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MODELLING OF MECHANICAL SYSTEM VOLUME 2 Episode 7

Tham khảo tài liệu 'modelling of mechanical system volume 2 episode 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 218 Structural elements 4.3.1 Equations of motion projected onto a modal basis Let us consider the forced dynamical problem governed by the linear partial differential equations of the general type K XX C XX M XX F e r t I.C. and B.C. 4.35 I.C. stands for the initial conditions and B .C. for the conservative boundary conditions. The stiffness damping and mass operators K C M can depend on the vector position r in the Euclidean space. To the forced problem governed by 4.35 we associate the modal problem 4.36 which satisfies the same boundary conditions K r - rn2M r y 0 B .C. 4.36 solutions of which are defining the following modal quantities ỹn Kn Mn n 1 2 3 . 4.37 As in the case of discrete systems the mode shapes can be used to determine an orthonormal basis with respect to the stiffness and mass operators in which the solution X r t of 4.35 can be expanded as the modal series cc XX r t 2 qn t f n r 4.38 where the time functions qn t n 1 2 . termed modal displacements are the components of the displacement field X r t in the modal coordinate system. Formal proof of such a statement is not straightforward and is omitted as already mentioned in the introduction. As the dimension of the functional vector space is infinite a delicate problem of convergence and space completeness arises. Substitution of 4.38 into the equation of motion 4.35 leads to K Vn r qn t C Vn r q_n t M Vn r qn t F e r t n 1 I .C. 4.39 The system of equations 4.39 is projected on the k-th mode shape f k by using the scalar product 1.43 and the orthogonality properties 4.2 . The transformation Modal analysis methods 219 results into the following ordinary differential equation Kkqk Cknịn Mkqk Qk t 4 40 where Ckn P C P V and Q P F e V The subscript V accounts for the space integration domain involved in the scalar product. Q t k 1 2 . termed modal forces are the components of the external force field F e r t as expressed in the modal coordinate system. As already discussed in AXI 04 Chapter 7 the .