The linearization method based on the equivalence of dissipated energies for nonlinearly damped structural systems

Even though the nonlinear damping characteristics inherently pertain to most structural systems, many useful dynamic models are still linear. As a consequence, a method that is able to equivalently linearize (EQL) the nonlinear damping so that it can be directly applied in these existing linear models is essential. In the present paper, a novel EQL method for nonlinearly damped and single-degree-of-freedom systems is developed. The method is theoretically derived by modulating the steady-state responses of the original nonlinear system. During the linearization, the new EQL method requires the dissipated energy of the target linear system to equal that of the original nonlinear one. In effect, this criterion is equivalent to forcing both the phase angles and amplitudes of the two systems to be equal, or at least to within a small allowable error. Furthermore, the present paper proves that it is possible for the dissipative energy to be expressed in terms of the Fourier coefficients of the modulated signal. Thus, the equivalent viscous damping ratio can be computed from these Fourier coefficients. This new EQL method is numerically tested using examples of bi-linear damping models, and subjected to experimental measurements. Both the simulation results and the experimental data verify the validity of the method. They also prove that the current method gives the equivalent viscous damping with good accuracy. In addition, a quality index that signifies how well the EQL system reaches is appropriately added.