Dynamic Transformation Method for Modal Synthesis

Abstract

This paper presents a condensation method for large discrete parameter vibration analysis of complex structures that greatly reduces truncation errors and provides accurate definition of modes in a selected frequency range. A dynamic transformation is obtained from the partitioned equations of motion that relates modes not explicitly in the condensed solution to the retained modes at a selected system frequency. The generalized mass and stiffness matrices, obtained with existing modal synthesis methods, are reduced using this transformation and solved. Revised solutions are then obtained using new transformations at the calculated eigenvalues and are also used to assess the accuracy of the results. If all the modes of interest have not been obtained, the results are used to select a new set of retained coordinates and a new transformation frequency and the procedure repeated for another group of modes. Computations are made tractable by simplified forms of the transformation that result with various modal synthesis methods. Three examples using the dynamic transformation in conjunction with a General Electric stiffness coupling method and the method of Craig and Bampton indicate large reductions in truncation errors and demonstrate the method for sequential groups of modes. Comparisons with truncated results using current methods indicate that two to three times as many accurate modes are obtained from solutions keeping less than half the component modes. Nomenclature [/c] = stiffness matrix for total structure in {x} physical coordinates [X] = generalized stiffness matrix for total structure in {q} modal coordinates [/CCPL] = stiffness matrix in {x} coordinates for coupling substructures [m] = mass matrix for total structure in {x} physical coordinates [M] = generalized mass matrix for total structure in {q} modal coordinates [Am] = incremental mass matrix of coupling structures in {x} coordinates