Linear and non-linear dynamic analysis of multi degree of freedom systems by the method of harmonic differential quadrature (HDQ)

Real physical systems or engineering problems are often described by partial differential equations, either linear or nonlinear and in most cases, their closed form solutions are extremely difficult to establish. As a result, approximate numerical methods have been widely used to solve partial differential equations which arise in almost all engineering disciplines. The most commonly used numerical methods for such applications are the finite difference and boundary element methods, and most engineering problems can be solved by these methods to satisfactory accuracy if a proper and sufficient number of grid points are used. However, in a large number of practical applications where only reasonably accurate solutions at few specified physical coordinates are of interest, the finite difference method becomes inappropriate since they still require a large number of grid points and so large a computer capacity, especially in the cases of nonlinear problems where iteration becomes inevitable. Consequently, both CPU time and storage requirements are often considerable for the standard methods. With the modern computer technology, various numerical methods were well developed and widely used to solve various kinds of engineering and science problems, which are described by the partial or ordinary differential equations. In seeking a more efficient numerical method which requires fewer grid points yet achieves acceptable accuracy, the method of differential quadrature (DQ) was introduced. The basic idea of the differential quadrature method is that the derivative of a function, with respect to a space variable at a given sampling point, is approximated as a weighted linear sum of the sampling points in the domain of that variable. As with other numerical analysis techniques, such as finite element or finiteVII difference methods, the DQM also transforms the given differential equation into a set of analogous algebraic equations in terms of the unknown function values at the reselected sampling points in the field domain. In this thesis, free and forced vibration analysis of a single and multi degree of freedom systems are made by the harmonic differential quadrature (HDQ) method. Unlike the differential quadrature and generalized differential quadrature that use the polynomial functions, harmonic differential quadrature uses trigonometric functions as the test functions. As the name of the test function suggested, this method is called the HDQ method. Both the linear and the nonlinear behavior are taken into consideration in the numerical applications. The accuracy, efficiency and convenience of HDQ are demonstrated throughout the numerical examples. The obtained results are compared with existing solutions available from other numerical methods and analytical results. The method presented gives efficient accurate results in a gradually little CPU time and grid points for the linear and nonlinear dynamic analysis of the single and multi degree of freedom systems. Key Words: Harmonic differential quadrature; Non-linear dynamic analysis; Single- degree-of-freedom systems, Multi degree-of-freedom systems, Structural dynamic.