Second-order eigensensitivity analysis of asymmetric damped systems using Nelsonʹs method

First-order eigensensitivity analysis using Nelsonʹs method is first reviewed. Then, Nelsonʹs approach is extended to the computation of the second-order derivatives of the eigenvalues and eigenvectors of symmetric and asymmetric damped systems. The computation of second-order derivatives may be required for large variation of design parameters and for some optimization algorithms. Nelsonʹs method has the advantage of requiring only the knowledge of the eigenvector to be differentiated. Only systems with distinct eigenvalues are considered in this paper. Two numerical examples have been selected to illustrate the application and the utility of the derived expressions of the first- and second-order eigensensitivities. A four degree of freedom symmetric damped system is chosen for the first example whereas a finite element asymmetric 20 dof damped rotor model is considered in the second example.