A Generalized Finite Element Approach towards the Free Vibration

Analysis of Non-uniform Axially Functionally Graded Beam

Ramya Prakash Sahu, Mihir Kumar Sutar*, Sarojrani Pattnaik

Department of Mechanical Engineering, VSSUT Burla, Burla-768018,

Sambalpur, Odisha, India

*Email ID of Corresponding Author: mihirsutar05@gmail.com,

ABSTRACT:

A generalized finite element approach, for the free vibration analysis of an axially

functionally graded (AFG) beam, having non-uniform thickness, has been presented in

the current analysis. The use of non-uniform beam element and the way of assembling the

same, make the finite element model, a generalized one. The current approach can be used

for beams of both uniform and non-uniform thickness, with any of the homogenous and

inhomogeneous material variation. The governing equation for free vibration of beam has

been derived considering Euler-Bernoulli beam theory and by using Euler-Lagrange's

equation. The cross-section of the of the beam is decreasing along the length depending

upon the exponential function considered for variation in thickness. The material

inhomogeneity is as per the Power and Exponential law of material variation along the

axial direction, taken from the literature. Mathematical modelling of geometric non-

uniformity, material inhomogeneity and finite element analysis of the AFG beam, have

been performed using MATLAB. The effect of geometric non-uniformity and material

gradient parameters on the fundamental frequencies of vibration in different classical

boundary conditions have been investigated. The efficacy of the current method has been

ascertained by comparing the result of available literature.

Keywords: Generalized finite element, Free vibration, AFG beam, Power law,

Exponential law, Fundamental frequency.

1. INTRODUCTION

Beam as an inevitable part of any mechanical structure finds its numerous applications

ranging from micro-beams used in MEMS devices to large structures like aircraft carriers.

The recent advancements in the manufacturing of composite materials have made it

possible to replace the conventional beam materials with the composite ones because of

their high strength to weight ratio. Functionally Graded Materials (FGM) are one of the

results of these research advancements in the field of composite materials. The idea of

FGM first came into picture in the mid 80’s in order to solve a problem of space plane

project in Japan, where material capability to withstand an external temperature of 2100

K and a gradient of 1600 K was needed for a very small thickness [1]. Since then FGMs

have gained a widespread popularity as thermal barrier materials in various engineering

applications such as nuclear reactors, defense systems, automobiles, aerospace, spacecraft

etc.

So FGM can be defined as a relatively new class of composite material having a

continuous variation of composition and/or microstructure in a particular direction and

possessing no specific material interface unlike conventional laminated composites, thus

lessening the stress concentrations found in them [2]. The variation in composition also

called as Grading, can be done for any of the material properties such as porosity, density,

elasticity, thermal or electrical conductivity etc. Our discussion in this paper is limited to

the mechanical properties such as density, elasticity which have direct impact on the

mechanical behaviors such as bending, buckling and vibration of the beams or any other

mechanical element of a structure for that matter.

Considering the kind of extreme dynamic conditions present in the wide range of FGM

applications like aerospace and defense, it becomes quite essential to understand the

dynamics of any mechanical system made of it. And in order to understand the dynamics

especially from the design perspective, the study of natural frequencies is fundamental

and a lot of researchers have put their effort on it. Kapuria et al. [3] studied the free

vibration responses of functionally graded beams of layered type with two different

combinations i.e. Aluminum/Silicon Carbide and Nickel/Alumina, at theoretical level

with experimental validation as well. The response prediction was performed by

considering zig zag theory in conjunction with modified rule of mixture. Free vibration

and buckling behavior of functionally graded beams having open edge cracks were studied

by Yang and Chen [4] using the analytical approach based on the rotational-spring model.

Alshorbagy et al. [5] have investigated the effect of different material distributions in the

thickness direction of a beam of uniform thickness using finite element method

(FEM).Their numerical results depicted the variation in material variation parameter in

the longitudinal direction certainly affected the natural frequency and mode shape of the

beam. Simsek and Kocaturk [6] have analyzed free and forced vibration characteristics

and the dynamic behavior of a functionally graded simply-supported beam under a

concentrated moving harmonic load. They considered functional grading along thickness

direction using the Power law and the Exponential law and as a result it was established

that different material distribution, velocity of the moving harmonic load, the excitation

frequency are pretty important in determining the dynamic behavior of the FG beam.

Pradhan and Chakraverty [7] have also investigated vibration characteristics of Euler-

bernoulli FGM beams using the integral approach of Rayleigh–Ritz method and

concluded the frequency parameters in downhill manner with both end fixed support being

the highest followed by one end fixed and one simply supported and both end simply

supported FGM beams follow the same pattern as that of isotropic beams independent of

the complexity in material variations. Rao and Ganesan [8] parametrically studied on the

influence of taper profile and taper parameter on the harmonic response of a beam tapered

in several manner, concluding that various type of taper provided in the beam leads to the

inter laminar stress development at free edges. Karami et al. [9] employed the differential

quadrature element method (DQEM) based on the theory of shear deformable beams to

analyze the free vibration of beams with non-uniform thickness. It was proved that for

beams which were neither uniform nor continuous in thickness profile, the differential

quadrature technique could come out as quite an accurate tool considering its practical

applicability. Aydogdu and Taskin [10] analyzed the free vibration response of a simply

supported FGM beam having material gradient in the thickness direction and following

the Exponential law unlike most of the other works cited above. Through Hamilton's

principle they have derived the system of equations of motion and finally the Navier type

solution was assumed. Various higher order shear deformation theory (HSDT) and

classical beam theory (CBT) have been used in analysis. They have concluded that CBT

gives higher results as compared to HSDT and the difference between the results of two

theories increases with the increase in mode number. Nguyen and Quoc [11] used a new

shear deformation theory to analyze the free vibration of the functionally graded

rectangular plates through finite element approach. By employing Hamiltonian approach

for deriving governing equations of motion the influence of material gradation parameter

and geometric parameter on the natural vibration frequencies of the functionally graded

plate was thoroughly investigated.

It is apparent that a lot of effort has been made but most of these works are confined to

the analysis of beams having uniform thickness and grading along the thickness direction.

In the recent past many of the researchers have analyzed FGM in axial direction well.

Huang and Li. [12] considered a new approach to solve free vibration of axially functional

graded (AFG) Euler-Bernoulli beams of non-uniform cross section and relied upon

Fredholm’s integral approach rather solving a relatively complex differential equation

consisting of variable coefficients with order four. Cao et al. [13] presented an

approximate analytical solution methodology for power law based AFG beam, using

Euler-Bernoulli beam theory, by asymptotic development method (ADM). The non-

linearity in the vibration of tapered AFG beam was investigated by Ghayesh [14] through

the utilization of third order shear deformation theory (TSDT). Huang et al. [15] studied

the whirling frequency and critical speed of a spinning AFG beam and modelled it in

accord with the Timoshenko beam theory. By applying spectral-Tchebychev method for

the derivation of governing differential equation as well as for solution it was found out

that axial grading had a profound impact on the whirling frequency and critical speed

specifically for cantilever boundary condition. A new computationally efficient

modification of the well-known initial parameter in differential form (IPDF) method, was

proposed by Salinic et al. [16] i.e., the symbolic numeric method of initial parameters

(SNMIP), for the vibration analysis of tapered and segmented type Euler-Bernoulli AFG

beam. Zheng et al. [17] opted for the modified couple stress theory and the Euler-Bernoulli

beam theory considering the geometric non-linearity of von-Karman type, for the non-

linear finite element modelling of axially graded tapered micro beam undergoing non-

linear vibration. Sahin et al. [18] have chosen Timoshenko beam theory and completed

the finite element modelling based on Hamilton principle for the free vibration

investigation of double tapered AFGB. Xie et al. [19] carried out the dynamic analysis of

AFG beam considering both classical beam theory (CBT) and the Timoshenko beam

theory (TBT). The governing equation of motion was derived using Lagrange’s theorem,

recognizing the longitudinal and transverse coupling effect of vibration. The numerical

technique of Newmark along with the direct iteration method was employed for the

solution. In the analysis of transverse and longitudinal beam vibration under combined

axial loads, Sun and Li [20] relied on Timoshenko beam theory considering it as an initial

value problem and employed the Fourth order Runge-Kutta method for solution.

Isogeometric analysis (IGA) of functionally graded materials is a leap in the numerical

analysis process. It acts like a connection between the FGM model and FEA. Several

attempts by researchers in this domain have been reported in the literature. The idea of

IGA have been introduced by Hughes et al. [21]. Nguyen et al. [22]. have proposed a

novel quasi-3D shear deformation theory in association with isogeometric analysis. The

proposed approach has been successfully used for analysis of rectangular and circular

functionally graded micro-plates, for static bending, free vibration and buckling analysis.

Thanh et al.[23] have proposed a refined modified couple stress theory based on IGA and

Hamilton's principle, for analysis of laminated composite micro-plates. Three material-

length scale parameters have been considered instead of one, using asymmetric couple

stress curvature tensor. The approach found to be more accurate. The behavior of porous

micro plates in thermal and post buckling have been analyzed by Thanh et al. [24], using

modified couple stress theory along with IGA, at different boundary conditions. IGA

along with modified couple stress theory (MCST) has also been reportedly used in carbon

nanotube reinforced composite nanoplates by Thanh et al. [25]. Van et al. [26] have

proposed a shear deformation theory in association with IGA for nonlinear transient

analysis of piezoelectric FGM plates. The proposed methodology was found to be

effective compared to the standard ones. IGA also found to be applied for analysis of

porous FG nanoplates by Van et al. [27]. It has been observed that the nonlinear response

in the form of deflection has been influenced by porosity, material properties etc. IGA

along with MCST have also been applied in analysis of composite laminate microplates

[28] for studying the buckling behavior under thermal loading conditions. it has been

observed from the analysis that by increasing the material length scale ratio, parameters

like thermal central deflection, shear and axial stress are decreased and on the other hand

the critical temperature has been decreased. Nguyen et al. [29] have analyzed a thin walled

functionally graded beam having non-uniform polygonal cross section. In the analysis

HSDT has been used to model the complex geometry and ABAQUS platform was

employed for the finite element analysis. Finding a computationally efficient process even

after considering the practical aspects of distortion, warping they have opened new

pathway for research in the area of shell-like structure analysis. Most of the research works

discussed till now are focused on unidirectional variation of properties but Nguyen and

Lee have considered the bi-directional material properties variation for the flexural-

torsional vibration analysis of thin-walled beams. They found out that the location of the

center of gravity and shear center are imperatively sensitive to the type of variation in

geometry as well as material distribution [30,31].

In this paper, an attempt was made to present a simplified finite element approach for the

free vibration of non-uniform AFG beam. The main difficulty with the analysis of non-

uniform beams is the definition beam element which itself changes every time which in

turn causes a major problem during the assembly of elements. So, the problems of

different kinds of non-uniformity as well as material gradation schemes have been

properly acknowledged and results have been conformed to the existing literature.

2. MATHEMATICAL MODELLING OF FGB

2.1 Geometry of the FGB

In the present analysis of free transverse vibration, a beam having length L, width b and

thickness g(x) have been considered, as shown in Fig.1. Here g(x) is assumed as an

exponential function of the axial location ‘x’ with respect to the origin at O making the

thickness and thus the cross-section variable along the axial direction of the beam

In the problem XZ-plane was taken as the plane of vibration. So g(x) is given by

0( )x

Lg x h e−

=

(1)

2.2 Material Gradation

In this work material properties of the beam such as Young’s Modulus E(x) and mass

density ρ(x) vary axially following exponential law of gradation used by Cao et al. [13]

and the power law of material gradation as given by Rajasekaran and Khaniki [32]. The

effective material property P(x) at a certain distance x from the origin along the axial

direction is determined by rule of mixture [33] as follows;

( ) L L R RP x P V P V= +

For 1L RV V+ =

(2)

(3)

Here VL is the volume fraction of the material used in left most interface of the beam i.e.

at x = 0 and VR is the volume fraction of the material used in right most part of the beam

i.e. at x = L. PL and PR stand for the material properties of left and right most surfaces of

the beam.

( ) ( )L R L RP x P P P V= + − (4)

For Exponential law of grading ref. [13]:

1

1

x

L

R

eV

e

−=

−, when, 0 (5)

R

xV

L

=

, when, 0 = (6)

For Power law of grading ref. [32]:

n

R

xV

L

=

, where 0n

(7)

A graphical representation of the variation in material properties along the beam length

was presented for different values of gradient parameters ‘α’ for exponential law and ‘n’

for the power law of grading in Fig-2 and Fig-3 respectively. For the exponential law of

variation with the increase in ‘α’, the richness of property of right end material goes on

decreasing. The same happens with the increase in power law exponent ‘n’.

3. THE FINITE ELEMENT ANALYSIS:

3.1 Beam element description:

For the finite element analysis a two-node non-uniform beam element having two degrees

of freedom at each node i.e. displacement along z-direction and rotation in xz-plane as

shown in the Fig-4 was considered. W1 and W2 are the transverse displacements of node1

and node2, whereas θ1 and θ2 are the rotations at node1 and node2 respectively as shown

in Fig.4.

3.2 Derivation of governing dynamic equation

In this problem formulation transverse displacement W was defined as;

( ) w N d=

(8)

Here [N (ζ)] is the shape function vector and depends only on local co-ordinate ζ. {d}

on the other hand is the vector containing nodal variables and is dependent only upon

time. The local co-ordinate ζ is defined for a beam element as follows;

1

i i

i i

x l x l

l l s

+

− −= =

− (9)

Where x is the axial co-ordinate in the global co-ordinate system. li is the axial distance

of the ith node from the origin. Now s is constant for the same element spans and depends

upon the number of elements taken for the analysis. So the value of ζ varies from 0 to 1

for a given element.

According to Euler-Bernoulli beam theory the bending moment M is related to the

transverse displacement W(x, t) of the beam as follows;

( ) ( )2

2

wM E x I x

x

=

(10)

Where E(x) young’s modulus and I(x) is the area moment of inertia of the cross section of

beam located at a distance x from the origin O. In terms of local co-ordinate eqn. (10) can

be written as;

( ) ( )2

2 2

1 wM E I

s

=

(11)

Strain energy ‘U’ of beam element can be found out as;

( ) ( )

2

02

LM

U dxE x I x

=

(12)

Now putting the value of M from eqn. (10) strain energy can be expressed as

( ) ( )2

2

2

0

1

2

Lw

U E x I x dxx

=

(13)

Again in terms ζ, strain energy can be expressed as;

( ) ( )21 2

3 2

0

1

2

wU E I d

s

=

(14)

Now [ N(ζ)] = [ N1 N2 N3 N4 ] and {d} = [w1 θ1 w2 θ2]T

. Ni here is called as shape

function and these are defined in terms of Hermite’s polynomials as follows;

( )

( )

2 3 2 3

1 2

2 3 3 2

3 4

1 3 2 ; 2

3 2 ;

N N s

N N s

= − + = − +

= − = −

(15)

Then

22 2 22

31 2 4

2 2 2 2 2,

NN N Nwd N d

= =

(16)

And

2

2

2, ,

TTwd N N d

=

(17)

So eqn. (14) can be written as;

( ) ( ) 1

3

0

1 , ,

2

TTU d E I N N d d

s = (18)

The strain energy of the beam element can also be expressed as;

1

2

TU d k d= (19)

Here [k] is the element stiffness matrix thus can be defined as;

( ) ( )1

3

0

1, ,

T

k E I N N ds

=

(20)

In this similar manner the consistent element mass matrix [m] of the beam can be found

out. The kinetic energy T of the beam can be calculated as;

( ) ( ) 2

0

1

2

L

T x A x w dx=

(21)

Now ( ) w N d= and 2T T

w d N N d=

(22)

The kinetic energy of the beam element can also be defined as

1

2

T

T d m d=

(23)

Hence ( ) ( ) 1

0

Tm s A N N d =

(24)

Since the strain energy and the kinetic energy of the beam element are known, using the

Euler-Lagrange equation of motion [6] of the form

0g gL Ld

d dt d

− =

(25)

Where the Lagrangian functional Lg is defined as: - Lg = T – V (26)

Now using the equations (19), (23) and (26) in eqn. (25) the transverse motion of the beam

element can be expressed as;

0m d k d+ = (27)

Since {d} is a function of time only, it can be expressed as;

1 1

1 2

2 3

2 4

i t i t

w d

dd e D e

w d

d

= = =

(28)

Where ω is the fundamental frequency of the beam element undergoing transverse

vibration. Now putting {d} in eqn. (27) will yield an eigenvalue problem as follows;

2 0i tk m D e − = (29)

3.3 Assembly of elements

Equation (29) is for a single element but as it is known that in finite element analysis for

better accuracy of the results number of elements must be increased considering all other

factors fixed. Hence there is a need of increment in the number of elements for sure but

again there arises the problem of assembling the elements, since each of the elements is

different from other in terms of geometry as well as material properties.

In order to solve the problem every time before the assembly we need to find out the

element stiffness and mass matrix of each element, by solving eqn. (20) and (24)

respectively, unlike the assembly of homogeneous uniform beam elements.

For a single element problem:

Material properties are:

( ) ( )L R L RE x E E E V= + − (30)

( ) ( )L R L Rx V = + − (31)

Where the value of VR can be calculated from eqn. (5) and (6) or eqn. (7)

Geometric properties are:

( ) ( ). A x b g x= (32)

( )( )

3

12

b g xI x = (33)

Where value of ‘g(x)’ can be found out from eqn. (1). So as the number of elements

increases the global co-ordinate ‘x’ in each of the 4 equations above has to be replaced by

the expression given below and then the element stiffness and mass matrices can be found

out for the assembly.

ix l s= + (34)

Where all the symbols have the same meaning defined earlier. Since a lot of tedious

calculations, integration etc. are involved, MATLAB coding platform was used to solve

the problem.

Details of the steps involved in the mathematical modeling and Finite Element Analysis

are depicted in Fig.5. Any unsatisfactory results obtained in the FEA for the natural

frequency will repeat the process, till the satisfactory result is obtained.

4. Numerical results & discussion:

The section is mainly devoted to the numerical results for the natural frequencies for the

free transverse vibration of uniform and non-uniform beams in different boundary

conditions. Firstly, results are found out for uniform and then non-uniform beam.

Aluminum (Al) and zirconia (ZrO2) have been used as the metal and ceramic constituents

respectively as shown in Table-1. For the calculation purpose b = 0.02, h0 = 0.001 and L

= 0.2 have been considered for uniform beam from ref. [13] in order to compare the results

found in the present method. For non-uniform beam L=1, b= 0.02, h0 = 0.01. In this study

material used at the left most surface (at x = 0) of the beam is Al and ZrO2 is used at the

right most surface (at x = L).

For non-dimensional natural frequency (λ) calculation the formula given below is used

for comparison with the ref. [13];

2 L L

L L

AL

E I

=

Where;

ω is the angular natural frequency and depends upon the mode of vibration (rad/s)

ρL is density of the material used at the left most surface of the beam (kg/m3)

AL is the area of cross-section of the beam at the left most part of the beam (m2)

EL is Young’s modulus of the material present at left most part of the beam (Pa)

IL is the area moment of inertia at the left most part of the beam (m4)

L is the length of the beam (m)

After using all the similar conditions of ref. [13] results of the present method are found

to have a good conformation with the results of the same as shown in Table-2, in fact it is

lesser than the results of ADM used in ref. [13] and slightly greater than the results of

FEM using ANSYS in ref. [13]. The mass of uniform beam for exponential law and power

law, with same dimensions have been compared and α =3 gives the same amount of mass

in exponential grading as that of power law grading when n = 2.5. It can be observed from

Table-3 that for the uniform beam of same mass both in exponential law grading and

power law grading produces more or less the same natural frequencies in different

boundary conditions.

In case of non-uniform beam the fundamental frequency results are lesser compared to

that of uniform beam which is evident from Table-4. It is expected because the beam with

exponentially decreasing thickness was considered and with the reduction in thickness the

beam loses some material which leads to decrease in its mass as well as stiffness as

Zirconia with higher stiffness was considered as material at the right most surface of the

beam. So this could have been the cause of decrement in the fundamental frequencies.

Again Table-5 shows, there is not much of difference between the results of power law

gradation and exponential law of gradation provided same mass is considered.

From the above Table-1 it is evident that the numerical outcomes of the present method

are reasonably convergent with those of ADM and software package-based FEM of [13].

To understand the convergence in a better manner a graphical comparison between the

results of current method and the average of ADM and FEM results of [13] have been

taken together, and are presented for each boundary condition as shown in Fig.6. The

percentage of difference A graph (Fig.7) depicting the Absolute Mean Deviation

percentage from the average value considered from the results of ref [13] is also presented

in order to show the effect of geometric constraint (i.e. the applied boundary conditions)

on the current analysis.

Table-6 to 9 are dedicated to show the effect of geometry parameter and the gradation

parameter on the fundamental frequency of non-uniform beam in different boundary

conditions. It can be clearly seen from the Table-6 and 7 with the increase in geometry

parameter the non-dimensional frequencies are reducing. Fig.8 also provides graphical

representation of monotonously decreasing characteristics of fundamental frequencies

with respect to increase in the geometric parameter β. After observing the results with

different values of β, it is understood that for comparable frequency results with the

uniform AFG beam β should be less than 0.1. Hence in order to check the effect of gradient

parameters on the fundamental frequencies the value of β = 1 was considered. There are

some unpredictable trend which might be coming because with the increase in β the beam

gets softer but at the same time the mass of the beam is also getting reduced, so we can’t

apprehend the trend in a usual manner. From Fig.2 and 3 it can be clearly observed that

with the increase in gradation parameters the overall material quantity is reduced even if

it is having same dimension. This in turn leads to reduction in stiffness as well as mass of

the beam. So, it is naturally expected to have a direct impact on the fundamental

frequencies of the beam. In case of power law grading the fundamental frequencies go on

decreasing continuously up to n = 5 (Fig.9) and then becomes constant implying that the

further increase in the power law exponent has no effect on the fundamental frequencies.

This is unlike exponential law of grading where the changes mostly occur nearby α = 0

and remains constant before as well as after (Fig.10). Table 8 and 9 again make it evident

that with the increase of gradation parameter α and n the fundamental frequencies

decrease, with the exception of 1st Mode in clamped-free and clamped-clamped boundary

conditions. Although the increase in constraint results in higher fundamental frequencies,

the trend is of non-decreasing nature in these two boundary conditions.

5. Conclusion:

In the analysis a generalized finite element approach based on Euler-Bernoulli beam

theory, has been considered to solve the free vibration problems of beams with non-

uniform thickness and inhomogeneous material distribution along its axis. The approach

is infact a generalized one because it can solve free vibration problems of beams with

various kinds of non-uniformities in thickness as well as inhomogeneity in material

distribution, the only requirement being they have to be a function of axial distance from

origin. In order to validate the effectiveness of current approach the fundamental

frequencies obtained were compared with that of uniform inhomogeneous beams analysis

done by Cao et al. [13] for the respective boundary conditions, by taking the geometry

parameter as 0 for the non-uniform beam model. After finding a decent level of

conformation the present method was utilized to assess the effect of the variation in

exponential geometric parameter on the fundamental frequencies of the non-uniform AFG

beam. It was found that the fundamental frequency results are directly dependent upon the

geometric non-uniformity parameter. After observing the impact of variation in material

gradient parameters on the numerical outcomes of the analysis the following points can

be concluded;

• With the increase in material gradient parameters the fundamental frequencies in each

mode got reduced for different boundary conditions. This might be due to the fact

that with increase in these parameters the dominance of Aluminum increases in the

volume fraction over zirconia which makes the system more flexible and the lesser

value of overall stiffness results in lower fundamental frequency. Both the gradation

schemes give almost equal results when considered in equal masses.

• The increase in boundary constraints lead to increase in the values of fundamental

frequencies as expected because initial modes of less constrained systems get

suppressed with the increase in constraints. Hence in order to vibrate naturally a

higher value of frequency is needed, even if the system is to vibrate in the first mode.

Conflict of Interest: The authors declare that they have no conflict of interest.

Nomenclature

L Axial length of the beam

b Width of the beam

x Axial distance from the origin

( )g x Exponentially variable thickness function

0h Thickness of the beam at the origin

Exponential geometry parameter

( )E x Young’s modulus function

( )x Mass density function

( )A x Cross-sectional area function

( )I x Area moment of inertia function

( )P x Effective material property function

LP Effective material property at the left end of the beam

RP Effective material property at the right end of the beam

LV Volume fraction at the left end of the beam

RV Volume fraction at the right end of the beam

Exponential law parameter

n Power law exponent

w Transverse displacement function

Local co-ordinate for an element

s Span of an element

M Bending moment

U Strain energy of the beam element

T Kinetic energy of the beam element

gL Lagrangian function

Fundamental angular frequency of the beam element

Non-dimensional natural frequency of the beam

E Young’s modulus of the material

Mass density of the material

C F− Clamped-free boundary condition

H H− Hinged-hinged boundary condition

C H− Clamped-hinged boundary condition

C C− Clamped-clamped boundary condition

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Figure Captions:

Figure No. Caption

Figure 1 The non-uniform geometry of the beam with variable thickness g(x)

Figure 2 Exponential property variation with respect to beam length for PR = 5

PL

Figure 3 Property variation based on Power law with respect to beam length for

PR = 5 PL.

Figure 4 A two-node non-uniform beam element

Figure 5 Flow-chart for the steps involved in Mathematical modelling and Finite

Element Analysis

Figure 6 Comparison between the Non-dimensional natural frequencies (λ) for

different in different modes of vibration for (a) Clamped-Free (b)

Hinged-Hinged (c) Clamped-Hinged (d) Clamped-Clamped boundary

conditions.

Figure 7 Absolute Mean Deviation Percentage of Non-dimensional natural

frequency (λ) from ref [13] in respective different boundary conditions

Figure 8 Effect of geometric non-uniformity parameter β on the fundamental

frequencies of the AFG beam considering n=2.5 in different boundary

conditions (a) Clamped - Free boundary condition (b) Hinged - Hinged

boundary condition (c) Clamped – Hinged boundary condition (d)

Clamped – Clamped boundary condition at the left and right ends of the

AFG non-uniform beam.

Figure 9 Effect of power Law exponent n on the fundamental frequencies of the

AFG beam in different boundary conditions (a) Clamped - Free

boundary condition (b) Hinged - Hinged boundary condition (c)

Clamped – Hinged boundary condition (d) Clamped – Clamped

boundary condition at the left and right ends of the AFG non-uniform

beam respectively.

Figure 10 Effect of the Exponential Law parameter α on the fundamental

frequencies of the AFG beam in different boundary conditions (a)

Clamped - Free boundary condition (b) Hinged - Hinged boundary

condition (c) Clamped – Hinged boundary condition (d) Clamped –

Clamped boundary condition at the left and right ends of the AFG non-

uniform beam respectively.

Table Captions:

Table No. Caption

Table 1 Properties of the constituent materials used in ref. [13]

Table 2 Comparison of non-dimensional fundamental frequencies (λ) of uniform

AFG (Al-ZrO2) beam in different boundary conditions for α = 3.

Table 3 Comparison of non-dimensional fundamental frequencies of uniform

AFG (Al-ZrO2) beam for power law and exponential law of gradation

in different boundary conditions for α = 3 and n = 2.5.

Table 4 Comparison of non-dimensional fundamental frequencies of AFG (Al-

ZrO2) uniform beam vs. non-uniform beam in different boundary

conditions for α = 3 and β = 0.1.

Table 5 Comparison of non-dimensional fundamental frequencies of AFG (Al-

ZrO2) non-uniform beam in different boundary conditions for α = 3, n =

2.5 and β = 0.1.

Table 6 Variation of non-dimensional fundamental frequencies of AFG (Al-

ZrO2) non-uniform beam with respect to change in geometry parameter

(β) with power law grading (n=2.5) in different boundary conditions.

Table 7 Variation of non-dimensional fundamental frequencies of AFG (Al-

ZrO2) non-uniform beam with respect to change in geometry parameter

(β) with exponential law of grading (α=3) in different boundary

conditions.

Table 8 Variation of non-dimensional fundamental frequencies of AFG (Al-

ZrO2) non-uniform beam (β = 0.1) with respect to power law exponent

(n) in different boundary conditions.

Table 9 Variation of non-dimensional fundamental frequencies of AFG (Al-

ZrO2) non-uniform beam (β = 0.1) with respect to exponential law

parameter (α) in different boundary conditions.

Figures:

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

(a)

(b)

(c)

(d)

Figure 6

Figure 7

(a)

(b)

(c)

(d)

Figure 8

(a)

(b)

(c)

(d)

Figure 9

(a)

(b)

1

(c)

(

d

)

2

Fig.10

Tables:

Table 1

Properties Unit Aluminum Zirconia (ZrO2)

E GPa 70 200

ρ Kg/m3 2702 5700

Table-2

Boundary

condition Method used 1st mode 2nd mode 3rd mode 4th mode

C-F

Present 2.854463 21.4957915 63.68948972 126.7057

Ref.[13] 2.863 21.394 63.712 127.21

FEM Ref.[13] 2.852 21.494 63.673 126.575

H-H

Present 10.36696 41.9729437 94.5556821 168.2817

Ref.[13] 10.5 42.376 95.525 169.937

FEM Ref.[13] 10.368 41.973 94.51 167.997

C-H

Present 15.71721 52.8117755 110.689377 189.7663

Ref.[13] 15.952 53.417 111.903 191.646

FEM Ref.[13] 15.718 52.807 110.611 189.356

C-C

Present 24.93755 67.1232031 130.3613409 214.8374

Ref.[13] 25.625 68.537 132.592 217.742

FEM Ref.[13] 24.942 67.113 130.236 214.258

Table-3

Boundary

Conditions Gradation Scheme Mode1 Mode2 Mode3 Mode4

C-F

Power Law Ref.[21] 2.788372 21.46919 63.90569 126.9603

Exponential Law

Ref.[13] 2.854463 21.49579 63.68949 126.7057

H-H Power Law Ref.[21] 10.32393 41.95427 94.46947 168.0887

3

Exponential Law

Ref.[13] 10.36696 41.97294 94.55568 168.2817

C-H

Power Law Ref.[21] 15.68958 52.92778 110.7761 189.7579

Exponential Law

Ref.[13] 15.71721 52.81178 110.6894 189.7663

C-C

Power Law Ref.[21] 24.96248 67.07427 130.1854 214.5121

Exponential Law

Ref.[13] 24.93755 67.1232 130.3613 214.8374

Table 4

Boundary condition Method used Mode1 Mode2 Mode3 Mode4

C-F Present 2.89199 20.8508 60.97785 120.9449

Ref.[13] 2.863 21.394 63.712 127.21

H-H Present 9.894899 39.92297 89.94498 160.0938

Ref.[13] 10.5 42.376 95.525 169.937

C-H Present 15.17266 50.42204 105.4868 180.7283

Ref.[13] 15.952 53.417 111.903 191.646

C-C Present 23.65832 63.78144 123.9482 204.3266

Ref.[13] 25.625 68.537 132.592 217.742

Table 5

Boundary condition Gradation type Mode1 Mode2 Mode3 Mode4

C-F

Exponential law 2.89199 20.8508 60.97785 120.9449

Power law 2.824893 20.8322 61.18508 121.1914

H-H

Exponential law 9.894899 39.92297 89.94498 160.0938

Power law 9.849662 39.86223 89.78602 159.7856

C-H

Exponential law 15.17266 50.42204 105.4868 180.7283

Power law 15.14726 50.49737 105.495 180.5936

4

C-C

Exponential law 23.65832 63.78144 123.9482 204.3266

Power law 23.68861 63.71877 123.7262 203.9088

Table 6

Boundary

Condition Mode

Exponential geometry parameter (β)

0.1 0.2 0.4 0.6 0.8 1

C-F

1 2.8 2.9 2.9 3 3 3

2 20.8 20.2 19 17.8 16.6 15.6

3 61.2 58.6 53.6 49 44.8 40.9

4 121.2 115.6 105.2 95.6 86.8 78.7

5 201.4 192 174.2 157.9 142.9 129.2

H-H

1 9.9 9.4 8.5 7.7 6.9 6.2

2 40 38 34.3 30.9 27.9 25.1

3 90.5 86.1 77.7 70.2 63.3 57

4 163.1 155 140.2 126.9 115 104.1

5 276.7 263.4 237.7 213.1 190.1 169.1

C-H

1 15.2 14.6 13.6 12.652 11.7531 10.9143

2 50.7 48.3 44 40.1055 36.5207 33.2545

3 106.6 101.6 92.2 83.6131 75.8374 68.7797

4 184.2 175.6 159.9 145.8132 133.019 121.1962

5 308.6 292.1 261 232.6496 207.2354 184.8249

C-C

1 23.7 22.5 20.3 18.3 16.6 15.1

2 63.8 60.6 54.7 49.4 44.7 40.4

3 124.2 118 106.6 96.3 86.9 78.5

4 205.6 195.5 176.6 159.5 144 129.8

5 309.3 294.2 265.9 240 216.5 195.4

Table 7

Boundary

condition

Mode Exponential geometry parameter (β)

0.1 0.2 0.4 0.6 0.8 1

C-F 1 2.9 2.9 3 3 3.0723 3.0909

5

2 20.9 20.2 19 17.8 16.6391 15.5417

3 61.2 58.6 53.6 49 44.8122 40.9153

4 122.4 116.7 106.1 96.3 87.35 79.1836

5 203.8 193.9 176.1 160.4 146.4825 133.9566

H-H

1 9.9 9.4 8.5 7.7 6.9 6.1

2 40 38 34.3 31 27.9 25.2

3 90.6 86.1 77.8 70.3 63.4 57.1

4 163.2 155.2 140.4 127.2 115.2 104.4

5 277.1 263.8 237.9 213.2 190.2 169.2

C-H

1 15.2 14.6 13.6 12.6668 11.7702 10.9362

2 50.5 48.3 44 40.0451 36.4733 33.2177

3 106.5 101.5 92.1 83.5308 75.7661 68.7225

4 184.3 175.8 160.1 145.9887 133.1571 121.2849

5 308.8 292.2 261 232.6166 207.2262 184.8701

C-C

1 23.7 22.5 20.3 18.3 16.6 15.1

2 63.8 60.7 54.8 49.5 44.7 40.4

3 124.3 118.2 106.8 96.4 87 78.5

4 205.9 195.7 176.9 159.7 144.1 129.9

5 309.8 294.6 266.1 240.1 216.6 195.5

Table 8

Boundary

condition

Mode Power law exponent n

0.1 0.2 0.5 1 2 5 10

C-F

1 3.9 3.7 3.3 3 2.9 2.9 3.1

2 24.2 23.6 22.5 21.7 21 20.3 20

3 67.8 66.9 65 63.6 62 59.2 57.6

4 134 132.6 129.8 127.1 123.8 118.8 115.5

5 222.9 221.2 217.2 212.5 206.1 197.5 192.9

H-H

1 10.9 10.8 10.6 10.4 10 9.6 9.4

2 43.5 43.3 42.6 41.6 40.4 38.8 38.1

3 98.5 98 96.3 94.1 91.3 88.2 86.5

6

4 177.8 176.7 173.8 169.9 164.7 158.7 155.9

5 300.9 298.8 293.5 287.1 279.2 270.2 265.9

C-H

1 16.8 16.4 15.7 15.3 15.2 15 14.9

2 54.6 53.9 52.4 51.6 50.9 49.4 48.4

3 114.7 113.4 111 109 106.9 103.6 101.6

4 198.2 196.4 192.6 188.9 184.4 178.6 175.2

5 307.5 305 299.4 292.8 284.2 274.6 270.6

C-C

1 24.3 23.9 23.2 23.1 23.5 23.9 23.6

2 67.3 66.5 65 64.3 63.9 63.4 63.1

3 132.7 131.3 128.8 126.8 124.8 122.5 121.9

4 221.1 219.1 215.1 211.3 207 202.1 200.7

5 334 331.4 325.5 319.2 311.6 303.8 300.6

Table 9

Boundary

Condition Mode

Exponential law parameter (α)

-10 -5 -3 0 3 5 10

C-F

1 3.6 3.3 3.2 3 2.9 2.9 3.1

2 23.1 22.7 22.4 21.7 20.9 20.5 20.1

3 65.9 65.4 65 63.6 61.2 59.8 57.9

4 131.8 130.9 130.1 127.1 122.4 119.7 116

5 221.3 220.1 218.4 212.5 203.8 199.2 193.5

H-H

1 10.9 10.8 10.7 10.4 9.9 9.7 9.5

2 43.5 43.1 42.7 41.6 40 39.2 38.2

3 98.4 97.6 96.7 94.1 90.6 88.9 86.8

4 177.4 176.1 174.6 169.9 163.2 160 156.4

5 298.7 296.5 294.3 287.1 277.1 272.2 266.7

C-H

1 15.9 15.6 15.5 15.3 15.2 15.1 15

2 52.9 52.5 52.2 51.6 50.5 49.8 48.7

3 113 112.3 111.7 109.7 106.5 104.7 102.3

7

4 198.9 197.6 196.1 191.2 184.3 181 177.4

5 331.3 329.9 327.9 320.3 308.8 302.8 295.6

C-C

1 22.9 22.7 22.7 23.1 23.7 23.8 23.6

2 65 64.6 64.5 64.3 63.8 63.5 63.1

3 129.7 129 128.5 126.8 124.3 123.1 122

4 217.7 216.5 215.2 211.3 205.9 203.3 201

5 330.8 328.6 326.3 319.2 309.8 305.6 301.2

Technical Biography

Ramya Prakash Sahu is a M.Tech. Scholar in Machine Design and Analysis specialization,

in Mechanical Engineering Department of Veer Surendra Sai University of Technology Burla,

Odisha, India. He completed in B.Tech. from IGIT Sarang in the year 2017 and his M.Tech.

from Veer Surendra Sai University in the year 2020.

Dr. Mihir Kumar Sutar has completed his PhD from Indian Institute of Technology Roorkee

in 2015 and currently working as Assistant Professor in the Mechanical Engineering

Department of Veer Surendra Sai University of Technology Burla, Odisha, India. His expertise

area of interest includes Machine Design and Analysis and Robotics.

Dr. Sarojrani Pattnaik has completed her PhD from Indian Institute of Technology Roorkee

in 2014 and currently working as Associate Professor in the Mechanical Engineering

8

Department of Veer Surendra Sai University of Technology Burla, Odisha, India. Her expertise

area of interest includes Production Engineering.