A new perturbation approach, based on the second-order theory, has been developed for application to nonproportionally damped, asymmetric vibration systems. The approach significantly truncates the computational procedure of the first- and second-order perturbations that are necessary for calculating the perturbed eigenproperties. This is accomplished by calculating these perturbations from the outcomes of solving the eigenproblem of a single, symmetric, positive-definite matrix representing the unperturbed system. Two numerical examples in addition to a practical rotor-bearing system are used to demonstrate the capabilities of the proposed method. The results of the exemplified problem show that the proposed method is comparable to other existing methods that implement exact techniques.