Finite element beam model for piezoelectric energy harvesting using higher order shear deformation theory

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1 Clemson University TigerPrints All Theses Theses Finite element beam model for piezoelectric energy harvesting using higher order shear deformation theory Akhilesh Surabhi Clemson University, Follow this and additional works at: Part of the Mechanical Engineering Commons Recommended Citation Surabhi, Akhilesh, "Finite element beam model for piezoelectric energy harvesting using higher order shear deformation theory" (2014). All Theses This Thesis is brought to you for free and open access by the Theses at TigerPrints. It has been accepted for inclusion in All Theses by an authorized administrator of TigerPrints. For more information, please contact

2 Finite element beam model for piezoelectric energy harvesting using higher order shear deformation theory A Thesis Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Master of Science Mechanical Engineering by Akhilesh Surabhi August 2014 Accepted by: Dr. Lonny L.Thompson, Committee Chair Dr. Mohammed Daqaq Dr. Gang Li

3 Abstract Piezoelectric energy harvesting devices convert mechanical energy to usable electrical energy, which can be used to power other electronic devices and sensors. Typical piezoelectric harvesters are unimorph cantilever composite beams, which have a single active piezoceramic layer and a passive substrate or a bimorph that has a passive substrate sandwiched between two piezoceramic layers. Power is captured across a coupled load resistor circuit in either a series or parallel connection. The mathematical modeling approaches for piezoelectric beam harvesters present in literature range from analytical distributed parameter modeling, to approximate distributed parameter, Rayleigh- Ritz global discretization or finite element local discretization. For slender electromechanical beam devices, the Classical Beam Theory, which assumes that transverse shear strain is zero, predicts natural frequencies accurately for lower frequencies. First Order Beam Theory accounts for transverse shear deformation in beam bending, but assumes that the shear strain and stress is constant through the thickness and the shear stiffness must be adjusted with a shear correction factor as an approximation. The shear correction factor depends on the lamina material properties and so for composite beams, a model, which does not require the use of shear correction factor, is desirable. In the present work, a beam finite element model based on a high-order parabolic shear deformation theory for multi-layered composite piezoelectric beam ii

4 energy harvesting device is developed. The proposed mathematical model based on the Higher Order Shear Deformation Theory accounts not only for transverse shear strains, but also for a parabolic variation of the transverse shear strains through thickness. This satisfies the zero transverse shear stresses condition on the boundary planes and consequently, there is no need for a shear correction factor. A layerwise theory is used to model the electric potential in the thickness direction, with a fully coupled load resistor circuit in both series and parallel configurations. The beam element uses four mechanical degrees-of-freedom per node, axial displacement, transverse displacement, slope, and independent section rotation angle. Comparisons of the natural frequencies, steady-state power and voltage values from time-harmonic base excitation obtained from piezoelectric bimorph cantilever beams using the Euler- Bernoulli, Timoshenko and the higher order shear deformation theory are presented. Comparisons for the different shear deformation theories are presented for different length-to-depth aspect ratios. The results show increased accuracy for steady-state power solutions using the higher-order beam elements for moderately thick beams at higher frequencies. iii

5 Dedication Dedicated to my family and friends. iv

6 Acknowledgments First and foremost, I would like to acknowledge the immense help provided by my advisor Dr. Lonny Thompson in helping me with my research. I would like to thank Dr. Mohammed Daqaq and Dr. Gang Li for serving on my committee and helping me build a strong foundation in the finite element method and structural dynamics. Finally, a special thanks to all those amazing friends who have always been kind and helpful. v

7 Table of Contents Title Page i Abstract ii Dedication Acknowledgments iv v List of Tables viii List of Figures x 1 Introduction Literature Review: Thesis Organization and Objectives: Classical Beam Theory Introduction: Series and Parallel connections: Internal Moment : Energy stored in the PZT: Kinetic energy: Finite element discretization: Element stiffness matrix: Element coupling matrix, θ e : Element capacitance matrix, c p : Element mass matrix, m e : Potential Constraint: Electromechanical equaion: Resistive load: Base excitation: Steady state solution of the electromechanical equations: Natural Frequencies Case Study: vi

8 2.18 Steady state voltage in series connection: Steady state current in series connection: Steady state power in series: Steady state values for the parallel connection: Timoshenko beam theory Introduction: Finite element discretization: Element stiffness matrix: Element coupling matrix, θ e : Element capacitance matrix, c p : Element mass matrix, m e : Case Study: Steady state values in parallel connection: Higher Order beam theory Introduction: Displacement field equations: Finite element discretization: Element stiffness matrix: Element coupling matrix, θ e : Element capacitance matrix, c p : Element mass matrix, m e : Comparison study: Conclusion: Appendices A Variation of Voltage and power FRF s with depth-length aspect ratio. 89 vii

9 List of Tables 2.1 Material properties of Aluminum Material properties of PZT-5A Geometric properties of the bimorph cantilever beam Comparison of natural frequencies for the first three modes of vibration Peak voltage at resonance frequencies for the first mode of vibration for various load resistance values The peak values of Current Peak power for various resistance values Power at the short circuit resonance frequency for various resistance values Power at the open circuit resonance frequencies for various resistance values Matetial properties of Brass Matetial properties of PZT-5A Geometric properties of the bimorph cantilever beam The short and open circuit natural frequencies for the first three modes of vibration Percentage relative error in the classical and higher order theory fundamental short circuit resonance frequency values for different lengths of the beam Percentage relative error in the first order and higher order theory fundamental short circuit resonance frequency values for different lengths of the beam Percentage relative error between the higher order theoy and the classical, first order theory voltage values for L = 100mm at the first fundamental short circuit resonance frequency (f sc = 63Hz) Percentage relative error between the higher order theory and the classical, first order theory voltage values for L = 39mm at the first fundamental short circuit resonance frequency (f sc = 413.7Hz) Percentage relative error between the higher order theory and the classical, first order theory voltage values for L = 15mm at the first fundamental short circuit resonance frequency (f sc = Hz) viii

10 4.6 Percentage relative error between the higher order theory and the classical, first order theory voltage values for L = 10mm at the first fundamental short circuit resonance frequency (f sc = Hz) Percentage relative error between the higher order theory and the classical, first order theory power values for L = 100mm at the first fundamental short circuit resonance frequency (f sc = 63Hz) Percentage relative error between the higher order theory and the classical, first order theory power values for L = 39mm at the first fundamental short circuit resonance frequency (f sc = 413.7Hz) Percentage relative error between the higher order theory and the classical, first order theory power values for L = 15mm at the first fundamental short circuit resonance frequency (f sc = Hz) Percentage relative error between the higher order theory and the classical, first order theory power values for L = 10mm at the first fundamental short circuit resonance frequency (f sc = Hz) ix

11 List of Figures 2.1 A piezobimorph cantilever beam in series connection A piezobimorph cantilever beam in parallel connection A 2 node Euler-Bernoulli element Variation of voltage with frequency in series connection for a range of load resistance values Variation of voltage with frequency in series connection for the mode Variation of Voltage with resistance for the short and open circuit natural frequencies corresponding to mode Variation of current with frequency for a bimorph in series connection for a range of resistance values Variation of current with frequency for a bimorph in series connection for a range of resistance values.mode 1 is shown Variation of Current with resistance for the short and open circuit natural frequencies corresponding to mode Variation of power with frequency in series connection for a range of load resistance values Variation of power with frequency around the first vibration mode Variation of power with load resistance values Voltage FRF s for the parallel connection case(first mode is shown) Current FRF s for the parallel connection case(the first mode is shown.) Voltage FRF s for the parallel connection case(first mode is shown) Current versus Resistance for the parallel connection case for excitation at the fundamental short circuit frequency A 2 node Timoshenko beam element Voltage FRF s for the parallel connection of piezoceramic layers in a bimorph for various values of load resistance (a) frequency sweep, (b) Mode Current FRF s for the parallel connection of piezoceramic layers in a bimorph for various values of load resistance (a) frequency sweep, (b) Mode Power FRF s for the parallel connection of piezoceramic layers in a bimorph for various values of load resistance (a) frequency sweep, (b) Mode x

12 4.1 A 2 node beam element based on the higher order shear deformation theory Percentage relative error between the Higher order deformation theory and the (a) Classical beam theory,(b) First order deformation theory fundamental short circuit resonance frequency(f sc ) versus the length of the beam Comparison Voltage FRF s around the first vibration mode for bimorph in series connection for (a) L = 100mm,(b) L = 39mm, (c) L = 15mm, (d) L = 10mm Comparison Power FRF s around the first vibration mode for bimorph in series connection for (a) L = 100mm,(b) L = 39mm, (c) L = 15mm, (d) L = 10mm Percentage relative error between the Higher order deformation theory and the (a) Classical beam theory,(b) First order deformation theory voltage values at the fundamental resonance frequency (f sc ) Percentage relative error between the Higher order deformation theory and the (a) Classical beam theory,(b) First order deformation theory power values at the fundamental resonance frequency versus resistance (f sc ) Voltage FRF s around the first vibration mode for bimorph of L = 100mm in series connection for (a) R = 1KΩ,(b) R = 10KΩ, (c) R = 100KΩ, (d) R = 1MΩ Power FRF s around the first vibration mode for bimorph of L = 100mm in series connection for (a) R = 1KΩ,(b) R = 10KΩ, (c) R = 100KΩ, (d) R = 1MΩ Voltage FRF s around the first vibration mode for bimorph of L = 39mm in series connection for (a) R = 1KΩ,(b) R = 10KΩ, (c) R = 100KΩ, (d) R = 1MΩ Power FRF s around the first vibration mode for bimorph of L = 39mm in series connection for (a) R = 1KΩ,(b) R = 10KΩ, (c) R = 100KΩ, (d) R = 1MΩ Voltage FRF s around the first vibration mode for bimorph of L = 15mm in series connection for (a) R = 1KΩ,(b) R = 10KΩ, (c) R = 100KΩ, (d) R = 1MΩ Power FRF s around the first vibration mode for bimorph of L = 15mm in series connection for (a) R = 1KΩ,(b) R = 10KΩ, (c) R = 100KΩ, (d) R = 1MΩ Voltage FRF s around the first vibration mode for bimorph of L = 10mm in series connection for (a) R = 1KΩ,(b) R = 10KΩ, (c) R = 100KΩ, (d) R = 1MΩ xi

13 8 Power FRF s around the first vibration mode for bimorph of L = 10mm in series connection for (a) R = 1KΩ,(b) R = 10KΩ, (c) R = 100KΩ, (d) R = 1MΩ xii

14 Chapter 1 Introduction Piezoelectricity is the property of certain crystals to develop charges on their surfaces when deformed by the application of external loads. It was discovered in 1880 by French physicists Jacques and Pierre Curie with tourmalin crystals. This property was observed even before the curie brothers discovered the piezoelectric effect. In 1823, Henry Becquerel published his experimental work with piezoelectric materials.in 1877, Lord Kelvin established the correlation between pyroelectricity and piezoelectricity. Common materials that possess the piezoelectric effect are quartz, Rochelle salt, topaz and certain ceramics such as Barium Titanate, Lead Zirconate and Lead Titanate. The inverse piezoelectric effect, which is the deformation of the material when placed in an external electric field was predicted by Lippmann [1]. Piezoelectric effect is the result of inherent dipole moment in the material. The electric dipole moment is due to the absence of symmetry in their crystal lattice. Piezoelectric materials are ferroelectric and exhibit hysteresis. When insulators are placed in an external electric field, they get polarized. The dipoles in the material align themselves along the direction of the electric field resulting in a net charge on the surfaces of the material. When the electric field is removed, dipoles arrange themselves such that the 1

15 material is electrically neutral. But, in the case of piezoelectric materials polarization persists even after the external electric field is removed. This process of polarizing a piezoelectric material is known as poling. 1.1 Literature Review: Reddy [18] proposed a higher order deformation theory for anisotropic laminated composite plates under transverse loading using the principle of virtual displacements that accounts for not only transverse shear strains, but also for a parabolic distribution of transverse shear strain. A common plate theory which includes a linear first-order shear deformation contribution attributed to Timoshenko, Reissner and Mindlin improves accuracy for moderately thick plates. The inclusion of shear deformation is considered crucial for layered composite plates, especially when the middle layers are relatively flexible compared to outer layers. Comparisons of static and vibration problems concluded that the classical theory which neglects shear deformation for thin plate bending, under-predicts deflections and over-predicts natural frequencies. The Reddy parabolic shear deformation improves on the classical theory and the first order deformation theory in that it accommodates the exact parabolic shear distribution for single layer plates and satisfied zero shear strain boundary condition on the outer surfaces of the plate, and also eliminates the need for a shear correction factor present in the linear first-order theory, resulting in a more accurate prediction of deflection and stresses. Rastgaar Aagaah [17], used the higher order shear deformation theory to develop a 6-node quadratic triangle finite element to calculate statics deformations of a laminated composite plate due to sinusoidal distributed mechanical loads. Chee et al[5] developed a finite element model based on the higher order shear 2

16 deformation theory for thick composite beams. The displacement field presented had a different form of rotation angle. For thin beams, the rotation angle is negligible and the displacement field reduced to that assumed by the classical theory. A linear Layerwise model [18] was adopted for electric potential to facilitate embedding piezoelectric sensors and actuators anywhere in the structure. They considered the problem of static sensing using piezoelectric materials. The piezoelectric coupled equations of motion were developed using the Hamilton s variational principle and the formulation was robust in the sense that it incorporated flexibility in the model such that the piezoelectric material can function as a sensor or an actuator. Erturk and Tekinalp [12], presented a finite element model for multilayer plates with piezoelectric layers. The model included adhesive bonded layers. The model uses a multi-layered element driver from a stack of 4-node classic plate elements. While this results in improved accuracy over a classical plate element for composite plates representing deformation at a neutral mid surface, a significantly larger number of degrees-of-freedom are introduced, increasing computational expense and memory requirements. Studies on partially delaminated PZT patches were presented and compared with other results present in literature. Bendary et al [4] developed a finite element model for distributed actuators and sensors based on the classical laminate theory. Results showed that as expected for thin plates, the classical theory gives acceptable accuracy with minimal computational effort. Shimpi and Ainapure [19] developed a beam finite element base on the layerwise trigonometric shear deformation theory. The formulation used the virtual work principle for multi-layered plates, but did not include the electrical-mechanical coupling effect in piezoelectric materials. A two node element with three mechanical degrees of freedom per node was considered. The element accounted for a sinusoidal 3

17 distribution of displacement in the plane of the beam, and enforces the condition that transverse shear is zero on exposed surface of outer layers. Similar to the highorder parabolic shear deformation in the Reddy theory, the trigonometric theory also eliminates the need for a shear correction factor. Elshafei et al [2] developed a finite element model for piezoelectric composite beams based on the Reddy higher order shear deformation theory for analyzing isotropic as well as orthotropic composite beams with distributed piezoelectric actuators subjected to both mechanical and electrical loads. This model is valid for continuous as well as segmented piezoelectric elements. A two node element with four mechanical degrees of freedom and one electrical degree of freedom per node was considered. The axial displacement was constructed using hermit cubic polynomials and the electric field was assumed to be a function of the length and the thickness of the beam. Their results were compared with other researchers, such as [12]. Vincent et al [16] presented a finite element formulation for piezoelectrically coupled systems. Piezoelectric finite elements were developed based on Mindlin shell elements and integrated in the FE package Samcef. A shear actuation device was modeled and the interfacing with a control oriented software environment was discussed and applications in noise and vibration control were presented. Wang and Cross [20] derived the constitutive equations of the cantilever mounted triple layer piezoelectric bender under different excitation conditions. The constitutive equations of the triple layer bender, relating displacement, rotation, and charge degrees-of freedom in terms of excitation conditions included an external moment at the free end of the bender, an external force perpendicularly acting at the tip, a uniform body force and an electric voltage applied across the thickness. In particular the degrees-of-freedom induced in the piezoelectric bender are angular tip deflection, tip displacement, volume displacement, and an electrical charge. The coupling ma- 4

18 trix was derived based on the total internal energy under these four standard static excitation conditions. All of the above papers considered piezoelectric composite beam and plate bending devices for actuator and sensing applications, but did not consider energy harvesting where an electric circuit is included to collect power. Erturk and Inman [9] derived a distributed parameter electromechanical model for a unimorph piezoelectric energy harvesting beam bending device, undergoing base excitation, based on the classic Euler-Bernoulli assumption. The analytical expressions obtained from a modal solution were used to perform parametric studies on piezoelectric unimorph cantilever beams in order to observe the frequency response of the voltage, power and tip displacement. Case studies with the piezoceramic layers covered completely and partially with electrodes were also presented. The effect of strain nodes of vibration mode shapes and cancellation of electrical outputs due to using continuous electrodes were also described. Erturk and Inman et al [10] applied an analytical modal series solution for a unimorph cantilever beam under base excitation to bimorph cantilever beams in series and parallel connection of piezoceramic layers. The analytical electromechanical expressions were first obtained for the steady state response to harmonic excitation at arbitrary frequencies. The resulting expressions were then reduced to single-mode expressions by assuming excitation at the fundamental mode. An experimental study was presented for a bimorph cantilever with a tip mass. It was shown that the singlemode frequency response functions (FRFs) obtained from the simple single mode expressions approximates the voltage output and the vibration response FRFs of the bimorph obtained by the full series modal solution. Hou et al [14] proposed a PZT-based smart aggregate for compressive seismic stress monitoring. The smart aggregate consists of a piece of PZT patch sandwiched 5

19 between a pair of marble cubes using epoxy. Three-dimensional finite element analysis using ABAQUS was conducted to investigate the stress distribution in the smart aggregate under compression. The smart aggregate had capability of monitoring the seismic stresses of low- and middle-rise buildings subject to moderate earthquakes. Zhu et al [21] presented a coupled piezoelectric-circuit three-dimensional finite element model using ANSYS to analyze the power output of a vibration-based piezoelectric energy-harvesting device when it is connected to a load resistor. It was found that the electrical and mechanical outputs of the devices have a significant dependence on the value of the load resistor, rather than being independent. They noted that the maximum power output of a piezoelectric EHD does not appear at the maximum vibrational displacement, because the power is determined by the product of current and voltage. Both short-circuit and open-circuit limits were also discussed and shown that approximate solutions for the coupled load resister circuit based on these limits are not accurate when the electromechanical coupling is large. Carlos De Marqui Junior [7], presented an electromechanically coupled finite element plate for predicting the electrical power output of piezoelectric energy harvester unimorph and bimorph plate devices. The finite element model was derived based on the classic Kirchhoff plate theory, which neglects shear deformation. Finally optimization studies were carried out using this model on a UAV wing spar with embedded piezoelectrics. Dietl et al [8] derived the piezoelectric coupled equations using the Timoshenko beam theory which includes first-order shear deformation and presented a modal series solution. It was concluded that that the predicted responses converge towards classical Euler-Bernoulli beam models under the limiting condition of small depth to length ratio. It was found that the Euler-Bernoulli model severely over-predicts the tip displacement and consequently the power transduction of a cantilevered piezoelectric 6

20 bimorph at high depth-to-length aspect ratios. Renno et al [11] used an L-shaped piezoelectric beam-mass structure for harvesting energy. The structure can be tuned to have the first two natural frequencies relatively close to each other, resulting in the possibility of a broader band energy harvesting system. The L- shaped beam was used as landing gears in unmanned air vehicle applications and the electrical power generation was investigated and found favorable against the power from a curved beam configuration used for the same purpose. 1.2 Thesis Organization and Objectives: The primary objective is to develop a beam element based on the Reddy highorder parabolic shear deformation theory for multi-layered composite piezoelectric beam energy harvesting devices. The developed beam uses four mechanical degreesof-freedom per node, axial displacement, transverse displacement, slope, and independent section rotation angle. Comparisons of the natural frequencies, steady-state power and voltage values from time-harmonic base excitation obtained from piezoelectric cantilever beams using the Euler-Bernoulli, Timoshenko and the higher order shear deformation theory are presented. The piezoelectric composite beam element formulation includes electro-mechanical material coupling with an fully coupled load resistance circuit for voltage and power output. Both series and parallel load resistance circuit connections are considered. Voltage and power across the load in the circuit as a function of base excitation frequency including maximum values at the fundamental resonance frequencies are compared. Parameter study of varying load resistance between short and open circuit limits are given. Comparisons for the different shear deformation theories are presented for different length-to-depth aspect 7

21 ratios. The solutions presented are based on the finite element beam models. Finite element analysis (FEA) is an alternative to solving exact analytical closed-form solutions [15]. For simple structures, the closed form analytical solutions are relatively simple to obtain, but as the structural complexity increases, the application of boundary conditions in the conventional way is difficult to implement. Therefore, for complex structures, a global series basis approximation or local basis finite element approximation based on variational or energy based formulation is an effective method because the mechanical equilibrium equations do not need to be solved explicitly. Using this approach, the important physics of the structure is accounted for in the formulation without a closed-form solution. The finite element method has the advantage of easily handle combinations of unimorph and bimorph piezoelectric devices in multi-dimensional frame structures coupling axial and bending and shear effects, imbedded actuators and harvesters, and other multi-material, and geometric configurations. The process of obtaining closed form analytical equations becomes more complicated when included transverse shear deformation. Although, analytical solutions for the electromechanical coupling problems using the first order shear deformation theory for cantilever beams are available, obtaining analytical solutions for highorder shear deformation is very difficult the additional coupling terms in the electromechanical equations of motion. Chapter 2 reviews the Euler-Bernoulli beam theory for piezoelectric electrical coupling beams and finite elements for steady state response of a bimorph power harvesting device with a fully coupled load resistance circuit; both series and parallel connections are considered. The excitation is due to its base motion in the form of translation. Chapter 3 presents the steady state formulation using Timoshenko 8

22 beam theory and develops a fully coupled beam finite element for energy harvesting. The effect of shear modulus is considered and the transverse shear stress is found by applying a shear correction factor. Chapter 4 presents a steady state formulation for the higher order beam elements based the Reddy parabolic transverse shear deformation theory. The layerwise theory is used to model the electric potential in the thickness direction, with a fully coupled load resistor circuit in both series and parallel configurations. Comparisons of voltage and power obtained for a bimorph cantilever energy harvesting device using the finite elements based on the different beam theories are then presented together with parameter studies on varying load resistance and length-to-thickness aspect ratios. 9

23 Chapter 2 Classical Beam Theory The aim of this chapter is to find the steady state response of a piezoelectric symmetric cantilever bimorph subjected to base excitation, using the finite element method.this process involves modeling the cantilever beam using beam elements based on the classical beam theory.the extended hamilton s principle is then used to find the governing coupled electromechanical equations of motion of the cantilever beam.a MATLAB code is developed to compute the static deformation and free vibration parameters of the beams with distributed piezoelectric actuators. The obtained results from the proposed model are compared with the available analytical results and the finite element results of other researchers. 2.1 Introduction: The Classical Beam Theory(CBT), popularly referred to as the Euler- Bernoulli beam theory,describes the kinematics of thin beams (beams for which the ratio of length to the in-plane thickness L/b > 10). This theory assumes that the area of cross section is rigid in its own plane. So, no deformation of the cross section oc- 10

24 curs in its plane, and the cross section remains normal to the deformed axis of the beam.the axial displacement can be written in terms of rigid body translation and rotation. u(x, z, t) = u 0 (x, t) zθ(x, t) (2.1.1) v(x, t) = 0, (2.1.2) w(x, z, t) = w 0 (x, t) (2.1.3) Where u 0 (x, t), v(x, t) and w 0 (x, t) are the displacements of the point of interest along the x, y and z directions respectively and θ(x, t) is the rotation of the cross sectional area about the y axis. The strains developed in the beam can be found using the kinematic quantities defined in equations (2.2.1), (2.2.2) and (2.2.3). ɛ xx = u x = u 0 x z θ x (2.1.4) ɛ zz = w z = 0 (2.1.5) γ xz = u z + w x = θ + w x = 0 (2.1.6) Where ɛ xx,ɛ zz are the normal strains in the x and y directions respectively and γ xz is the shear strain in the xz plane. From equation (2.2.6), it is clear that the shear strain at any point in the beam 11

25 is zero and therefore an important relation can be derived. w x = θ (2.1.7) For convenience, the following notations are used, ɛ 0 = u 0 x, κ0 = 2 w x 2 (2.1.8) Substituting equations (2.2.8) and (2.2.7) in (2.2.4), the normal strain in the x direction can finally be written as ɛ xx = ɛ 0 z 2 w x 2 = ɛ0 zκ 0 (2.1.9) [ ] ɛ 0 = 1 z (2.1.10) It should be noted that the above equations for strains are used to calculate the internal moments and forces developed in the beams and also to choose the right interpolation functions for finite element formulation. κ Series and Parallel connections: A piezoelectric bimorph in series and parallel is shown in figures 1 and 2 respectively. Electrodes are placed on top and bottom of the piezostrips and they are assumed to be perfectly conductive so that a uniform electric field exists across them.a resistor R is connected across the bimorph. In the series connection,the two piezoelectric layers have opposite polarization directions and an electric field is 12

26 Figure 2.1: A piezobimorph cantilever beam in series connection applied across the total thickness of the bimorph. While in parallel connection, the two piezoelectric layers have the same polarization directions, and the electric field is applied across each individual layer with opposite polarity Piezoelectric constitutive equations for the series connection: For the bimorph in series connection, the constitutive equations for the top layer are σ p 11 = c 11 ɛ p 11 + e 31 E 3 (2.2.1) D p 3 = e 31 ɛ p 11 + ε s 33E 3 (2.2.2) and for the bottom layer are, σ p 11 = c 11 ɛ p 11 e 31 E 3 (2.2.3) 13

27 D p 3 = e 31 ɛ p 11 + ε s 33E 3 (2.2.4) Piezoelectric constitutive equations for the parallel connection: Figure 2.2: A piezobimorph cantilever beam in parallel connection For the parallel connection, since the polarization direction is parallel to the electric field in both the top and bottom layers, the constitutive equations are the same and are written as σ p 11 = c 11 ɛ p 11 + e 31 E 3 (2.2.5) D p 3 = e 31 ɛ p 11 + ε s 33E 3 (2.2.6) The substrate shown above is usually made of an isotropic material. The stress, strain relationship for it can be expressed as σ s 11 = Eɛ p 11 (2.2.7) 14

28 2.3 Internal Moment : The internal moment is the first moment of bending stress over the cross section and is expressed as M(x, t) = zσ p da + zσ p da + zσ s da (2.3.1) A p1 A p2 A s Substituting (2.3.1) in the above equation and assuming that ɛ 0 is zero for the symmetric bimorph = Q 11 (I 1 ɛ 0 I 2 κ 0 ) + I 1 e 31 E 3 (2.3.2) 2.4 Energy stored in the PZT: The energy stored in the PZT is due to internal stresses and electrostatic field developed as a result of external loads. The total internal energy U is thus the combination of elastic and electro static energy Elastic energy of deformation: The energy stored in a body as it is being deformed under the action of an external load is the elastic energy of deformation and is given by [6] W = v 1 σεdv (2.4.1) 2 Where σ, and ε are the stress and strains developed at a location in the body. 15

29 2.4.2 Electrostatic energy density: The energy stored in electrostatic field is equal to the work done by an external force in moving a charge from a cathode to an anode. If dw e is the differential work done by an external force in moving a charge dq across a differential potential difference V, then the work done is given by [13] dw e = V dq (2.4.2) The charge deposited on the capacitor plates, Q is related to the voltage across them is given by [13] Q = CV (2.4.3) Here C is the capacitance of the capacitor. Therefore, the work done W e is W e = 1 2 CV 2 = 1 QV (2.4.4) 2 If ρ is the electric charge density on the capacitor plates, eq can be rewritten as W e = 1 V ρdω (2.4.5) 2 The relation between the charge density ρ and the electric displacement vector D can shown using maxwell s equation v ρ = D (2.4.6) 16

30 Therefore, the external work done is W e = 1 V (.D)dΩ (2.4.7) 2 = 1 D).dA 2 A(V 1 2 v v D. V dω (2.4.8) = 1 D. V dω (2.4.9) 2 v W e = 1 D.EdΩ (2.4.10) 2 v U = U strain + U electrostatic (2.4.11) = 1 σ 11 ɛ 11 dv 1 2 vp 2 vp E 3 D 3 dv (2.4.12) = 1 2 c 11 ɛ 2 11dv + e 31 E 3 ɛ 11 dv 1 2 ε s 33E 2 3 (2.4.13) Variation of internal energy: The variation of the internal energy U is δu = δɛ 11 c 11 ɛ 11 dv + δɛ 11 e 31 E 3 dv + δe 3 e 31 ɛ 11 dv δe 3 ε s 33E 3 dv (2.4.14) 17

31 2.5 Kinetic energy: The total kinetic energy of the beam is given by T = 1 2 ρ( u 2 + ẇ 2 )dv (2.5.1) The variation of kinetic energy is δt = ρ(δuü + δwẅ)dv (2.5.2) In the absence of mechanical dissipative effects, the extended Hamiltons principle with internal electrical energy is t2 t1 (δt δu + δw nc )dt = 0 (2.5.3) Here, δw nc represents the the first variation of work done by external loads and charges. In the case of beam subjected to base excitation, the only non conservative force acting on the system is due to electrical charge Q. The first variation of work due to these charges is give as δw nc = Qδv (2.5.4) 2.6 Finite element discretization: The element shown in figure has two nodes labeled 1 and 2. Node 1 is positioned at x = x 1 and node 2 at x = x 2. Three mechanical degrees of freedom(dof)(u, w, θ), which are the axial nodal, transverse nodal displacements and the rotation angle respectively are defined at each node. 18

32 Figure 2.3: A 2 node Euler-Bernoulli element. The axial displacement along the element u 0 (x) is assumed to be varying in a linear fashion, u 0 (x) = a 0 + a 1 x. The constants a 0 and a 1 are expressed in terms of u 1 and u 2 by requiring that u 0 (x 1 ) = u 1 and u 0 (x 2 ) = u 2. The transverse displacement field, w(x), is expressed using a cubic polynomial. Therefore w(x) = c 0 + c 1 x + c 2 x 2 + c 3 x 3. Here, the constants c 0 through c 3 are determined using the condition that w(x 1, x 2 ) = (w 1, w 2 ) and θ = w x (x 1, x 2 ) = (θ 1, θ 2 ). The axial and transverse displacements can finally be expressed in terms of shape functions as u 0 (x) = N 1 u 1 + N 2 u 2 (2.6.1) and w(x) = N 3 w 1 + N 4 θ 1 + N 5 w 2 + N 6 θ 2 (2.6.2) For convenience, the strain on the neutral axis ɛ 0 and the curvature κ 0 is shown 19

33 in matrix form ɛ0 κ 0 = N 1(x) 0 0 N 2(x) N 3 (x) N 4 (x) 0 N 5 (x) N 6 (x) u 1 w 1 θ 1 u 2 w 2 θ 2 (2.6.3) = B e (x)d e (2.6.4) The axial,transverse displacements and the rotation angles u 0 (x), w(x), θ(x) can be written in similar fashion u 0 N 1 (x) 0 0 N 2 (x) 0 0 w = 0 N 3 (x) N 4 (x) 0 N 5 (x) N 6 (x) w 0 N x 3(x) N 4(x) 0 N 5(x) N 6(x) u 1 w 1 θ 1 u 2 w 2 θ 2 (2.6.5) = N e (x)d e (2.6.6) For finite element calculations, the physical element shown in Fig. is mapped to a master element with limits ξ = [ 1, 1]. The relation between x and ξ is given by 20

34 x = x 1 + ( 1+ξ 2 )L. Here L = x 2 x 1 is the length of the element. The axial and transverse displacements can now be expressed in terms of the master element shape functions as u 0 (ξ) = N 1 (ξ)u 1 + N 2 (ξ)u 2 (2.6.7) and w(ξ) = N 3 (ξ)w 1 + N 4 (ξ)θ 1 + N 5 (ξ)w 2 + N 6 (ξ)θ 2 (2.6.8) Where, N 1 = 1 ξ 2 (2.6.9) N 2 = 1 + ξ 2 (2.6.10) N 3 = 1 4 (2 + ξ)(1 ξ)2 (2.6.11) N 4 = L 8 (1 + ξ)(1 ξ)2 (2.6.12) N 5 = 1 4 (1 + ξ)2 (2 ξ) (2.6.13) N 6 = L 8 (1 + ξ)2 (1 ξ) (2.6.14) 21

35 2.7 Element stiffness matrix: The elemental strain is ] ɛ 11 [1 ɛ 0 z (2.7.1) κ 0 [ ] = 1 z B e (x)d e (2.7.2) is The first variation of strain energy in an individual element due to strain ɛ 11 v δɛ 11 Q 11 ɛ 11 dv = = x2 v δd T e B ex) ( 1 [ ] c z B e (x)d e dv (2.7.3) z x 1 I 1 I 2 δd T e B e (x) T c 11 I 0 I 1 B e (x)d e dx (2.7.4) Where, I 0 is the area of cross section, I 1 and I 2 are the first and second moments of cross sectional area about the neutral axis. (I 0, I 1, I 2 ) = (z 0, z 1, z 2 )da (2.7.5) To be able to use gaussian quadrature, the coordinates in the physical element have to be mapped to a master element. The master element is chosen to have limits between -1 and 1. The transformation used here is x x 1 = ( 1+ξ )L, which implies 2 22

36 dx = L dξ. The strain energy is, therefore 2 v δɛ 11 Q 11 ɛ 11 dv = 1 1 δd T e B e (ξ) T c 11 I 0 I 1 L B e (ξ)d e dξ (2.7.6) I 1 I 2 2 = δd T e k e d e (2.7.7) where, k e = 1 Be T (ξ)cb e (ξ)( L n g 2 )dξ = w i Be T (ξ i )CB e (ξ i )( L 2 ) (2.7.8) 1 i=1 is the element stiffness matrix. 2.8 Element coupling matrix, θ e : fields is The integral which contributes to the coupling between strain and electrostatic v δɛ 11 e 31 E 3 dv = v δd T e Be T e 31 1 E 3 dv (2.8.1) z In general, the electric filed E is the negative gradient of electric potential, φ, E = φ. In this case, the electric field exists only in the z direction. Therefore, E 3 = ve t p. Here, v e is the potential difference between the two electrodes of a 23

37 piezoelectric beam element and t p, is its thickness in the z direction. v x2 δɛ 11 e 31 E 3 dv = δd T e Be T e 31 x 1 1 = 1 δd T e Be T (ξ)e 31 I 0 I 1 I 0 I 1 Lv e v e t p dx (2.8.2) 2t p dξ (2.8.3) = δd T e θ e v e (2.8.4) Where θ e is the electromechanical coupling matrix. It determines the strength of coupling between the strain and the electrostatic fields. A coupling matrix consisting of all zeros only means that no coupling can exist. Similarly, δe 3 e 31 ɛ 11 = δve T θe T d e (2.8.5) v 2.9 Element capacitance matrix, c p : The first variation of the electrostatic energy is δe 3 ε s 33E 3 dv = v v v e δv e ε s 33 dv (2.9.1) t p t p = δv e ε s 33A p t p v e (2.9.2) = δv e c p v e (2.9.3) 24

38 Where c p is the capacitance associated with an element, and A p is the cross sectional area of the piezoceramic element Element mass matrix, m e : form as For convenience, the axial and transverse displacements are arranged in matrix u = 1 0 z u 0 w w w x (2.10.1) The first variation of kinetic energy is [ δt = ρ δu v ] ü δw dv (2.10.2) ẅ [ ] 1 0 z ü = ρ δu 0 δw δw x ẅ dv (2.10.3) v z 0 z 2 x2 [ ] I 0 0 I 1 ü = ρ δu 0 δw δw x 0 I 0 0 ẅ dx (2.10.4) x 1 I 1 0 I 2 ẅ x ẅ x x2 = ρδd T e N e (x) T DN e (x) d e dx (2.10.5) x 1 = δd T e m e de (2.10.6) 25

39 The variation equation for an element is δd T e m e de + δd T e k e d e δd T e θ e v e δv T e θ T e d e δv T e c p v e = δd T e f δv T e q (2.10.7) δd T M d + δd T Kd δd T Θv δv T Θ T d δv T C p v = δd T F δv T Q (2.10.8) M is the global matrix (3n n), K is the global stiffness matrix (3n n), θ is the global coupling matrix (3n n),c p is the global diagonal capacitance matrix [ ] T (n n). d = u 1 w 1 θ 1... u n+1 w n+1 θ n+1 is the global mechanical DOF [ ] T matrix (n 1), v = v 1 v 2... v n is the global electrical DOF matrix (n 1) [ ] T and Q = q 1 q 2... q n is the matrix containing charge outputs from individual elements Potential Constraint: By defining a potential degree of freedom for each element, it is assumed that there are as many electrodes as the number of elements and that these elements are insulated from each other. So, the potential values are not the same. In reality, a continuous electrode is placed on top of the piezoceramic layer and a single potential difference v 0 exists across the thickness of the piezoceramic layer. The potential is constrained by as [ ] v = v 0 (2.11.1) 1 n 26

40 As a consequence of constraining potential, the dimensions of a few other matrices also change. [ T Θv = Θ ] v 0 (2.11.2) = Θv 0 (2.11.3) Here Θ is the coupling matrix (n 1) for the piezoceramic layer. δv T C p v = δv 0 diag(cp )v 0 (2.11.4) = δv T C p v = δv 0 C p v 0 (2.11.5) Here C p is the capacitance of the piezoceramic layer. and ] δv T Q = δv 0 [ Q (2.11.6) = δv 0 Q (2.11.7) The global mechanical damping matrix is assumed to be proportional to the global mass and stiffness matrices C = αm + βk (2.11.8) 27

41 2.12 Electromechanical equaion: Applying the boundary conditions and the potential constraint, the electromechanical equations of motion can be expressed as M d + Cḋ + Kd Θv 0 = 0 (2.12.1) Θ T d + C p v 0 + Q = 0 (2.12.2) 2.13 Resistive load: A resistive load R, added across the piezoceramic layer results in a current I = dq, in the circuit. Differentiating equation , dt Θ T ḋ + C p v 0 + Q = 0 (2.13.1) The current in the circuit can be expressed using Ohm s law as I = v 0 R Θ T ḋ + C p v 0 + v 0 R = 0 (2.13.2) 2.14 Base excitation: The axial and transverse displacements of any point in the beam can be represented as w(x, t) = w b (x, t) + w r (x, t), u = u(x, t) (2.14.1) 28

42 Here, w b is the displacement of the base with respect to a coordinate system fixed to ground and w r is the transverse displacement with respect to a material coordinate system. The first variation of w(x, t) and u(x, t) can therefore be written as δw(x, t) = δw r (x, t), δu = δu(x, t) (2.14.2) Recall, the first variation of kinetic energy for layer i was written as δt i = (ρ i δuü + ρ i δwẅ)dv (2.14.3) vi [ ] = ρ i δu δw r ü dv vi The variation of kinetic energy for the composite is ẅ r vi ρ i δw r ẅ b dv (2.14.4) δt = δd T M d + δd T F (2.14.5) Where F is the force due to base excitation Steady state solution of the electromechanical equations: Reviewing the electromechanical equations of motion M d + Cḋ + Kd Θv 0 = F (2.15.1) 29

43 Θ T ḋ + C p v 0 + v 0 R = 0 (2.15.2) Assuming base excitation of the form w b = w b0 e jωt, the base acceleration is a b = a b0 e jωt. Here, a b0 = ω 2 w b0 The force due to base excitation is therefore F = ρ i δw r dv(a b0 e jωt ) (2.15.3) vi = a b0 m e jωt (2.15.4) Assuming harmonic response for,d = d 0 e jωt and v 0 = ve jωt Substituting d and v 0 in eq.(2.15.1) ω 2 Md 0 e jωt + jωcd 0 e jωt + Kd 0 e jωt Θve jωt = F 0 e jωt (2.15.5) [K Mω 2 + jωc]d 0 Θv = F 0 (2.15.6) Substituting d and v 0 in eq.(2.15.2), we get ( 1 R + jωc p)v + jω Θd 0 = 0 (2.15.7) The voltage due to base excitation is therefore v = jω Θ T d 0 ( 1 R + jωc p) (2.15.8) 30

44 and the steady state voltage is v ss = ω Θ T d 0 ( 1 R + jωc p) (2.15.9) Substituting eq.(2.15.8) in eq.(2.15.6), we get the value of displacement [3] by solving [K Mω 2 + jωc + (jω Θ Θ T )( 1 R + jωc p) 1 ]d 0 = F 0 ( ) 2.16 Natural Frequencies Natural frequencies are calculated by solving the free vibration problem. In the case of piezoelectric energy harvesting, two kinds of natural frequencies are defined. The short circuit natural frequency (f sc ) is calculated by assuming that no potential difference exists across the beam in free vibration. As a result, the coupling in the beam is zero at all times. The short circuit natural frequencies can then be calculated using the equation M d + Kd = 0 (2.16.1) The open circuit natural frequencies (f oc ) are calculated by assuming that the resistive load connected in the circuit is infinity (R ). No charge flows in the circuit. The open circuit natural frequencies can be calculated using the equation M d + K d = 0 (2.16.2) 31

45 where K T Θ Θ = K + C p 2.17 Case Study: This section presents a case study using the Euler Bernoulli finite element model. The results obtained are verified against the ones given by Erturk and Inman [15].The model considered here is a symmetric piezoelectric bimorph cantilever beam. Its geometric properties are given in Table 2.3, the material properties of the substrate and PZT are given in Tables 2.1 and 2.2 respectively. Perfectly conductive electrodes are placed on top and bottom of the piezoelectric strips so that a single potential difference is defined across them. The results for the series and parallel connection are presented here. Material properties: Aluminum: Table 2.1: Material properties of Aluminum Mod.of elasticity E 70 GPa Shear modulus G GPa Poisson ratio 0.33 Density 2700 Kg/m 3 PZT 5A : The short circuit and the open circuit natural frequency values for the first three modes of vibration are compared against the values obtained analytically. These values are listed in table

46 Table 2.2: Material properties of PZT-5A C GPa C GPa ε s F/m e C/m 2 Density 7750 Kg/m 3 Table 2.3: Geometric properties of the bimorph cantilever beam piezo substructure length(mm) width(mm) 5 5 Thickness(mm),t p Table 2.4: Comparison of natural frequencies for the first three modes of vibration Mode f sc (Hz)(EB) f sc (Hz)(Erturk,Inman,[15]) f oc (Hz)(EB) f oc (Hz)(Erturk,Inman,[15]) Steady state voltage in series connection: The steady state voltage Frequency response function, for the series connection is shown in Figure 2.4. Only the first three vibration modes are shown. The voltage value peaks at frequencies that are the resonant frequencies for the first three modes of vibration respectively. For higher resonant frequencies, the peak value of voltage is lower. This is because of the appearance of strain nodes that result in charge cancellations. The short circuit resonant frequency of the beam is the resonant frequency when the load resistance, R, is zero and the open circuit resonant frequency is the 33