1 Introduction

Recently,the wireless electronic devices (WEDs),which have developed rapidly,have been widely used in many fields. Providing WEDs with energy without a wired power source is an interesting issue. Harvesting energy from the surrounding environment is an effective way^[1]. Owing to the strong electromechanical coupling property,piezoelectric materials are used to manufacture high-performance piezoelectric power harvesters that are good at scavenging energy from mechanical vibrations^{[1, 2, 3, 4]}.

This paper is concerned with the converting mechanical energy from flow-induced vibrations into electric energy. Piezoelectric power harvesters that are composed of flexible piezoelectric films or strips are used for this purpose^{[5, 6, 7, 8]}. These harvesters are fitted behind a blunt body in a flow from which the asymmetric vortex sheds,as excites the harvesters into transversal vibrations. Due to the transversal vibration,an electrical output voltage is generated. Resorting to the theory of linear piezoelectricity,the basic performances of the piezoelectric devices can be conducted^{[4, 5, 6, 7, 8, 9]}. To extract energy as much as possible,the harvesters are usually designed as resonant devices operating near resonance. There are some nonlinear effects responsible for some behaviors such as jump and multivaluedness under that situation. Obviously,these nonlinear behaviors,e.g.,multivaluedness and jump phenomena,which are usually considered undesirable,are to be avoided in design. A nonlinear analysis is necessary for the purpose of determining the linear operating range of the devices. Recently,Hu et al.^[10] analyzed the nonlinear performance of a piezoelectric power harvester consisting of a mass and a plate in thickness-shear vibration. Xue and Hu^[11] analyzed the geometrically nonlinear characteristics of a harvester consisting of circular piezoelectric plate by axially symmetrical vibration mode. In this paper,we study a circular-cylinder piezoelectric power harvester (CCPPH) based on the flow-induced flexural vibration mode. The resonant frequency range of the CCPPH is considered to be relatively low in deflection mode. Therefore,the energy-scavenging structure usually resonates easily with ambient vibrations^{[12, 13]}. Therefore,we study the nonlinear characteristics of the harvester,taking into account the in-plane extension incidental to large transversal deflection. In Section 2,we establish the motion equations of the CCPPH near resonance with flow-induced flexural vibration mode. Because the polarization charges generated by the in-plane extension deformation produce no effect to the output current,the boundary electric charges are carefully divided into two parts according to the different deformation modes. After re-establishing the relationship between output current and boundary electric charges,a lot of computations on output powers are conducted. Numerical results are shown in Section 3,and some conclusions are drawn in Section 4. 2 Analysis of CCPPH with geometrical nonlinearity

The structure of a CCPPH is shown in Fig. 1,where the flow is along the x3-direction to induce the beam flexurally vibrating in the x₁x₂-plane. For the tube,L ＞ d ＞ h,the length is 2L,the diameter is d = 2R,and the thickness is h. The electrodes of the CCPPH cover its whole length with the electrode configuration at the cross section shown in Fig. 1(b). The whole inner surface is regarded as an electrode that is grounded. The top of the outer surface is designed as an electrode. Like the top,the bottom has a similar electrode. Both the top electrode and the bottom electrode sweep out an equal angle that is 2α. α cannot be larger than π/2 for the purpose of avoiding the overlap between the top electrode and the bottom electrode. The CCPPH is poled along the R-direction. The output voltage between the top electrode and the bottom electrode is regarded as 2V ,which loads on an resistance Z. Considering that the two ends x₁ = ±L are simply-supported,it is not allowable to occur any in-plane displacements u₁ = ±L.

When the energy-scavenging structure is driven by the flow vortex shedding into flexural vibration near resonance,large deflection induces an incidental in-plane extension together with an axial force N in the x₁-direction^[14]. Then,an additional electric-potential difference V₀ between the outer and the central electrodes is excited by the in-plane extension with appearance of the corresponding electric charge Q₀ at the top/bottom of the tube. Both flexural motion of the CCPPH in the x₂-direction with the deflection u₂(x₁,t) and extension motion in the x₁-direction with the extension u₁(x₁,t) are considered. Thus,the in-plane normal strain for the beam can be written as^{[15, 16]}

According to the electrodes shown in Fig. 1(b),the electric voltage can be written as

where V stands for the electric-potential difference between the outer and the central electrodes generated by the deflection,and V₀ is the electric-potential difference generated by the in-plane extension. The electric field can be written as The constitutive relations of the CCPPH can be written as^[8] where T₁ and D_r stand for the stress and the electric displacement. d₃₁,ε₃₃,and s₁₁ are the piezoelectric coefficient constant,the dielectric constant,and the elastic compliance constant, respectively. In the above,we have denoted that ε₃₃ = ε₃₃(1−k² ₃₁) and k² ₃₁ = d² ₃₁/(ε₃₃s₁₁). An axial force N corresponding to the in-plane extension can be obtained using the expression for T₁ in (4) as Taking an integral at N along the tube length leads to We therefore obtain the average axial force N as The bending moment M can be obtained by using the expression for T₁ in (4) as The dynamic effect in the x₂-direction is much stronger than that in the x₁-direction. Hence,the latter can be reasonably ignored for simplicity,and thus,the motion equation in the x₁-direction reduces to an equilibrium condition. Under this case,the axial force N can be approximately replaced by N. The motion equation of a beam in the x₂-direction can be written as^[16] where m = 2πρRh stands for the mass per unit length of the CCPPH. F(x₁,t) stands for the load per unit length acted at the CCPPH. Substituting (8) into (9),we obtain For a simply-supported beam,we have the boundary conditions for u₂(x₁,t) as

From Ref. ^[14],the flow load on the CCPPH is independent of x₁ and can be written as

where f,ρ_L,U,and C_L0 are the load amplitude,the fluid density,the speed of the flow,and a known constant,respectively. Using the Galerkin method,we approximately write u₂(x₁,t) as follows: which satisfies the boundary condition (11) in advance^[11]. From (13) and (11),we have Taking an integral at (4) on the cross section of the beam yields

Applying the Galerkin technique to (10) yields

Furthermore,we obtain the governing equation for '(t) as follows:

The electric charges at the top and bottom electrodes are

It is well-known that no current presents for the piezoceramic tube subjected to in-plane deformation only because there does not exist any electric difference between the two outer electrodes under that situation. Thus,the output current flowing out from the top to the bottom electrode would be obtained from with the electric charge Q_e decided by We note from (18) and (19) that V₀ has influence on the output current. This is easily understood because the electric-field generated by V₀ still produces driving effect on the boundary electric charge Q_e. In the external circuit,V and I are related through the Ohm law Substituting (18) and (19) into (20),we get Further,we obtain the following equation for the output voltage from (13) and (21): where U₁ and U₂ are undetermined constants. Then,we obtain

We substitute (23) and (24) into (16),collect both the sine and cosine term coefficients, respectively,and neglect the higher-order terms^{[10, 11]}. Then,we obtain

where ω² ₀ = c₁ − c₃k₃,and ω₀ is the resonant frequency of harvester. Equation (26) can be solved numerically. Therefore,we can get U₁ and U₂. ψ(t) and V can be obtained from (23) and (24). The output power P can be given as 3 Numerical results and discussion

In this section,the numerical results of CCPPH are presented. In the calculations,we consider CCPPH of polarized ceramics PZT-5H with ρ = 7 500 kg/m³ and^[17]

The geometric parameters of the tube are d = 2R = 10mm,h = 0.5mm,2L = 200mm,and α = 60◦ unless stated otherwise. In addition,we denote Δω = ω − ω₀,where ω₀ = 2 330 rad/s. For the fluid,we consider air at 20◦C with the density ρ_L = 1.204 7 kg/m³. The air viscosity μ = 1.81 × 10⁻⁵ N · sm⁻²,and other related parameters are ν = μ/ρ_L = 1.502 4 × 10⁻⁵m² · s⁻¹, C_L0 = 0.63,and U = d kω,where k = 1.2^[14]. We choose Z = 1 000K unless stated otherwise.

Figures 2 and 3 show the dependence of P upon Δ! for different tube diameters and different resistances. At the left of resonance,output power is multivalued with jump. Thus,we should design CCPPH structure to operate on the right side adjacent to the resonance to avoid the multivaluedness and jump,otherwise,output power of CCPPH will be either very low or unstable. From Fig. 2,we can also find that the location of jump moves left and the graph of P-Δω becomes vertical gradually with the increase in the tube diameter. This is understandable because the increase in the tube diameter makes the energy-scavenging structure tend to a linear resonance from a nonlinear resonance. Figure 3 shows the influence of impedance on output power with d = 10mm. The magnitude of output power decreases when impedance increases. However,the location of jump has less dependence on impedance. This indicates that the range of nonlinear region is independent of external-circuit.

Figure 4 shows the dependence of output power upon the tube diameter for different electric resistances with Δω = −20 rad/s. After the tube diameter is designed at the right of FGHK, multi-valuedness disappears and the energy-scavenging structure gets a relatively large output power. The same as above,external resistance does not influence the location of FGHK although it produces large effect on the output power magnitude.

To further realize the multivaluedness and jumps of a power harvester near resonance,we calculate the dependence of P upon the tube diameter d for different Δω in Fig. 5. Numerical results show that a larger value of |Δω| corresponds to a larger nonlinear P-d loop. When |Δω| decreases,the P-d loop reduces,too,and gradually shrinks to a point which implies a linear resonance as discussed in Fig. 2.

4 Conclusions

The nonlinear performance of CCPPH near resonance is presented with consideration of in-plane extension induced by large deflection. The boundary electric charges corresponding to two deformation modes,deflection and in-plane extension,are distinguished with each other.Basic nonlinear behaviors of the harvesting structure near resonance are predicted. The multivaluedness and jump phenomena is shown by numerical results,which are useful in CCPPH design. Acknowledgements The authors would like to thank Professor Jia-shi YANG at University of Nebraska for helpful discussion.