Skip to main content Accessibility help

An Analytical Solution for Free Vibration of Piezoelectric Nanobeams Based on a Nonlocal Elasticity Theory

  • A. A. Jandaghian (a1) and O. Rahmani (a1)


In the present study, an exact solution for free vibration analysis of piezoelectric nanobeams based on the nonlocal theory is obtained. The Euler beam model for a long and thin beam structure is employed, together with the electric potential satisfying the surface free charge condition for free vibration analysis. The governing equations and the boundary conditions are derived using Hamilton's principle. These equations are solved analytically for the vibration frequencies of beams with various end conditions. The model has been verified with the previously published works and found a good agreement with them. A detailed parametric study is conducted to discuss the influences of the nonlocal parameter, on the vibration characteristics of piezoelectric nanobeams. The exact vibration solutions should serve as benchmark results for verifying numerically obtained solutions based on other beam models and solution techniques.


Corresponding author

*Corresponding author (


Hide All
1.Casadei, F., Dozio, L., Ruzzene, M. and Cunefare, K. A., “Periodic Shunted Arrays for the Control of Noise Radiation in an Enclosure,” Journal of Sound and Vibration 329, pp. 36323646 (2010).
2.Cortes, D. H., Datta, S. K. and Mukdadi, O. M., “Elastic Guided Wave Propagation in a Periodic Array of Multi-Layered Piezoelectric Plates with Finite Cross-Sections,” Ultrasonics, 50, pp. 347356 (2010).
3.Sumali, H., Meissner, K. and Cudney, H. H., “A Piezoelectric Array for Sensing Vibration Modal Coordinates,” Sensors and Actuators A: Physical, 93, pp. 123131 (2001).
4.Wu, T., “Modeling and Design of a Novel Cooling Device for Microelectronics Using Piezoelectric Resonating Beams,” Ph. D. Dissertation, Department of Mechanical and Aerospace Engineering, NC State University, Raleigh, U.S.A. (2003).
5.Hao, Z. and Liao, B., “An Analytical Study on Interfacial Dissipation in Piezoelectric Rectangular Block Resonators with In-Plane Longitudinal-Mode Vibrations,” Sensors and Actuators A: Physical, 163, pp. 401409 (2010).
6.Lazarus, A., Thomas, O. and Deü, J.-F., “Finite Element Reduced Order Models for Nonlinear Vibrations of Piezoelectric Layered Beams with Applications to NEMS,” Finite Elements in Analysis and Design, 49, pp. 3551 (2012).
7.Tanner, S. M., Gray, J., Rogers, C. and Bertness, K., “Sanford, N., High-Q Gan Nanowire Resonators and Oscillators,” Applied Physics Letters, 91, p. 203117 (2007).
8.Wan, Q., Li, Q., Chen, Y., Wang, T.-H., He, X., Li, J. and Lin, C., “Fabrication and Ethanol Sensing Characteristics of Zno Nanowire Gas Sensors,” Applied Physics Letters, 84, pp. 36543656 (2004).
9.Murmu, T. and Adhikari, S., “Nonlocal Frequency Analysis of Nanoscale Biosensors,” Sensors and Actuators A: Physical, 173, pp. 4148 (2012).
10.Wang, Z. L. and Song, J., “Piezoelectric Nanogener-ators Based on Zinc Oxide Nanowire Arrays,” Science, 312, pp. 242246 (2006).
11.Feng, X., Yang, B. D., Liu, Y., Wang, Y., Dagdeviren, C., Liu, Z., Carlson, A., Li, J., Huang, Y. and Rogers, J. A., “Stretchable Ferroelectric Nanoribbons with Wavy Configurations on Elastomeric Substrates,” ACS Nano, 5, pp. 33263332 (2011).
12.Park, K.-I., Xu, S., Liu, Y., Hwang, G.-T., Kang, S.-J. L., Wang, Z. L. and Lee, K. J., “Piezoelectric Ba-TiO3 Thin Film Nanogenerator on Plastic Substrates,” Nano Letters, 10, pp. 49394943 (2010).
13.Qi, Y., Kim, J., Nguyen, T. D., Lisko, B., Purohit, P. K. and McAlpine, M. C., “Enhanced Piezoelectricity and Stretchability in Energy Harvesting Devices Fabricated from Buckled PZT Ribbons,” Nano letters, 11, pp. 13311336 (2011).
14.Wang, X., Song, J., Liu, J. and Wang, Z. L., “Direct-Current Nanogenerator Driven by Ultrasonic Waves,” Science, 316, pp. 102105 (2007).
15.Eringen, A. C., “On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves,” Journal of Applied Physics, 54, pp. 47034710 (1983).
16.Eringen, A. C., Nonlocal Continuum Field Theories, Springer, New York, U.S.A. (2002)
17.Eringen, A. C. and Edelen, D., “On Nonlocal Elasticity,” International Journal of Engineering Science, 10, pp. 233248 (1972).
18.Narendar, S., “Buckling Analysis of Micro-/Nano-Scale Plates Based on Two-Variable Refined Plate Theory Incorporating Nonlocal Scale Effects,” Composite Structures, 93, pp. 30933103 (2011).
19.Pradhan, S. and Murmu, T., “Small Scale Effect on the Buckling Analysis of Single-Layered Graphene Sheet Embedded in an Elastic Medium Based on Nonlocal Plate Theory,” Physica E: Low-Dimensional Systems and Nanostructures, 42, pp. 12931301 (2010).
20.Aydogdu, M., “A General Nonlocal Beam Theory: Its Application to Nanobeam Bending, Buckling and Vibration,” Physica E: Low-Dimensional Systems and Nanostructures, 41, pp. 16511655 (2009).
21.Civalek, Ö. and Demir, Ç., “Bending Analysis of Microtubules Using Nonlocal Euler-Bernoulli Beam Theory,” Applied Mathematical Modelling, 35, pp. 20532067 (2011).
22.Rahmani, O. and Pedram, O., “Analysis and Modeling the Size Effect on Vibration of Functionally Graded Nanobeams Based on Nonlocal Timoshenko Beam Theory,” International Journal of Engineering Science, 77, pp. 5570 (2014)..
23.Rahmani, O. and Ghaffari, S., “Frequency Analysis of Nano Sandwich Structure with Nonlocal Effect,” Advanced Materials Research, 829, pp. 231235 (2014).
24.Rahmani, O., “On the Flexural Vibration of Pre-Stressed Nanobeams Based on a Nonlocal Theory,” Acta Physica Polonica A, 125, pp. 532533 (2014).
25.Pirmohammadi, A., Pourseifi, M., Rahmani, O. and Hoseini, S., “Modeling and Active Vibration Suppression of a Single-Walled Carbon Nanotube Subjected to a Moving Harmonic Load Based on a Nonlocal Elasticity Theory,” Applied Physics A, 117, pp. 15471555 (2014).
26.Setoodeh, A., Khosrownejad, M. and Malekzadeh, P., “Exact Nonlocal Solution for Postbuckling of Single-Walled Carbon Nanotubes,” Physica E: Low-dimensional Systems and Nanostructures, 43, pp. 17301737 (2011).
27.Shen, H.-S. and Zhang, C.-L., “Torsional Buckling and Postbuckling of Double-Walled Carbon Nano-tubes by Nonlocal Shear Deformable Shell Model,” Composite Structures, 92, pp. 10731084 (2010).
28.Heireche, H., Tounsi, A., Benzair, A., Maachou, M. and Adda Bedia, E., “Sound Wave Propagation in Single-Walled Carbon Nanotubes Using Nonlocal Elasticity,” Physica E: Low-dimensional Systems and Nanostructures, 40, pp. 27912799 (2008).
29.Wang, Q. and Varadan, V., “Application of Nonlocal Elastic Shell Theory in Wave Propagation Analysis of Carbon Nanotubes,” Smart Materials and Structures, 16, 178 (2007).
30.Ansari, R., Sahmani, S. and Arash, B., “Nonlocal Plate Model for Free Vibrations of Single-Layered Graphene Sheets,” Physics Letters A, 375, pp. 5362 (2010).
31.Wang, Y.-Z., Li, F.-M. and Kishimoto, K., “Thermal Effects on Vibration Properties of Double-Layered Nanoplates at Small Scales,” Composites Part B: Engineering, 42, pp. 13111317 (2011).
32.Yan, Z. and Jiang, L., “The Vibrational and Buckling Behaviors of Piezoelectric Nanobeams with Surface Effects,” Nanotechnology, 22, p. 245703 (2011).
33.Ghorbanpour Arani, A., Atabakhshian, V., Loghman, A., Shajari, A. R. and Amir, S., “Nonlinear Vibration of Embedded Swbnnts Based on Nonlocal Timo-shenko Beam Theory Using DQ Method,” Physica B: Condensed Matter, 407, pp. 25492555 (2012).
34.Ghorbanpour Arani, A., Shokravi, M., Amir, S. and Mozdianfard, M. R., “Nonlocal Electro-Thermal Transverse Vibration of Embedded Fluid-Conveying Dwbnnts,” Journal of Mechanical Science and Technology, 26, pp. 14551462 (2012).
35.Ke, L.-L. and Wang, Y.-S., “Thermoelectric-Mechanical Vibration of Piezoelectric Nanobeams Based on the Nonlocal Theory,” Smart Materials and Structures, 21, p. 025018 (2012).
36.Ke, L.-L., Wang, Y.-S. and Wang, Z.-D., “Nonlinear Vibration of the Piezoelectric Nanobeams Based on the Nonlocal Theory,” Composite Structures, 94, pp. 20382047 (2012).
37.Rahmani, O. and Noroozi Moghaddam, M. H., “On the Vibrational Behavior of Piezoelectric Nano-Beams,” Advanced Materials Research, 829, pp. 790794 (2014).
38.Liu, C., Ke, L.-L., Wang, Y., Yang, J. and Kitiporn-chai, S., “Buckling and Post-Buckling of Size-Dependent Piezoelectric Timoshenko Nanobeams Subject to Thermo-Electro-Mechanical Loadings,” International Journal of Structural Stability and Dynamics, 14, p. 1350067 (2014).
39.Ke, L.-L., Liu, C. and Wang, Y.-S., “Free Vibration of Nonlocal Piezoelectric Nanoplates Under Various Boundary Conditions,” Physica E: Low-Dimensional Systems and Nanostructures, 66, pp. 93106 (2015).
40.Asemi, H. R., Asemi, S. R., Farajpour, A. and Mohammadi, M., “Nanoscale Mass Detection Based on Vibrating Piezoelectric Ultrathin Films Under Thermo-Electro-Mechanical Loads,” Physica E: Low-Dimensional Systems and Nanostructures, 68, pp. 112122 (2015).
41.Reddy, J., “Nonlocal Theories for Bending, Buckling and Vibration of Beams,” International Journal of Engineering Science, 45, pp. 288307 (2007).
42.Wang, Q., “On Buckling of Column Structures with a Pair of Piezoelectric Layers,” Engineering Structures, 24, pp. 199205 (2002).
43.Eringen, A. C., “Linear Theory of Nonlocal Elasticity and Dispersion of Plane Waves,” International Journal of Engineering Science, 10, pp. 425435 (1972).
44.Wang, Q., “Wave Propagation in Carbon Nanotubes Via Nonlocal Continuum Mechanics,” Journal of Applied Physics, 98, p. 124301 (2005).


Related content

Powered by UNSILO

An Analytical Solution for Free Vibration of Piezoelectric Nanobeams Based on a Nonlocal Elasticity Theory

  • A. A. Jandaghian (a1) and O. Rahmani (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.