Investigation Of A Well-Posed Wavelet Method To Analyse Flexural Beams/Plates And Nonlinear Vibrations

For wavelet methods in solving initial-boundary-value problems on finite domains, the construction of wavelet formation of solutions and the treatment of boundary conditions of the problem are the key to solve problems. However, no general method has been found to handle the construction of wavelet formations and the treatment of boundary conditions of the problem. Therefore, the following works are carried out in this thesis:1). A modified wavelet method is proposed for the first time, which is applicable for both initial- and boundary-value problems on finite domains. This method presents a general form of wavelet expansion for the solutions of initial- and boundary-value problems, by extrapolating external wavelet coefficients by boundary and inner ones, in which way the continuity of the solutions' wavelet expansions are preserved near the domain boundaries. Based on this, the values of the solution and its first-order derivatives at the domain boundaries are explicitly combined into the newly presented wavelet expansions of solutions, by which many mechanic problems can be solved with wavelet methods.2). Based on the modified wavelet formation mentioned above, wavelet-variational methods are established for the static/dynamic problems of beams and plates, and discrete static/dynamic equations and characteristic equations are derived in a general form, respectively, in which all types of homogeneous and non-homogeneous boundary conditions and boundary support conditions are treated in a general way. Because the wavelet formation for the deflection of beams and plates is general for all boundary conditions, not only the form of discrete static/dynamic equations and the characteristic equations, but also the the coefficient matrices of these equations are invariant to boundary conditions; on the other hand, the wavelet coefficients of the proposed modified wavelet formation are independent from each other, so the derived discrete static/dynamic equations and characteristic equations are well-posed, respectively, and for any given boundary condition the discrete static/dynamic equation has unique solution. The proposed method overcomes the defficiencies of current wavelet-Galerkin methods and wavelet-FEMs for the static/dynamic problems of beams and plates: the non-uniformity of the discrete static/dynamic equations for different types of boundary conditions, and the ill-posedness of equations as non-homogeneous boundary conditions are considered. 3). On the base of the proposed modified wavelet formation in this thesis, an initial-value problem of the nonlinear vibration of a MDOF system on time domain [0, T] is equivalently reduced to a group of nonlinear algebraic equations by applying collocation scheme; thus, a modified multi-resolution and a modified adaptive wavelet collocation method are established, respectively, for the initial-value problems of the nonlinear vibration of MDOF systems; and an adaptive algorithm based on homotopic algorithm is specially designed to cope with the nonlinearity of the problem. Based on the proposed modified wavelet formation, the nonlinear vibration equation(generally second-order time-dependent ODEs) of a MDOF system are directly transformed to a group of well-posed nonlinear algebraic equations, without been tranformed to state equations in advance. Hence, the number of unkowns in the presented method in this thesis is only 1/2 of that in most current wavelet collocation methods that are based on state equations. Therefore, compared with current wavelet collocation methods, the amount of storage required by the proposed modified wavelet collocation methods is greatly reduced, about 1/2 of the former at most, while the computation efficiency is greatly impoved, at least twice faster that the former.4). Ctihsiderihg that many control systems can be described by time-delayed MDOF systems, we have tried to applying wavelets in the stability analysis of time-delayed systems. An adaptive wavelet formula is obtained for numerical inverse (?)aplace transformation, based on a current adaptive decomposition and re-construction algorithm; then time-delayed linear MDOF systems are tranformed to discrete dynamic systems, by using numerical inverse (?)aplace transformation. Therefore, the solution of the linear time-delayed system can be solved, and its right-most eigen-value can be approximately calculated, from which its approximate stable regions can be recognized in a selected parameter plane.