On the use of complex numbers in equations of nonlinear structural dynamics

Abstract

Abstract In many technical applications, non-linearities play an important role and must be taken into account as early as the design phase. In the field of dynamics and vibration, nonlinear effects can dramatically change the forced response behavior. For example, several vibration states can be possible for one excitation frequency. The Harmonic Balance Method (HBM) is widely used to solve differential equations resulting from nonlinear dynamic problems. Periodic stationary oscillations are efficiently approximated by a truncated Fourier series, where the Fourier coefficients are to be calculated from a nonlinear algebraic system of equations. As in linear theory, the harmonic signals can be defined with real or complex arithmetic and so HBM can be applied in real or complex form. However, since the resulting equation system for the Fourier coefficients is usually solved numerically with Newton’s method, one encounters a problem when using the complex representation. For the complex residual function, the complex derivative simply does not exist, which is why it must be separated into real and imaginary parts. The present paper deals with the problem of non-differentiable complex functions by first briefly discussing the history of complex numbers. Complex differentiability is explained using simple mathematical examples. For a Duffing oscillator and for a Friction oscillator it is shown in detail that no complex derivative exists for the associated HBM residual function. Furthermore, general amplitude-dependent nonlinear terms are shown to end up in complex non-differentiability with the need of splitting the problem into real and imaginary parts.