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


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.


Tunable Shape memory alloy Energy harvester PZT Cantilever 

List of symbols


Modal amplitude

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

Mass densities

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

Axial stress components


Undamped natural frequency

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



Electric field in 3-direction

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

Electrical charge


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


Resistive load

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

Maximum residual strain


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


Piezoelectric stress constant

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

Strain rate damping coefficient and viscous air damping coefficient

\(d_{s} I\)

Damping due to structural viscoelasticity


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


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


Modal coupling


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


Electric displacement vector


Moment of inertia


Beam length

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

Internal bending moment


Temperature and initial temperature

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

Voltage, current and power FRFs


Beam width

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

Translation and small rotation


Unit outward normal


Mass per unit length


Time constant

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


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

Smooth test function

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

Dirac delta function


Kroneker delta

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

Total, stress induced and temperature induced martensite volume fractions


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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