Second-order parameter perturbation method for dynamic structures with interval parameters

When considering the problem of the dynamic responses of structures with interval parameters, previous interval analysis methods are mostly restricted to its first-order. But if the uncertainties of the parameters are fairly large, the response region obtained using the first-order interval analysis method would fail to contain the real region of the dynamic response of uncertain structures. Therefore, the second-order analysis method should be considered. However, the second-order analysis method relating to operations of interval may result in an exorbitantly overestimated dynamic response region, which makes the result useless for practical engineering problems. To circumvent this drawback, firstly the general function of the dynamic response of structures in terms of structural parameters is obtained based on the second-order parameter perturbation method. Then via solving the maximum and minimum of the function, the problem of determining the bounds of the dynamic response of uncertain structures is changed into a series of low dimensional box constrained quadratic problems, and these box constrained quadratic programming problems can be solved using the DC algorithm (difference of convex functions algorithm) effectively. The proposed method can avoid the exorbitant overestimate of the dynamic response region of uncertain structures, while does not introduce much more computational expense. A numerical example is used to illustrate the accuracy and the efficiency of the proposed method when comparing with other methods.