变速轴向运动三参数模型黏弹性梁的稳定性

Stability of Axially Accelerating Viscoelastic Beams with 3-Parameter Model

本文研究了速度变化的轴向运动三参数模型黏弹性梁在主参数共振以及组合参数共振范围内的稳定性。轴向运动梁的黏弹性本构关系采用三参数模型并引入了物质时间导数。运用渐进摄动法,直接求解梁的控制微分方程并导出了当运动参数激励频率接近某一阶固有频率2倍或接近某两阶固有频率之和时主参数共振和组合参数共振的稳定性条件。在解谐参数和激励振幅平面上,可以找出由于共振而产生的失稳区域。数值结果给出了梁的刚度系数、黏弹性系数及轴向平均速度对失稳区域的影响。在发生组合共振和主共振时,随着刚度系数E_1的变大,失稳区域变小;刚度系数E_2的变大,失稳区域变大。随着黏弹性系数的变大,失稳区域变小。发生组合共振时,随着平均速度的变大,失稳区域变小;发生主共振时,随着平均速度的变大,失稳区域变大。

Stability of an axially accelerating viscoelastic beam constituted by 3-parameter model was investigated in the range of combination and principal parametric resonance. The material time derivative was used in the viscoelastic constitutive relation. The governing equation and stability condition of beam were determined for combination and principal parametric resonance via the asymptotic analysis when the variation frequency approaches the two times any natural frequency or the sum of any two natural frequencies. In the plane of detuning parameter and excited amplitudes, the instability regions can be figured out. Numerical examples show the effects of the stiffness constants of the beam, viscoelasticity and the mean axial speed in instability regions. With the stiffness coefficient E1 increasing, the instability regions will become larg. On the contrary, the increasing stiffness coefficient E2 leads the instability regions to become narrow. In addition, the instability regions will become narrow with the increase of viscoelasticity. With the mean axial speed increasing, the instability regions become narrow in the summation parametric resonance but there is an opposite effect in the principle resonance.