Eigenvalue sensitivity analysis of structures using lagrange multiplier method and Its Applications

Abstract

An eigenvalue sensitivity analysis method of a structure is developed. This method uses constrained domain and/or shape as design variables. Firstly, a free vibration equation of the considering structure is formulated. The constraint equations are explicitly included in the governing equation using Lagrange multipliers. Then, the eigenvalue sensitivity is derived as changing constrained domain by taking its variations. The resulting sensitivity equation shows that it is proportional to the gradient of eigenfunction and the constraint force. The eigenvalue sensitivity due to structural shape change is also studied by making proper assumptions of the shape variation of a structure. The structural shape change was decomposed into two steps, the change of constrained boundary and the change of free boundary. The proposed method is applied to derive eigenvalue sensitivity equation for a combined beam structure with respect to its joint coordinates. The change of joint coordinates is given by its length and orientation changes of each sub-beam. Then, the corresponding sensitivity equations are simply obtained using the developed method. It is also found that there is a conservative property on a uniform beam section which determines the pattern of eigenvalue sensitivity equations. An optimization problem is provided to determine optimal positions of resilient supports of a structure to get a maximum fundamental natural frequency. Firstly, the eigenvalue sensitivity equation with changing support positions is derived. Then, some valuable characteristics concerning optimal support positions are discussed. Those are limit eigenvalue, critical stiffness, and separation point. Based on those characteristics, a general procedure is proposed to build up the loci of optimal support positions for a beam and plate structure as increasing their supporting stiffnesses. The loci start from the maximum displacement position of its original fundamental eigenfunction, and, in man...