estimation of nonlinear structures in the presence of multi-harmonic response and force. Mechanical Systems and Signal Processing, 162, 108057 (2022)
HALL, K. C., THOMAS, J. P., and CLARK, W. S. Computation of unsteady nonlinear flows in cascades using a harmonic balance technique. AIAA Journal, 40, 879–886 (2002)
ZHOU, S. H., SONG, G. Q., LI, Y. M., HUANG, Z. L., and REN, Z. H. Dynamic and steady analysis of a 2-DOF vehicle system by modified incremental harmonic balance method. Nonlinear Dynamics, 98, 75–94 (2019)
GROLL, G. and EWINS, D. J. The harmonic balance method with arc-length continuation in rotor/stator contact problems. Journal of Sound and Vibration, 241, 223–233 (2001)
DETROUX, T., RENSON, L., MASSET, L., and KERSCHEN, G. The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems. Computer Methods in Applied Mechanics and Engineering, 296, 18–38 (2015)
GULLAPALLI, K. K. and GOURARY, M. M. A new computational approach to simulate highly nonlinear systems by harmonic balance method. Proceedings of the 16th IMACS World Congress on Scientific Computation, Applied Mathematics and Simulation, Lausanne, Switzerland (2000)
MAPLE, R. C., KING, P. I., and OXLEY, M. E. Adaptive harmonic balance solutions to Euler’s equation. AIAA Journal, 41, 1705–1714 (2003)
MAPLE, R. C., KING, P. I., ORKWIS, P. D., and WOLFF, J. M. Adaptive harmonic balance method for nonlinear time-periodic flows. Journal of Computational Physics, 193, 620–641 (2004)
ZHU, L. and CHRISTOFFERSEN, C. E. Adaptive harmonic balance analysis of oscillators using multiple time scales. The 3rd International IEEE-NEWCAS Conference, IEEE Xplore, New York, 187–190 (2005)
JAUMOUILLE, V., SINOU, J. J., and PETITJEAN, B. An adaptive harmonic balance method for predicting the nonlinear dynamic responses of mechanical systems-application to bolted structures. Journal of Sound and Vibration, 329, 4048–4067 (2010)
GROLET, A. and THOUVEREZ, F. On a new harmonic selection technique for harmonic balance method. Mechanical Systems and Signal Processing, 30, 43–60 (2012)
SUESS, D., JERSCHL, M., and WILLNER, K. Adaptive harmonic balance analysis of dry friction damped systems. Nonlinear Dynamics, 1, 405–414 (2016)
GASTALDI, C. and BERRUTI, T. M. A method to solve the efficiency-accuracy trade-off of multi-harmonic balance calculation of structures with friction contacts. International Journal of Non-Linear Mechanics, 92, 25–40 (2017)
SERT, O. and CIGEROGLU, E. A novel two-step pseudo-response based adaptive harmonic balance method for dynamic analysis of nonlinear structures. Mechanical Systems and Signal Processing, 130, 610–631 (2019)
CAMERON, T. M. and GRIFFIN, J. H. An alternating frequency/time domain method for calculating the steady-state response of nonlinear dynamic systems. Journal of Applied Mechanics-Transactions of the ASME, 56, 149–154 (1989)
WANG, X. F. and ZHU, W. D. A modified incremental harmonic balance method based on the fast Fourier transform and Broyden’s method. Nonlinear Dynamics, 81, 981–989 (2015)
GUSKOV, M. and THOUVEREZ, F. Harmonic balance-based approach for quasi-periodic motions and stability analysis. Journal of Vibration and Acoustics-Transactions of the ASME, 134, 031003 (2012)
CHOI, S. and NOAH, S. T. Response and stability analysis of piecewise-linear oscillators under multi-forcing frequencies. Nonlinear Dynamics, 3, 105–121 (1991)
LINDBLAD, D., FREY, C., JUNGE, L., ASHCROFT, G., and ANDERSSON, N. Minimizing aliasing in multiple frequency harmonic balance computations. Journal of Scientific Computing, 91, 65 (2022)
GAO, P., HOU, L., YANG, R., and CHEN, Y. S. Local defect modelling and nonlinear dynamic analysis for the inter-shaft bearing in a dual-rotor system. Applied Mathematical Modelling, 68, 29–47 (2019)
CHEN, Y., HOU, L., CHEN, G., SONG, H. Y., LIN, R. Z., JIN, Y. H., and CHEN, Y. S. Nonlinear dynamics analysis of a dual-rotor-bearing-casing system based on a modified HB-AFT method. Mechanical Systems and Signal Processing, 185, 109805 (2023)
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Conflict of interest Yushu CHEN is an editorial board member for Applied Mathematics and Mechanics (English Edition) and was not involved in the editorial review or the decision to publish this article. The authors declare no conflict of interest.
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Project supported by the National Natural Science Foundation of China (Nos. 11972129 and 12372008), the National Major Science and Technology Projects of China (No. 2017-IV-0008-0045), the Natural Science Foundation of Heilongjiang Province of China (No. YQ2022A008), the Fundamental Research Funds for the Central Universities of China (No. HIT.OCEF.2023006), the Polish National Science Centre of Poland under the OPUS 18 grant (No. 2019/35/B/ST8/00980), and the Tianjin University Independent Innovation Foundation of China (No. 2023XJS-0038)
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Lin, R., Hou, L., Chen, Y. et al. A novel adaptive harmonic balance method with an asymptotic harmonic selection. Appl. Math. Mech.-Engl. Ed. 44, 1887–1910 (2023). https://doi.org/10.1007/s10483-023-3047-6
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DOI: https://doi.org/10.1007/s10483-023-3047-6
Key words
- harmonic balance method (HBM)
- adaptive harmonic balance (AHB) method
- harmonic selection
- nonlinear vibration
- multi-frequency excitation