Investigation of the Size Effect on the Nano-beam Type Piezoelectric Low Power Energy Harvesting


1 Department of Textile Engineering, Isfahan University of Technology, Isfahan, Iran

2 Department of Mechanical Engineering, Shahrekord University, Shahrekord, Iran


In this paper, size dependent beam type peizoelectric energy hardvester is investigated. For this goal, first nonlinear formulation of isotropic piezoelectric Euler-Bernoulli nano-beam is developed based on the size-dependent piezoelectricity theory then special beam type piezoelectric energy hardvester is probed for different parameters. Basic nonlinear equations of piezoelectric nano-beam are derived using principle of minimum of potential energy and variational method. To evaluate the formulation derived, static deformation and free vibration of the clamped-clamped piezoelectric nano-beam is investigated in the special case. The results of the formulation derived are investigated under different parameters, and particularly, the ability and performance of the beam type piezoelectric low power energy harvesting was evaluated in nanoscale.


1.     Qin, Y., Wang, X., Wang, Z.L., “Microfibre–nanowire hybrid structure for energy scavenging”,  Nature, Vol. 451, No.7180, (2008), 809–813.
2.     Erturk, A., Inman, D.J., “An experimentally validated bimorph cantilever model for piezoelectric energy harvesting from base excitations”, Smart  Materials and  Structures., Vol. 18, No. 2, (2009), 025009.
3.     DuToit, N.E., Wardle, B.L., “Experimental verification of models for microfabricated piezoelectric vibration energy harvesters”,  AIAA Journal, Vol. 45, No. 5, (2007) , 1126–1137.
4.     Shu, Y.C., Lien, I.C., “Analysis of power output for piezoelectric energy harvesting systems”, Smart  Materials and  Structures., Vol. 15, (2006) ,1499-1512.
5.     Karami, M.A., Inman, D.J., “Equivalent damping and frequency change for linear and nonlinear hybrid vibrational energy harvesting systems”, Journal of Sound and Vibration, Vol. 330, No. 23, (2011), 5583-5597.
6.     Mohammadpour, M., Dardel, M., Ghasemi, M. H., Pashaei, M. H., ”Nonlinear energy harvesting through a multimodal electro-mechanical system”, Journal of Theoretical and Applied Vibration and Acoustics, Vol. 1, No. 2, (2015), 73-84.
7.     Tadi Beni Y., Jafari A., Razavi H. “Size Effect on Free Transverse Vibration of Cracked Nano-beams using Couple Stress Theory”, International Journal of Engineerig- Transactions B: Applications, Vol. 28, No. 2, (2015), 296-304.
8.     Fattahian Dehkordi S., Tadi Beni Y.,”Electro-mechanical free vibration of single-walled piezoelectric/flexoelectric nano cones using consistent couple stress theory”, International Journal of Mechanical Sciences, 128–129 (2017), 125–139.
9.     Simsek, M., ”Dynamic analysis of an embedded microbeam carrying a moving microparticle based on the modified couple stress theory”, International Journal of Engineering Science, Vol. 48, (2010), 1721-1732.
10.   Bakhshi Khaniki, H., Hosseini Hashemi, S., “Free Vibration Analysis of Nonuniform Microbeams Based on Modified Couple Stress Theory: an Analytical Solution”, International Journal of Engineering-Transactions B: Applications, Vol. 30, No. 2, (2017), 311-320.
11.   Akbari, M.R., Nimafar, M., Ganji, D.D., Karimi Chalmiani, H., “Investigation on non-linear vibration in arched beam for bridges construction via AGM method” ,Applied Mathematics and Computation, Vol. 298, (2017), 95-110 ,
12.   Hatami, M., Vahdani, S., Ganji, D.D., ”Deflection prediction of a cantilever beam subjected to static co-planar loading by analytical methods”, HBRC Journal, Vol. 10, No. 2, (2014), 191-197.

13.   Gürses, M., Civalek,  O., Korkmaz, A., Ersoy, H., “Free vibration analysis of symmetric laminated skew plates by discrete singular convolution technique based on first‐order shear deformation theory”, International Journal for Numerical Methods in Engineering, Vol. 79, No.3,( 2009), 290-313.

14.   Demir, C, Mercan, K, Civalek, O. “Determination of critical buckling loads of isotropic, FGM and laminated truncated conical panel”, Composites Part B, Vol. 94, (2016), 1-10.

15.   Mercan, K, Civalek, O., “DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix”, Composite Structures, Vol. 143, (2016), 300–309.

16.   Akgoz, B., Civalek, O., “free vibration analysis of axially functionally graded tapered bernoulli-Euler microbeams based on the modified couple stress theory”, Composite Structures, Vol. 98, (2013), 314-322.

17.   Mehralian, F., Tadi Beni, Y., Ansari, R., “Size dependent buckling analysis of functionally graded piezoelectric cylindrical nanoshell”, Composite Structures, Vol. 152, ( 2016), 45–61.

18.   Baltacıoglu, A., K., Akgoz, B., Civalek, O., “Nonlinear static response of laminated composite plates by discrete singular convolution method”, Composite Structures ,Vol. 93, (2010) ,153–161.

19.   Beni, Y. T., Karimipour, I.,  Abadyan, M. “Modeling the effect of intermolecular force on the size-dependent pull-in behavior of beam-type NEMS using modified couple stress theory”, Journal of Mechanical Science and Technology, Vol. 28, No. 9, ( 2014) , 3749-3757.
20.   Tadi Beni,Y.,” Size-dependent analysis of piezoelectric nanobeams including electro-mechanical coupling”, Mechanics Research Communications, Vol. 75, (2016), 67–80.

21.   Tadi Beni Y., “Size-dependent electromechanical bending, buckling, and free vibration analysis of functionally graded piezoelectric nanobeams”, Journal of Intelligent Material Systems and Structures, , Vol. 27, No. 16,( 2016), 2199–2215.

22.   Tadi Beni, Y., “A nonlinear electro-mechanical analysis of nanobeams based on the size-dependent piezoelectricity theory”, Journal of Mechanics, Vol. 65, ( 2016), 1-13.
23.   Shah-Mohammadi-Azar, A., Khanchehgardan,  A., Rezazadeh, G., Shabani R., “Mechanical response of a piezoelectrically sandwiched nano-beam based on the nonlocal theory”, International Journal of Engineering-Transactions C: Aspects, Vol. 26, No. 12, (2013), 1515-1524.
24.   Wang, K.F., Wang, B.L., “an analytical model for nanoscale unimorph piezoelectric energy harvesters with flexoelectric effect”, Composite Structures, Vol. 153,(2016), 253-261.
25.   Liang, Xu., Hu, S., Shen, S.,”size dependent buckling and vibration behaviors of piezoelectric nanostructures due to flexoelectricity”, Smart  Materials and  Structures. , Vol. 24, (2015), 105012.
26.   Shams Nateri, M., Azizollah Ganji, B.,” The Effect of Material Properties on Sensitivity of the Microelectromechanical Systems Piezoelectric Hydrophone”, International Journal of Engineering-Transactions C: Aspects, Vol. 30, No. 12, (2017) 1848-1855.

27.   Mehralian, F., Tadi Beni, Y., Ansari, R., “On the size dependent buckling of anisotropic piezoelectric cylindrical shells under combined axial compression and lateral pressure”, International Journal of Mechanical Sciences, Vol. 119, (2016), 155–169.

28.   Ebrahimi, N., Tadi Beni, Y., “Electro-mechanical vibration of nanoshells using consistent size-dependent piezoelectric theory”, Steel and Composite Structures, Vol. 22, No. 6, (2016), 1301-1336.
29.   Kheibari F., Tadi Beni Y., “Size dependent electro-mechanical vibration of single-walled piezoelectric nanotubes using thin shell model”, Materials & Design,Vol. 114, (2017), 572–583.
30.   Alibeigi B., Beni Y.T., Mehralian F.,”On the thermal buckling of magneto-electro-elastic piezoelectric nanobeams” The European Physical Journal Plus Vol.133, No. 3, (2016), 133.

31.   Shen, S.P., Hu, S.L”, A theory of flexoelectricity with surface effect for elastic dielectrics”, Journal of the Mechanics and Physics of Solids, Vol. 58, (2010), 665–677.

32.   Hadjesfandiary, A.R., “size dependent piezoelectricity”, International Journal of Solids and Structures, Vol. 50, (2013), 2781-2791.

33.   Ghorbanpour Arani, A., Kolahchi, R., Vossough, H.,” Buckling analysis and smart control of SLGS using elastically coupled PVDF nanoplate based on the nonlocal Mindlin plate theory”, Physica B, Vol. 407, (2012), 4458–4465.