Electromechanical analytical model of shape memory alloy based tunable cantilevered piezoelectric energy harvester

  • M. Senthilkumar
  • M. G. VasundharaEmail author
  • G. K. Kalavathi


The resonant frequency of electromechanical energy harvester should be tuned to ambient frequency so as to maximize the harvester power. In this paper, it is proposed to tune the natural frequency of the energy harvester by varying the martensite volume fraction (0–1) of shape memory alloy (SMA). It is found that the shifting of natural frequency is around 8–10% for first three modes of vibration. This paper presents the exact analytical solution for composite cantilevered energy harvester composed of piezoelectric energy harvester layer, substructure layer and SMA layer using the assumption of Euler–Bernoulli beam. The beam base displacement in transverse direction with small angular rotation is due to base excitation. The electromechanical coupled equations of response are derived and then reduced to harmonic behavior. The expressions for frequency response of voltage output, current output and power output are obtained. Finally, a parametric case study of composite cantilevered energy harvester with shape memory alloy and important behavior of the electromechanical coupled system for short circuit and open circuit behaviors are studied in detail.


Cantilever beam Energy harvester Piezoelectric Shape memory alloy 

List of symbols


Modal amplitude

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

Mass densities

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

Axial stress components


Undamped natural frequency

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

PZT, substructure and SMA thickness


Electric field in 3-direction


Piezoelectric constant

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

Martensite and austenite elastic modulus


Elastic modulus of substructure


Young’s modulus of austenite and martensite mixture

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

Modal mechanical forcing function and amplitude


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 relative

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

Amplitude translation and angular motion


Effective piezoelectric stress constant


Piezoelectric strain constants

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

Strain rate and viscous air damping coefficient

\(d_{s} I\)

Damping due to structural viscoelasticity


Modulus of piezoelectric at constant electric field

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

\(k{\text{th}}\) mass normalized Eigen function and modal coordinate


Dimensionless frequency

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

Complex terms of steady state response

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

Constant strain and stress permittivity


Martensite volume fraction


Modal coupling


Damping ratio

\(\sigma_{MS} , \sigma_{Mf}\)

Start and final martensite stress

\(p\)’, ‘\(s\)’, ‘\(sm\)’ and ‘\(k\)

PZT, substructure, SMA and number of modes


Vector of electric displacement


Moment of inertia


Length of the beam

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

Internal bending moment

\(V\left( t \right)\)

Voltage cross the resistive load


Width of beam

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

Translation and small rotation


Unit outward normal


Mass per unit length


Time constant



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

Smooth test function

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

Dirac delta function


Frequency of the excitation


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

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

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