Hamiltonian Solution Approach For The Vibration Of Composite Plate

Composite plate has a very wide range of applications in the aerospace, civil defense, automobile manufacturing and so on. Because of its superior performance, composite plate gets more and more attention. In the practical app lication of the composite plate, the problem of vibration cannot be avoided. Therefore, it is important to analyze the vibration of composite plate.Because there are so many parameters, when the composite plate gets motivation and vibrates, different composite plate has different characteristics. Young's modulus of fiber and matrix, the direction of fiber and matrix to the border, and the laying angle of composite laminates all could affect the vibration. Analyzing the vibration of plate is very important. Based on the published papers, this paper will use Symplectic Mathematics which gets more and more attention to analyze the vibration problem of composite plates.In the published papers, they often use Newton mechanics or Lagrange mechanics in Euclidean space to solve mechanical proble ms. This paper will use Hamilton mechanics and the Symplectic Mathematics, which is used to solve the Hamilton mechanics equation, to analyze the vibration problem. Symplectic is a new mathematical method. It comes from the Symplectic relationship of Hamilton canonical equations and establishes a state space by Symplectic geometry. Because this method maintains the symmetrical feature of Hamilton system, it can make sure that the result is accuracy and reliability when we use it to analyze any problems. This characteristic of Symplectic mathematic is conservation symplecticness.At first, this paper analyzes the free vibration of composite plate. According to Kirchhoff theory, this paper translates the old system of vibration problems into Hamilton system. Then this paper use Symplectic mathematical methods to analyze the vibration of composite plate and calculate the natural frequency. By comparing the result with others of the published papers, this paper verifies the feasibility and accuracy of Symplectic mathematical method in this area, and discusses its advantages and disadvantages.Then this paper will use the same method to analyze the vibration of laminated composite plates. Based on vibration equations of the three-dimensional structure, this paper translates the old system to the Hamilton system. Then this paper uses the Symplectic mathematics and Fourier transform to solve the canonical equation. According to the result, it analyzes the factors that affect the vibration. For example, each layer of the elastic constants and the laying angle all can affect the vibration of laminated composite plates. At last, the result will be compared with the existing result and experimental results in the published papers. It can be used to verify the accuracy and feasibility of Symplectic mathematics for this issue in and discuss the advantages and disadvantages.