### Forced vibration analysis of inhomogeneous rods with non-uniform cross-section

#### Abstract

Forced vibration analysis of cantilever rods is presented that have material properties and crosssection

areas that arbitrarily vary in the axial direction, solved using Laplace transform in time

domain and complementary functions method (CFM) in the spatial domain. Under the Laplace

transformation, the partial differential equation is transformed into time-independent boundary

value problem in the axial direction, which is solved by CFM. Then, inverse transform is taken

by modified Durbin’s method into the time domain. In the end, the non-dimensional displacement

results are compared with both benchmark and finite element method (FEM) solutions available in

the literature. In addition to satisfying a fair amount of accuracy with small computational costs,

the approach presented in this study is well-structured, simple, and efficient.

#### Keywords

#### Full Text:

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Clough, R. W. & Penzien, J. 1993. Dynamics of Structures, Mc Graw-Hill

Tong, X., Tabarrok, B. & Yeh, K. Y. 1995. Vibration analysis of Timoshenko beams with non-homogeneity

and varying cross-section. J. of Sound Vib. 186(5): 821- 835.

Elishakoff, I. & Candan, S. 2001. Apparently first closed-form solution for vibrating: inhomogeneous

beams. Int. J of Sol. and Struct. 38(19): 3411- 3441.

Avcar, M. 2015. Effects of rotary inertia shear deformation and non-homogeneity on frequencies of beam.

Struct. Eng. and Mech. 55(4): 871 -884.

Avcar, M. 2016. Effects of material non-homogeneity and two parameter elastic foundation on fundamental

frequency parameters of Timoshenko beams. Acta Physica Polonica A 130(1): 375- 378.

Yang, J., Chen, Y., Xiang, Y. & Jia, X. L. 2008. Free and forced vibration of cracked inhomogeneous beams

under an axial force and a moving load. J of Sound Vib. 312(1): 166- 181.

Shahba, A., Attarnejad, R., Marvi, M. T. & Hajilar, S. 2011. Free vibration and stability analysis of axially

functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions.

Comp. Part B: Eng. 42(4): 801- 808.

Huang, Y. & Luo, Q. Z. 2011. A simple method to determine the critical buckling loads for axially

inhomogeneous beams with elastic restraint. Comp. & Math. with Appl. 61(9): 2510- 2517.

Horgan, C.O. 1999. The pressurized hollow cylinder or disk problem for functionally graded isotropic

linearly elastic materials. J. Elasticity 55(1): 43–59.

Efraim, E. & Eisenberger, M. 2007. Exact solution analysis of variable thickness thick annular isotropic and

FGM plates. J. Sound Vib. 299(4–5): 720–738.

Maalawi, K.Y. 2011. Functionally graded bars with enhanced dynamic performance. J. Mech. Mater.

Structures. 6(1–4): 377–93.

Celebi, K., Keles, I. & Tutuncu, N. 2012. Closed-form solutions for forced vibration analysis of

inhomogeneous rod. Journal of the Faculty of Engineering and Architecture of Gazi University 27(4):

- 763.

Akgoz, B. & Civalek, Ö. 2013. Longitudinal vibration analysis of strain gradient bars made of functionally

graded materials (FGM). Comp.: Part B: Eng. 55: 263- 268.

Hong, M., Park, Il. & Lee, U. 2014. Dynamics and waves characteristics of the FGM axial bars by using

spectral element method. Composite Structures 107: 585- 593.

Hong, M. & Lee, U. 2015. Dynamics of a functionally graded material axial bar: Spectral element modeling

and analysis. Comp.: Part B 69: 427- 434.

Mecitoglu, Z. 1996. Governing equations of a stiffened laminated inhomogeneous conical shell. AIAA

journal 34(10): 2118- 2125.

Heyliger, P. R. & Jilani, A. 1992. The free vibrations of inhomogeneous elastic cylinders and spheres. Int. J.

of Sol. and Struct. 29(22): 2689 -2708.

Abrate, S. 1995. Vibration of non-uniform rods and beams. J. Sound Vib. 185: 703–716.

Kumar, B. M. & Sujith, R. I. 1997. Exact solutions for the longitudinal vibration of non-uniform rods. J.

Sound Vib. 207(5): 721–729.

Li, Q. S. 2000. Exact solutions for free longitudinal vibration of non-uniform rods. J. Sound Vib. 234(1):

–19.

Yardimoglu, B. & Aydin, L. 2011. Exact longitudinal vibration characteristics of rods with variable crosssections.

Shock and Vibration 18: 555 -562.

Celebi, K., Keles, I. & Tutuncu, N. 2011. Exact solutions for forced vibration of non-uniform

rods by Laplace transformation. Gazi University Journal of Science 24(2): 347- 353.

Shokrollahi, M. & Nejad, A. Z. B. 2014. Numerical Analysis of Free Longitudinal Vibration of Nonuniform

Rods: Discrete Singular Convolution Approach. J. Eng. Mech. 140(8): 06014007.

Conway, H. D., Becker, E. C. H. & Dubil, J. F. 1964. Vibration frequencies of tapered bars and circular

plates. ASME Journal of Applied Mechanics 31: 329–31.

Candan, S. & Elishakoff, I. 2001. Constructing the axial stiffness of longitudinally vibrating rod from

fundamental mode shape. Int. J. of Solids and Struct. 38: 3443–3452.

Elishakoff, I. & Perez, A. 2006. Design of a polynomially inhomogeneous bar with a tip mass for specified

mode shape and natural frequency. J. Sound Vib. 295: 458 -460.

Nachum, S. & Altus, E. 2007. Natural frequencies and mode shapes of deterministic and stochastic nonhomogeneous

rods and beams. J. of Sound and Vib. 302: 903 -924.

Calio, I. & Elishakoff, I. 2008. Vibration tailoring of inhomogeneous rod that possesses a trigonometric

fundamental mode shape. J. of Sound and Vib. 309: 838 -842.

Menaa, R. et al. 2012. Analytical solutions for static shear correction factor of functionally graded rectangular

beams. Mech. Adv. Mater. Struct. 19(8): 641–652.

Huang, Y., Yang, L.E. & Luo, Q.Z. 2013. Free vibration of axially functionally graded Timoshenko beams

with non-uniform cross-section. Compos. Part B Eng. 42: 1493 -1498.

Celebi, K. & Tutuncu, N. 2014. Free vibration analysis of functionally graded beams using an exact plane

elasticity approach. Proc. I. MechE Part C 228(14): 2488- 2494.

Murin, J., Aminbaghai, M. & Kutis, V. 2010. Exact solution of the bending vibration problem of FGM

beams with variation of material properties. Eng. Struct. 32(6): 1631–40.

Murin, J., Aminbaghai, M., Hrabovsky, J., Kutis, V. & Kugler, S. 2013. Modal analysis of the FGM beams

with effect of the shear correction function. Composites Part B 45(1): 1575–82.

Calim, F.F. 2009. Free and forced vibration of non-uniform composite beams. Composite Structures. 88:

-423.

Horgan, C.O. 2007. On the torsion of functionally graded anisotropic linearly elastic bars. IMA J Appl. Math.

(5): 556–62.

Chakraborty, A., Gopalakrishinan, S. & Reddy, J.N. 2003. A new beam finite element for the analysis of

functionally graded materials. Int. J. Mech. Sci. 45(3): 519–539.

Piovan, M.T. & Sampaio, R. 2008. Vibrations of axially moving flexible beams made of functionally graded

materials. Thin Wall Struct. 46(2): 112–21.

Alshorbagy, A.E., Eltaher, M.A. & Mahmoud, F.F. 2011. Free vibration characteristics of a functionally

graded beam by finite element method. Appl. Math. Model. 35(1): 412–25.

Shahba, A., Attarnejad, R. & Hajilar, S. 2011. Free vibration and stability of axially functionally graded

tapered Euler–Bernoulli beams. Shock Vib. 18(5): 683–96.

Tutuncu, N. & Temel, B. 2009. A novel approach to stress analysis of pressurized FGM cylinders, disks and

spheres. Composite Structure 91: 385- 390.

Celebi, K., Yarimpabuc, D. & Keles, I. 2016. A unified method for stresses in FGM sphere with

exponentially-varying properties. Structural Engineering and Mechanics 57: 823- 835.

Celebi, K., Yarimpabuc, D. & Keles, I. 2017. A novel approach to thermal and mechanical stresses in

FGM cylinder with exponentially-varying properties. Journal of Theoretical and Applied Mechanics 55

(1): 343 -351.

Narayanan, G.V. 1979. Numerical operational methods in structural dynamics. Minneapolis, University of

Minnesota.

Durbin, F. 1974. Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate’s

method. The Computer Journal, 17: 371- 376.

Thompson, M.K. & Thompson, J.M. 2017. ANSYS Mechanical APDL for Finite Element Analysis,

Elsevier.

Abell, M.L. & Braselton, J.P. 2004. Differential Equations with Mathematica, Elsevier.

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