A new development of the harmonic balance method: The hybrid harmonic balance method

Abstract

The harmonic balance (HB) method is efficient in predicting the limit cycle oscillations for many dynamic systems including very high dimensional systems in fluid dynamics. Recently, some variations and extensions of the HB method have been developed and employed in the computation of large systems. This study focuses on a new development of the HB method. The new approach, the hybrid harmonic balance method (HHB), is developed based on the ideas in the conventional HB approach and the high dimensional harmonic balance (HDHB) approach. To demonstrate the applications and advantages, the new method is employed on a prototypical dynamic system in comparison with the results from both the HB and HDHB approaches. It is theoretically proved that when twice the number of harmonics in HB are included in the HHB derivation the results from HHB achieve the accuracy of the HB method for the cubic nonlinearity. Numerical simulations reveal that HHB combines the advantages of both the conventional HB and HDHB methods: Ease of implementation for very high dimensional systems regardless of the complexity of the nonlinearities and generation of meaningful solutions while the spurious solutions (from HDHB) are diminished. This new method is more accurate when compared with the HDHB method and much more computationally efficient when compared with the conventional HB method.