Vibro-Impact Dynamics of a Piezoelectric Energy Harvester
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1 Proceedings of the IMAC-XXVIII February 1 4, 1, Jacksonville, Florida USA 1 Society for Experimental Mechanics Inc. Vibro-Impact Dynamics of a Piezoelectric Energy Harvester K.H. Mak *, S. McWilliam, A.A. Popov, C.H.J. Fox Materials, Mechanics & Structures Division, Faculty of Engineering, ITRC Building, University Park, University of Nottingham, NG7 RD, UK Abstract A common design of piezoelectric energy harvester consists of a piezoelectric monomorph cantilever which converts the ambient vibration into electrical energy. However, if the level of the ambient vibration is high, large bending stresses will develop in the monomorph, which can cause fatigue and mechanical failure. A stop is introduced into the harvester to reduce the bending stress by limiting the maximum amplitude of oscillation of the cantilever. The dynamics of such a system is complex and involves considerations of vibro-impact mechanics as well as electromechanical interactions. A theoretical model of a piezoelectric-vibro-impact system is demonstrated in this study. The theoretical model is able to predict the dynamical and electrical responses of an energy harvester. It also estimates the contact force between the cantilever and a stop. Typical simulation results are presented and the physical meaning of the results is explained. The simulation results also show that moving the position of a stop can significantly affect the electrical output from the monomorph. Keywords: Energy harvesting, Vibration, Piezoelectricity, Impact dynamics 1. Introduction The harvesting of vibration energy is a feasible solution to supplying electrical energy to small electronic devices, such as wireless sensors for monitoring. In practice, energy harvesters are subjected to periodic and random vibrations during operation. They may also be subjected to shock accelerations which can reduce service life. For a cantilever type of harvester, the maximum bending stress always acts at the clamped end. The use of a stop can restrict the displacement of a beam, thus limiting the bending stress, including the maximum stress at the root of the beam. However, there is a concern that limiting the displacement of a cantilever monomorph can lead to a reduction in generated electrical power from the energy harvester. Figure 1 Illustration of the impact configuration between a cantilever beam and a stop Mechanical impact is not a new area of research and a variety of studies have been reported for different kinds of impact systems [1]. Exact solutions are not normally possible and most studies use approximate solutions [] [5]. There are two basic approaches to modelling an impact system, like the one shown in Figure 1. One * Corresponding author: Tel: address: epxkhm@nottingham.ac.uk (K.H. Mak)
2 approach is to employ Newton s coefficient of restitution to predict the velocity after impact [6]. The benefit of this approach is to simplify the analysis by assuming a known relationship between the velocities before and after impact. This approach does not require the contact force to be determined [7]. The value of the coefficient of restitution can only be determined through experiments and it may vary if the contact duration varies and more high modes are excited [6]. The second approach is to estimate the contact force and use this to predict the dynamics after impact [] [5]. Higher mode transient waves will be induced in the beam by the impact force, excluding these will obscure some of the fine detail of the dynamics during contact. A theoretical model for a piezoelectric monomorph cantilever with impact is derived in Section. Numerical simulation results will be presented and discussed in Section 3 and broad conclusions are provided in Section 4. The dynamical and electrical responses for the impact system are determined and the simulation results obtained offer an insight into the chattering impact and the contact force. This paper also suggests a simple way to minimise the reduction in electrical output from impact.. Impact model for a piezoelectric monomorph cantilever The cantilever beam consists of a substrate layer on which a layer of piezoelectric material (e.g. PZT) is mounted. The structure is treated as an Euler-Bernoulli beam with a stop as shown in Figure. The stop can be modelled as a linear spring or a rod capable of longitudinal vibration. The displacement of a linear spring is always zero unless there is a contact force acting on it, so it is not possible for the spring to vibrate when the beam is out of contact with the stop. For this reason, it is more realistic to model the stop as a longitudinal rod, and this approach is used in the theoretical model. y x L b b ( t) s X c z, ξ L r Figure Illustration of the impact configuration between a cantilever beam and a rod The governing equation for the transverse motion of the beam responding to the contact force, F(t), between the beam and the stop, and to the base acceleration, b s ( t), is: 4 y( x, t) y( x, t) y( x, t) d bs ( t) EbIb C ( ) ( ) 4 b b Ab F t x X c b Ab (1) x t t dt Similarly, the governing equation for the motion of the rod (stop) to the contact force and the external acceleration is: z(, t) z(, t) z(, t) d bs ( t) Er Ar C ( ) ( ) r r Ar F t L r r Ar () t t dt In equations (1) and (), y(x,t) is the transverse displacement of the beam and z(ξ,t) is the longitudinal displacement of the rod. The subscripts b and r denote the beam and the rod respectively and E and ρ are the Young s modulus and density. L and A are the length and uniform cross sectional area and I b is the second moment of area about the neutral axis of the beam. C is the structural damping coefficient. F(t) is the contact force acting between the beam and the stop. X c is the position of impact along the beam. δ(x X c ) is a Dirac delta function and is used to specify the position of the contact force. Notice that the piezoelectric effect is not considered in (1), but it will be introduced into the equation of motion once the impact model is developed.
3 Theoretically, the solutions to equations (1) and () can be expressed as an infinite series of eigenfunctions and associated generalized coordinates: where ( x) and r( t) y( x, t) ( x) r ( t) (3) i 1 i 1 bi bi z(, t) ( ) r ( t) (4) ri ri are the mode shape functions and generalised coordinates respectively. The basic solution procedure for the impact system defined by equations (1) and () is as follows. When the beam and the stop are out of contact, F(t) will be zero in both equations (1) and (), which can therefore be solved separately. When the beam comes into contact with the stop, F(t), is non-zero in (1) and () (see Fig.3) and the motions of the beam and the stop are then coupled. Equations (1) and () must be solved simultaneously to determine the size and the duration of the contact force. In this situation, the nature of the coupled equations means that they must be solved numerically using a time-stepping method..1 Determining of contact force To determine the unknown contact force, the motion of the beam has to be coupled with the motion of the stop because their displacements at the point of contact are identical when they are in contact. Before estimating the contact force, the time at which contact begins must be determined by using the following inequalities: Out-of-contact: y( X, t) z( L, t) F( t) (5) c r In-contact: y( X, t) z( L, t) F( t) (6) c r These equations are used to check whether the beam and the stop are in contact. If contact is detected, the time at which contact takes place, t e, must be determined. This can be done by equating the displacements of the beam and rod as equation (7). y( X, t ) z( L, t ) (7) c e r e This equation is solved in the time domain to calculate t e. Once this has been achieved, the contact force can be estimated by coupling the displacements of the beam and stop at the contact position. The motion of the coupled system is then tracked until the contact force becomes negative, indicating that the beam has separated from the stop. The time at which separation takes place is determined by the same method as used to determine the time of contact. y x z, ξ F(t) Figure 3 The contact force acts between the beam and the stop at the impact position The displacements of the beam and the stop are separated into two parts as follows: y( x, t) y ( x, t) y ( x, t) (8) s imp z(, t) z (, t) z (, t) (9) s imp
4 In both equations, the first term on the right hand side is the response to the base excitation and the second term is the response to the contact force. The displacements are such that y=z+δ during contact and this is used to calculate the contact force, F(t). These terms are expressed in the form of Duhamel s integral to facilitate estimation of the contact force [5].. Harvesting electrical energy The monomorph is designed to work in the d31 mode to generate electrical charge. The piezoelectric material is modelled as a current source I(t) in parallel with a capacitor, C P. To withdraw the generated electrical charge from the piezoelectric material, the energy harvester can be connected to a resistor, R, in series, as shown in Figure 4. Based on the electrical circuit in Figure 4, Kirchhoff s law is used to deduce the following governing equation. v( t) C pv ( t) ir bi ( t) (1) R i 1 The electromechanical coupling coefficient, Θ, is a measure of the conversion of mechanical energy from the beam shape function into electrical energy. It is introduced to obtain with the voltage, v(t), across the harvester, and it describes the piezoelectric effect in the theoretical model [8]. Energy Harvester I(t) Cp R Vm Voltmeter Figure 4 The energy harvester is connected in series to a resistor The procedure and logic of the theoretical model are summarised in the flow chart provided in Figure 5. These procedures were used in a computer program that was developed. Yes Contact No Evaluate F(t k ) y(x c,t k ), v(t) z(l r,t k ) F(t k ) = Evaluate y(x c,t k ),v(t) z(l r,t k ) F(t k ) < No Find the exact time of contact / separation, t e No y < z+δ Yes Set F(t k ) = y(x c,t) = z(l r,t)+δ Yes t k+1 =t k +Δt Figure 5 A flow chart for the theoretical model of a piezoelectric impact system
5 3. Numerical simulation results Numerical simulation results, based on the model developed in Section, are presented here. A harmonic support motion is applied such that b s ( t) Bmax sin ft where B max is the maximum acceleration and f is the excitation frequency. A damping ratio of.1 is assumed for all beam and rod modes considered. The stop is located at the end of cantilever beam. Table 1 shows the dimensions and mechanical properties of the energy harvester considered. The numbers of modes used for the beam and the rod were determined from an investigation of system convergence. Contact forces can excite high frequency modes significantly and these modes affect the predicted interaction between the beam and the stop. However, all the modes cannot be included in the simulation, so the minimum number of modes needed for practically useful predictions must be determined. A convergence study was also performed to calculate the required time step. There are restrictions on the size of time step for such an impact system [], [3]. All of the key parameters used in the results are listed in Table. L (mm) w (mm) t (mm) ρ (kg/m 3 ) E (Gpa) C p (nf) Substrate PZT Stop Table 1 The dimensions and mechanical properties of the beam and the rod used in the simulations Number of mode time step Δ F B max X c R Beam Rod (s) (mm) (Hz) (m/s ) (mm) (MΩ) 1 1 x L b 1 Table The key parameters of the impact model An example from a numerical simulation is presented in Figure 6. It can be seen that the beam is excited by the support motion and it repeatedly impacts the stop. When impact occurs, the displacement of the beam at the impact position is limited by the stop and the contact force reacts against the beam. The calculated contact force is shown in Figure 6(b). The beam does not rest continuously on the stop during contact; instead a chattering impact occurs as shown in Figure 7. The duration of contact is relatively short compared to the first few oscillating periods of the beam, indicating that the higher frequency modes are excited by the contact force. These components can be seen in the electrical response (Figure 6(c)). (a) 4 x 1-5 (b) (c) y (m) F (N) Voltage (V) Figure An example of simulation for an impact system (a) displacements of the beam; (b) contact force; (c) voltage
6 Displacement (m) 3 1 x 1-7 Beam Rod Figure 7 Chattering impact between the beam and the stop The developed model can be used to assess the influence of the impact on the generated voltage. Figure 8 shows a comparison between the simulated voltages with and without (i.e. no stop) impact taking place. It is obvious that the voltage is reduced during the contact phase, but is maintained when the beam and the stop are out of contact. Clearly the overall generated electrical power would be reduced by the impact, which is undesirable. One way to alleviate this reduction in voltage might be to move the position of the stop, subject to consideration of the maximum induced stress. If the stop is relocated so that it is not at the free end of the beam, the deflected shape will be different. Figure 9 compares the generated voltage with the stop located at the free end of the beam, at the mid-span, and without a stop. It shows that higher peak voltages are generated when the stop is located at the mid span of the beam, compared to the tip. The level of generated power from the harvester greatly relies on the deflected shape of the monomorph. Moving the position of the stop changes the deflected shape in the contact phase. Figure 1 shows the possible deflected shapes of the monomorph with the stop located at the end and the mid-span. The monomorph in Figure 1 (a) looks to have a clamped-pinned boundary condition and the bending stress across the entire monomorph is less than the one shown in Figure 1 (b). This is one of the reasons why the generated voltage is less when the stop is at the free end (i.e. X c = L b ), but this is a subject for more detailed investigation. 1 with impact without impact Voltage (V) Figure 8 Comparison of generated voltages when impact is included and excluded in the simulation 1 X c = L b Voltage (V) 5-5 X c =.5L b w ithout impact Figure 9 Generated voltage with different stop locations
7 (a) (b) Figure 1 Illustrations of the beam deflection during contact (a) X c = L b; (b) X c =.5L b 4. Conclusions A computer model of a piezoelectric vibro-impact system has been developed. The simulation is able to predict dynamical and electrical responses, as well as the impact interaction between a beam and a stop. The simulation gives insight into a beam-stop chattering impact. It is important to perform convergence conditions studies to ensure that the simulation results reflect the important responses in the system. It is also worth mentioning that one set of convergence conditions will not necessarily be valid for every combination of dimensions and parameters. It was found that moving the position of the stop can affect the electrical output from the energy harvester, and this is a matter for further investigation. Further work is in progress to validate the simulation results against experimental measurements. This will include connection of the energy harvester to a charging circuit as shown in Figure 11, so that a comparison can be made for the energy stored in the storage capacitor, C s, within a given period of time. Energy Harvester I(t) Cp Cs Vm Figure 11 Energy harvester is connected AC-DC circuit to charge a capacitor Acknowledgement The authors gratefully acknowledge Atlantic Inertial Systems (AIS) for their financial and technical support for the work reported here. Reference [1] Babitsky V.I., Theory of Vibro-Impact Systems and Applications, 1998, Springer, Pages 45-55, March 198 [] Lo C.C., A cantilever beam chattering against a stop, Journal of Sound and Vibration, Volume 69, Issue, Pages 45-55, March 198
8 [3] Fathi A., Popplewell N., Improved Approximations for a Beam Impacting a Stop, Journal of Sound and Vibration, Volume 17, Issue 3, Pages , 4 February 1994 [4] Tsai H. C., Wu M.K., Methods to compute dynamic response of a cantilever with a stop to limit motion, Computers & Structures, Volume 58, Issue 5, Pages , 3 March 1996 [5] Wang C., Kim J., New Analysis Method for a Thin Beam Impacting Against a Stop based on the Full Continuous Model, Journal of Sound and Vibration, Volume 191, Issue 5, Pages 89-83, 18 April 1996 [6] Wagg D.J., Bishop S. R., Application of Non-Smooth Modelling Techniques to the Dynamics of a Flexible Impacting Beam, Journal of Sound and Vibration, Volume 56, Issue 5, Pages 83-8, 3 October [7] Yin X.C., Qin Y., Zou H., Transient responses of repeated impact of a beam against a stop, International Journal of Solids and Structures, Volume 44, Issues -3, Pages , November 7 [8] Sodano H.A., Park G. and Inman D.J., Estimation of Electric Charge Output for Piezoelectric Energy Harvesting, Strain (4), Pages 49 58, 4 [9] Se J.A., Weui B.J., Wan S.Y., Improvement of impulse response spectrum and its application, Journal of Sound and Vibration, Volume 88, Issues 4-5, Pages ,, December 5
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