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

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 ac-curately 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 frac-tional-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 α of the in-fluencing time factor has a very obvious effect on the mode shape.