Linearized Harmonic Balance Method for Seeking the Periodic Vibrations of Second- and Third-Order Nonlinear Oscillators
To solve the nonlinear vibration problems of second- and third-order nonlinear oscillators, a modified harmonic balance method (HBM) is developed in this paper. In the linearized technique, we decompose the nonlinear terms of the governing equation on two sides via a constant weight factor; then, th...
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2025-01-01
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| author | Chein-Shan Liu Chung-Lun Kuo Chih-Wen Chang |
| author_facet | Chein-Shan Liu Chung-Lun Kuo Chih-Wen Chang |
| author_sort | Chein-Shan Liu |
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| description | To solve the nonlinear vibration problems of second- and third-order nonlinear oscillators, a modified harmonic balance method (HBM) is developed in this paper. In the linearized technique, we decompose the nonlinear terms of the governing equation on two sides via a constant weight factor; then, they are linearized with respect to a fundamental periodic function satisfying the specified initial conditions. The periodicity of nonlinear oscillation is reflected in the Mathieu-type ordinary differential equation (ODE) with periodic forcing terms appeared on the right-hand side. In each iteration of the linearized harmonic balance method (LHBM), we simply solve a small-size linear system to determine the Fourier coefficients and the vibration frequency. Because the algebraic manipulations required for the LHBM are quite saving, it converges fast with a few iterations. For the Duffing oscillator, a frequency–amplitude formula is derived in closed form, which improves the accuracy of frequency by about three orders compared to that obtained by the Hamiltonian-based frequency–amplitude formula. To reduce the computational cost of analytically solving the third-order nonlinear jerk equations, the LHBM invoking a linearization technique results in the Mathieu-type ODE again, of which the harmonic balance equations are easily deduced and solved. The LHBM can achieve quite accurate periodic solutions, whose accuracy is assessed by using the fourth-order Runge–Kutta numerical integration method. The optimal value of weight factor is chosen such that the absolute error of the periodic solution is minimized. |
| format | Article |
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| institution | Kabale University |
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| language | English |
| publishDate | 2025-01-01 |
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| spelling | doaj-art-4bc32ec4983048ee9e377f736372cbe12025-01-10T13:18:28ZengMDPI AGMathematics2227-73902025-01-0113116210.3390/math13010162Linearized Harmonic Balance Method for Seeking the Periodic Vibrations of Second- and Third-Order Nonlinear OscillatorsChein-Shan Liu0Chung-Lun Kuo1Chih-Wen Chang2Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, TaiwanCenter of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, TaiwanDepartment of Mechanical Engineering, National United University, Miaoli 36063, TaiwanTo solve the nonlinear vibration problems of second- and third-order nonlinear oscillators, a modified harmonic balance method (HBM) is developed in this paper. In the linearized technique, we decompose the nonlinear terms of the governing equation on two sides via a constant weight factor; then, they are linearized with respect to a fundamental periodic function satisfying the specified initial conditions. The periodicity of nonlinear oscillation is reflected in the Mathieu-type ordinary differential equation (ODE) with periodic forcing terms appeared on the right-hand side. In each iteration of the linearized harmonic balance method (LHBM), we simply solve a small-size linear system to determine the Fourier coefficients and the vibration frequency. Because the algebraic manipulations required for the LHBM are quite saving, it converges fast with a few iterations. For the Duffing oscillator, a frequency–amplitude formula is derived in closed form, which improves the accuracy of frequency by about three orders compared to that obtained by the Hamiltonian-based frequency–amplitude formula. To reduce the computational cost of analytically solving the third-order nonlinear jerk equations, the LHBM invoking a linearization technique results in the Mathieu-type ODE again, of which the harmonic balance equations are easily deduced and solved. The LHBM can achieve quite accurate periodic solutions, whose accuracy is assessed by using the fourth-order Runge–Kutta numerical integration method. The optimal value of weight factor is chosen such that the absolute error of the periodic solution is minimized.https://www.mdpi.com/2227-7390/13/1/162strongly nonlinear oscillatorsanalytic periodic solutionharmonic balance methodjerk equationDuffing equation |
| spellingShingle | Chein-Shan Liu Chung-Lun Kuo Chih-Wen Chang Linearized Harmonic Balance Method for Seeking the Periodic Vibrations of Second- and Third-Order Nonlinear Oscillators Mathematics strongly nonlinear oscillators analytic periodic solution harmonic balance method jerk equation Duffing equation |
| title | Linearized Harmonic Balance Method for Seeking the Periodic Vibrations of Second- and Third-Order Nonlinear Oscillators |
| title_full | Linearized Harmonic Balance Method for Seeking the Periodic Vibrations of Second- and Third-Order Nonlinear Oscillators |
| title_fullStr | Linearized Harmonic Balance Method for Seeking the Periodic Vibrations of Second- and Third-Order Nonlinear Oscillators |
| title_full_unstemmed | Linearized Harmonic Balance Method for Seeking the Periodic Vibrations of Second- and Third-Order Nonlinear Oscillators |
| title_short | Linearized Harmonic Balance Method for Seeking the Periodic Vibrations of Second- and Third-Order Nonlinear Oscillators |
| title_sort | linearized harmonic balance method for seeking the periodic vibrations of second and third order nonlinear oscillators |
| topic | strongly nonlinear oscillators analytic periodic solution harmonic balance method jerk equation Duffing equation |
| url | https://www.mdpi.com/2227-7390/13/1/162 |
| work_keys_str_mv | AT cheinshanliu linearizedharmonicbalancemethodforseekingtheperiodicvibrationsofsecondandthirdordernonlinearoscillators AT chunglunkuo linearizedharmonicbalancemethodforseekingtheperiodicvibrationsofsecondandthirdordernonlinearoscillators AT chihwenchang linearizedharmonicbalancemethodforseekingtheperiodicvibrationsofsecondandthirdordernonlinearoscillators |