Subscribe

Subscribe to our Newsletter and get informed about new publication regulary and special discounts for subscribers!

IFSL > IFSL Volume 9 > Natural Frequency and Mode Shapes of Exponential...
Natural Frequency and Mode Shapes of Exponential Tapered AFG Beams on Elastic Foundation
< Back to Volume

Natural Frequency and Mode Shapes of Exponential Tapered AFG Beams on Elastic Foundation

Full Text PDF

Abstract:

A displacement based semi-analytical method is utilized to study non-linear free vibration and mode shapes of an exponential tapered axially functionally graded (AFG) beam resting on an elastic foundation. In the present study geometric nonlinearity induced through large displacement is taken care of by non-linear strain-displacement relations. The beam is considered to be slender to neglect the rotary inertia and shear deformation effects. In the present paper at first the static problem is solved through an iterative scheme using a relaxation parameter and later on the subsequent dynamic analysis is carried out as a standard eigen value problem. Energy principles are used for the formulation of both the problems. The static problem is solved by using minimum potential energy principle whereas in case of dynamic problem Hamilton’s principle is employed. The free vibrational frequencies are tabulated for exponential taper profile subject to various boundary conditions and foundation stiffness. The dynamic behaviour of the system is presented in the form of backbone curves in dimensionless frequency-amplitude plane and in some particular case the mode shape results are furnished.

Info:

Periodical:
International Frontier Science Letters (Volume 9)
Pages:
9-25
DOI:
https://doi.org/10.18052/www.scipress.com/IFSL.9.9
Citation:
H. Lohar et al., "Natural Frequency and Mode Shapes of Exponential Tapered AFG Beams on Elastic Foundation", International Frontier Science Letters, Vol. 9, pp. 9-25, 2016
Online since:
August 2016
Authors:
Hareram Lohar, Anirban Mitra, Sarmila Sahoo
Keywords:
Axially Functionally Graded Beam, Backbone Curve, Elastic Foundation, Energy Principles, Geometric Nonlinearity, Mode Shape
Export:
RIS, BibTeX, Text
Distribution:
Open Access

Creative Commons License
This work is licensed under a
Creative Commons Attribution 4.0 International License

References:

[1] M. Koizumi, FGM activities in Japan, Composites, Part B: Engineering. 28(1) (1997) 1–4.

[2] T. Nguyen, T. Vo, H. Thai, Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory, Compos. Part B. 55 (2013) 147-157.

[3] S. Suresh, A. Mortensen, Fundamentals of Functionally Graded Materials, IOM Communications Limited, London, (1998).

[4] H. Su, J.R. Banerjee, C.W. Cheung, Dynamic stiffness formulation and free vibration analysis of functionally graded beams, Composite Structures. 106 (2013) 854-862.

[5] R. Kadoli, K. Akhtar, N. Ganesan, Static analysis of functionally graded beams using higher order shear deformation theory, Applied Mathematical Modelling. 32 (2008) 2509-2525.

[6] C. Jin, X. Wang, Accurate free vibration analysis of Euler functionally graded beams by the weak form quadrature element method, Composite Structures. 125 (2015) 41-50.

[7] H. Yaghoobi, A. Fereidoon, Influence of neutral surface position on deflection of functionally graded beam under uniformly distributed load, World Applied Science Journal. 10 (2010) 337-341.

[8] X.L. Jia et al., Size effect on the free vibration of geometrically nonlinear functionally graded micro-beams under electrical actuation and temperature change, Composite Structures. 133 (2015) 1137-1148.

[9] F. Ebrahimi, E. Salari, Thermal buckling and free vibration analysis of size dependent timoshenko FG nanobeams, in thermal environments, Composite Structures. 128 (2015) 363-380.

[10] K. K Pradhan, S. Chakraverty, Effects of different shear deformation theories on free vibration of functionally graded beams, International Journal of Mechanical Sciences. 82 (2014) 149-160.

[11] D. Chen, J. Yang, S. Kitipornchai, Elastic buckling and static bending of shear deformable functionally graded porous beam, Composite Structures. 133 (2015) 54-61.

[12] F. Ebrahimi, M. Zia, Large amplitude nonlinear vibration analysis of functionally graded timoshenko beams with porosities, Acta Astronautica. 116 (2015) 117-125.

[13] M. Hemmatnezhad, R. Ansari, G. H. Rahimi, Large-amplitude free vibrations of functionally graded beams by means of a finite element formulation, Applied Mathematical Modelling. 37 (2013) 8495–8504.

[14] X.F. Li, Y.A. Kang, J.X. Wu, Exact frequency equations of free vibration of exponentially functionally graded beams, Applied Acoustics. 74 (2013) 413-420.

[15] Y. Huang, Q. Luo, A simple method to determine the critical buckling loads for axially inhomogeneous beams with elastic restraint, Computers and Mathematics with Applications. 61 (2011) 2510-2517.

[16] B. Akgöz, Ö. Civalek, Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory, Composite Structures. 98 (2013) 314-322.

[17] A. Shahba et al., Free vibration and stability analysis of axially functionally graded tapered timoshenko beams with classical and non-classical boundary conditions, Composites: Part B 42 (2011) 801-808.

[18] Y. Huang, L. Yang, Q. Luo, Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section, Composites: Part B. 45 (2013) 1493-1498.

[19] A. Shahba, S. Rajasekaran, Free vibration and stability of tapered euler–bernoulli beams made of axially functionally graded materials, Applied Mathematical Modelling. 36 (2012) 3094-3111.

[20] H. Zeighampour, Y.T. Beni, Free vibration analysis of axially functionally graded nanobeam with radius varies along the length based on strain gradient theory, Applied Mathematical Modelling. 39 (2015) 5354-5369.

[21] N. Kien, Large displacement response of tapered cantilever beams made of axially functionally graded material, Composites: Part B. 55 (2013) 298–305.

[22] S. Kumar, A. Mitra, H. Roy, Geometrically nonlinear free vibration analysis of axially functionally graded taper beams, Engineering Science and Technology, an International Journal. 18 (2015) 579-593.

[23] K. Sarkar, R. Ganguli, Closed-form solutions for axially functionally graded timoshenko beams having uniform cross-section and fixed–fixed boundary condition, Composites: Part B. 58 (2014) 361-370.

[24] M.H. Yas, N. Samadi, Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation, International Journal of Pressure Vessels and Piping. 98 (2012) 119-128.

[25] M. Simsek, S. Cansız, Dynamics of elastically connected double-functionally graded beam systems with different boundary conditions under action of a moving harmonic load, Composite Structures. 94 (2012) 2861-2878.

[26] J. Murin et al., Modal analysis of the FGM beams with effect of axial force under longitudinal variable elastic winkler foundation, Engineering Structures. 49 (2013) 234-247.

[27] M. Simsek, J.N. Reddy, A unified higher order beam theory for buckling of a functionally graded microbeam embedded in elastic medium using modified couple stress theory, Composite Structures. 101 (2013) 47-58.

[28] B. Akgöz, Ö. Civalek, Bending analysis of fg microbeams resting on winkler elastic foundation via strain gradient elasticity, Composite Structures. 134 (2015) 294-301.

[29] A.S. Kanani et al., Effect of nonlinear elastic foundation on large amplitude free and forced vibration of functionally graded beam, Composite Structures. 115 (2014) 60-68.

[30] H. Niknam, M.M. Aghdam, A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation, Composite Structures. 119 (2015) 452-462.

[31] R. K. Gupta et al., Relatively simple finite element formulation for the large amplitude free vibrations of uniform beams, Fin. El. Anal. Des. 45 (2009) 624-631.

Show More Hide
Cited By:

[1] H. Lohar, A. Mitra, S. Sahoo, "Large amplitude forced vibration analysis of an axially functionally graded tapered beam resting on elastic foundation", Materials Today: Proceedings, Vol. 5, p. 5303, 2018

DOI: https://doi.org/10.1016/j.matpr.2017.12.114

[2] H. Lohar, A. Mitra, S. Sahoo, "Geometrically Non-Linear Frequency Response of Axially Functionally Graded Beams Resting on Elastic Foundation Under Harmonic Excitation", International Journal of Manufacturing, Materials, and Mechanical Engineering, Vol. 8, p. 23, 2018

DOI: https://doi.org/10.4018/IJMMME.2018070103