Efficient Energy Balancing Across Multiple Harmonics of Nonlinear Normal Modes

Predicting the forced responses of nonlinear systems is a topic that attracts extensive studies. The energy balancing method considers the net energy transfer in and out of the system over one period, and establishes connections between forced responses and nonlinear normal modes (NNMs). In this paper, we consider the energy balancing across multiple harmonics of NNMs for predicting forced resonances. This technique is constructed by combining the energy balancing mechanism with restrictions (established via excitation scenarios) on external forcing and harmonic phase-shifts; a semi-analytical framework is derived to achieve both accurate/robust results and efficient computations. With known inputs from NNM solutions, the required forcing amplitudes to reach NNMs at resonances, along with their discrepancy, i.e. the harmonic phase-shifts, are computed via a one-step scheme. Several examples are presented for different excitation scenarios to demonstrate the applicability of this method, and to show its capability in accurately predicting the existence of an isola where multiple harmonics play a significant part in the response.

Relative rotor phasing for multicopter vibratory load minimisation

This study focuses on vibration reduction for quadcopters and octocopters with elastic, two-bladed, fixed-speed, variable-pitch rotors through the use of relative rotor phasing. The study defines phase modes such as a pitch phase mode with relative phasing between the front and aft rotors, a roll phase mode with relative phasing between the left and right rotors, and a differential phase mode with relative phasing between the clockwise and counter-clockwise spinning rotors for both the quadcopter and the octocopter, as well as additional higher harmonic phase modes for the octocopter. Parametric studies on individual phase modes indicate that, for the quadcopter in forward flight, the pitch and roll phase modes can almost entirely eliminate the 2/rev vibratory forces (at the aircraft level), but the 2/rev vibratory moments cannot be minimised at the same time. By simultaneously using multiple phase modes, a Pareto front can be generated and a solution selected based on the relative emphasis on force or moment vibration reduction. For the octocopter, it was observed that individual higher harmonic modes (specifically the 2c or 2s modes) could almost entirely eliminate both the 2/rev vibratory forces and moments, simultaneously. Compared with vibration levels in forward flight that might, on average, be expected if the rotors were randomly phased, a 62% reduction of a composite vibration index can be achieved on a quadcopter, and complete elimination of vibration was achievable on an octocopter, with appropriate rotor phasing.