Vibrational power flow analysis of nonlinear dynamic systems and applications

Jian Yang. Supervisors: Dr. Ye Ping Xiong and Prof. Jing Tang XingFaculty of Engineering and the Environment, University of Southampton, UK.

jian.yang@soton.ac.uk

FSI Away Day 2012

Fluid Structure Interactions Research Group

Background

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.

Equation of energy flow balance can be obtained by multiply both sides of Eq.(1) with velocity

● 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.

Kinetic energy change rate

Potential energy change rate

Dissipated Power

Instantaneous

input power

Typical nonlinear dynamic systems

Van der Pol’s (VDP) oscillator -Nonlinear damping

Duffing’s oscillator -Nonlinear stiffness

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.

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.

Future work 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.

Fig. 1 Nonlinear seat suspension system

Dynamic equation for a single degree-of-freedom system

Fundamental PFA Theory

Fig. 2 Nonlinear energy harvesting using a flapping

foil[1]

. (3)

(4)

Fig. 3 Limit cycle oscillation of VDP oscillator

Fig. 4 Chaotic motion of Duffing’s oscillator.

Instantaneous power flow

Time-averaged power flow

. (1)

. (2)

Fig.5 (a) Instantaneous input power and (b) frequency components in the input power of Duffing’s oscillator

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).

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

Fig.6(a) shows the forced response of VDP oscillator may be either periodic or non-periodic 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.

Fig.6 (a) Bifurcation diagram and (b) time averaged input power of VDP oscillator

𝑝𝑖𝑛=1𝑇∫

0

𝑇

𝑝𝑖𝑛 d 𝑡 .

(a) (b)

(a) (b)

�̇� 𝑝𝑑 �̇� 𝑝𝑖𝑛++ =