Interval analysis of dynamic response of structures using Laplace transform

In this paper, an interval based method for dynamic analysis of structures with uncertain parameters using Laplace transform is proposed. The structural physical parameters and the external loads are considered as interval variables. The structural stiffness matrix, mass matrix and loading vectors are thus described as the sum of two parts corresponding to the deterministic matrix and the uncertainty matrix of the interval parameters. The Laplace transform is used to convert the dynamic equations into a linear system of equations. The matrix perturbation technique is then utilized to remove the higher order terms, and the inverse Laplace transform is employed to obtain the structural dynamic responses. In addition, the element-by-element (EBE) idea used previously in static analysis is extended to dynamic analysis. A special matrix treatment is also used for both the EBE and non-EBE cases to reduce the overestimation of interval analysis and to facilitate the inverse Laplace transform. Finally, the effectiveness of the proposed method is demonstrated by numerical examples compared with the vertex solutions and other researchers’ work. An interesting finding is the divergent response of the undamped system, which is verified by the vertex solution. The proposed method is suitable for small uncertainties of system parameters since the formulation is limited to the first order terms, which results in an inner enclosure of the system response.