变速旋转运动功能梯度圆板热弹主参数共振

THERMO-ELASTIC PRIMARY PARAMETER RESONANCE OF FUNCTIONALLY GRADED CIRCULAR PLATE WITH VARIABLE-SPEED ROTARY MOTION

  • 摘要: 对温度影响下变速旋转运动功能梯度圆板主参数共振进行研究。给出热传导方程和依赖于空间位置与温度变化的物性参数表达式,基于薄板理论,考虑几何非线性,得到功能梯度圆板动能和势能表达式,运用哈密顿变分原理推导出非线性动力学控制方程。使用伽辽金方法离散时间与空间,给出变速旋转运动功能梯度圆板主参数非线性振动微分方程,得到由于旋转及温度引起的径向、横向静挠度方程组。应用多尺度法求非线性振动微分方程,得出幅频响应方程,并对系统进行稳定性分析。通过算例分析,得到系统的幅频曲线图、参数-振幅图、动相平面轨迹图与时程图,从中得到系统主参数共振的非线性变化规律,结果表明,当无量纲后的固有频率分别接近于1和2时,系统会出现两种共振现象;随着温度场温度升高及转速增大,系统振幅增大,共振区域减小。

     

    Abstract: The study investigates the primary parameter resonance of a functionally graded circular plate with variable-speed rotational motion under the influence of temperature. Developed are the heat conduction equation and physical parameter expressions depend on the spatial position and on temperature variation. Based on the thin plate theory and considering the geometric nonlinearity, the expressions of kinetic and potential energies for the circular plate are obtained. The nonlinear dynamic governing equations are derived by applying the Hamiltonian principle. The Galerkin method is employed to discretize time and space, the nonlinear vibration differential equation is derived, and the control equations of radial and lateral static deflections generated by temperature field and rotational motion are obtained. The amplitude-frequency response equations are derived by solving the differential equation upon the multi-scale method, and the stability of steady-state motion is discriminated. Through example analyses, obtained are the amplitude-frequency curves, parameter-amplitude diagrams, dynamic phase plane trajectory diagrams and time-history diagrams, revealing nonlinear dynamic characteristics of the primary parameter resonance. The analysis results indicate that: when the dimensionless natural frequency is close to 1 and 2, respectively, two forms of primary parameter resonance can be activated, and the temperature rises and speed improves, the response amplitude increases, and resonance region narrows.

     

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