| Periodic structure,containing a lot of basic unit cells which usually be arranged in a certain rule,has a wide range of applications in practical engineering and new materials.In practical engineering,the periodic structure inevitably has defects which will destroy the periodicity of the structure and lead to nonlinearity.Serious defects can even affect the mechanical properties and service life of periodic structure.Therefore,it is of great significance to study the transient response of periodic structure with nonlinear defects under external excitation.Generally,the size of periodic structure with nonlinear defects is relatively large.After discretizing periodic structure in space,the entire structure contains many degrees of freedom.When we use traditional numerical methods to solve its dynamic response,the scale of nonlinear algebraic equations is very large,that makes the calculation of the dynamic response of the structure timeconsuming.Based on the Newmark method,this paper proposes a numerical method for efficiently solving the dynamic response of periodic structures with nonlinear defects through condensation techniques,the dynamic characteristics of periodic structure with nonlinear defects and the Sherman-Morrison-Woodbury formula.The main idea of the proposed algorithm is to convert the dynamic response of the entire nonlinear structure into the dynamic responses of several equivalent small-scale substructures.Numerical examples show that,compared with the method of directly solving the dynamic response of the whole structure,the method proposed in this paper has higher efficiency.The main work of my research includes:(1)A decomposition method for the dynamic response of periodic structure with nonlinear defects is established.This paper systematically analyzes the energy propagation characteristics in a periodic structure with nonlinear defects,discusses the influence domain of any local equivalent external force applied on the nonlinear system,and proposes an algorithm to quickly solve the local displacement response of any local continuous substructure within in a time step.Based on this algorithm,the dynamic response of the periodic structure with nonlinear defects can be converted into several small-scale substructure dynamics.Firstly,the dynamic response of the whole structure can be decomposed into the response of a small-scale substructure with nonlinear defect unit cells and the response of a complete periodic structure.The dynamic response of a complete periodic structure can also be converted into the response of a series of small-scale substructures.These substructures are identical in size and relatively small in scale,so their dynamic response can be solved efficiently.We only need to store the equivalent stiffness matrix and response of a small-scale equivalent substructure,which greatly saves computer memory during calculation.(2)An efficient iterative method for the dynamic response of small-scale substructures with nonlinear defects is established.When solving the dynamic response of the substructure with nonlinear defect unit cells,the equivalent stiffness matrix of the substructure in each iteration process is related to the iterative displacement of the previous step,so the stiffness matrix needs to be updated,which requires a large amount of calculation resources.Using the relatively limited number of non-linear defect unit cells in the substructure and the local variation of the structural equivalent stiffness matrix,a fast method for calculating the inverse matrix of the equivalent stiffness matrix can be established based on the Sherman-MorrisonWooodbury formula.Through this method,we can improve the efficiency of the iterative solution and avoid generating the equivalent stiffness matrix of the entire substructure to save computer memory. |
