Vibration Analysis of Timoshenko Beam reinforced with Boron-Nitride Nanotube on Elastic Bed

Document Type: Persian


1 MSc Student, Department of Mechanical Engineering, Razi University, Kermanshah, Iran.

2 Assistant Prof., Department of Mechanical Engineering, Razi University, Kermanshah, Iran.


In this paper, free vibration analysis of a polymer-based nano-composite beam reinforced by boron-nitride nanotubes and subjected on elastic foundation, is studied. Smooth and defect-free nanotubes with uniform and directly- orientated in matrix are intended. Also, nanotubes’ distribution in the thickness direction of beam is regarded as a uniform distribution of the three different targeted ones. The properties of nano-beam are obtained by using a micromechanical model. The governing equations based on Timoshenko beam theory are derived by using the Hamilton principle. The equations are solved by the extended differential Quadrature and the natural frequencies are obtained. The effect of various parameters such as volume fraction of nanotubes, carbon-nanotube (CNT) distribution in the thickness direction of the beam, elastic media, boundary conditions and the aspect ratio is investigated on the natural frequency. The results show that a change on these parameters has a significant impact on the natural frequency.


[1]Zhi C., Bando Y., Tang C.,GolbergBoron, D., nitride nanotubes, Materials Science and Engineering, Vol. 70,2010, pp.92–111.

 [2]Dolati S., Fereidoon A.,Kashyzadeh K.R., A Comparison Study between Boron nitride Nanotubes and Carbon Nanotubes, International Journal of Emerging Technology and Advanced Engineering ISSN 2250-2459, Vol. 2, Issue 10, 2012.

[3] Verma V.,Jindal V.K.,DharamvirK., Elastic moduli of a boron nitride nanotube, Nanotechnology,Vol. 18,2007, 435711 ,6pp.

[4] Golberg, D., Bando, Y.,Tang,C.,Zhi, C., Functional Boron Nitride Nanotubes, Materials Science and Engineering, Vol. 70,2010, pp. 92–111.

[5] TerronesM., Romo-HerreraJ. M., Cruz-SilvaE., López-UríasF., Muñoz-SandovalE., Velázquez-SalazarJ.J., TerronesH., BandoY., GolbergD., Pure and doped boron and nitride nanotubes, materialstoday, Vol. 10 ,No.5, 2007, pp.30-38.

[6]Zhi C.Y.,   Bando Y., Wang W.L., TangC.C., Kuwahara H., Golberg D., Mechanical and Thermal Properties of Polymethyl Methacrylate-BN Nanotube Composites, Journal of Nanomaterials, 2008, Article ID 642036, 5 pages.

 [7]Yang J., ChenY., XiangY., JiaX.L., Free and forced vibration of crackedinhomogeneous beams under an axial force and a moving load, Journal of Sound andVibration, Vol. 312, pp.166-181.

 [8]Ke L.L., Yang J., Kitipornchai S., Xiang Y., Flexural Vibration and Elastic Buckling of a Cracked Timoshenko Beam Made of Functionally Graded Materials. Mechanics of Advanced Materials and Structures, Vol. 16, 2009, pp. 488-502.

 [9]YingJ., LuC.F., ChenW.Q., Two-dimensional elasticity solutions for functionallygraded beams resting on elastic foundations,Composite Structures,Vol. 84, No.3, 2008, pp. 209–219.



[10]YanT., KitipornchaiS., YangJ., QiaoH., Dynamic behaviour of edge-cracked sheardeformable functionally graded beams on an elastic foundation under a moving load,Composite Structures, Vol. 93,2011, pp.2992–3001.

[11] YasM.H., SamadiN., Free vibrations and buckling analysis of carbon nanotube-reinforced compositeTimoshenko beams on elastic foundation, International Journal of Pressure Vessels and Piping,Vol. 98 ,2012, pp.119-128.

[12]HeshmatiM., YasaM.H., Dynamic analysis of functionally graded multi-walled carbon nanotube-polystyrene nanocomposite beams subjected to multi-moving loads,Materials and Design,Vol. 49,2013, pp. 894–904.

[13]Mallick P.K., Fiber-reinforced composites:materials, manufacturing, and designbyTaylor & Francis Group, LLC, 2008.

 [14]Reddy J.N., Mechanics of Laminated Composite Plates. CRC Press, New York,1997.

 [15] ShuC., Differential Quadrature and Its Application in Engineering, Springer,Berlin, 2000.

 [16]Atlihan G., ÇalliogluH., Free Vibration Analysis of the Laminated compositeBeams by Using DQM, Plastics and Composites, Vol. 28, 1998, pp.881-890.