Research On Free Vibration Modeling Of Elliptic Plates Based On Conformal Mapping Method

Thin plate structure is very common in various engineering fields.So,it is very important to analyze its vibration characteristics in order to avoid the fatigue damage.However,in recent years,it has been still focused on rectangular or circular plates(including those with cutouts),cylindrical shells,and some other those have special shapes about the research of the vibration modeling of plates and shells more than elliptic plates and some other singular shapes.Based on above facts,it is also necessary to study the vibration characteristics about elliptic plates.While in terms of research methods,mainly include Mathieu method,Galerkin method,Rayleigh-Ritz method,etc.But all of these traditional methods have their own limitation and drawbacks generally.Recently,the conformal mapping method which is used to solve the wave motion about infinite thin plates with cutouts has been widely focused on.In fact,the only difference between wave motion and vibration is the magnitude about study environment.Therefore,this approach can also be used to study the vibration of finite thin plates with complicated shapes.The main idea is finding a certain analytic function which can map an area with complicated shape into an area consisting of a unit circle,then we will use the traditional method of studying the vibration of a circular plate to acquire the answer.Obviously,this approach may has some accuracy and universality.Based on above facts,this article have contents as follows:1)Review the Kirchhoff theory of plate and the solution of the displacement function of the vibration of the circular plate based on mathematical and physical methods;then introduce the theory of complex function in order to put these above mathematical models into complex plane and convert the variables into complex forms;after that,the conformal mapping method is brought up so that all of the formulas about generalized forces and boundary conditions are transformed.2)Acquire the natural frequencies and corresponding mode shapes in various boundary conditions by analytical method;meanwhile,complete numerical results are showed which convergence precision will be compared with FEM and another approach in another reference so as to verify the accuracy and practicality of this method.3)Experimental setups of elliptic plates under free and clamped are built up by pickingup a single point.The natural frequencies and corresponding mode shapes for experimental setups will be obtained and then compared with theoretical results so as to verify the accuracy of this method further.