Structural Dynamics Optimization Design With Constraints Of Multiple Natural Frequencies And Modal Shapes

In this paper, the Sequential Linear Programming is applied to structural dynamics optimization design. The aim of this study is to solve the optimization problems which should meet the requirements of nodal positions of multiple modal shapes, by changing the sizes of the structure, with multiple natural frequencies constraints. The engineering background of this kind of problem comes from various engineering fields: In the design process of airplane, rocket and missile, there will be some specific demands of structural natural frequencies, but also demands of nodal locations (or nodal lines) of modal shapes, in order to place apparatus and instruments which are sensitive to vibration. This kind of engineering problem falls into the category of structural dynamics optimization design problem with constraints of multiple natural frequencies and modal shapes.One of the key points of structural dynamics optimization design is the sensitivity analysis of natural frequency, modal shape and objective function. Structures are optimally designed for the prescribed purpose by shifting the sizes of structural elements.It is found that the complexity of modal shape sensitivity calculation varies with the element type. When the dimension of element increases, the deduction of derivative of state variable with respect to design parameters becomes very difficult.To avoid the complicated deduction in the sensitivity analysis of natural frequency, modal shape and objective function, the first order difference calculation is used to substitute for the first order differential analysis. It simplifies the sensitivity calculation and can be widely used for any element without the effect of high order and complexity of element matrix.The relationship between nodal position and structural sizes is established with size optimization method firstly, and then with given constraints the optimal solutions are obtained by optimization method.Using the Parametric Design Language (APDL) of ANSYS software, thestructural dynamics optimization program is developed. With the constraints of multiple natural frequencies and nodal positions of modal shapes, structural optimization design is performed on a cantilever beam, a planar frame structure and a five floor space frame. To verify the efficiency and feasibility of the optimization method, two cantilever beams are manufactured with the sizes before and after optimization respectively and the beams are tested to compare their nodal positions of the optimized modal shape. The results show that using size optimization the nodal position can be moved to the desired position with satisfactory accuracy.Optimization calculations and model experiments also show that using theapproach proposed in this paper, it is effective to solve this kind of optimizationproblems with the constraints of multiple natural frequencies and nodal positions ofmodal shapes.This research was funded by the Natural Science Foundation of China undergrant number 50375122, 50575181 and the Aeronautical Science Foundation of Chinaunder grant number 04153072.