Damping is generally considered to be linear in the mathematical analysis of lumped parameter systems by their associated modal equations of motion. Considering all sources of damping, the damping force should be represented by both linear and nonlinear functions of the modal displacement and velocity and/or a hysteresis curve. One procedure for studying the various representations of damping is based on the energy loss per cycle for each representation. For transient systems, the energy loss per cycle includes an exponential decay factor. For both steady-state and transient systems, the energy loss is a function of the response amplitude to a power and frequency to a different power. To represent “gross” damping, an empirical equation with energy loss equal to a polynomial in modal amplitude and frequency is used. Coefficients of this polynomial are determined from a curve-fit program using experimental data. Applying the curve-fit program requires accurate experimental data. An alternate approach for studying various representations of damping is based on the amplitude versus frequency curve for steady-state response. The shape of this curve changes as the damping changes. To obtain an amplitude versus frequency curve from the modal equation of motion, the Ritz averaging method is used. Similarly for transient systems, differences occur in the amplitude-versus-time curves (ringout curves). The approximate analytical solution for a transient modal equation is obtained using the exponential series method. Finally, a discussion of what experimentation and research should be initiated to permit additional insight into nonlinear damping and transient systems is included.