Describing function-based modal analysis

Nonlinear modal analysis provides a framework for characterising amplitude-dependent dynamic behaviour in nonlinear mechanical systems. Among existing approaches, describing function-based formulations approximate nonlinear effects through equivalent linear representations evaluated at prescribed response levels. This paper extends existing describing function-based nonlinear modal concepts to damped complex modal analysis. Nonlinear restoring and dissipative forces are represented by a complex stiffness matrix that depends on amplitude, so that the quasi-linearised structural system retains a complex characteristic equation and is treated directly as a complex eigenvalue problem that varies with amplitude. Within a fundamental-harmonic approximation, the complex eigensolution directly provides the nonlinear natural frequency, modal loss factor, and complex mode shape. Nested fixed-point and Rayleigh quotient iterations with amplitude continuation are used to trace these modal quantities at prescribed amplitudes. The formulation is assessed on representative benchmark systems using the extended periodic motion concept (EPMC) as a reference. It is further shown that both approaches yield equivalent nonlinear modal solutions under the fundamental-harmonic and single-mode assumptions adopted in this work. Overall, the framework provides a direct link between nonlinear modal analysis and classical complex eigenvalue methods, offering a physically interpretable formulation for damped nonlinear modal analysis.