Performance enhancement of cantilever piezoelectric energy harvesters by sizing analysis

This paper presents the results of the performance of piezoelectric cantilever beams in relation to their size. The total produced power represents the main indicator of performance of a piezoelectric harvesting system while the area of the beams stays constant. Lightweight design is an important aspect in any industry, mainly in the aerospace. In this study, the effects of non-uniformity on the efficiency and power output are studied. Finite element method (FEM) with the application of superconvergent element (SCE) is adopted here to solve the equations. It is observed that the trapezoidal geometry (converging beam) provides a higher output power while the efficiency decreases. Moreover, in order to prove that the power enhancement is achievable while the amount of piezoelectric material consumed is constant the new configuration is proposed. In the configuration, an array of uniform beams connected in series is used instead of one single rectangular beam. The proposed setting generates an output power of 1.817 mW at a resonant frequency of 284.6 Hz when excited by an input acceleration of 1 g. The only challenge is the fundamental frequency difference which is met with the application of proof mass and thinner substrate and piezoelectric layers. Abbreviations: $$a\left(t \right)$$: Generalized coordinates vector; $${A_p}$$: The electrode area; A: The amplitude vector of generalized coordinates; $$b\left(x \right)$$: Width; $${b_0}$$: Initial width; C: Damping matrix; $${C_p}$$: Capacitance of the one piezoelectric layer; $${\bar C_p}$$: Effective (equivalent) capacitance of the piezoelectric layers; $${d₃₁}$$: Piezoelectric strain coefficient; $${D_3}$$: The electric displacement; $$E$$: Young’s modulus; $${E_3}$$: The electrical field along the thickness direction; F: Dynamic force vector; $$g\left(t \right)$$: The translation part of base motion; $$G$$: Shear modulus; $${h_p}$$: Piezoelectric layers’ thickness; $${h_0}$$: Initial thickness of substrate layer; $$i$$: Imaginary number; $$I\left(t \right)$$: Current output; $${k_s}$$: Shear correction factor; K: Global stiffness matrix; $$L$$: Length of the beam; $${l_e}$$: The length of one element; M: Global mass matrix; $$n$$: Polynomial’s degree; $$ \boldsymbol{N_w}, {N_\phi }$$: Matrices of shape functions (1×4 for one element); $${P_in}$$: The input mechanical power; $${P_out}$$: The output electrical power; $$Q\left(t \right)$$: Electric charge output; R: Displacement vectors of the generic point S (3×1); $${R_l}$$: External load resistance; $${\dot R}$$: Velocity vector of the generic point S (3×1); $$t$$: Time; $$T$$: Kinetic energy; $$u$$: Axial displacement; $$U$$: Potential energy; $$v, w$$: Transversal displacements; $$V$$: Volume; $$v\left(t \right)$$: The generated piezoelectric voltage$${\ }{v_0},{\ }{w_0}$$: Displacements on the middle-plane; $$x,{\ }y,{\ }z$$: Positions of a general point in the relative coordinates system; $$X$$: The amplitude of the effective displacement, $${w_b}$$; $$\varepsilon ₃₃^T$$: The dielectric permittivity of piezoelectric layer at constant stress; $$\varepsilon ₃₃^s$$: The dielectric permittivity of piezoelectric layer at constant strain; $$\boldsymbol {\theta}$$: The electromechanical coupling vector (3×1 for one element); $${\phi _x}$$: The binding rotation of the cross section; $$\gamma $$: Transverse shear strain; $$\varepsilon $$: Axial strain; $$\sigma $$: Axial stress; $$\tau $$: Transverse shear stress$${\ }\psi _j^e,{\ }\varphi _j^e$$: Shape functions of superconvergent element, $$j = 1,{\ } \ldots {\ },4$$; $${\ }W_i^e,{\ }\phi _i^e, W_{\rm{i}}^"e, \break{\ }\phi _{\rm{i}}^"e\left(t \right)$$: Generalized coordinate elements, $$i = 1,{\ }2$$; $$\rho $$: Mass density; $${\rm{\Gamma }}$$: The efficiency of harvesting; $$\omega $$: Excitation frequency; $$\mu $$: Constant of mass proportionality; Constant of stiffness proportionality; $$\zeta $$: Damping ratio; $${\alpha _b}$$: Width taper ratio; $${\alpha _h}$$: Height taper ratio; $$\delta $$: The symbol of virtual work; $${W_IE}$$: The internal electrical energy; $${W_nc}$$: The non-conservative mechanical force; RD: Relative difference; PEHs: Piezoelectric energy harvesters; FEM: Finite element method; FRFs: Frequency response functions; DOF: Degree-of–freedom; MEMS: Micro-electromechanical system; MPGs: Micro-power generators; SCE: Superconvergent element; $${()_p}$$: Piezoelectric layer properties; $${()_s}$$: Substrate layer properties; $$\mathop {()}\limits"$$ : Partial differentiation with respect to x; $$\mathop {()}\limits"$$: Partial differentiation with respect to t.