We are interested in the quadratic eigenvalue problem of damped oscillations where the damping matrix has dimension one. This describes systems with one point damper. A generic example is a linearn-mass oscillator fixed on one end and damped on the other end. We prove that in this case the system parameters (mass and spring constants) are uniquely (up to a multiplicative constant) determined by any given set of the eigenvalues in the left half plane. We also design an effective construction of the system parameters from the spectral data. We next propose an efficient method for solving the Ljapunov equation generated by arbitrary stiffness and mass matrices and a one dimensional damping matrix. The method is particularly efficient if the Ljapunov equation has to be solved many times where only the damping dyadic is varied. In particular, the method finds an optimal position of a damper in some 60n3 operations. We apply this method to our generic example and show, at least numerically, that the damping is optimal (in the sense that the solution of a corresponding Ljapunov equation has a minimal trace) if all eigenvalues are brought together. We include some perturbation results concerning the damping factor as the varying parameter. The results are hoped to be of some help in studying damping matrices of the rank much smaller than the dimension of the problem.
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