Applied Mathematics and Mechanics (English Edition) ›› 2020, Vol. 41 ›› Issue (1): 1-14.doi: https://doi.org/10.1007/s10483-020-2560-6

• 论文 •    下一篇

Vibration reduction evaluation of a linear system with a nonlinear energy sink under a harmonic and random excitation

Jiren XUE1, Yewei ZHANG1,2, Hu DING1, Liqun CHEN1   

  1. 1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200072, China;
    2. Faculty of Aerospace Engineering, Shenyang Aerospace University, Shenyang 110136, China
  • 收稿日期:2019-06-25 修回日期:2019-08-20 出版日期:2020-01-01 发布日期:2019-12-14
  • 通讯作者: Yewei ZHANG E-mail:zhangyewei1218@126.com
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Nos. 11772205 and 11572182) and the Liaoning Revitalization Talents Program of China (No. XLYC1807172)

Vibration reduction evaluation of a linear system with a nonlinear energy sink under a harmonic and random excitation

Jiren XUE1, Yewei ZHANG1,2, Hu DING1, Liqun CHEN1   

  1. 1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, School of Mechanics and Engineering Science, Shanghai University, Shanghai 200072, China;
    2. Faculty of Aerospace Engineering, Shenyang Aerospace University, Shenyang 110136, China
  • Received:2019-06-25 Revised:2019-08-20 Online:2020-01-01 Published:2019-12-14
  • Contact: Yewei ZHANG E-mail:zhangyewei1218@126.com
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Nos. 11772205 and 11572182) and the Liaoning Revitalization Talents Program of China (No. XLYC1807172)

摘要: The nonlinear behaviors and vibration reduction of a linear system with a nonlinear energy sink (NES) are investigated. The linear system is excited by a harmonic and random base excitation, consisting of a mass block, a linear spring, and a linear viscous damper. The NES is composed of a mass block, a linear viscous damper, and a spring with ideal cubic nonlinear stiffness. Based on the generalized harmonic function method, the steady-state Fokker-Planck-Kolmogorov equation is presented to reveal the response of the system. The path integral method based on the Gauss-Legendre polynomial is used to achieve the numerical solutions. The performance of vibration reduction is evaluated by the displacement and velocity transition probability densities, the transmissibility transition probability density, and the percentage of the energy absorption transition probability density of the linear oscillator. The sensitivity of the parameters is analyzed for varying the nonlinear stiffness coefficient and the damper ratio. The investigation illustrates that a linear system with NES can also realize great vibration reduction under harmonic and random base excitations and random bifurcation may appear under different parameters, which will affect the stability of the system.

关键词: nonlinear energy sink (NES), Gauss-Legendre polynomial, transmissibility, percentage of energy absorption

Abstract: The nonlinear behaviors and vibration reduction of a linear system with a nonlinear energy sink (NES) are investigated. The linear system is excited by a harmonic and random base excitation, consisting of a mass block, a linear spring, and a linear viscous damper. The NES is composed of a mass block, a linear viscous damper, and a spring with ideal cubic nonlinear stiffness. Based on the generalized harmonic function method, the steady-state Fokker-Planck-Kolmogorov equation is presented to reveal the response of the system. The path integral method based on the Gauss-Legendre polynomial is used to achieve the numerical solutions. The performance of vibration reduction is evaluated by the displacement and velocity transition probability densities, the transmissibility transition probability density, and the percentage of the energy absorption transition probability density of the linear oscillator. The sensitivity of the parameters is analyzed for varying the nonlinear stiffness coefficient and the damper ratio. The investigation illustrates that a linear system with NES can also realize great vibration reduction under harmonic and random base excitations and random bifurcation may appear under different parameters, which will affect the stability of the system.

Key words: nonlinear energy sink (NES), Gauss-Legendre polynomial, transmissibility, percentage of energy absorption

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