Closed Solution To Nonlinear Problem Of Beams/Plates Based On Wavelet

Nonlinear differential equations are mathematical models of many nonlinear problems in natural sciences and engineering. Method of solving nonlinear differential equations cann't be avoided in the studying of nonlinear science. Since the beginning of nonlinear science, many methods including analytical and numerical methods have been developed, but the existing numerical methods in solving nonlinear differential equations, especially in the quantitative solution of strongly nonlinear problems, there are still many deficiencies. One of the most important reason is that the existing numerical methods can't make high-level information and low-level information decoupling in the nonlinear equation, leading to that the discarding of high-order terms have influence in the low-order approximate solution, therefore, when the nonlinear effection increases to a certain extent, the accuracy of existing numerical methods for solution is not enough and even the solution doesn't converge. So, how to obtain a high-precision approximate solution for strongly nonlinear systems is an important topic in nonlinear science.Wavelet analysis is a new branch of mathematics, and has excellent recognition capabilities in the time domain and frequency domain. It has been used in many fields such as image processing, fault diagnosis and the numerical solution of equations, which have demonstrated the superiority and vitality. Specific to the field of numerical solution of differential equations, based on multi-resolution analysis, wavelet series has special approximation characteristics of the layers of nested space, in addition, the wavelet function has characteristics like compactly supported, orthogonal, smooth conductivity and so on. Therefore, the wavelet approximation of an arbitrary square-integrable function based on wavelet function or scaling function has stable and fast approach advantages. Further, the generalized-Gaussian-quadrature method in wavelet analysis solve the problem that the high-level information and the low-level information cann't be decoupled in the existing numerical methods, which makes a theoretical foundation in order to obtain the closed solution of equations. This dissertation presents closed solution of nonlinear equations of the beams/plates based on wavelet.First, construct a coordinated improved wavelet scaling function associated with the arbitrary boundary conditions through the basis of the Taylor series expansion on boundarys. Wavelet approximation based on the modified scaling function, can be an effective approximation of a function defined on a finite interval, avoid the undesired jump or wiggle phenomenon near the boundary points when the wavelet-based method is employed to solve a boundary-value problem. After that, according to the generalized-Gaussian-quadrature method in wavelet analysis, it has the properties of making the symbols of nonlinear function on the expanded coefficients when approximate arbitrary nonlinear functions. Then, a wavelet approximation of the nonlinear terms with explicit expansion of an arbitrary function defined on a bounded interval is obtained.Apply the modified wavelet approximation with explicit expansion to the wavelet Galerkin method, which solved the problem about bending of circular thin plate with nonlinear characterization and the transitional problem of film theory successfully. With the approximation, boundary values and boundary derivatives of the unknown function to be approximated can be explicitly embedded in the resulting scaling function expansions, which avoid the trouble of handling the boundary conditions in traditional Galerkin method. And the expansion coefficients in the approximation of arbitrary nonlinear term can be explicitly expressed by the unknown coefficients, laid the foundation to obtain the closed solution of equations. Finally, by comparison with the exact solution, show that the modified wavelet Galerkin method has good numerical accuracy in solving strongly nonlinear equations, and the accauracy of the approximate solution isn't sensitive to the nonlinear characteristics. It means that the modified wavelet Galerkin method still has high numerical accuracy even the nonlinearity is strong.Ones need to calculate the multiplied of scaling function and its derivative in the classical wavelet Galerkin method, which is inevitable to introduce numerical errors. To overcome this shortcoming, by means of that differential equations can be transformed into integral equations and the interval wavelet numerical integration method proposed, we can obtain a high-precision approximate solution of equations. Numerical example shows that this method greatly reduces the amount of computation and improves the numerical accuracy comparing to the classical wavelet Galerkin method.Consider that there are singular integral in differential equations with fractional order, we use the Laplace transform and inverse transform to convert them into the second type Voltera integral equations with non-singular kernels. During the Laplace transform process, it is difficult to obtain the explicit expression of the nonlinear terms in the Laplace image space, but by the inverse transform it can be eliminate, which means the nonlinear terms are expressed by the symbols of their Laplace transform. By the interval wavelet numerical integration method proposed, and by using the direct numerical integration method in time, while the collocation method is using for discrete space, the original nonlinear differential equation can be changed into nonlinear algebraic equations. At the same time, most of the problems in mechanics, such as the vibration problem, diffusion problem, due to its conveolution kernel function is continuous at zero and the corresponding function value is zero, so the results of current solution step can be explicitly expressed by the results of the aforementioned solution, the proposed method does not involve any matrix inversions and can be implemented as simple as the linear multi-step methods for solving the nonlinear fractional equations, which improved the efficiency of the computation.The wavelet direct iterative method is applied for numerical solution of nonlinear fractional vibration, diffusion and wave equations, and compared the numerical results with the exact solution, results obtained by the Adomain decomposition method and the difference method. The quantitative numerical results obtained in solving fractional differential equations with either integer order or non-integer order nonlinear terms or both, and the effectiveness of the proposed wavelet-based method is analyzed and discussed with different nonlinear characterization, indicate that this method has high accuracy no matter the nonlinearity is weak or strong.Based on the modified scaling function transform of the Coiflets wavelet theory, an one-to-one explicit relation between the electric signals measured in piezoelectric sensors and deflection of the laminated plate is established in this paper, and proposes a hybrid active-passive control strategy for suppressing vibrations of laminated rectangular plates bonded with distributed piezoelectric sensors and actuators via thin viscoelastic bonding layers. Due to that the reconstruction function of the scaling function has the ability of automatical filtering high order frequency signals of vibration or disturbance, the phenomenon of instability which caused from the spilling over of measurement and controller, doesn't occur in the control strategy proposed. Numerical simulation results shows that the hybrid active-passive control strategy can effectively control the low frequency vibration in the control loop of the system. Moreover, the existence of thin viscoelastic bonding layers can further improve robustness and reliability of the system through dissipating the energy of any other possible noise partially induced by numerical errors during the control process.