Report

Fluid Structure Interactions Research Group Vibrational power flow analysis of nonlinear dynamic systems and applications Jian Yang. Supervisors: Dr. Ye Ping Xiong and Prof. Jing Tang Xing Faculty of Engineering and the Environment, University of Southampton, UK. [email protected] Background ● Predicting the dynamic responses of complex systems, such as aircrafts, ships and cars, to high frequency vibrations is a difficult task. Addressing such problems using Finite Element Analysis (FEA) leads to a significant numerical difficulty. ● Power flow analysis (PFA) approach provides a powerful technique to characterise the dynamic behaviour of various structures and coupled systems, based on the universal principle of energy balance and conservation. ● PFA is extensively studied for linear systems, but much less for nonlinear systems, while many systems in engineering are inherently nonlinear or designed deliberately to be nonlinear for a better dynamic performance. Aims ● Reveal energy generation, transmission and dissipation mechanisms in nonlinear dynamic systems. ● Develop effective PFA techniques for nonlinear vibrating systems . ● Apply PFA to vibration analysis and control of marine appliances, such as comfortable seat and energy harvesting device design. Fig. 2 Nonlinear energy harvesting using a flapping foil[1] Fig. 1 Nonlinear seat suspension system Fundamental PFA Theory Instantaneous power flow Dynamic equation for a single degree-of-freedom system Fig.5 shows the instantaneous input power flow of Duffing’s oscillator when it exhibits chaotic motion. The irregularity in input power pattern shown in Fig.5(a) results from the incorporated infinite frequency components which is demonstrated by Fig.5(b). + , + = cos . (1) Equation of energy flow balance can be obtained by multiply both sides of Eq.(1) with velocity + , 2 + = cos . + Kinetic energy change rate Dissipated Power + Potential energy change rate (2) = Instantaneous input power (a) Typical nonlinear dynamic systems Fig.5 (a) Instantaneous input power and (b) frequency components in the input power of Duffing’s oscillator( = 0.02, = −1, = 1, = 1, = 0.6). Van der Pol’s (VDP) oscillator -Nonlinear damping + 2 − 1 + = cos . (3) Time-averaged power flow (4) Time averaged input power of the system can be employed to incorporate the effects of multiple frequency components in the response, which can expressed as 1 = d . 0 Duffing’s oscillator -Nonlinear stiffness + 2 + + 3 = cos . (b) These nonlinear systems behave differently compared with their linear counterparts as the former may exhibit inherently nonlinear phenomenon such as limit cycle oscillation, sub- or super- harmonic resonances , quasi-periodic or even chaotic motion. Their responses may also be sensitive to initial conditions when multiple solutions exist. Although their nonlinear dynamics have been extensively investigated. The corresponding nonlinear power flow behaviours remains largely unexplored. Fig.6(a) shows the forced response of VDP oscillator may be either periodic or nonperiodic for different excitation frequencies. In this situation, the time averaged input power provides a good performance indicator of input power level by using a long time span for averaging. It can be seen that the averaged input power value of VDP oscillator can be negative, which is different from that of linear systems. (a) (b) Fig.6 (a) Bifurcation diagram and (b) time averaged input power of VDP oscillator ( = 0.5, = 1.0). Future work Fig. 3 Limit cycle oscillation of VDP oscillator( = 0.5, = 0) Fig. 4 Chaotic motion of Duffing’s oscillator. = 0.02, = −1, = 1, = 1, = 0.6. 1. To study power flow behaviours of systems exhibiting inherent nonlinear phenomenon; 2. To develop effective power flow techniques for nonlinear systems; 3. Apply nonlinear power flow theory to vibration control as well as energy harvester design. Reference [1] J.Yang, Y.P.Xiong and J.T.Xing, Investigations on a nonlinear energy harvesting system consists of a flapping foil and electro-magnetic generator using power flow analysis, 23rd Biennial Conference on Mechanical Sound and Vibration, ASME, Aug 28-31, Washington, US, 2011. FSI Away Day 2012