Forced Vibration Responses of Axially Functionally Graded Beams … · 1 day ago · presented...

J. Appl. Comput. Mech., 7(1) (2021) 109-115 DOI: 10.22055/JACM.2020.34865.2491

ISSN: 2383-4536 jacm.scu.ac.ir

Published online: September 22 2020

Forced Vibration Responses of Axially Functionally Graded Beams

by using Ritz Method

Şeref Doğuşcan Akbaş

Bursa Technical University, Department of Civil Engineering, Yıldırım Campus, 16330, Yıldırım/Bursa, Turkey

Received September 01 2020; Revised September 20 2020; Accepted for publication September 21 2020. Corresponding author: S.D. Akbaş ([email protected]) © 2020 Published by Shahid Chamran University of Ahvaz

Abstract. This work presents forced vibration responses of a cantilever beam made of functionally graded material under a harmonic load. The material properties of beam vary along the axial direction. The kinematics of the beam are considered within Timoshenko beam theory. The governing equations of problem are derived by using the Lagrange procedure. In the solution of the problem the Ritz method is used and algebraic polynomials are used with the trivial functions for the Ritz method. In the solution of the forced vibration problem, the Newmark average acceleration method is used in the time history. In this study, free and forced vibration responses of the axially functionally graded beam are investigated in detail. In the numerical examples, the effects of material graduation, geometric and dynamic parameters on the free and forced vibration response of axially graded beam are investigated.

Keywords: Axially Functionally Graded Material; Forced Vibration Analysis; Timoshenko Beams; Ritz Method.

1. Introduction

Functionally graded materials are a type of composites whose material properties vary gradually though direction. In generally, functionally graded materials consist of a mixture of ceramic and metal materials. In the last years, the functionally graded materials have been found in engineering applications, such as aircrafts, space vehicles and biomedical sectors. In the future, it is estimated that the using of functionally graded materials will increase in many engineering projects. By increasing functionally graded structures, many researchers investigated the mechanical behavior of functionally graded structures in recent times.

In the literature, investigations of dynamics of axially functionally graded beams are presented in briefly as follows; Wu et al. [1] presented dynamic investigations of axially functionally graded beams by using the semi-inverse method. Huang and Li [2] investigated free vibration of axially functionally non-uniform graded beams. Hein and Feklistova [3] investigated vibration of axially functionally graded beams with different cross-sections and boundary conditions by using the Haar wavelet series. Alshorbgy et al. [4] presented free vibration analysis of of non-uniform axially or transversally graded beams. Eltaher et al. [5] presented free vibration analysis of functionally graded nanobeams based on nonlocal elasticity theory by using finite element method. Shahba et al. [6] and Shahba and Rajasekaran [7] analyzed free vibration and stability of axially functionally graded beams by using finite element method. Şimşek et al. [8] investiated dynamic analysis of axially functionally graded simply supported beam subjected to moving harmonic load. Huang et al. [9] investigated vibration behaviors of axially functionally graded Timoshenko beams with non-uniform cross-section. Rajasekaran [10,11] presented vibration analysis of axially functionally graded tapered and non-uniform beams by using differential transformation and differential quadrature element methods. Akgöz and Civalek [12] presented vibration responses of axially functionally graded tapered microbeams based on modified couple stress theory. Nguyen [13] studied large displacements of tapered an axially functionally graded cantilever beam. Babilio [14] investigated the dynamics of an axially functionally graded simply supported beam under axial time-dependent load. Akbaş [15,16,17,18] presented free vibration of functionally graded beams with crack, elastic foundation, nonlocal problems. Avcar [19] investigated free vibration of non-homogeneous beams. Hadji et al. [20] analyzed wave propagation of functionally graded beams with higher order shear deformation theory. Mehar et al. [21] investigated nonlinear dynamics of functionally graded carbon nanotube reinforced sandwich panels under temperature effects by using finite element method. Ghayesh [22] analyzed forced nonlinear vibration of axially functionally graded micro beams by using coupled stress theory. Akbaş [23,24] studied effects of porosity on the dynamic behavior of functionally graded beams. Liu et al. [25] studied free vibration of axially functionally graded tapered beams by using the spline finite point method. Çalım [26] presented transient analysis of axially functionally graded beams with variable cross-section. Alimoradzadeh et al. [27] investigated nonlinear vibration analysis of axially functionally graded beams under moving harmonic load. Cao and Gao [28] investigated free vibration of non-uniform axially functionally graded beams with the asymptotic development method. Sharma et al. [29] presented the modal analysis of an axially functionally graded beam under hygrothermal effect. Yaylı et al. [30] investigated free vibration of functionally graded nanobeams by using finite element method. Shahba [31] and

110 Şeref Doğuşcan Akbaş, Vol. 7, No. 1, 2021

Journal of Applied and Computational Mechanics, Vol. 7, No. 1, (2021), 109-115

Cao [32] investigated free vibration of axially functionally graded beams. In addition, different mechanical analyses of functionally graded beams with different graduation disturubution were investigated in the literature [33-52]

It is seen from literature survey, the forced vibration studies of axially functionally graded beams have not been investigated broadly. The novelty of this study is to investigate forced vibration responses of the axially functionally graded beam in detail. In the solution of the paper, Timoshenko beam theory and Ritz method are used. The governing equations of problem are obtained by using the Lagrange procedure. In the solution of the forced vibration problem, the Newmark average acceleration method is used in the time history. In the numerical results, effects of the material graduation parameter, geometry parameter and dynamic parameters on the free and forced vibration response of the axially functionally graded beam are investigated.

2. Formulations

Figure 1 show a cantilever axially functionally graded beam under a dynamic point load �� at free end with the length L, the height h and width b. The dynamic point load �� is assumed to be sinusoidal harmonic in time domain as following:

�� = ��sin �ω��, 0 ≤ � ≪ ∞ (1) In eq. 1, �� and ω indicate the amplitude and the frequency of the dynamic load. The material properties (P) of the beam in case of functionally graded material change though the axial direction based on

following power-law function distribution;

P�Z� = �P� − P�� 1 − ZL�� + P� (2)

where, �� and �� are the material properties of the left and the right surfaces of the beam and n is the non-negative power-law exponent which indictates the material graduation through the axially direction. In Eq. (2), � = �� at Z = 0 and � = �� at Z = L. When n = 0 material of beam gets homogenous full left material, and when n ∞ material of beam gets homogenous full rigth material.

The axial strain (ε�) and shear strain (γ�!) are given according to the Timoshenko beam theory as follows; "# $ %&%' − ( %∅%' (3a)

*#+ = ∂-∂z − ∅ (3b) where, &, - and ∅ are axial displacement, vertical displacement and rotation, respectively. The strain energy (U), the kinetic

energy (K) and potential energy of the external loads (/0) are presented as follows; / = 12 2 34�'�5 �%&%6�

7 + 4�'�8 �%∅%9�7 + :�'�5;< =�∂-∂Z�

7 − 2 ∂-∂Z ∅ + ∅7>?�� @6 (4a)

A = 12 2 =B�'�5 �%&%� �7 + B�'�8 �%∅%� �

7 + B�'�5 �%-%� �7>�� @6 (4b)

/0 = −�� -C'D, �F (4c) where, E, G, B, A, I are the Young‘s modulus, shear modulus, mass density, area of cross section and he second moment of area

of the beam, respectively. Also, ks indicates the shear correction factor. In equations, t indicates the time.

Fig. 1. A cantilever axially functionally graded beam under a dynamic point load at free end.

Forced Vibration Responses of Axially Functionally Graded Beams by using Ritz Method 111


The Lagrangian functional of the problem is presented as follows;

8 = A − �/G + /0� (5) In the solution of the problem in Ritz method, approximate solution is given as a series of i terms of the following form:

&�', �� = H aG ��JG�'�KG$L (6a)

-�', �� = H bG ��NG�'�KG$L (6b)

∅�', �� = H cG ��*G�'�KG$L (6c) where ai, bi and ci are the unknown coefficients, JG�', ��, NG�', ��, *G�', �� are the coordinate functions depend on the boundary

conditions over the interval [0,L]. The coordinate functions for the cantilever beam are given as algebraic polynomials:

JG�'� = 'G (7a) NG�'� = '�GPL� (7b)

*G�'� = 'G (7c) where i indicates the number of polynomials involved in the admissible functions. After substituting equations (6) into energy

equations (4), and then using the Lagrange’s equation gives the following equation;

%8%QG − %%� %8%QR G = 0 (8) where qi is the unknown coefficients which are ai, bi and ci. After implementing the Lagrange procedure, the motion equation of

the problem is obtained as follows;

SKUVq�t�Y + SMUVq[ �t�Y = VF�t�Y (9) where SAU, S]U and VF�t�Y are the stiffness matrix, the mass matrix and the load vector, respectively. When the VF�t�Y = 0 in eq.

(9), the problem reduces the free vibration problem. The detail of these expressions are given as follows;

SAU = ^ALL AL7 AL_A7L A77 A7_A_L A_7 A__` (10) where

AGaLL = ∑ ∑ c 5� defd# degd#��ha$LhG$L @' , AGaL7 = AGa7L = AGaL_ = AGa_L = 0 , AGa77 = ∑ ∑ c :�'�5;< difd# digd#��ha$LhG$L @', AGa7_ = − H H 2 :�'�5;< %NG%' *a��

ha$L

hG$L

@',

AGa_7 = − H H 2 :�'�5;



S]U = ^]LL ]L7 ]L_]7L ]77 ]7_]_L ]_7 ]__` (12) where

]LL = ∑ ∑ c B�'�5�� JGJadz�k$L�l$L , ]L7 = ]7L = ]L_ = ]_L = ]7_ = ]_7 = 0 ]77 = H H 2 B�'�5�� NGNadz

�k$L

�l$L

]__ = H H 2 B�'�8�� *G*adz�

k$L�

l$L

(13)

VF�t�Y = �Na (14) The governing equation of motions Eq. (9) is solved numerically by using implicit Newmark average acceleration method in the

time domain �J = 0.5 , N = 0.25�. By this procedure, the dynamic problem is transferred to system of static problem in each step as following

SAo��, 9�UVQhYaPL = Vpq��Y (15) in which

SAo��, 9�U = SAU + S]UN∆�7 (16a)

Vpq��Y = Vp��YaPL + sLVQhYa + s7VQRhYa + s_VQ[hYa (16b) where

sL = StUi∆uv , s7 = StUi∆u , s_ = S]U w L7i − 1x (17) After evaluating VQhYaPL at a time �aPL = �a + ∆� , the acceleration and velocity vectors can be evaluated by

VQhYaPL = 1N∆�7 CVQhYaPL − V@hYaF − S]UN∆� VQRhYa − � J2N − 1� VQ[hYa (18a)

VQRhYaPL = VQRhYa + ∆� �1 − J�VQ[hYa + ∆� J VQ[hYaPL (18b) 3. Numerical Results

In this section, dynamical displacements of the axially functionally graded cantilever beam under the sinusoidal harmonic load are presented and discussed according to material, geometry and load parameters. The material parameters of beam are given as follows; at the left side is fully Zirconia (E=151 GPa, ν=0.2882) and at the right side is fully Aluminum Oxide (E=70 GPa, ν=0.31). The geometry values are selected as b=0.1 m, h=0.1 m and L=1 m. The magnitude of load is selected as Q0=100 kN. In the numerical results, number of the series term is taken as 8.

In order to verify the using theory and formulations, a comparison study is presented with the special results of a published paper whose topic is related with this paper. For this purpose, the dimensionless natural frequency of axially functionally graded cantilever beam is obtained from the eigenvalue solution of eq. (9) and compare with results of [31] and [32] in table 1. The dimensionless natural frequency is calculated as Ω = ωhy7zB�/4� . Where, ωh indicates the natural frequency. It is seen from Table 1 that obtained from this study is very close to the results of [31] and [32].

In figure 2, effects of the material graduation parameter (n) on the lateral dynamical free-end displacements of the axially functionally graded beam are presented for different values of load frequency (ω�in time history. Also, natural frequencies of the axially functionally graded beam are obtained and plotted in figure 3 with effects of the material graduation parameter (n) for different values of L/h ratios.

Table 1. Comparision study for dimensionless natural frequeny of axially functionally graded cantilever beam.

The dimensionless natural frequeny (Ω� Present 2.447 Ref. [31] 2.426 Ref. [32] 2.439



Fig. 2. Time history of dynamic displacements at free end of axially functionally graded beam with different values of material graduation parameter (n) for the load frequency a) ω = 2 rd/s, b) ω = 20 rd/s and ω = 100 rd/s.

Fig. 3. Natural frequencies of axially functionally graded beam with different values of material graduation parameter (n) for a) L/h=8 b) L/h=10 and c)

L/h=15.

It is observed from figures 2-3, the dynamical displacements of the axially functionally graded beams increase and the natural frequencies of the axially functionally graded beam decrase with increasing of the material graduation parameter (n). This is because that with increasing the material graduation parameter, the stiffness of the beam decreases due to used material properties and the equation (2). Because when the material graduation parameter (n) increase, the material of the beam get close to Aluminum Oxide (right side) according to Eq. 2. Due to the Young’s modulus of Zirconia is approximately two times greater than that of Aluminum Oxide, so the strength of the material decreases and the dynamically displacements increase and natural frequencies decrase by increasing n parameter. Also, the load frequency has important role on the dynamical responses of axially functionally graded beams. With increasing of the load frequency, difference among of results of n parameter decrease significiantly. In higher values of the load frequency, the dynamically displacements for n parameter are very close to each other.

In figure 4, the relationship between of the maximum displacements and the frequency of the dynamic load (ω) of axially functionally graded beam is presented for different values the material graduation parameter (n) for t=0.2 s for different values of L/h ratios.. It is seen from figure 4 that the resonance frequencies of the axially functionally graded beam decrease with increasing of material graduation parameter (n). The resonance frequencies can be observed in the values of amplitude hit. Because of increasing the material graduation parameter (n), the axially functionally beam gets more flexible and so, the resonance frequencies decrease naturally. Another result of figure 4 that increasing of L/h ratio cause to decrase increasing of L/h ratio and, so the resonance frequencies decrease naturally. The dynamic responses of axially functionally graded beam change considerably with different L/h ratio. The effects of material graduation on the dynamic responses vary according to L/h ratio. It shows that the geormetry parameter has important role on the dynamic responses. Also, it is seen from figure 4 that high fluctuations can be observed in the resonance values. In higher values of L/h ratios, more fluctuations occurs because the beam gets more flexible and sensitive in the dynamic responses.



Fig. 4. The relationship between of the maximum displacements and the frequency of the dynamic load (ωo) of axially functionally graded beam beam with different values of material graduation parameter (n) for a) L/h=8 b) L/h=10 and c) L/h=15.

4. Conclusion

The dynamic analysis of an axially functionally graded cantilever beam under a sinusoidal dynamic load are investigated based on the Timoshenko beam theory by using the Ritz methodwithin Newmark average acceleration method. Effects of material graduation, L/h ratio and frequency of dynamic load on the dynamically displacements and natural frequencies of the axially functionally graded beam are obtained and discussed. It is concluded from results that the material graduation has important role on the dynamical responses of axially functionally graded. The load frequency is very effective on the influence of material graduation on the dynamic responses of the beam. With load frequency, the dynamic responses of the axially functionally graded beams change significiantly. Also, the geometry parameter has important role on the dynamic responses of the axially functionally graded beams.

Conflict of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and publication of this article.

Funding

The author received no financial support for the research, authorship, and publication of this article.

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ORCID iD

Şeref Doğuşcan Akbaş https://orcid.org/0000-0001-5327-3406

© 2020 by the authors. Licensee SCU, Ahvaz, Iran. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0 license) (http://creativecommons.org/licenses/by-nc/4.0/).

How to cite this article: Akbaş Ş.D. Forced Vibration Responses of Axially Functionally Graded Beams by using Ritz Method, J. Appl. Comput. Mech., 7(1), 2021, 109–115. https://doi.org/10.22055/JACM.2020.34865.2491

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