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Smart Material Systems and MEMS - Vijay K. Varadan Part 7

Tham khảo tài liệu 'smart material systems and mems - vijay k. varadan part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Introduction to the Finite Element Method 175 Here d is the best-guess displacement profile of the structure. In principle the static solution can form the best initial guess and this solution can be iterated to get the correct eigenvalue. It is apparent that the extraction of eigenvalues vectors is the most computationally costly activity in the entire analysis process. The computational time and the memory cost involved in the various schemes are dealt with in great detail in Bathe 12 . For a system with n degrees of freedom only the first m natural frequencies and mode shapes are computed where m n. After obtaining the first m eigenvalues vectors these are put in the matrix form as and A . The former is called the modal matrix which is of size n X m. In this matrix the modes are stored column-wise. The latter is a diagonal matrix of size m X m containing the natural frequencies of the computed m modes. This matrix is also called the spectral matrix. The modal matrix is orthogonal with respect to both the stiffness and mass matrix. These two matrices along with the orthogonality conditions are used to estimate the dynamic response. There are two orthogonality conditions which can be stated as K A Mp I 7.138 In general modal methods use similarity transformation to convert the actual degrees of freedom d of size n X 1 to generalized degree of freedom Z of size m X 1. This similarity transformation is given by d t nx1 Pkm Z t mx1 7.139 There are two different modal methods by which the response can be computed. These are Normal Mode method or Mode Displacement method Mode Acceleration method In the Normal Mode method the orthogonality relations are used to uncouple the governing differential equation. This is done in the following manner. The FE differential equation is given by M d C d K d F In this equation let us use Rayleigh s proportional damping of the form C a K b M . The reason for using such a damping scheme will become clear in the next few steps. Now we