分数阶单自由度间隙振子的受迫振动

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振动与冲击 ›› 2020, Vol. 39 ›› Issue (14) : 251-256.

论文

分数阶单自由度间隙振子的受迫振动

牛江川1,2，赵志爽2，邢海军1,2，申永军1,2

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Forced vibration of a fractional-order single degree-of-freedom oscillator with clearance

NIU Jiangchuan1,2, ZHAO Zhishuang2, XING Haijun1,2, SHEN Yongjun1,2

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摘要

研究了含有分数阶微分项的单自由度间隙振子的受迫振动，利用KBM渐近法获得了系统的近似解析解。分析了分段线性系统的主共振，得到了分数阶阶次在0~ 2时分数阶项的统一表达式；发现分数阶微分项在分段系统中以等效线性阻尼和等效线性刚度的形式影响着系统的动力学特性，而间隙以等效非线性刚度的形式影响着系统的动力学特性。获得了主共振幅频响应的表达式，并得到了系统的稳定性条件；比较了系统主共振幅频响应的近似解析解和数值解，发现两者符合程度较高，验证了近似解析解的正确性；详细分析了分数阶项和间隙对系统主共振幅频响应的影响。研究表明KBM渐近法是分析分数阶分段光滑系统动力学的有效方法。

Abstract

The forced vibration of a single degree-of-freedom piecewise linear oscillator with a clearance and a fractional-order derivative term was investigated.The approximate analytical solution for its primary resonance was obtained by the Krylov-Bogoliubov-Mitropoisky (KBM) asymptotic method.The primary resonance of the piecewise linear system was analyzed, and a unified expression of the fractional-order differential term was obtained, where the fractional order was restricted in 0 to 2.The effects of the fractional-order differential term on the dynamic characteristics of the piecewise system were expressed as an equivalent linear damping and an equivalent linear stiffness, while that of the clearance was an equivalent nonlinear stiffness.The expression of the amplitude-frequency response of the primary resonance was obtained, and the stability condition of the system was also achieved.The approximate analytical solutions and numerical solutions of the primary resonance amplitude-frequency responses were compared, which shows both are in good agreement.The effects of the fractional-order term and clearance on the amplitude-frequency response of the primary resonance were analyzed in detail.It concludes that the KBM asymptotic method is an effective method to analyze the dynamics of fractional-order piecewise smooth systems.

导出引用

牛江川1,2，赵志爽2，邢海军1,2，申永军1,2. 分数阶单自由度间隙振子的受迫振动[J]. 振动与冲击, 2020, 39(14): 251-256

NIU Jiangchuan1,2, ZHAO Zhishuang2, XING Haijun1,2, SHEN Yongjun1,2. Forced vibration of a fractional-order single degree-of-freedom oscillator with clearance[J]. Journal of Vibration and Shock, 2020, 39(14): 251-256

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