Transient response of general one-dimensional distributed systems through eigenfunction expansion with an implicit filter scheme

The current work is concerned with eigenfunction expansion of the transient response of general one-dimensional non-self-adjoint distributed dynamic systems. The main goal of the current work is to propose an alternative eigenfunction expansion approach for transient analysis of general one-dimensional non-self-adjoint systems which avoids complex-valued eigenvalue problems. With the implicit filter scheme proposed here one splits the solution into two additive components, namely, a filtered solution for which all boundary conditions are homogeneous and a carefully selected filter. Moreover, the eigenvalue problem associated with the differential equation for the filtered solution is self-adjoint and, therefore, all eigenvalues are real-valued and the eigenfunctions satisfy the classical orthogonality relations. The unknown time-dependent coefficients of the filter are then solved simultaneously with the modal co-ordinates of the filtered solution in order to satisfy the original boundary conditions. The proposed approach is demonstrated on a distributed system composed of a cantilever beam with end mass, viscous damper and linear spring.