分数阶微积分理论下粘弹性矩形薄膜自由振动

Free Vibration of the Viscoelastic Rectangular Films under the Fractional Calculus Theory

摘要: 薄膜振动问题一直都备受学者关注，经典的薄膜理论是建立在整数阶微积分上来刻画薄膜振动的波动方程，其在建模过程中忽略了材料记忆性的特征。本文针对粘弹性薄膜材料特征具有时间记忆性，提出了一种新的薄膜振动模型，解决了建立在整数阶微积分理论下波动方程无法准确刻画材料的时间记忆性难题。分数阶微积分模型所反映出来的性质与其整个发展史密切相关，本文基于分数阶微积分理论。将Westerlund提出的分数阶模型运用到粘弹性薄膜自由振动中，得到薄膜自由振动的分数阶波动方程。结合矩形和圆形薄膜初边界条件，建立了粘弹性薄膜自由振动的场系方程。运用分离变量法，拉普拉斯变换对矩形薄膜波动方程进行求解。结果表明：在矩形薄膜的自由振动中，体现影响时间因子的分数阶阶数 对振型的影响非常明显。

Abstract: The problem of thin film vibration has always attracted scholars. The classical thin film theory is based on the integer order calculus to characterize the wave equation of thin film vibration, which ignores the characteristics of material memory in the modeling process. In this paper, a new film vibration model is proposed for the temporal memory of viscoelastic film materials, which solves the problem of time memory that the wave equations based on integer calculus theory can not accurately characterize materials. The properties reflected by the fractional-order calculus model are closely related to its entire development history, and this paper is based on the theory of fractional-order calculus. The fractional order model proposed by Westerlund is applied to the free vibration of viscoelastic film to obtain the fractional order wave equation of free vibration of thin film. Combining the initial conditions and boundary conditions of rectangular and circular films, a field system equation for the free vibration of viscoelastic films is established. Using the separation variable method, the Laplace transform solves the rectangular thin film wave equation. The results show that in the free vibration of the rectangular film, the fractional order   of the influencing time factor has a very obvious effect on the mode shape.