A method of estimating damping parameters for multi-degree-of-freedom vibration systems is outlined, involving a balance of dissipated and supplied energies over a cycle of periodic vibration. The power is formulated as the inner product between the velocity and force terms and integrated over a cycle. Conservative terms (mass and stiffness) drop out of the formulation. The displacement response and the input are measured, and the damping coefficients are estimated without knowledge of the mass and stiffness, which can be nonlinear, as illustrated in one example. The identification equations are also obtained with a modal reduction based on proper orthogonal decomposition. The method can be applied with a harmonic motion assumption or by simple numerical integration. The method is illustrated with linear and quadratic damping, in simulations of a four degree-of-freedom system, and in a string with and without noise.