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A distributed parametric model of Brinson shape memory alloy based resonant frequency tunable cantilevered PZT energy harvester

  • M. G. VasundharaEmail author
  • M. Senthilkumar
  • G. K. Kalavathi
Article
  • 131 Downloads

Abstract

This paper presents, an analytical model of piezoelectric vibration energy harvester consists of Brinson shape memory alloy (SMA) plate which can tune the resonant frequency. As the energy harvester should be tuned to excitation frequency in order to drive the maximum power, the temperature of SMA is varied to tune the natural frequency of the composite beam. In addition to SMA it also consists of piezoelectric layer and substructure layer. Using Euler–Bernoulli beam assumption, the expressions for frequency response of voltage, current and power outputs with temperature are obtained. From parametric study, it is observed that the tuning of natural frequency is 25–26%, for first three modes of vibration in short and open circuit conditions.

Keywords

Tunable Shape memory alloy Energy harvester PZT Cantilever 

List of symbols

\(A_{k}\)

Modal amplitude

\(\rho_{s}\), \(\rho_{p}\) and \(\rho_{sm}\)

Mass densities

\(\sigma_{1}^{s} , \sigma_{1}^{p}\) and \(\sigma_{1}^{sm}\)

Axial stress components

\(\emptyset_{k}\)

Undamped natural frequency

\(h_{p} , h_{s}\) and \(h_{sm}\)

Thickness

\(B_{3}\)

Electric field in 3-direction

\(q\left( t \right)\)

Electrical charge

\(C_{31}\)

Piezoelectric constant

\(C_{M} ,C_{A}\)

Slopes between temperature and stress

\(E_{M} , E_{A} ,\varPhi\)

Martensite, austenite elastic modulus and thermoelastic modulus

\(E_{s}\), \(E_{sm}\)

Young’s modui

\(f_{k} \left( t \right), M_{k}\)

Modal mechanical forcing function and amplitude

\(M_{s} ,M_{f} ,A_{s} ,A_{f}\)

Martensite start, martensite  finish, austenite start  and austenite finish temperatures

\(R_{l}\)

Resistive load

\(S_{{1_{l} }}^{sm}\)

Maximum residual strain

\(S_{11}^{E}\)

Electric compliance at constant electric field

\(S_{1}^{p} ,S_{1}^{s} ,S_{1}^{sm}\)

Axial strain components

\(U_{b} \left( {x,t} \right)\)

Base motion of the beam

\(U_{rel} \left( {x,t} \right)\)

Transverse displacement of the beam

\(Z_{0} , \theta_{0}\)

Magnitude of translation motion and angular motion

\(d_{31}\)

Piezoelectric stress constant

\(d_{s} ,d_{a}\)

Strain rate damping coefficient and viscous air damping coefficient

\(d_{s} I\)

Damping due to structural viscoelasticity

\(e_{11}^{E}\)

Modulus of piezoelectric at constant electric field

\(w_{k} \left( x \right),y_{k} \left( t \right)\)

kth mass normalized eigen function and modal coordinate

\(\beta_{k}\)

Dimensionless frequency

\(\gamma_{k} , V_{0}\)

Complex terms of steady state responses

\(\varepsilon_{33}^{s} ,\varepsilon_{33}^{T}\)

Constant strain and stress permittivity

\(\zeta_{o} ,\zeta_{so} ,\zeta_{ro}\)

Total, stress induced, temperature induced initial state martensite volume fraction

\(\theta_{k}\)

Modal coupling

\(\xi_{k}\)

Damping ratio

\(\sigma_{S}^{cr} , \sigma_{f}^{cr}\)

Start martensite critical stress and final martensite critical stress

p’, ‘s’, ‘sm’ and ‘k

Denotes PZT, substructure, SMA & mode number

\(E\)

Electric displacement vector

\(I\)

Moment of inertia

\(L\)

Beam length

\(N\left( {x,t} \right)\)

Internal bending moment

\(T,T_{o}\)

Temperature and initial temperature

\(V\left( t \right), i\left( t \right)\) and \(P\left( t \right)\)

Voltage, current and power FRFs

\(a\)

Beam width

\(g\left( t \right),h\left( t \right)\)

Translation and small rotation

\(i\)

Unit outward normal

\(m\)

Mass per unit length

\(t\)

Time constant

\(x\left[ 1 \right]\) and \(y\left[ 3 \right]\)

Directions

\(\gamma \left( x \right)\)

Smooth test function

\(\delta \left( x \right)\)

Dirac delta function

\(\delta_{ks}\)

Kroneker delta

\(\zeta ,\zeta_{s} ,\zeta_{r}\)

Total, stress induced and temperature induced martensite volume fractions

\(\omega\)

Frequency of the excitation

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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • M. G. Vasundhara
    • 1
    • 2
    Email author
  • M. Senthilkumar
    • 1
  • G. K. Kalavathi
    • 3
  1. 1.Department of Production EngineeringPSG College of TechnologyPeelamedu, CoimbatoreIndia
  2. 2.Department of Mechanical EngineeringMalnad College of EngineeringHassanIndia
  3. 3.Department of MathematicsMalnad College of EngineeringHassanIndia

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