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.
jian.yang@soton.ac.uk
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).
𝑚𝑥 + 𝑐 𝑥, 𝑥 𝑥 + 𝑘 𝑥 𝑥 = 𝑓 cos 𝜔𝑡.
(1)
Equation of energy flow balance can be obtained by multiply both sides of Eq.(1)
with velocity
𝑚𝑥 𝑥 + 𝑐 𝑥, 𝑥 𝑥 2 + 𝑘 𝑥 𝑥𝑥 = 𝑓𝑥 cos 𝜔𝑡.
𝑇
+
Kinetic energy
change rate
𝑝𝑑
Dissipated
Power
+
𝑈
Potential energy
change rate
(2)
=
𝑝𝑖𝑛
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(𝜉 = 0.02, 𝛼 = −1, 𝛽 = 1, 𝑓 = 1, 𝜔 = 0.6).
Van der Pol’s (VDP) oscillator -Nonlinear damping
𝑥 + 𝛼 𝑥 2 − 1 𝑥 + 𝑥 = 𝑓 cos 𝜔𝑡.
(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 𝑇
𝑝𝑖𝑛 =
𝑝𝑖𝑛 d𝑡 .
𝑇 0
Duffing’s oscillator -Nonlinear stiffness
𝑥 + 2𝜉 𝑥 + 𝛼𝑥 + 𝛽𝑥 3 = 𝑓 cos 𝜔𝑡 .
(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
(𝛼 = 0.5, 𝑓 = 1.0).
Future work
Fig. 3 Limit cycle oscillation of VDP
oscillator(𝛼 = 0.5, 𝑓 = 0)
Fig. 4 Chaotic motion of Duffing’s oscillator.
𝜉 = 0.02, 𝛼 = −1, 𝛽 = 1, 𝑓 = 1, 𝜔 = 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