  The harmonic balance method is the most commonly used method for solving
periodic solutions of nonlinear dynamic systems, but the high-order
approximation of nonlinear terms requires sophisticated formula derivation,
which limits its ultra-high accuracy. The authors' team proposed the
reconstruction harmonic balance (RHB) method through the equivalent
reconstruction of the frequency domain nonlinear quantity in the time domain,
which settled the problem of ultra-high-order calculation of the classical
harmonic balance method. However, both methods require the dynamical system to
be polynomial nonlinear, and cannot be directly used to solve the
quasi-periodic solution of the nonlinear system. In view of the above problems,
this paper proposes a computational method that combines the RHB method and the
recast technique for complex nonlinear systems. First, the general nonlinear
problem is non-destructively recast into a polynomial nonlinear system, and
then the RHB method is used for high-precision solutions. Aiming at computing
the quasi-periodic response, the RHB method based on the idea of "supplemental
frequency" is derived. By optimizing and selecting base frequencies, the fast
and accurate capture of quasi-periodic response is achieved. The typical
systems such as nonlinear pendulum, relativistic harmonic oscillator, and
nonlinear coupling asymmetric pendulum are selected for simulation. The
simulation results show that the accuracy of the RHB-recast method for solving
nonpolynomial nonlinear systems is on the order of 10^(-12), reaching the
computer accuracy, far exceeding state-of-the-art methods. The supplemental
frequency RHB method achieves the efficient solution of quasi-periodic
problems.
