基于非局部理论的轴向运动黏弹性纳米板的参数振动及其稳定性

Parametric vibration and stability of an axially moving viscoelastic nanoplate based on the nonlocal theory

研究了轴向运动黏弹性二维纳米板结构的非局部横向参数振动及其稳态响应。利用哈密顿原理推导了问题模型的控制方程,应用多尺度法分析了带有周期脉动成分的变速运动黏弹性纳米板的失稳现象。根据边界条件及复模态法可确定模态函数的表达,讨论了其特例匀速运动时固有频率与小尺度参数的关系,重点探讨了当脉动频率为两阶固有频率之和或者为某阶固有频率二倍时所发生的和型组合参数共振及主参数共振。结果表明,小尺度参数的存在使得轴向运动黏弹性纳米板的弯曲刚度及固有频率减小,并导致组合参数共振失稳区域减小但主参数共振区域增大,同时削弱了黏弹性系数对主参数共振区域的影响。同等条件下,黏弹性系数对组合共振区域的影响更为明显。

The nonlocal transverse parametric vibration and steady-state response of axially moving viscoelastic two-dimensional nanoplate-like structures are studied. The Hamilton’s principle is employed to derive the governing partial differential equations of the mathematical model. The instable behaviors of axially moving viscoelastic nanoplate with a periodic pulsation velocity are addressed using the method of multiple scales. The modal functions are determined by some specific boundary conditions and the method of complex mode, and then the effects of small-scale parameter on the natural frequencies of the axially moving nanoplates with uniform velocity are discussed. Subsequently, the analyses are mainly focused on the instable regions caused by summation and principal parametric resonances, respectively, of which the summation parametric resonance occurs when the harmonic frequency approaches the sum of any two mode natural frequencies, while the principal parametric resonance occurs when the harmonic frequency approaches two times of certain mode natural frequency. It is shown that the existence of small-scale parameter contributes to reduce the bending stiffness and natural frequencies of axially moving viscoelastic nanoplates, and further decreases the instable regions of summation parametric resonance, while increases the instable regions of principal parametric resonance. On the other hand, the small-scale parameter softens the influence of viscoelasticity on the instable regions of principal parameter resonance. Moreover, the effect of viscoelasticity on the instability of summation parametric resonance is more obvious, ceteris paribus.