Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T14:08:37.541Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  15 July 2015

A. A. Jandaghian  and
O. Rahmani
A. A. Jandaghian
Affiliation:
Smart Structures and New Advanced Materials Laboratory, Department of Mechanical Engineering, University of Zanjan, Zanjan, Iran
O. Rahmani*
Affiliation:
Smart Structures and New Advanced Materials Laboratory, Department of Mechanical Engineering, University of Zanjan, Zanjan, Iran
*
*Corresponding author ([email protected])
Get access

Abstract

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.

Keywords

NanostructuresPiezoelectricVibrationAnalytical Solution
Type
Research Article
Information
Journal of Mechanics , Volume 32 , Issue 2 , April 2016 , pp. 143 - 151
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

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. 3632–3646 (2010).Google Scholar
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. 347–356 (2010).Google Scholar
3.Sumali, H., Meissner, K. and Cudney, H. H., “A Piezoelectric Array for Sensing Vibration Modal Coordinates,” Sensors and Actuators A: Physical, 93, pp. 123–131 (2001).Google Scholar
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).Google Scholar
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. 401–409 (2010).Google Scholar
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. 35–51 (2012).Google Scholar
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).CrossRefGoogle Scholar
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. 3654–3656 (2004).CrossRefGoogle Scholar
9.Murmu, T. and Adhikari, S., “Nonlocal Frequency Analysis of Nanoscale Biosensors,” Sensors and Actuators A: Physical, 173, pp. 41–48 (2012).Google Scholar
10.Wang, Z. L. and Song, J., “Piezoelectric Nanogener-ators Based on Zinc Oxide Nanowire Arrays,” Science, 312, pp. 242–246 (2006).CrossRefGoogle ScholarPubMed
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. 3326–3332 (2011).CrossRefGoogle ScholarPubMed
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. 4939–4943 (2010).Google Scholar
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. 1331–1336 (2011).CrossRefGoogle ScholarPubMed
14.Wang, X., Song, J., Liu, J. and Wang, Z. L., “Direct-Current Nanogenerator Driven by Ultrasonic Waves,” Science, 316, pp. 102–105 (2007).CrossRefGoogle ScholarPubMed
15.Eringen, A. C., “On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves,” Journal of Applied Physics, 54, pp. 4703–4710 (1983).Google Scholar
16.Eringen, A. C., Nonlocal Continuum Field Theories, Springer, New York, U.S.A. (2002)Google Scholar
17.Eringen, A. C. and Edelen, D., “On Nonlocal Elasticity,” International Journal of Engineering Science, 10, pp. 233–248 (1972).Google Scholar
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. 3093–3103 (2011).Google Scholar
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. 1293–1301 (2010).CrossRefGoogle Scholar
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. 1651–1655 (2009).CrossRefGoogle Scholar
21.Civalek, Ö. and Demir, Ç., “Bending Analysis of Microtubules Using Nonlocal Euler-Bernoulli Beam Theory,” Applied Mathematical Modelling, 35, pp. 2053–2067 (2011).CrossRefGoogle Scholar
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. 55–70 (2014)..Google Scholar
23.Rahmani, O. and Ghaffari, S., “Frequency Analysis of Nano Sandwich Structure with Nonlocal Effect,” Advanced Materials Research, 829, pp. 231–235 (2014).Google Scholar
24.Rahmani, O., “On the Flexural Vibration of Pre-Stressed Nanobeams Based on a Nonlocal Theory,” Acta Physica Polonica A, 125, pp. 532–533 (2014).CrossRefGoogle Scholar
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. 1547–1555 (2014).CrossRefGoogle Scholar
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. 1730–1737 (2011).CrossRefGoogle Scholar
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. 1073–1084 (2010).CrossRefGoogle Scholar
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. 2791–2799 (2008).Google Scholar
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).Google Scholar
30.Ansari, R., Sahmani, S. and Arash, B., “Nonlocal Plate Model for Free Vibrations of Single-Layered Graphene Sheets,” Physics Letters A, 375, pp. 53–62 (2010).CrossRefGoogle Scholar
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. 1311–1317 (2011).Google Scholar
32.Yan, Z. and Jiang, L., “The Vibrational and Buckling Behaviors of Piezoelectric Nanobeams with Surface Effects,” Nanotechnology, 22, p. 245703 (2011).Google Scholar
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. 2549–2555 (2012).Google Scholar
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. 1455–1462 (2012).Google Scholar
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).Google Scholar
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. 2038–2047 (2012).CrossRefGoogle Scholar
37.Rahmani, O. and Noroozi Moghaddam, M. H., “On the Vibrational Behavior of Piezoelectric Nano-Beams,” Advanced Materials Research, 829, pp. 790–794 (2014).Google Scholar
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).Google Scholar
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. 93–106 (2015).Google Scholar
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. 112–122 (2015).Google Scholar
41.Reddy, J., “Nonlocal Theories for Bending, Buckling and Vibration of Beams,” International Journal of Engineering Science, 45, pp. 288–307 (2007).CrossRefGoogle Scholar
42.Wang, Q., “On Buckling of Column Structures with a Pair of Piezoelectric Layers,” Engineering Structures, 24, pp. 199–205 (2002).Google Scholar
43.Eringen, A. C., “Linear Theory of Nonlocal Elasticity and Dispersion of Plane Waves,” International Journal of Engineering Science, 10, pp. 425–435 (1972).Google Scholar
44.Wang, Q., “Wave Propagation in Carbon Nanotubes Via Nonlocal Continuum Mechanics,” Journal of Applied Physics, 98, p. 124301 (2005).CrossRefGoogle Scholar