Natural modes of complex stiffness systems

For a vibrating dissipative system with hysteresis loop, complex (dynamic) stiffness is widely used. Although there are eigenvalue solvers available to solve the symmetric complex eigenvalue problems being produced, the physical interpretation of the eigensolutions which are no longer in complex conjugate pairs is generally impossible. An approximate method is introduced so that the natural modes are real while the natural frequencies are still complex. For certain forms of complex stiffness, the method is exact. In general, the imaginary part of the complex stiffness is transformed according to the requirement that the energy dissipated in each mode is the same under the transformation. The advantage of the present method is in its ability to produce real modal vectors and not necessary in its computational efficiency. The model vectors produced are orthogonal so that a realistic modal analysis is possible.