# Free Transverse Vibration Analysis of Axially Functionally Graded Tapered Euler-Bernoulli Beams through Spline Finite Point Method

Authors: Peng Liu; Kun Lin; Hongjun Liu; Rong Qin
Journal: Shock and Vibration
 (2016)

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## Abstract

A new model for the free transverse vibration of axially functionally graded (FG) tapered Euler-Bernoulli beams is developed through the spline finite point method (SFPM) by investigating the effects of the variation of cross-sectional and material properties along the longitudinal directions. In the proposed method, the beam is discretized with a set of uniformly scattered spline nodes along the beam axis instead of meshes, and the displacement field is approximated by the particularly constructed cubic B-spline interpolation functions with good adaptability for various boundary conditions. Unlike traditional discretization and modeling methods, the global structural stiffness and mass matrices for beams of the proposed model are directly generated after spline discretization without needing element meshes, generation, and assembling. The proposed method shows the distinguished features of high modeling efficiency, low computational cost, and convenience for boundary condition treatment. The performance of the proposed method is verified through numerical examples available in the published literature. All results demonstrate that the proposed method can analyze the free vibration of axially FG tapered Euler-Bernoulli beams with various boundary conditions. Moreover, high accuracy and efficiency can be achieved.


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## Body

## 1. Introduction

Functionally graded (FG) materials were initially proposed by material scientists from Japan in 1984 as thermal barrier materials; since then, the development of engineering application and theoretical research on structures with FG materials has made a great breakthrough because of the particular properties of these materials, such as high stiffness and thermal resistance [1–3]. FG materials are multiphase composites with the volume fraction of phase varying through any desired direction [4]. FG material gradation along an arbitrary direction can be expressed with power (polynomial) laws and exponential laws, which are the most renowned theories for FG material description. The gradient variation for FG beams may also be oriented in the thickness direction, longitudinal direction, and bidirection (oriented in both the thickness and lengthwise directions).During the past decade, the problems of the stability, dynamic response, and linear/nonlinear free vibration behavior of the FG structures with material properties varying along the thickness direction have been thoroughly investigated by researchers around the world, and lots of numerical and analytical solutions have been obtained [5–11]. However, given the existence of variable coefficients on the governing equation, the free vibration analysis of beams with nonuniform cross sections and variable material properties along the longitudinal direction becomes more complicated than the beams with uniform cross sections and homogenous materials [3]. And so far, few analytical solutions are found for the free vibration of the axially FG tapered beams due to the difficulty of mathematical treatment of the problem; for example, Elishakoff et al. [12, 13] proposed the close-form solutions for axially FG tapered beam with material gradation that follows the polynomial law using the semi-inverse method; however, the proposed semi-inverse method cannot be applied for graded beams of any axial nonhomogeneity. Therefore, a large number of numerical methods have been proposed in the reported literatures by other researchers, according to which lots of numerical models, based on different beam theories, namely, the Euler-Bernoulli beam theory (EBT) [2, 3, 14–20] and the Timoshenko beam theory (TBT) [21–25], have been constructed to investigate the free vibration of the tapered beams considering the material property homogeneity or nonhomogeneity. However, it is well understood that the effects of the shear deformation and rotary inertia of the cross section for the long slender beams can be neglected; thus the EBT models are capable of analyzing the vibration problems of the long slender beams and the reliable results with high accuracy can be achieved. In this paper, the authors mainly engage in establishing a new EBT model for the free vibration of axially tapered beams (long slender beams).Based on EBT, Shahba et al. [2] firstly investigated the stability and free vibration of the axially FG Euler beams considering cross-sectional breadth and height both linearly tapered along the longitudinal direction by using the finite element method (FEM); then, in order to improve computational efficiency and modify the convergence conditions, Shahba and Rajasekaran [3] developed other models by using the differential transform element method (DTEM) and differential quadrature element method of the lowest order (DQEL) to investigate the same problems. Results show that the DTEM and DQEL models obtained better convergence and had fewer elements than the FEM model. However, with the increase of the taper ratio, all models proposed by Shahba et al. needed to solve a larger scale of eigenvalue equations in order to achieve a stable accuracy. Huang and Li [14] studied free vibration problems of axially FG tapered Euler beams through a new approach that transformed the governing equation into the Fredholm integral equation with consideration of the case of material properties in polynomial and exponential variations along longitudinal direction. Li et al. [15] developed the exact frequency equation for axially exponentially FG Euler beams, according to which the results showed that the natural frequencies noticeably depend on exponential gradient parameter and boundary conditions. Akgöz and Civalek [16] developed a model combined with the EBT and modified coupled stress theory to study the free vibration of the axially FG micro-Euler beams, in which the Rayleigh-Ritz solution method was utilized to obtain the natural transverse frequencies. Mao and Pietrzko [17] studied the free vibration of tapered Euler beams with a continuously exponential variation of breadth and a constant height along the length through Adomian decomposition method. Hsu et al. [18] developed a model to investigate the free vibration of homogenous tapered Euler beams with elastic supports through Adomian modified decomposition method. Rajasekaran [4] and Özdemir and Kaya [19] applied differential transform method (DTM) to study the free bending vibration of rotating tapered Euler beams by considering material nonhomogeneity and homogeneity along the longitudinal direction. Banerjee et al. [20] applied the dynamic stiffness method (DSM) to analyze the free bending vibration of the rotating homogeneous tapered Euler beams; though the high accuracy was obtained, the modeling process was not convenient. Other models have been investigated based on TBT; Shahba et al. [21] developed the FEM model to study the free vibration of the axially FG taper Timoshenko beams. Rajasekaran and Norouzzadeh Tochaei [22, 23] developed different models through DTM and DQEL to study axially FG Timoshenko beams by considering the effects of rotating and nonrotating. Ozgumus and Kaya [24] investigated model through DTM for the free bending vibration of rotating homogeneous tapered Timoshenko beams by considering the effects of taper ratio, rotating speed, and slenderness ratio. Yardimoglu [25] investigated a new FEM model with displacement functions derived from the coupled displacement field of transverse displacement and cross-sectional rotation to study the free bending vibration of the rotating Timoshenko beams.Among the aforementioned numerical models for the axially tapered beams, such as FEM, DTM, DTEM, and DSM models, the disadvantages are obvious. (1) For the FEM models, the element generation and assembling are both needed in the modeling process; the displacement interpolation functions of FEM models were unable to fully capture the variation of the cross section and material properties along the longitudinal direction due to the homogenous uniform beams elements were utilized, and with the increase of the taper ratio and gradient parameter, the results’ error and elements numbers increase noticeably [2, 21, 25]. (2) For the DTM and DTEM models, although both models obtained high accuracy, these two numerical models needed to solve a large scale of eigenvalue equations, thereby leading to a high computational cost; what is more, the DTM model encountered convergence problems in some cases [3, 4, 19]. (3) For the DSM models, the most common disadvantages were their complicated modeling process [20]. Therefore, the aforementioned numerical models need to be improved, and a new model with convenience modeling process, high accuracy, and low computational cost will benefit the analysis of the axially tapered Euler beams.Spline finite point method (SFPM) was firstly proposed by Professor Qin in 1979 [26, 27], who proposed a cubic B-spline function-based method for structural analysis. Given that the interpolation of structural displacement is constructed through a set of scattered spline nodes along one direction instead of meshes, the element generations and assembling are both unnecessary for the modeling process of SFPM; thus, the SFPM contains the excellent feature of no mesh distortion. The process of modeling and computer programming can be achieved easily, which leads to high computation accuracy and efficiency because the interpolation functions used in SFPM have explicit forms. The performances of SFPM on the structures with uniform cross section and material property that keeps constant along the longitudinal direction were validated by reported works on the FGM beams/plates, laminates, and other structures [28–32]. Li et al. [28] investigated the free vibration of piezoelectric smart FGM plates using SFPM. Then, Li et al. [29] investigated a new model through bidirectional B-spline finite point method (B-SFPM) to analyze the displacement and stress distribution of a piezoelectric laminate composite plate combined with the problems of material parameter identification; Zhou and Li [30] studied the free vibrations of asymmetrical sandwich plates with laminated faces and orthotropic core through SFPM. Qin et al. [31, 32] developed different models through SFPM to study the nonlinear behaviors of composite plates and the seismic behavior of a box arch bridge. Other researchers applied the spline strip method to the free vibration of rectangular composite laminates [33, 34], which also validated the advantages of application of the cubic B-spline functions. However, to the best of the authors’ knowledge, the SFPM have not been applied to analyze the free vibration of the axially FG tapered beams before, and similar works or models have been rarely found in the reported literatures as well. In this study, the authors introduce the SFPM to analyze the vibration behavior of the axially FG tapered Euler-Bernoulli beams for the first time, which provide a novelty approach for solving such problems.The main objective of this study is to develop new simple models to study the free vibrations of axially FG tapered Euler-Bernoulli beams by using SFPM. The new models are able to achieve faster modeling process, higher accuracy, and lower computational cost compared with those of reported numerical models. On the basis of SFPM, the axially FG tapered beam is discretized with a set of uniformly scattered spline nodes along the beam axis instead of meshes, and the displacement field is approximated by particularly constructed cubic B-spline interpolation functions, which possess good adaptability for various boundary conditions. Both the global structural stiffness and mass matrices for beams are directly generated after spline discretization, thereby making the modeling convenient. Because of the employment of presented displacement interpolation functions, the derived structural stiffness and mass matrices being in the explicit forms are capable of directly taking account of any variations of the material properties and cross-section profile for the axially FG beams directly. A suitable spline node number is discussed and selected to maintain high computational accuracy based on the convergence analysis. The processes of boundary condition treatment and derivation of the structural matrices are introduced in detail. The effects of the variation of cross-sectional and material properties along the longitudinal direction are investigated. The performance of the proposed method is verified with numerical examples available in the published literature. All results demonstrate that the proposed method can analyze the free vibration of axially FG tapered Euler-Bernoulli beams with various boundary conditions with quick convergence and high accuracy.


## 2. Structural Model

Figure1 shows the axially FG tapered beams in length L. The beam with cross-sectional parameters, such as the cross-sectional area A
(
x
) and the second moment of inertia I
(
x
), and with material properties, such as Young’s modulus E
(
x
) and the mass density ρ
(
x
), continuously vary along the lengthwise direction, which are the functions of longitudinal coordinate x.Figure 1
Schematic of an axially FG tapered beam.According to EBT, the following assumptions are made: (1) the cross section of FG beams remains plane after deformation; (2) the slenderness ratio (
L
/
h
0
) is big enough that the effects of rotary inertia and shear deformation in x
-
z plane can be ignored; (3) the transverse deformation is small enough that only the linear vibration response is considered. Given that the shear deformation and rotary inertia are not considered, the axial and transverse displacement fields for axially FG tapered beams are presented, respectively, as follows:(1)
u
=
u
x
,
y
,
z
,
t
=
-
z
∂
w
x
,
t
∂
x
,
w
=
w
x
,
y
,
z
,
t
=
w
x
,
t
,where x, y, and z are the spatial coordinates and w is the transverse displacement. The axial and shear strains are, respectively, obtained as follows by using (1):(2)
ε
x
x
=
-
z
∂
2
w
∂
x
2
,
γ
x
z
=
0
.The stresses can be given as (3)
σ
x
x
=
E
x
ε
x
x
,
τ
x
z
=
0
,where σ
x
x is the axial stress. The shear stress τ
x
z is equal to zero because shear deformation is not considered in the model. The strain and kinetic energies of the axially FG tapered Euler beams can be written, respectively, as (4)
U
=
1
2
∫
0
L
∫
A
x
σ
x
x
ε
x
x
d
A
d
x
=
1
2
∫
0
L
E
x
I
x
∂
2
w
∂
x
2
2
d
x
,
T
=
1
2
∫
0
L
∫
A
x
ρ
x
∂
w
∂
t
2
d
A
d
x
=
1
2
∫
0
L
ρ
x
A
x
∂
w
∂
t
2
d
x
.The following can be obtained by applying the Hamilton principle: (5)
δ
Π
=
δ
∫
t
1
t
2
U
-
T
=
0
,where Π is the total potential energy function for the axially FG tapered beam, which can be derived by some manipulations from (4) and (5). The derivation of Π can be seen in a reported work [19] and can be expressed by(6)
Π
=
1
2
∫
0
L
E
x
I
x
∂
2
w
∂
x
2
2
-
2
w
-
ρ
x
A
x
∂
2
w
∂
t
2
d
x
.The governing equations of motion for an axially FG tapered Euler beam undergoing transvers free vibration are derived as follows through the variation principle:(7)
∂
2
∂
x
2
E
x
I
x
∂
2
w
∂
x
2
-
ρ
x
A
x
∂
2
w
∂
t
2
=
0
.


## 3. SFPM Formulation

### 3.1. Definition of Cubic B-Spline Function

The cubic B-spline functionφ
3
(
x
) is adopted in this study and is defined by the following expressions [27]: (8)
φ
3
x
=
∑
k
=
0
4
-
1
k
4
k
x
-
x
k
+
3
3
!
x
∈
-
2
2
,
(9)
x
-
x
k
+
3
=
x
-
x
k
3
x
-
x
k
≥
0
0
x
-
x
k
<
0
x
k
=
k
-
2
.According to the characteristics of φ
3
(
x
), its derivative and integration can be expressed by the following:(10)
φ
3
j
x
=
∑
k
=
0
4
-
1
k
4
k
x
-
x
k
+
3
-
j
3
-
j
!
x
∈
-
2
2
,where φ
3
(
j
)
(
x
) denotes the jth derivative (
j
>
0
) or jth integration (
j
<
0
) for φ
3
(
x
) with the variation of “j”; if j
=
0, φ
3
(
x
) is obtained; if j
>
3, φ
3
(
j
)
(
x
)
=
0. Moreover, if j
=
3, φ
3
(
3
)
(
x
) is maintained as a stepped constant value within the interval -
2
2 as shown in Figure 2.Figure 2
Variation value of functionφ
3
(
3
)
(
x
).


### 3.2. SFPM Modeling

Figure3 illustrates that the beam is modeled via SFPM considering an axially FG tapered Euler beam described in Figure 1. The beam is discretized along the beam axis through uniformly scattered spline nodes on the entire beam. The tapered beam can be divided by the interval h
x in x direction, and the local coordinates of the corresponding nodes in the axis of the beam can be presented by(11)
x
i
=
x
0
+
i
h
x
i
=
1
,
…
,
N
,
(12)
h
x
=
L
N
,
(13)
0
=
x
0
<
x
1
<
x
2
<
⋯
<
x
N
-
1
<
x
N
=
L
,where N stands for the number of subdivisions of beams in x direction, which is also called the maximum number of spline nodes; x
0 denotes the x coordinate of left end of beam; and h
x is the length between two spline nodes.Figure 3
Spline discretization of axially FG tapered beams.In accordance with the modeling process for SFPM, the linear combination of cubic B-spline functions defined in (8), also called spline interpolation functions ϕ
i
(
x
), is applied to approximate the transverse displacement w
(
x
,
t
) of the beam, which can be expressed by(14)
w
=
w
x
,
t
=
ϕ
i
x
a
i
t
=
ϕ
a
,
(15)
ϕ
=
ϕ
-
1
ϕ
0
ϕ
1
⋯
ϕ
N
ϕ
N
+
1
1
×
N
+
3
,
(16)
a
=
a
-
1
a
0
a
1
⋯
a
N
a
N
+
1
N
+
3
×
1
T
,where [
ϕ
] denotes the spline interpolation functions formed by the linear combination of cubic B-spline functions and {
a
} represents the vector of unknown generalized displacement parameters.In this paper, the spline interpolation functions[
ϕ
], which are given in the literature [27], can be constructed by (17)
ϕ
=
φ
3
,
i
Q
s
,where [
Q
s
] is (
N
+
3
)
×
(
N
+
3
) transformation matrix and [
φ
3
,
i
] are a group of cubic B-spline functions, which is expressed by(18)
Q
s
=
diag
⁡
g
,
I
,
h
,
(19)
φ
3
,
i
=
φ
3
,
-
1
φ
3,0
⋯
φ
3
,
i
⋯
φ
3
,
N
φ
3
,
N
+
1
=
φ
3
,
i
1
×
N
×
3
,
(20)
φ
3
,
i
=
φ
3
x
-
x
0
h
x
-
i
i
=
-
1,0
,
…
,
N
,
N
+
1
,where [
I
] is (
N
-
3
)
×
(
N
-
3
) identity matrix and(21)
g
=
1
-
4
1
0
1
-
1
2
0
0
1
,
h
=
1
0
0
-
1
2
1
0
1
-
4
1
.In this study, spline interpolation functions [
ϕ
] are characterized as the following boundary conditions: (22)
ϕ
i
′
0
=
0
,
i
≠
-
1,0
,
ϕ
i
0
=
0
,
i
≠
-
1
,
ϕ
i
L
=
0
,
i
≠
N
+
1
,
ϕ
i
′
L
=
0
,
i
≠
N
,
N
+
1
,where the subscript “a
′” denotes the derivative of ϕ
i with respect to x.With (14) substituted into (6), the total potential energy function Π for the free transverse vibration of the axially FG tapered Euler beams can be expressed by (23)
Π
=
1
2
a
T
K
a
-
a
T
-
M
a
¨
,in which(24)
K
=
∫
0
L
E
x
I
x
ϕ
′
′
T
ϕ
′
′
d
x
,
M
=
∫
0
L
ρ
x
A
x
ϕ
T
ϕ
d
x
,where the subscript double dot of {
a
¨
} denotes the differentiation with respect to time; [
K
] and [
M
] are, respectively, the (
N
+
3
)
×
(
N
+
3
) stiffness and mass matrices; and {
a
¨
} stands for the (
N
+
3
)
×
1 vector of unknown generalized acceleration parameters.The following is obtained through the variation principle:(25)
δ
Π
=
∂
Π
∂
a
=
0
.Moreover, the free vibration equation for axially FG tapered beam can be expressed as (26)
M
a
¨
+
K
a
=
0
.The following eigenvalue equation must be solved to perform free vibration analysis: (27)
K
-
ω
2
M
φ
=
0
,where ω is the natural frequency and {
φ
} is the mode shape parameter vectors. However, the boundary conditions must be manipulated first before solving (27).


### 3.3. Calculations of Stiffness and Mass Matrices

#### 3.3.1. Cross-Sectional Variation

As shown in Figure1, the axially FG tapered beam with a rectangular cross section, whose breath and height are tapered linearly, is considered. Thus, the cross-sectional area A
(
x
) and second moment of inertia I
(
x
) vary along the beam axis:(28)
A
x
=
A
0
1
-
c
b
x
L
1
-
c
h
x
L
,
I
x
=
I
0
1
-
c
b
x
L
1
-
c
h
x
L
3
.In (28), c
b and c
h are the breadth and height taper ratios, respectively; and A
0 and I
0 are the cross-sectional area and second moment of inertia, respectively, at x
=
0. The tapered beam becomes uniform when c
b
=
c
h
=
0, and the tapered beam theoretically tapers to be a point at x
=
L when c
b
=
c
h
=
1.After some manipulations, (28) can be simplified as (29)
A
x
=
∑
i
=
0
2
A
~
i
x
i
;
I
x
=
∑
i
=
0
4
I
~
i
x
i
,in which(30)
A
~
0
=
A
0
,
A
~
1
=
-
c
b
+
c
h
L
A
0
,
A
~
2
=
c
b
c
h
L
2
A
0
,
I
~
0
=
I
0
,
I
~
1
=
-
c
b
+
3
c
h
L
I
0
,
I
~
2
=
3
c
b
c
h
+
c
h
2
L
2
I
0
,
I
~
3
=
-
3
c
b
c
h
2
+
c
h
3
L
3
I
0
,
I
~
4
=
c
b
c
h
3
L
4
I
0
,where A
~
i
(
i
=
0,1
,
2
) and I
~
i
(
i
=
0
,
…
,
4
) are the constant coefficients for the cross-sectional area and second moment of inertia, respectively, which are equal to the polynomial coefficients of the expanded forms of A
(
x
) and I
(
x
).


#### 3.3.2. Material Property Variation

In this study, two laws are considered for the variation of material property along the lengthwise direction of the beam in accordance with the reported literature [3, 14].(i) Material Type I: Power Law or Polynomial Law. The material properties of the axially FG beam are considered to vary along the lengthwise direction following the power law. The graded functions are simplified into a generalized form [3]:(31)
E
x
=
∑
i
=
0
n
1
E
~
i
x
i
;
ρ
x
=
∑
i
=
0
n
2
ρ
~
i
x
i
,where E
~
i
(
i
=
1
,
…
,
n
1
) and ρ
~
i
(
i
=
1
,
…
,
n
2
) stand for the constant coefficients derived from the polynomial expansion of E
(
x
) and ρ
(
x
) and n
1 and n
2 represent the maximum power number for the expansion of E
(
x
) and ρ
(
x
). For homogenous beams, only the items E
~
0
=
E
0 and ρ
~
0
=
ρ
0 are not zero in (31).(ii) Material Type II: Exponential Law. The material properties of the axially FG beam are considered to vary along the lengthwise direction following the exponential law, which can be simplified as(32)
Y
x
=
Y
~
1
+
Y
~
2
e
α
x
/
L
,where Y
~
1 and Y
~
2 correspond to the coefficient of the simplified form of the exponential functions, such as Young’s modulus E
(
x
) and mass density ρ
(
x
), respectively. α is the gradient parameter that describes the volume fraction change of both constituents involved. For homogenous beams, only the item Y
~
1 is not zero in (32).


#### 3.3.3. Stiffness and Mass Matrices

According to (24), the calculation of [
K
] and [
M
] depends on the variation of the cross section and the graded functions of the material property.(i) The cross sections are considered to be defined by (28), whereas the material properties are defined by (31) (polynomial law). Thus, the following is obtained: (33)
E
x
I
x
=
∑
i
=
0
n
1
+
4
k
~
i
x
i
;
ρ
x
A
x
=
∑
i
=
0
n
2
+
2
m
~
i
x
i
,where k
~
i
(
i
=
0
,
…
,
n
1
+
4
) and m
~
i
(
i
=
1
,
…
,
n
2
+
2
) are constant coefficients of the polynomial expansion for the product of E
(
x
)
I
(
x
) and ρ
(
x
)
A
(
x
), respectively.Then, (33) is substituted into (24). The stiffness and mass matrices are obtained:(34)
K
=
∫
0
L
E
x
I
x
ϕ
′
′
T
ϕ
′
′
d
x
=
∑
i
=
0
n
1
+
4
k
~
i
A
x
,
i
,
M
=
∫
0
L
ρ
x
A
x
ϕ
T
ϕ
d
x
=
∑
i
=
0
n
2
+
2
m
~
i
F
x
,
i
,in which(35)
A
x
,
m
=
∫
0
L
x
m
ϕ
′
′
T
ϕ
′
′
d
x
=
Q
s
T
A
x
,
m
Q
s
,
F
x
,
n
=
∫
0
L
x
n
ϕ
T
ϕ
d
x
=
Q
s
T
F
x
,
n
Q
s
,
A
x
,
m
=
∫
0
L
x
m
φ
3
,
i
′
′
T
φ
3
,
k
′
′
d
x
m
=
0,1
,
…
,
n
1
+
4
,
F
x
,
n
=
∫
0
L
x
n
φ
3
,
i
T
φ
3
,
k
d
x
n
=
0,1
,
…
,
n
2
+
2
,
φ
3
,
i
′
′
=
φ
3
′
′
x
h
x
-
i
,
φ
3
,
k
′
′
=
φ
3
′
′
x
h
x
-
k
,
φ
3
,
i
=
φ
3
x
h
x
-
i
,
φ
3
,
k
=
φ
3
x
h
x
-
k
i
,
k
=
-
1,1
,
0
,
…
,
N
,
N
+
1
.(ii) The cross sections are considered to be defined by (28), and the material properties are defined by (32) (exponential law). Thus, the following is obtained: (36)
E
x
I
x
=
∑
i
=
0
4
k
~
11
,
i
x
i
+
∑
i
=
0
4
k
~
12
,
i
x
i
e
α
x
/
L
,
ρ
x
A
x
=
∑
i
=
0
2
m
~
11
,
i
x
i
+
∑
i
=
0
2
m
~
12
,
i
x
i
e
α
x
/
L
,where α represents the gradient index of the exponentially FG materials. Moreover k
~
11
,
i
,
k
~
12
,
i
(
i
=
0
,
…
,
4
) and m
~
11
,
i
,
m
~
12
,
i
(
i
=
0,1
,
2
) stand for the constant coefficients of the expanded form for the product of E
(
x
)
I
(
x
) and ρ
(
x
)
A
(
x
), respectively, with which the polynomial items and exponential items should be distinguished.Then, (36) is substituted into (24), and the stiffness and mass matrices are obtained: (37)
K
=
∫
0
L
E
x
I
x
ϕ
′
′
T
ϕ
′
′
d
x
=
∑
i
=
0
4
k
~
11
,
i
A
x
,
i
+
∑
i
=
0
4
k
~
12
,
i
A
x
,
e
x
p
,
i
,
M
=
∫
0
L
ρ
x
A
x
ϕ
T
ϕ
d
x
=
∑
i
=
0
2
m
~
11
,
i
F
x
,
i
+
∑
i
=
0
2
m
~
12
,
i
F
x
,
e
x
p
,
i
,in which (38)
A
x
,
p
=
∫
0
L
x
p
ϕ
′
′
T
ϕ
′
′
d
x
=
Q
s
T
A
x
,
p
Q
s
,
F
x
,
q
=
∫
0
L
x
q
ϕ
T
ϕ
d
x
=
Q
s
T
F
x
,
q
Q
s
,
A
x
,
p
=
∫
0
L
x
p
φ
3
,
i
′
′
T
φ
3
,
k
′
′
d
x
p
=
0,1
,
…
,
4
,
F
x
,
q
=
∫
0
L
x
q
φ
3
,
i
T
φ
3
,
k
d
x
q
=
0,1
,
2
,
A
x
,
e
x
p
,
p
=
∫
0
L
x
p
e
α
x
/
L
ϕ
′
′
T
ϕ
′
′
d
x
=
Q
s
T
A
x
,
e
x
p
,
p
Q
s
,
F
x
,
e
x
p
,
q
=
∫
0
L
x
q
e
α
x
/
L
ϕ
T
ϕ
d
x
=
Q
s
T
F
x
,
e
x
p
,
n
Q
s
,
A
x
,
e
x
p
,
p
=
∫
0
L
x
p
e
α
x
/
L
φ
3
,
i
′
′
T
φ
3
,
k
′
′
d
x
p
=
0,1
,
…
,
4
,
F
x
,
e
x
p
,
q
=
∫
0
L
x
q
e
α
x
/
L
φ
3
,
i
T
φ
3
,
k
d
x
q
=
0,1
,
2
,
i
,
k
=
-
1,1
,
0
,
…
,
N
,
N
+
1
.The matrices defined as (35) and (38) are presented in detail in Appendix A, which are the basic matrices for the derivation of stiffness and mass matrices. The integration methods are also proposed.


### 3.4. Boundary Conditions

Boundary conditions should be imposed on[
K
] and [
M
] matrices before solving the eigenvalue equation of (27) to obtain the natural transverse frequencies for the beams. According to (14) and (22), the end constraint of the transverse displacement w
=
w
(
x
,
t
) for beams in various boundary conditions can be expressed by {
a
} in the proposed model as follows:(i)
The following is obtained for pined-pined (P-P) beams:(39)
w
x
=
0
=
w
x
=
L
=
0
,
ϕ
i
0
=
0
i
≠
-
1
,
ϕ
i
L
=
0
i
≠
N
+
1
⇓
a
-
1
=
0
,
a
N
+
1
=
0
.
(ii)
The following is obtained for the clamped-clamped (C-C) beams:(40)
w
x
=
0
=
w
x
=
0
′
=
w
x
=
L
=
w
x
=
L
′
=
0
,
ϕ
i
′
0
=
0
i
≠
-
1,0
,
ϕ
i
0
=
0
i
≠
-
1
,
ϕ
i
L
=
0
i
≠
N
+
1
,
ϕ
i
′
L
=
0
i
≠
N
,
N
+
1
⇓
a
-
1
=
a
0
=
0
,
a
N
=
a
N
+
1
=
0
.
(iii)
The following is obtained for clamped-pined (C-P) beams:(41)
w
x
=
0
=
w
x
=
0
′
=
w
x
=
L
=
0
,
ϕ
i
′
0
=
0
i
≠
-
1,0
,
ϕ
i
0
=
0
i
≠
-
1
ϕ
i
L
=
0
i
≠
N
+
1
⇓
a
-
1
=
a
0
=
0
,
a
N
+
1
=
0
.
(iv)
The following is obtained for clamped-free (C-F) beams:(42)
w
x
=
0
=
w
x
=
0
′
=
0
,
ϕ
i
′
0
=
0
i
≠
-
1,0
,
ϕ
i
0
=
0
i
≠
-
1
⇓
a
-
1
=
0
,
a
0
=
0
,where a
-
1, a
0, a
N, and a
N
+
1 are the parameters that denote the 1st, 2nd, (
N
+
2
)th, and (
N
+
3
)th element of {
a
}, which correspond to the elements on the 1st, 2nd, (
N
+
2
)th, and (
N
+
3
)th row, and the column on the [
K
] and [
M
] matrices according to the free vibration equation of (26). For the free vibration problem, reducing the scale of structural [
K
] and [
M
] matrices effectively prevents ill matrices to solve the eigenvalue equation of (27). Thus, the way of boundary conditions treatment for the beams of the proposed model can be readily achieved by (39)–(42) by deleting the elements of the rows and columns of the [
K
] and [
M
] matrices that correspond to parameters a
-
1
,
a
0
,
a
N, and a
N
+
1. Take the S-S beam, for example; the 1st and (
N
+
3
)th rows and columns of the [
K
] and [
M
] matrices need to be deleted according to (39) to meet the boundary conditions. Therefore, the new forms of [
K
] and [
M
] matrices for beams in various boundary conditions after applying boundary constraints can be expressed as follows:(i)
The following is obtained for P-P beams:(43)
k
m
P
-
P
=
k
m
2,2
k
m
2,3
⋯
k
m
2
,
N
+
1
k
m
2
,
N
+
2
k
m
3,2
k
m
3,3
⋯
k
m
3
,
N
+
1
k
m
3
,
N
+
2
⋮
⋮
⋱
⋮
⋮
k
m
N
+
1
,
2
k
m
N
+
1
,
3
⋯
k
m
N
+
1
,
N
+
1
k
m
N
+
1
,
N
+
2
k
m
N
+
2
,
2
k
m
N
+
2
,
3
⋯
k
m
N
+
2
,
N
+
1
k
m
N
+
2
,
N
+
2
N
+
1
×
N
+
1
.
(ii)
The following is obtained for C-C beams:(44)
k
m
C
-
C
=
k
m
3,2
k
m
3,3
⋯
k
m
3
,
N
k
m
3
,
N
+
1
k
m
4,2
k
m
4,4
⋯
k
m
4
,
N
k
m
4
,
N
+
1
⋮
⋮
⋱
⋮
⋮
k
m
N
,
2
k
m
N
,
3
⋯
k
m
N
,
N
k
m
N
,
N
+
1
k
m
N
+
1
,
2
k
m
N
+
1
,
3
⋯
k
m
N
+
1
,
N
k
m
N
+
1
,
N
+
1
N
-
1
×
N
-
1
.
(iii)
The following is obtained for C-P beams:(45)
k
m
C
-
P
=
k
m
3,2
k
m
3,3
⋯
k
m
3
,
N
+
1
k
m
3
,
N
+
1
k
m
4,2
k
m
4,4
⋯
k
m
4
,
N
+
1
k
m
4
,
N
+
1
⋮
⋮
⋱
⋮
⋮
k
m
N
+
1
,
2
k
m
N
+
1
,
3
⋯
k
m
N
+
1
,
N
+
1
k
m
N
+
1
,
N
+
2
k
m
N
+
2
,
2
k
m
N
+
2
,
3
⋯
k
m
N
+
2
,
N
+
1
k
m
N
+
2
,
N
+
2
N
×
N
.
(iv)
The following is obtained for C-F beams:(46)
k
m
C
-
F
=
k
m
3,2
k
m
3,3
⋯
k
m
3
,
N
k
m
3
,
N
+
3
k
m
4,2
k
m
4,4
⋯
k
m
4
,
N
k
m
4
,
N
+
3
⋮
⋮
⋱
⋮
⋮
k
m
N
+
2
,
2
k
m
N
+
2
,
3
⋯
k
m
N
+
2
,
N
+
2
k
m
N
+
2
,
N
+
3
k
m
N
+
3
,
2
k
m
N
+
3
,
3
⋯
k
m
N
+
3
,
N
+
2
k
m
N
+
3
,
N
+
3
N
+
1
×
N
+
1
,where [
k
m
]
P
-
P, [
k
m
]
C
-
C, [
k
m
]
C
-
P, and [
k
m
]
C
-
F are the new forms of matrices for [
K
] and [
M
] after applying the boundary conditions with regard to the four defined boundary conditions, namely, P-P, C-C, C-P, and C-F, respectively. These matrices are in the scale of (
N
+
1
)
×
(
N
+
1
)
,
(
N
-
1
)
×
(
N
-
1
)
,
(
N
)
×
(
N
), and (
N
+
1
)
×
(
N
+
1
). Moreover, k
m
i
,
j
(
i
,
j
=
1
,
…
,
N
+
3
) are the elements located in the ith row and jth column of the original structural [
K
] and [
M
] matrices of the proposed model, which are (
N
+
3
)
×
(
N
+
3
) matrices mentioned before. The boundary condition treatment process can be easily achieved in a computer program through (43)–(46).After applying the boundary conditions, the stiffness and mass matrices are substituted into (27). The natural frequencies ω
i and the corresponding generalized mode shape parameters {
φ
i
} are determined by solving (27) with the standard computer subroutines. Then, the mode displacement at the arbitrary location along the beam axis can be calculated by substituting {
φ
i
} into (14).


### 3.5. Flowchart for Analysis of the Axially FG Tapered Beam via SFPM

The flowchart for free vibration of axially FG tapered beam via SFPM is shown in Figure4 to state the proposed method visually. It is obvious that the proposed SFPM model is convenient for studying the vibration problems of the axially tapered Euler beams.Figure 4
Flowchart of free vibration analysis of axially FG tapered beam via SFPM.


## 3.1. Definition of Cubic B-Spline Function

The cubic B-spline functionφ
3
(
x
) is adopted in this study and is defined by the following expressions [27]: (8)
φ
3
x
=
∑
k
=
0
4
-
1
k
4
k
x
-
x
k
+
3
3
!
x
∈
-
2
2
,
(9)
x
-
x
k
+
3
=
x
-
x
k
3
x
-
x
k
≥
0
0
x
-
x
k
<
0
x
k
=
k
-
2
.According to the characteristics of φ
3
(
x
), its derivative and integration can be expressed by the following:(10)
φ
3
j
x
=
∑
k
=
0
4
-
1
k
4
k
x
-
x
k
+
3
-
j
3
-
j
!
x
∈
-
2
2
,where φ
3
(
j
)
(
x
) denotes the jth derivative (
j
>
0
) or jth integration (
j
<
0
) for φ
3
(
x
) with the variation of “j”; if j
=
0, φ
3
(
x
) is obtained; if j
>
3, φ
3
(
j
)
(
x
)
=
0. Moreover, if j
=
3, φ
3
(
3
)
(
x
) is maintained as a stepped constant value within the interval -
2
2 as shown in Figure 2.Figure 2
Variation value of functionφ
3
(
3
)
(
x
).


## 3.2. SFPM Modeling

Figure3 illustrates that the beam is modeled via SFPM considering an axially FG tapered Euler beam described in Figure 1. The beam is discretized along the beam axis through uniformly scattered spline nodes on the entire beam. The tapered beam can be divided by the interval h
x in x direction, and the local coordinates of the corresponding nodes in the axis of the beam can be presented by(11)
x
i
=
x
0
+
i
h
x
i
=
1
,
…
,
N
,
(12)
h
x
=
L
N
,
(13)
0
=
x
0
<
x
1
<
x
2
<
⋯
<
x
N
-
1
<
x
N
=
L
,where N stands for the number of subdivisions of beams in x direction, which is also called the maximum number of spline nodes; x
0 denotes the x coordinate of left end of beam; and h
x is the length between two spline nodes.Figure 3
Spline discretization of axially FG tapered beams.In accordance with the modeling process for SFPM, the linear combination of cubic B-spline functions defined in (8), also called spline interpolation functions ϕ
i
(
x
), is applied to approximate the transverse displacement w
(
x
,
t
) of the beam, which can be expressed by(14)
w
=
w
x
,
t
=
ϕ
i
x
a
i
t
=
ϕ
a
,
(15)
ϕ
=
ϕ
-
1
ϕ
0
ϕ
1
⋯
ϕ
N
ϕ
N
+
1
1
×
N
+
3
,
(16)
a
=
a
-
1
a
0
a
1
⋯
a
N
a
N
+
1
N
+
3
×
1
T
,where [
ϕ
] denotes the spline interpolation functions formed by the linear combination of cubic B-spline functions and {
a
} represents the vector of unknown generalized displacement parameters.In this paper, the spline interpolation functions[
ϕ
], which are given in the literature [27], can be constructed by (17)
ϕ
=
φ
3
,
i
Q
s
,where [
Q
s
] is (
N
+
3
)
×
(
N
+
3
) transformation matrix and [
φ
3
,
i
] are a group of cubic B-spline functions, which is expressed by(18)
Q
s
=
diag
⁡
g
,
I
,
h
,
(19)
φ
3
,
i
=
φ
3
,
-
1
φ
3,0
⋯
φ
3
,
i
⋯
φ
3
,
N
φ
3
,
N
+
1
=
φ
3
,
i
1
×
N
×
3
,
(20)
φ
3
,
i
=
φ
3
x
-
x
0
h
x
-
i
i
=
-
1,0
,
…
,
N
,
N
+
1
,where [
I
] is (
N
-
3
)
×
(
N
-
3
) identity matrix and(21)
g
=
1
-
4
1
0
1
-
1
2
0
0
1
,
h
=
1
0
0
-
1
2
1
0
1
-
4
1
.In this study, spline interpolation functions [
ϕ
] are characterized as the following boundary conditions: (22)
ϕ
i
′
0
=
0
,
i
≠
-
1,0
,
ϕ
i
0
=
0
,
i
≠
-
1
,
ϕ
i
L
=
0
,
i
≠
N
+
1
,
ϕ
i
′
L
=
0
,
i
≠
N
,
N
+
1
,where the subscript “a
′” denotes the derivative of ϕ
i with respect to x.With (14) substituted into (6), the total potential energy function Π for the free transverse vibration of the axially FG tapered Euler beams can be expressed by (23)
Π
=
1
2
a
T
K
a
-
a
T
-
M
a
¨
,in which(24)
K
=
∫
0
L
E
x
I
x
ϕ
′
′
T
ϕ
′
′
d
x
,
M
=
∫
0
L
ρ
x
A
x
ϕ
T
ϕ
d
x
,where the subscript double dot of {
a
¨
} denotes the differentiation with respect to time; [
K
] and [
M
] are, respectively, the (
N
+
3
)
×
(
N
+
3
) stiffness and mass matrices; and {
a
¨
} stands for the (
N
+
3
)
×
1 vector of unknown generalized acceleration parameters.The following is obtained through the variation principle:(25)
δ
Π
=
∂
Π
∂
a
=
0
.Moreover, the free vibration equation for axially FG tapered beam can be expressed as (26)
M
a
¨
+
K
a
=
0
.The following eigenvalue equation must be solved to perform free vibration analysis: (27)
K
-
ω
2
M
φ
=
0
,where ω is the natural frequency and {
φ
} is the mode shape parameter vectors. However, the boundary conditions must be manipulated first before solving (27).


## 3.3. Calculations of Stiffness and Mass Matrices

### 3.3.1. Cross-Sectional Variation

As shown in Figure1, the axially FG tapered beam with a rectangular cross section, whose breath and height are tapered linearly, is considered. Thus, the cross-sectional area A
(
x
) and second moment of inertia I
(
x
) vary along the beam axis:(28)
A
x
=
A
0
1
-
c
b
x
L
1
-
c
h
x
L
,
I
x
=
I
0
1
-
c
b
x
L
1
-
c
h
x
L
3
.In (28), c
b and c
h are the breadth and height taper ratios, respectively; and A
0 and I
0 are the cross-sectional area and second moment of inertia, respectively, at x
=
0. The tapered beam becomes uniform when c
b
=
c
h
=
0, and the tapered beam theoretically tapers to be a point at x
=
L when c
b
=
c
h
=
1.After some manipulations, (28) can be simplified as (29)
A
x
=
∑
i
=
0
2
A
~
i
x
i
;
I
x
=
∑
i
=
0
4
I
~
i
x
i
,in which(30)
A
~
0
=
A
0
,
A
~
1
=
-
c
b
+
c
h
L
A
0
,
A
~
2
=
c
b
c
h
L
2
A
0
,
I
~
0
=
I
0
,
I
~
1
=
-
c
b
+
3
c
h
L
I
0
,
I
~
2
=
3
c
b
c
h
+
c
h
2
L
2
I
0
,
I
~
3
=
-
3
c
b
c
h
2
+
c
h
3
L
3
I
0
,
I
~
4
=
c
b
c
h
3
L
4
I
0
,where A
~
i
(
i
=
0,1
,
2
) and I
~
i
(
i
=
0
,
…
,
4
) are the constant coefficients for the cross-sectional area and second moment of inertia, respectively, which are equal to the polynomial coefficients of the expanded forms of A
(
x
) and I
(
x
).


### 3.3.2. Material Property Variation

In this study, two laws are considered for the variation of material property along the lengthwise direction of the beam in accordance with the reported literature [3, 14].(i) Material Type I: Power Law or Polynomial Law. The material properties of the axially FG beam are considered to vary along the lengthwise direction following the power law. The graded functions are simplified into a generalized form [3]:(31)
E
x
=
∑
i
=
0
n
1
E
~
i
x
i
;
ρ
x
=
∑
i
=
0
n
2
ρ
~
i
x
i
,where E
~
i
(
i
=
1
,
…
,
n
1
) and ρ
~
i
(
i
=
1
,
…
,
n
2
) stand for the constant coefficients derived from the polynomial expansion of E
(
x
) and ρ
(
x
) and n
1 and n
2 represent the maximum power number for the expansion of E
(
x
) and ρ
(
x
). For homogenous beams, only the items E
~
0
=
E
0 and ρ
~
0
=
ρ
0 are not zero in (31).(ii) Material Type II: Exponential Law. The material properties of the axially FG beam are considered to vary along the lengthwise direction following the exponential law, which can be simplified as(32)
Y
x
=
Y
~
1
+
Y
~
2
e
α
x
/
L
,where Y
~
1 and Y
~
2 correspond to the coefficient of the simplified form of the exponential functions, such as Young’s modulus E
(
x
) and mass density ρ
(
x
), respectively. α is the gradient parameter that describes the volume fraction change of both constituents involved. For homogenous beams, only the item Y
~
1 is not zero in (32).


### 3.3.3. Stiffness and Mass Matrices

According to (24), the calculation of [
K
] and [
M
] depends on the variation of the cross section and the graded functions of the material property.(i) The cross sections are considered to be defined by (28), whereas the material properties are defined by (31) (polynomial law). Thus, the following is obtained: (33)
E
x
I
x
=
∑
i
=
0
n
1
+
4
k
~
i
x
i
;
ρ
x
A
x
=
∑
i
=
0
n
2
+
2
m
~
i
x
i
,where k
~
i
(
i
=
0
,
…
,
n
1
+
4
) and m
~
i
(
i
=
1
,
…
,
n
2
+
2
) are constant coefficients of the polynomial expansion for the product of E
(
x
)
I
(
x
) and ρ
(
x
)
A
(
x
), respectively.Then, (33) is substituted into (24). The stiffness and mass matrices are obtained:(34)
K
=
∫
0
L
E
x
I
x
ϕ
′
′
T
ϕ
′
′
d
x
=
∑
i
=
0
n
1
+
4
k
~
i
A
x
,
i
,
M
=
∫
0
L
ρ
x
A
x
ϕ
T
ϕ
d
x
=
∑
i
=
0
n
2
+
2
m
~
i
F
x
,
i
,in which(35)
A
x
,
m
=
∫
0
L
x
m
ϕ
′
′
T
ϕ
′
′
d
x
=
Q
s
T
A
x
,
m
Q
s
,
F
x
,
n
=
∫
0
L
x
n
ϕ
T
ϕ
d
x
=
Q
s
T
F
x
,
n
Q
s
,
A
x
,
m
=
∫
0
L
x
m
φ
3
,
i
′
′
T
φ
3
,
k
′
′
d
x
m
=
0,1
,
…
,
n
1
+
4
,
F
x
,
n
=
∫
0
L
x
n
φ
3
,
i
T
φ
3
,
k
d
x
n
=
0,1
,
…
,
n
2
+
2
,
φ
3
,
i
′
′
=
φ
3
′
′
x
h
x
-
i
,
φ
3
,
k
′
′
=
φ
3
′
′
x
h
x
-
k
,
φ
3
,
i
=
φ
3
x
h
x
-
i
,
φ
3
,
k
=
φ
3
x
h
x
-
k
i
,
k
=
-
1,1
,
0
,
…
,
N
,
N
+
1
.(ii) The cross sections are considered to be defined by (28), and the material properties are defined by (32) (exponential law). Thus, the following is obtained: (36)
E
x
I
x
=
∑
i
=
0
4
k
~
11
,
i
x
i
+
∑
i
=
0
4
k
~
12
,
i
x
i
e
α
x
/
L
,
ρ
x
A
x
=
∑
i
=
0
2
m
~
11
,
i
x
i
+
∑
i
=
0
2
m
~
12
,
i
x
i
e
α
x
/
L
,where α represents the gradient index of the exponentially FG materials. Moreover k
~
11
,
i
,
k
~
12
,
i
(
i
=
0
,
…
,
4
) and m
~
11
,
i
,
m
~
12
,
i
(
i
=
0,1
,
2
) stand for the constant coefficients of the expanded form for the product of E
(
x
)
I
(
x
) and ρ
(
x
)
A
(
x
), respectively, with which the polynomial items and exponential items should be distinguished.Then, (36) is substituted into (24), and the stiffness and mass matrices are obtained: (37)
K
=
∫
0
L
E
x
I
x
ϕ
′
′
T
ϕ
′
′
d
x
=
∑
i
=
0
4
k
~
11
,
i
A
x
,
i
+
∑
i
=
0
4
k
~
12
,
i
A
x
,
e
x
p
,
i
,
M
=
∫
0
L
ρ
x
A
x
ϕ
T
ϕ
d
x
=
∑
i
=
0
2
m
~
11
,
i
F
x
,
i
+
∑
i
=
0
2
m
~
12
,
i
F
x
,
e
x
p
,
i
,in which (38)
A
x
,
p
=
∫
0
L
x
p
ϕ
′
′
T
ϕ
′
′
d
x
=
Q
s
T
A
x
,
p
Q
s
,
F
x
,
q
=
∫
0
L
x
q
ϕ
T
ϕ
d
x
=
Q
s
T
F
x
,
q
Q
s
,
A
x
,
p
=
∫
0
L
x
p
φ
3
,
i
′
′
T
φ
3
,
k
′
′
d
x
p
=
0,1
,
…
,
4
,
F
x
,
q
=
∫
0
L
x
q
φ
3
,
i
T
φ
3
,
k
d
x
q
=
0,1
,
2
,
A
x
,
e
x
p
,
p
=
∫
0
L
x
p
e
α
x
/
L
ϕ
′
′
T
ϕ
′
′
d
x
=
Q
s
T
A
x
,
e
x
p
,
p
Q
s
,
F
x
,
e
x
p
,
q
=
∫
0
L
x
q
e
α
x
/
L
ϕ
T
ϕ
d
x
=
Q
s
T
F
x
,
e
x
p
,
n
Q
s
,
A
x
,
e
x
p
,
p
=
∫
0
L
x
p
e
α
x
/
L
φ
3
,
i
′
′
T
φ
3
,
k
′
′
d
x
p
=
0,1
,
…
,
4
,
F
x
,
e
x
p
,
q
=
∫
0
L
x
q
e
α
x
/
L
φ
3
,
i
T
φ
3
,
k
d
x
q
=
0,1
,
2
,
i
,
k
=
-
1,1
,
0
,
…
,
N
,
N
+
1
.The matrices defined as (35) and (38) are presented in detail in Appendix A, which are the basic matrices for the derivation of stiffness and mass matrices. The integration methods are also proposed.


## 3.3.1. Cross-Sectional Variation

As shown in Figure1, the axially FG tapered beam with a rectangular cross section, whose breath and height are tapered linearly, is considered. Thus, the cross-sectional area A
(
x
) and second moment of inertia I
(
x
) vary along the beam axis:(28)
A
x
=
A
0
1
-
c
b
x
L
1
-
c
h
x
L
,
I
x
=
I
0
1
-
c
b
x
L
1
-
c
h
x
L
3
.In (28), c
b and c
h are the breadth and height taper ratios, respectively; and A
0 and I
0 are the cross-sectional area and second moment of inertia, respectively, at x
=
0. The tapered beam becomes uniform when c
b
=
c
h
=
0, and the tapered beam theoretically tapers to be a point at x
=
L when c
b
=
c
h
=
1.After some manipulations, (28) can be simplified as (29)
A
x
=
∑
i
=
0
2
A
~
i
x
i
;
I
x
=
∑
i
=
0
4
I
~
i
x
i
,in which(30)
A
~
0
=
A
0
,
A
~
1
=
-
c
b
+
c
h
L
A
0
,
A
~
2
=
c
b
c
h
L
2
A
0
,
I
~
0
=
I
0
,
I
~
1
=
-
c
b
+
3
c
h
L
I
0
,
I
~
2
=
3
c
b
c
h
+
c
h
2
L
2
I
0
,
I
~
3
=
-
3
c
b
c
h
2
+
c
h
3
L
3
I
0
,
I
~
4
=
c
b
c
h
3
L
4
I
0
,where A
~
i
(
i
=
0,1
,
2
) and I
~
i
(
i
=
0
,
…
,
4
) are the constant coefficients for the cross-sectional area and second moment of inertia, respectively, which are equal to the polynomial coefficients of the expanded forms of A
(
x
) and I
(
x
).


## 3.3.2. Material Property Variation

In this study, two laws are considered for the variation of material property along the lengthwise direction of the beam in accordance with the reported literature [3, 14].(i) Material Type I: Power Law or Polynomial Law. The material properties of the axially FG beam are considered to vary along the lengthwise direction following the power law. The graded functions are simplified into a generalized form [3]:(31)
E
x
=
∑
i
=
0
n
1
E
~
i
x
i
;
ρ
x
=
∑
i
=
0
n
2
ρ
~
i
x
i
,where E
~
i
(
i
=
1
,
…
,
n
1
) and ρ
~
i
(
i
=
1
,
…
,
n
2
) stand for the constant coefficients derived from the polynomial expansion of E
(
x
) and ρ
(
x
) and n
1 and n
2 represent the maximum power number for the expansion of E
(
x
) and ρ
(
x
). For homogenous beams, only the items E
~
0
=
E
0 and ρ
~
0
=
ρ
0 are not zero in (31).(ii) Material Type II: Exponential Law. The material properties of the axially FG beam are considered to vary along the lengthwise direction following the exponential law, which can be simplified as(32)
Y
x
=
Y
~
1
+
Y
~
2
e
α
x
/
L
,where Y
~
1 and Y
~
2 correspond to the coefficient of the simplified form of the exponential functions, such as Young’s modulus E
(
x
) and mass density ρ
(
x
), respectively. α is the gradient parameter that describes the volume fraction change of both constituents involved. For homogenous beams, only the item Y
~
1 is not zero in (32).


## 3.3.3. Stiffness and Mass Matrices

According to (24), the calculation of [
K
] and [
M
] depends on the variation of the cross section and the graded functions of the material property.(i) The cross sections are considered to be defined by (28), whereas the material properties are defined by (31) (polynomial law). Thus, the following is obtained: (33)
E
x
I
x
=
∑
i
=
0
n
1
+
4
k
~
i
x
i
;
ρ
x
A
x
=
∑
i
=
0
n
2
+
2
m
~
i
x
i
,where k
~
i
(
i
=
0
,
…
,
n
1
+
4
) and m
~
i
(
i
=
1
,
…
,
n
2
+
2
) are constant coefficients of the polynomial expansion for the product of E
(
x
)
I
(
x
) and ρ
(
x
)
A
(
x
), respectively.Then, (33) is substituted into (24). The stiffness and mass matrices are obtained:(34)
K
=
∫
0
L
E
x
I
x
ϕ
′
′
T
ϕ
′
′
d
x
=
∑
i
=
0
n
1
+
4
k
~
i
A
x
,
i
,
M
=
∫
0
L
ρ
x
A
x
ϕ
T
ϕ
d
x
=
∑
i
=
0
n
2
+
2
m
~
i
F
x
,
i
,in which(35)
A
x
,
m
=
∫
0
L
x
m
ϕ
′
′
T
ϕ
′
′
d
x
=
Q
s
T
A
x
,
m
Q
s
,
F
x
,
n
=
∫
0
L
x
n
ϕ
T
ϕ
d
x
=
Q
s
T
F
x
,
n
Q
s
,
A
x
,
m
=
∫
0
L
x
m
φ
3
,
i
′
′
T
φ
3
,
k
′
′
d
x
m
=
0,1
,
…
,
n
1
+
4
,
F
x
,
n
=
∫
0
L
x
n
φ
3
,
i
T
φ
3
,
k
d
x
n
=
0,1
,
…
,
n
2
+
2
,
φ
3
,
i
′
′
=
φ
3
′
′
x
h
x
-
i
,
φ
3
,
k
′
′
=
φ
3
′
′
x
h
x
-
k
,
φ
3
,
i
=
φ
3
x
h
x
-
i
,
φ
3
,
k
=
φ
3
x
h
x
-
k
i
,
k
=
-
1,1
,
0
,
…
,
N
,
N
+
1
.(ii) The cross sections are considered to be defined by (28), and the material properties are defined by (32) (exponential law). Thus, the following is obtained: (36)
E
x
I
x
=
∑
i
=
0
4
k
~
11
,
i
x
i
+
∑
i
=
0
4
k
~
12
,
i
x
i
e
α
x
/
L
,
ρ
x
A
x
=
∑
i
=
0
2
m
~
11
,
i
x
i
+
∑
i
=
0
2
m
~
12
,
i
x
i
e
α
x
/
L
,where α represents the gradient index of the exponentially FG materials. Moreover k
~
11
,
i
,
k
~
12
,
i
(
i
=
0
,
…
,
4
) and m
~
11
,
i
,
m
~
12
,
i
(
i
=
0,1
,
2
) stand for the constant coefficients of the expanded form for the product of E
(
x
)
I
(
x
) and ρ
(
x
)
A
(
x
), respectively, with which the polynomial items and exponential items should be distinguished.Then, (36) is substituted into (24), and the stiffness and mass matrices are obtained: (37)
K
=
∫
0
L
E
x
I
x
ϕ
′
′
T
ϕ
′
′
d
x
=
∑
i
=
0
4
k
~
11
,
i
A
x
,
i
+
∑
i
=
0
4
k
~
12
,
i
A
x
,
e
x
p
,
i
,
M
=
∫
0
L
ρ
x
A
x
ϕ
T
ϕ
d
x
=
∑
i
=
0
2
m
~
11
,
i
F
x
,
i
+
∑
i
=
0
2
m
~
12
,
i
F
x
,
e
x
p
,
i
,in which (38)
A
x
,
p
=
∫
0
L
x
p
ϕ
′
′
T
ϕ
′
′
d
x
=
Q
s
T
A
x
,
p
Q
s
,
F
x
,
q
=
∫
0
L
x
q
ϕ
T
ϕ
d
x
=
Q
s
T
F
x
,
q
Q
s
,
A
x
,
p
=
∫
0
L
x
p
φ
3
,
i
′
′
T
φ
3
,
k
′
′
d
x
p
=
0,1
,
…
,
4
,
F
x
,
q
=
∫
0
L
x
q
φ
3
,
i
T
φ
3
,
k
d
x
q
=
0,1
,
2
,
A
x
,
e
x
p
,
p
=
∫
0
L
x
p
e
α
x
/
L
ϕ
′
′
T
ϕ
′
′
d
x
=
Q
s
T
A
x
,
e
x
p
,
p
Q
s
,
F
x
,
e
x
p
,
q
=
∫
0
L
x
q
e
α
x
/
L
ϕ
T
ϕ
d
x
=
Q
s
T
F
x
,
e
x
p
,
n
Q
s
,
A
x
,
e
x
p
,
p
=
∫
0
L
x
p
e
α
x
/
L
φ
3
,
i
′
′
T
φ
3
,
k
′
′
d
x
p
=
0,1
,
…
,
4
,
F
x
,
e
x
p
,
q
=
∫
0
L
x
q
e
α
x
/
L
φ
3
,
i
T
φ
3
,
k
d
x
q
=
0,1
,
2
,
i
,
k
=
-
1,1
,
0
,
…
,
N
,
N
+
1
.The matrices defined as (35) and (38) are presented in detail in Appendix A, which are the basic matrices for the derivation of stiffness and mass matrices. The integration methods are also proposed.


## 3.4. Boundary Conditions

Boundary conditions should be imposed on[
K
] and [
M
] matrices before solving the eigenvalue equation of (27) to obtain the natural transverse frequencies for the beams. According to (14) and (22), the end constraint of the transverse displacement w
=
w
(
x
,
t
) for beams in various boundary conditions can be expressed by {
a
} in the proposed model as follows:(i)
The following is obtained for pined-pined (P-P) beams:(39)
w
x
=
0
=
w
x
=
L
=
0
,
ϕ
i
0
=
0
i
≠
-
1
,
ϕ
i
L
=
0
i
≠
N
+
1
⇓
a
-
1
=
0
,
a
N
+
1
=
0
.
(ii)
The following is obtained for the clamped-clamped (C-C) beams:(40)
w
x
=
0
=
w
x
=
0
′
=
w
x
=
L
=
w
x
=
L
′
=
0
,
ϕ
i
′
0
=
0
i
≠
-
1,0
,
ϕ
i
0
=
0
i
≠
-
1
,
ϕ
i
L
=
0
i
≠
N
+
1
,
ϕ
i
′
L
=
0
i
≠
N
,
N
+
1
⇓
a
-
1
=
a
0
=
0
,
a
N
=
a
N
+
1
=
0
.
(iii)
The following is obtained for clamped-pined (C-P) beams:(41)
w
x
=
0
=
w
x
=
0
′
=
w
x
=
L
=
0
,
ϕ
i
′
0
=
0
i
≠
-
1,0
,
ϕ
i
0
=
0
i
≠
-
1
ϕ
i
L
=
0
i
≠
N
+
1
⇓
a
-
1
=
a
0
=
0
,
a
N
+
1
=
0
.
(iv)
The following is obtained for clamped-free (C-F) beams:(42)
w
x
=
0
=
w
x
=
0
′
=
0
,
ϕ
i
′
0
=
0
i
≠
-
1,0
,
ϕ
i
0
=
0
i
≠
-
1
⇓
a
-
1
=
0
,
a
0
=
0
,where a
-
1, a
0, a
N, and a
N
+
1 are the parameters that denote the 1st, 2nd, (
N
+
2
)th, and (
N
+
3
)th element of {
a
}, which correspond to the elements on the 1st, 2nd, (
N
+
2
)th, and (
N
+
3
)th row, and the column on the [
K
] and [
M
] matrices according to the free vibration equation of (26). For the free vibration problem, reducing the scale of structural [
K
] and [
M
] matrices effectively prevents ill matrices to solve the eigenvalue equation of (27). Thus, the way of boundary conditions treatment for the beams of the proposed model can be readily achieved by (39)–(42) by deleting the elements of the rows and columns of the [
K
] and [
M
] matrices that correspond to parameters a
-
1
,
a
0
,
a
N, and a
N
+
1. Take the S-S beam, for example; the 1st and (
N
+
3
)th rows and columns of the [
K
] and [
M
] matrices need to be deleted according to (39) to meet the boundary conditions. Therefore, the new forms of [
K
] and [
M
] matrices for beams in various boundary conditions after applying boundary constraints can be expressed as follows:(i)
The following is obtained for P-P beams:(43)
k
m
P
-
P
=
k
m
2,2
k
m
2,3
⋯
k
m
2
,
N
+
1
k
m
2
,
N
+
2
k
m
3,2
k
m
3,3
⋯
k
m
3
,
N
+
1
k
m
3
,
N
+
2
⋮
⋮
⋱
⋮
⋮
k
m
N
+
1
,
2
k
m
N
+
1
,
3
⋯
k
m
N
+
1
,
N
+
1
k
m
N
+
1
,
N
+
2
k
m
N
+
2
,
2
k
m
N
+
2
,
3
⋯
k
m
N
+
2
,
N
+
1
k
m
N
+
2
,
N
+
2
N
+
1
×
N
+
1
.
(ii)
The following is obtained for C-C beams:(44)
k
m
C
-
C
=
k
m
3,2
k
m
3,3
⋯
k
m
3
,
N
k
m
3
,
N
+
1
k
m
4,2
k
m
4,4
⋯
k
m
4
,
N
k
m
4
,
N
+
1
⋮
⋮
⋱
⋮
⋮
k
m
N
,
2
k
m
N
,
3
⋯
k
m
N
,
N
k
m
N
,
N
+
1
k
m
N
+
1
,
2
k
m
N
+
1
,
3
⋯
k
m
N
+
1
,
N
k
m
N
+
1
,
N
+
1
N
-
1
×
N
-
1
.
(iii)
The following is obtained for C-P beams:(45)
k
m
C
-
P
=
k
m
3,2
k
m
3,3
⋯
k
m
3
,
N
+
1
k
m
3
,
N
+
1
k
m
4,2
k
m
4,4
⋯
k
m
4
,
N
+
1
k
m
4
,
N
+
1
⋮
⋮
⋱
⋮
⋮
k
m
N
+
1
,
2
k
m
N
+
1
,
3
⋯
k
m
N
+
1
,
N
+
1
k
m
N
+
1
,
N
+
2
k
m
N
+
2
,
2
k
m
N
+
2
,
3
⋯
k
m
N
+
2
,
N
+
1
k
m
N
+
2
,
N
+
2
N
×
N
.
(iv)
The following is obtained for C-F beams:(46)
k
m
C
-
F
=
k
m
3,2
k
m
3,3
⋯
k
m
3
,
N
k
m
3
,
N
+
3
k
m
4,2
k
m
4,4
⋯
k
m
4
,
N
k
m
4
,
N
+
3
⋮
⋮
⋱
⋮
⋮
k
m
N
+
2
,
2
k
m
N
+
2
,
3
⋯
k
m
N
+
2
,
N
+
2
k
m
N
+
2
,
N
+
3
k
m
N
+
3
,
2
k
m
N
+
3
,
3
⋯
k
m
N
+
3
,
N
+
2
k
m
N
+
3
,
N
+
3
N
+
1
×
N
+
1
,where [
k
m
]
P
-
P, [
k
m
]
C
-
C, [
k
m
]
C
-
P, and [
k
m
]
C
-
F are the new forms of matrices for [
K
] and [
M
] after applying the boundary conditions with regard to the four defined boundary conditions, namely, P-P, C-C, C-P, and C-F, respectively. These matrices are in the scale of (
N
+
1
)
×
(
N
+
1
)
,
(
N
-
1
)
×
(
N
-
1
)
,
(
N
)
×
(
N
), and (
N
+
1
)
×
(
N
+
1
). Moreover, k
m
i
,
j
(
i
,
j
=
1
,
…
,
N
+
3
) are the elements located in the ith row and jth column of the original structural [
K
] and [
M
] matrices of the proposed model, which are (
N
+
3
)
×
(
N
+
3
) matrices mentioned before. The boundary condition treatment process can be easily achieved in a computer program through (43)–(46).After applying the boundary conditions, the stiffness and mass matrices are substituted into (27). The natural frequencies ω
i and the corresponding generalized mode shape parameters {
φ
i
} are determined by solving (27) with the standard computer subroutines. Then, the mode displacement at the arbitrary location along the beam axis can be calculated by substituting {
φ
i
} into (14).


## 3.5. Flowchart for Analysis of the Axially FG Tapered Beam via SFPM

The flowchart for free vibration of axially FG tapered beam via SFPM is shown in Figure4 to state the proposed method visually. It is obvious that the proposed SFPM model is convenient for studying the vibration problems of the axially tapered Euler beams.Figure 4
Flowchart of free vibration analysis of axially FG tapered beam via SFPM.


## 4. Numerical Examples

The nondimensional natural transverse frequencies are defined asω
-
=
ω
ρ
0
A
0
L
4
/
E
0
I
0, where A
0
,
I
0
,
ρ
0, and E
0 represent the cross-sectional and material properties of the beam at x
=
0. And the geometrical properties for the beams studied in this paper are given as follows: b
0
×
h
0
=
0.01 m × 0.03 m, A
0
=
3
×
10
-
2 m2, I
0
=
2.25
×
10
-
8 m4, and L
=
1 m; the material properties for all beams, except for the beams with material properties that vary in the exponential law (Section 4.3), are as follows: ρ
0
=
7900 kg/m3 and E
0
=
2.1
×
10
11 N/m4, and both the cross-sectional and material properties only vary along the longitudinal direction.


### 4.1. A Comparison of the Results for Uniform Homogenous Beams

As the first example, we consider the case of a homogeneous beam with uniform cross sections. For this case Young’s modulus(
E
(
x
)
=
E
0
), mass density (
ρ
(
x
)
=
ρ
0
), cross-sectional area (
A
(
x
)
=
A
0
), and second moment of inertia (
I
(
x
)
=
I
0
) are not changed. We calculate the nondimensional natural transverse frequencies, ω
-, of cantilever beams (C-F) by choosing a different spline node number (
N
) in (12) to check the convergence of the presented method (i.e., SFPM). The first four nondimensional natural frequencies, ω
-
i, are tabulated in Table 1 and compared with the exact solutions [35] and other numerical methods [14]. The comparison indicated that the SFPM solution has rapid convergence. As N increases from 5 to 25, the first three nondimensional natural frequencies are identical to the exact solutions. This finding indicates that the present method is efficient except for the case of N
=
5, in which the fourth and fifth nondimensional natural frequencies do not match the exact solution because of the insufficient order of the eigenvalue equation, that is, (27). However, the proposed model does not have any convergence problems. As N increases from 15 to 25, computational time and cost increase, but the results of the natural frequencies via SFPM vary slightly. Thus, we take spline node number N
=
15 in the following examples. Moreover, the first five nondimensional natural transverse frequencies for C-C, C-P, and P-P beams are calculated and tabulated in Table 2 with N
=
15. This finding verifies that SFPM can be applied to analyze the free vibration problem of beams with various boundary conditions.Table 1
First five nondimensional transverse natural frequenciesω
-
i for uniform beam; boundary condition: C-F.
ω
-
i
SFPM
Exact [35]
Reference [14]
N
=
5
N
=
10
N
=
15
N
=
20
N
=
25
ω
-
1
3.516
3.516
3.516
3.516
3.516
3.516
3.516
ω
-
2
22.049
22.035
22.035
22.035
22.035
22.034
22.0345
ω
-
3
62.11
61.716
61.701
61.698
61.698
61.701
61.6972
ω
-
4
124.526
121.062
120.929
120.91
120.905
120.909
120.9019
ω
-
5
205.179
200.702
199.994
199.899
199.875
199.855
—Table 2
First five nondimensional transverse natural frequenciesω
-
i for uniform beams with various boundary conditions.
ω
-
i
C-C
C-P
P-P
SFPM
Exact [35]
SFPM
Exact [35]
SFPM
Exact [35]
ω
-
1
22.373
22.372
15.418
15.413
9.87
9.869
ω
-
2
61.674
61.669
49.965
49.956
39.479
39.478
ω
-
3
120.912
120.912
104.253
104.244
88.828
88.826
ω
-
4
199.898
199.855
178.297
178.275
157.921
157.913
ω
-
5
298.69
298.529
272.132
272.019
246.769
246.739


### 4.2. Effect of the Variable Cross Section

The second example is devoted to the Euler-Bernoulli beam with a rectangular cross section and has a breadth and height that taper linearly, as shown in Figure2. Thus, the cross-sectional area A
(
x
) and the second moment of inertia I
(
x
) that vary along axis are defined as (28), as mentioned earlier.Firstly, the natural transverse frequencies of the tapered homogenous beams with various boundary conditions (C-F, C-C, and P-P) are calculated by using SFPM, for all beams,N
=
15 is selected, and two cases of cross-sectional variation, namely, the case of only the height tapered linearly (
c
h
=
c
0
) and the case of both breadth and height tapered linearly (
c
b
=
c
h
=
c
0
), are both investigated. In order to validate the performance of the proposed method, FEM models (3D), which are implemented in ANSYS (see in Figure 5), are employed to investigate the free vibration of the tapered homogenous beams. In the FEM models, 3D solid element SOLID64 is employed, and all beams are meshed into 1600 elements to ensure the computational accuracy for fundamental and high-order frequencies and to avoid producing the abnormal element due to the increase of taper ratios. The nondimensional natural transverse frequencies for C-F, C-C, and P-P beams are tabulated in Tables 3–5, which indicate that the results of SFPM model agree very well with those of FEM models and DTM model reported by Rajasekaran [4]. The taper ratio (c
0) plays an important role in the tapered homogenous beams. For all beams, the natural frequencies decrease with the increase of the taper ratio (c
0) except for the fundamental frequencies of C-F beams, which increase with c
0; for C-F beams, it is obvious that the frequencies of the case of height tapered linearly (
c
h
=
c
0
) are smaller than those of the case of both height and breadth tapered linearly (
c
b
=
c
h
=
c
0
), while, for C-C and P-P beams, the frequencies of the case of height tapered linearly (
c
h
=
c
0
) are equal to those of the case of both height and breadth tapered linearly (
c
b
=
c
h
=
c
0
). However, in this study, given N
=
15 is employed, the SFPM model shows a considerable decrease of the dimensions of the eigenvalue equations compared with the FEM and DTM models [4], leading to remarkable decrease in computational cost. Furthermore, FEM models implemented in ANSYS are unable to analyze the free vibration of tapered beams with material nonhomogeneity along the longitudinal direction.Table 3
Nondimensional transverse natural frequenciesω
-
i for tapered homogenous beam; boundary condition: C-F.
c
0
c
b
=
0, c
h
=
c
0
c
b
=
c
h
=
c
0
ω
-
1
ω
-
2
ω
-
3
ω
-
1
ω
-
2
ω
-
3
0.2
SFPM
3.6083
20.6212
56.1954
3.8551
21.0569
56.6335
Reference [4]
3.6083
20.621
56.1923
3.8551
21.0567
56.6303
FEM
3.6113
20.5759
55.8008
3.8580
21.0108
56.2356
0.4
SFPM
3.7371
19.114
50.3567
4.31878
20.0501
51.3378
Reference [4]
3.7371
19.1138
50.3537
4.3188
20.05
51.3346
FEM
3.7398
19.0857
50.0932
4.3213
20.0187
51.0670
0.6
SFPM
3.93428
17.488
44.0279
5.00904
19.0651
45.7417
Reference [4]
3.9343
17.4879
44.0248
5.009
19.0649
45.7384
FEM
3.9366
17.4717
43.8706
5.0110
19.0463
45.5747
0.8
SFPM
4.2925
15.7429
36.8881
6.19639
18.3858
39.8376
Reference [4]
4.2925
15.7427
36.8852
6.1964
18.3855
39.8339
FEM
4.2946
15.7366
36.8135
6.1978
18.3751
39.7517Table 4
Nondimensional transverse natural frequenciesω
-
i for tapered homogenous beam; boundary condition: C-C.
c
0
c
b
=
0, c
h
=
c
0
c
b
=
c
h
=
c
0
ω
-
1
ω
-
2
ω
-
3
ω
-
1
ω
-
2
ω
-
3
0.2
SFPM
20.0784
55.3432
108.5080
20.0968
55.3683
108.5357
FEM
20.0525
54.9593
106.9451
20.0708
54.9832
106.9690
0.4
SFPM
17.6357
48.5873
95.2438
17.7203
48.7027
95.3717
FEM
17.6321
48.3680
94.2691
17.7151
48.4791
94.3873
0.6
SFPM
14.9624
41.1721
80.6681
15.1914
41.4862
81.0196
FEM
14.9753
41.0829
80.1634
15.2019
41.3883
80.4955
0.8
SFPM
11.8486
32.5079
63.6182
12.3970
33.2794
64.5189
FEM
11.8684
32.4963
63.4193
12.4054
33.2308
64.2326Table 5
Nondimensional transverse natural frequenciesω
-
i for tapered homogenous beam; boundary condition: P-P.
c
0
c
b
=
0, c
h
=
c
0
c
b
=
c
h
=
c
0
ω
-
1
ω
-
2
ω
-
3
ω
-
1
ω
-
2
ω
-
3
0.2
SFPM
8.8462
35.4461
79.7399
8.8246
35.4665
79.7721
FEM
8.8370
35.2994
78.9997
8.8153
35.3191
79.0321
0.4
SFPM
7.7295
31.1945
70.0980
7.6314
31.2881
70.2453
FEM
7.7234
31.0975
69.6109
7.6251
31.1904
69.7559
0.6
SFPM
6.4667
26.6014
59.6076
6.2087
26.8534
60.0036
FEM
6.4631
26.5453
59.3230
6.2050
26.7957
59.7141
0.8
SFPM
4.9201
21.3485
47.5048
4.3536
21.9447
48.4357
FEM
4.9184
21.3218
47.3703
4.3515
21.9142
48.2878Figure 5
FEM model for homogenous tapered beams by using ANSYS.Then, we study the free vibration of axially FG tapered beam with Young’s modulus, and the material densities along the axis are defined as(47)
E
x
=
E
0
1
+
x
L
,
ρ
x
=
ρ
0
1
+
x
L
+
x
L
2
,where E
0 and ρ
0 are Young’s modulus and mass density, respectively, at x
=
0.The material properties defined by (47) follow the polynomial law distribution, and (47) can be simplified as (31), with n
1
=
1 and n
2
=
2: (48)
E
x
=
∑
i
=
0
1
E
~
i
x
i
,
ρ
x
=
∑
i
=
0
2
ρ
~
i
x
i
,in which(49)
E
~
i
=
E
0
L
i
i
=
0,1
;
ρ
~
i
=
ρ
0
L
i
i
=
0,1
,
2
.Using (48), the structural stiffness and mass matrices can be calculated by (34). In Figure 6, the convergence of natural frequencies for axially FG tapered cantilever beams (
c
b
=
c
h
=
0.8
) (material gradation following the polynomial law) with respect to the spline node number N is discussed; it is noticeable that the natural frequencies achieve a convergence value when N increase from 5 to 15 (about 4-digit precision), and when N increase from 15 to 25, the frequencies vary a little but the computational cost increased remarkably. Therefore, N
=
15 is a suitable choice for the proposed SFPM models to ensure both the accuracy and efficiency for the free vibration of the axially FG tapered beams with different taper ratios.Figure 6
Convergence of natural frequencies for axially FG tapered cantilever beams with respect to spline node numberN
(
c
b
=
c
h
=
0.8
).
(a)
(b)
(c)The natural transverse frequencies for axially FG tapered beams with different breadth taper ratio(
c
b
) and height taper ratio (
c
h
) are calculated by using the proposed method, and the results are tabulated in Tables 6–8. The results of SFPM agree with the results of DTEM reported by Shahba and Rajasekaran [3]. This observation indicates that the proposed method can be used to investigate the free vibration of axially FG tapered beams with high accuracy. The variations of the first two nondimensional natural frequencies of the axially FG beams with various boundary conditions for different values of taper ratios (
c
b
,
c
h
) are illustrated in Figures 7–9.Table 6
Nondimensional transverse natural frequenciesω
-
i for axially FG tapered beam; boundary condition: C-F.
c
h
c
b
0
0.2
0.4
0.6
0.8
ω
-
1
ω
-
2
ω
-
1
ω
-
2
ω
-
1
ω
-
2
ω
-
1
ω
-
2
ω
-
1
ω
-
2
0
SFPM
2.4256
18.6042
2.6054
19.0042
2.8508
19.5304
3.2137
20.2959
3.8310
21.6760
Reference [3]
2.4256
18.6041
2.6054
19.0041
2.8508
19.5303
3.2137
20.2958
3.8310
21.6759
0.2
SFPM
2.5051
17.3802
2.6863
17.7501
2.9336
18.2379
3.2993
18.9501
3.9219
20.2432
Reference [3]
2.5051
17.3802
2.6863
17.7501
2.9336
18.2379
3.2994
18.9501
3.9220
20.2432
0.4
SFPM
2.6155
16.0705
2.7987
16.4092
3.0486
16.8571
3.4181
17.5139
4.0471
18.7164
Reference [3]
2.6155
16.0705
2.7988
16.4092
3.0486
16.8571
3.4181
17.5139
4.0471
18.7164
0.6
SFPM
2.7835
14.6508
2.9699
14.9567
3.2237
15.3628
3.5985
15.9616
4.2355
17.0694
Reference [3]
2.7836
14.6508
2.9699
14.9567
3.2237
15.3627
3.5985
15.9616
4.2355
17.0694
0.8
SFPM
3.0871
13.1142
3.2794
13.3850
3.5401
13.7466
3.9232
14.2848
4.5695
15.2955
Reference [3]
3.0871
13.1142
3.2794
13.3849
3.5401
13.7466
3.9232
14.2848
4.5695
15.2954Table 7
Nondimensional transverse natural frequenciesω
-
i for axially FG tapered beam; boundary condition: C-C.
c
h
c
b
0
0.2
0.4
0.6
0.8
ω
-
1
ω
-
2
ω
-
1
ω
-
2
ω
-
1
ω
-
2
ω
-
1
ω
-
2
ω
-
1
ω
-
2
0
SFPM
20.4721
56.5492
20.4152
56.4716
20.2883
56.2981
20.0186
55.9206
19.3845
54.9713
Reference [3]
20.4721
56.5482
20.4151
56.4706
20.2883
56.2971
20.0186
55.9195
19.3844
54.9699
0.2
SFPM
18.2170
50.4796
18.1996
50.4560
18.1286
50.3593
17.9436
50.1011
17.4565
49.3718
Reference [3]
18.2170
50.4792
18.1995
50.4556
18.1286
50.3589
17.9436
50.1006
17.4564
49.3711
0.4
SFPM
15.8282
44.0246
15.8498
44.0553
15.8350
44.0371
15.7367
43.9027
15.4025
43.4066
Reference [3]
15.8281
44.0236
15.8497
44.0542
15.8349
44.0359
15.7366
43.9013
15.4021
43.4043
0.6
SFPM
13.2294
36.9654
13.2896
37.0509
13.3319
37.1137
13.3238
37.1105
13.1529
36.8680
Reference [3]
13.2291
36.9636
13.2894
37.0490
13.3316
37.1116
13.3234
37.1079
13.1521
36.8635
0.8
SFPM
10.2236
28.7498
10.3231
28.8919
10.4257
29.0416
10.5170
29.1851
10.5343
29.2419
Reference [3]
10.2217
28.7406
10.3211
28.8822
10.4234
29.0311
10.5143
29.1728
10.5301
29.2239Table 8
Nondimensional transverse natural frequenciesω
-
i for axially FG tapered beam; boundary condition: P-P.
c
h
c
b
0
0.2
0.4
0.6
0.8
ω
-
1
ω
-
2
ω
-
1
ω
-
2
ω
-
1
ω
-
2
ω
-
1
ω
-
2
ω
-
1
ω
-
2
0
SFPM
9.0286
36.3718
9.0599
36.3418
9.0867
36.3153
9.0994
36.2969
9.0685
36.2769
Reference [3]
9.0285
36.3715
9.0599
36.3415
9.0867
36.3151
9.0994
36.2966
9.0685
36.2766
0.2
SFPM
8.1341
32.5236
8.1462
32.5123
8.1498
32.5079
8.1336
32.5164
8.0646
32.5326
Reference [3]
8.1341
32.5234
8.1462
32.5121
8.1498
32.5077
8.1336
32.5161
8.0646
32.5323
0.4
SFPM
7.1531
28.4747
7.1455
28.4822
7.1254
28.5003
7.0794
28.5370
6.9703
28.5928
Reference [3]
7.1531
28.4744
7.1455
28.4819
7.1254
28.5001
7.0794
28.5367
6.9703
28.5924
0.6
SFPM
6.0357
24.1101
6.0082
24.1371
5.9638
24.1791
5.8868
24.2469
5.7351
24.3497
Reference [3]
6.0357
24.1098
6.0082
24.1367
5.9637
24.1787
5.8868
24.2465
5.7351
24.3491
0.8
SFPM
4.6520
19.1315
4.6046
19.1804
4.5355
19.2510
4.4264
19.3591
4.2284
19.5302
Reference [3]
4.6520
19.1304
4.6045
19.1792
4.5354
19.2496
4.4263
19.3576
4.2281
19.5281Figure 7
The variation of the first two nondimensional natural frequencies of axially FG beams with respect to taper ratios; boundary condition: C-F.
(a)
(b)Figure 8
The variation of the first two nondimensional natural frequencies of axially FG beams with respect to taper ratios; boundary condition: C-C.
(a)
(b)Figure 9
The variation of the first two nondimensional natural frequencies of axially FG beams with respect to taper ratios; boundary condition: P-P.
(a)
(b)Figures7–9 illustrate that the varying trend of the natural frequencies is distinguished from beams with different boundary conditions. In general, the natural transverse frequencies are considerably affected by the variation of height taper ratio c
h. Figure 7 and Table 6 indicate that the increase of taper ratios c
h and c
b leads to the increase of the fundamental natural frequencies for C-F beams, whose second frequencies decrease with c
h and slightly increase with c
b. For C-C and P-P beams, Figures 8 and 9 and Tables 7 and 8 indicate that all frequencies remarkably decrease with c
h, whereas the variations with the increase of c
b are not monotonic. Thus, predicting the natural frequencies for axially FG tapered beams of different boundary conditions with respect to the taper ratios is impossible.Notice that the proposed model needs considerably less computational cost and modeling time compared with the DTM and DTEM models developed by Rajasekaran [4] and Shahba et al. [2], respectively. In the DTM and DTEM models, 132 simple points and 12 elements are used, and the eigenvalue equation in the dimension of 578
×
578 needs to be solved. However, in the proposed model, as N
=
15 is selected based on the convergence study, the eigenvalue equation is only in the dimension of 18
×
18, which shows that the proposed method has higher remarkable ascending efficiency than the reported methods.Furthermore, the natural transverse frequencies for tapered beams with consideration of material nonhomogeneity and homogeneity are distinguished by comparing the results in Table3 with those in Table 4. The material nonhomogeneity defined by (47) decreases the natural frequencies for tapered C-F beams compared with homogenous beams. Thus, the material properties obviously affect the dynamic characteristics of the tapered beams.


### 4.3. Effect of Axial Gradient Parameter

Finally, the free vibration for tapered rectangular beams with axially exponentially graded material is investigated to verify the adaptability of the proposed method to different gradient functions. Similar works have been conducted by Huang and Li [14]. The material properties are defined as follows: (50)
Y
x
=
Y
1
1
-
e
α
x
/
L
-
1
e
α
-
1
+
Y
2
e
α
x
/
L
-
1
e
α
-
1
,
i
f
α
≠
0
,
Y
x
=
Y
1
1
-
x
L
+
Y
2
x
L
,
i
f
α
=
0
,in which Y
1 and Y
2 correspond to the material properties, such as Young’s modulus E
(
x
) and mass density ρ
(
x
) at x
=
0 and x
=
L, respectively, and α is the gradient parameter that describes the volume fraction change of both constituents involved. The variation of Y
(
x
) against x is shown in Figure 10 for Y
1
=
3
Y
2.Figure 10
Variation of gradation material properties defined by (50) with Y
1
=
3
Y
2.The material properties defined by (50) can be classified into two types of distribution that depends on the value of gradient index α; one is the polynomial type that is defined by (31) (α
=
0) with n
1
=
n
2
=
1 and the other is exponential type that is defined by (32) (α
≠
0), which can be simplified into the following equations:(51)
Y
x
=
Y
~
1
+
Y
~
2
e
α
x
/
L
,
i
f
α
≠
0
,
Y
x
=
Y
~
1
+
Y
~
2
x
,
i
f
α
=
0
,where(52)
Y
~
1
=
Y
1
-
Y
2
-
Y
1
e
α
-
1
,
Y
~
2
=
Y
2
-
Y
1
e
α
-
1
,
i
f
α
≠
0
,
Y
~
1
=
Y
1
,
Y
~
2
=
Y
2
-
Y
1
L
,
i
f
α
=
0
.Using (51), the stiffness and mass matrices can be calculated by (34) and (37) for α
=
0 and α
≠
0, respectively. In the following calculations, the two materials chosen are aluminum (Al) and zirconia (ZrO2). The material properties are presented by [14] (53)
Z
r
O
2
:
E
z
=
2
.
0
×
10
11
N
/
m
2
,
ρ
z
=
7900
k
g
/
m
3
,
A
l
:
E
a
=
7
.
0
×
10
1
0
N
/
m
2
,
ρ
a
=
2702
k
g
/
m
3
.The nondimensional frequencies are defined by (54)
ω
-
=
ω
ρ
a
A
0
L
4
E
a
I
0
.In consideration of the two cases in this paper, one is Al rich near the end x
=
0 and ZrO2 rich near the end at x
=
L (Case 1), whereas the other is Al rich near the end at x
=
L and ZrO2 rich near the end at x
=
0 (Case 2).The free vibration of axially FG beams with uniform cross-sectional(
c
b
=
c
h
=
0
) and material properties defined as (50) is calculated to verify the proposed method (i.e., SFPM). The results for the fundamental nondimensional natural transverse frequencies are tabulated in Table 9, and the high-order frequencies calculated with SFPM are tabulated in Table 10. Figure 11 gives the convergence of the natural frequencies for axially exponentially FG cantilever beams (C-F) (α
=
-
10) with respect to spline node number N; the variations of the first three nondimensional natural frequencies of axially FG beams with respect to gradient parameter α are depicted in Figures 12–15.Table 9
Nondimensional fundamental natural transverse frequenciesω
-
i for axially FG beams with respect to the gradient index.
α
P-P
C-C
C-P
C-F
Case  1
Case  2
Case  1
Case  2
Case  1
Case  2
Case  1
Case  2
−10
Reference [14]
11.4532
9.9358
24.0576
24.7949
16.4775
17.2993
3.5656
4.1800
SFPM
11.4560
9.9366
24.0660
24.8068
16.4111
17.3037
3.5340
4.1801
−3
Reference [14]
11.2443
10.3669
23.9456
24.9375
16.0307
17.8701
3.1421
4.8317
SFPM
11.2443
10.3669
23.9446
24.9366
16.0262
17.8701
3.1410
4.8317
0
Reference [14]
10.8663
10.8663
24.3752
24.3752
15.8734
17.9147
2.9256
5.0156
SFPM
10.8664
10.8664
24.3756
24.3756
15.8735
17.9148
2.9256
5.0156
3
Reference [14]
10.3669
11.2443
24.9375
23.9456
15.7171
17.8867
2.8544
4.8466
SFPM
10.3669
11.2443
24.9366
23.9446
15.7169
17.8874
2.8545
4.8466
10
Reference [14]
9.9358
11.4532
24.7949
23.9456
15.4930
17.9050
3.0431
4.4629
SFPM
9.9366
11.4560
24.8068
24.0660
15.4958
17.9155
3.0431
4.4629Table 10
Nondimensional higher-order transverse natural frequenciesω
-
i for axially FG beams with respect to the gradient index.
α
P-P
C-C
C-P
C-F
Case  1
Case  2
Case  1
Case  2
Case  1
Case  2
Case  1
Case  2
−10
ω
-
2
45.6887
40.0700
68.1328
66.2401
54.9276
53.9917
23.7075
24.4362
ω
-
3
102.6601
90.5367
135.4215
127.6871
116.4430
110.4619
68.1642
66.2609
ω
-
4
182.4345
161.3146
225.7493
209.1345
201.0000
186.8845
135.4187
127.6897
ω
-
5
285.1358
252.4647
339.2851
310.8793
308.7099
283.5206
225.7516
209.1407
−3
ω
-
2
44.8550
41.9691
67.7638
67.1081
54.4731
54.9320
23.0531
25.4924
ω
-
3
100.8244
94.5153
134.4815
130.2588
115.5467
112.8131
67.4843
67.7907
ω
-
4
179.1822
168.0682
223.7508
214.4078
199.1799
191.6794
134.1969
130.8653
ω
-
5
280.0397
262.7506
335.7204
319.7813
305.4568
291.7087
223.4883
214.9894
0
ω
-
2
43.6649
43.6649
67.5918
67.5918
53.9610
55.9731
22.3541
26.4650
ω
-
3
98.1296
98.1296
132.8706
132.8706
113.7904
115.8588
66.1736
70.2707
ω
-
4
174.3787
174.3787
220.0032
220.0032
195.4598
197.5581
131.4575
135.6415
ω
-
5
272.5370
272.5370
329.1761
329.1761
299.1037
301.2218
218.6135
222.8467
3
ω
-
2
41.9691
44.8550
67.1081
67.7638
52.8041
56.9021
21.4951
27.1414
ω
-
3
94.5153
100.8244
130.2588
134.4815
110.6243
118.4554
63.6755
72.7816
ω
-
4
168.0682
179.1822
214.4078
223.7508
189.4610
202.4128
126.6073
140.3662
ω
-
5
262.7506
280.0397
319.7813
335.7204
289.4704
308.9241
210.6519
230.2512
10
ω
-
2
40.0700
45.6887
66.2401
68.1328
50.7956
57.7893
20.7361
27.0569
ω
-
3
90.5367
102.6601
127.6871
135.4215
106.3245
120.4638
60.1871
74.3559
ω
-
4
161.3146
182.4345
209.1345
225.7493
182.1851
205.9533
120.0294
144.0360
ω
-
5
252.4647
285.1358
310.8793
339.2851
278.4567
314.4208
200.4314
236.3716Figure 11
Convergence of natural frequencies for axially exponentially FG cantilever beams with respect to spline node numberN
(
α
=
-
10
).
(a)
(b)
(c)Figure 12
Variation of first three nondimensional natural frequencies of an axially FG beam with respect toα; boundary condition: C-C.Figure 13
Variation of first three nondimensional natural frequencies of an axially FG beam with respect toα; boundary condition: P-P.Figure 14
Variation of first three nondimensional natural frequencies of an axially FG beam with respect toα; boundary condition: C-F.Figure 15
Variation of first three nondimensional natural frequencies of an axially FG beam with respect toα; boundary condition: C-P.Figure11 indicates that convergence solutions of natural frequencies with high accuracy will be obtained for the proposed model of the axially exponentially FG beams at N
=
15. Therefore, for axially FG beams with different gradient index investigated in this section, let N
=
15 be the suitable choice for all models to achieve the demands for both the high accuracy and low computational cost. Table 9 indicates that the results of fundamental frequencies of the proposed model agree with those reported by Huang and Li [14] very well. The comparison of the results for Cases 1 and 2 listed in Tables 9 and 10 indicates that the gradient index α is important for the natural frequencies.As shown in Figures12 and 13, the natural frequencies of P-P beams and C-C beams for Case 1 with index α are identical to those for Case 2 with index −α. However, the varying tendency of the natural frequencies for C-F and C-P beams with respect to gradient index is different because of the asymmetry of the boundary conditions, as shown in Figures 14 and 15. The fundamental natural transverse frequencies for Case 2 are larger than those for Case 1 with similar index α. However, the high-order natural frequencies, which show a complex variation tendency, do not always follow this law. For all beams, the natural frequencies for Case 2 show an ascending tendency with the increase of gradient index, whereas the natural frequencies for Case 1 show an opposite trend.The natural transverse frequencies of the tapered beams (c
b
=
c
h
=
c
0) with material nonhomogeneity defined by (50) (α
=
-
10), which belongs to exponential law, are calculated by the proposed model to investigate the effects of the taper ratio, and the results are tabulated in Table 11, which indicate that all natural transverse frequencies for the beams with different boundary conditions generally decrease with the taper ratio, except for the beams with C-F boundary, whose fundamental natural frequencies (as shown in italic bold fonts in Table 11) increase with the taper ratio. By contrast, the high-order frequencies show an opposite trend, and the same varying trends have been found for the tapered beams with material gradation following the polynomial law (Section 4.2) with respect to the taper ratios.Table 11
Nondimensional natural transverse frequenciesω
-
i for axially FG tapered beam with respect to taper ratio c
b
=
c
h
=
c
0, α
=
-
10.
c
0
P-P
C-C
ω
-
1
ω
-
2
ω
-
3
ω
-
4
ω
-
5
ω
-
1
ω
-
2
ω
-
3
ω
-
4
ω
-
5
Case  1
0
11.4560
45.6887
102.6601
182.4345
285.1358
24.0660
68.1328
135.4215
225.7493
339.2851
0.2
10.2635
41.0849
92.2393
163.8387
256.0119
21.5207
61.0451
121.4017
202.4339
304.2973
0.4
8.8941
36.2851
81.2897
144.1972
225.1632
18.8852
53.6029
106.6116
177.8075
267.3534
0.6
7.2513
31.1822
69.5172
122.9219
191.6076
16.1104
45.6006
90.5999
151.0965
227.2883
0.8
5.0932
25.5025
56.1391
98.3922
152.5382
13.0533
36.4076
71.8833
119.5672
179.6797
Case  2
0
9.9366
40.0700
90.5367
161.3146
252.4647
24.8068
66.2401
127.6871
209.1345
310.8793
0.2
8.8494
35.9365
81.2126
144.6952
226.4368
22.3802
59.6013
114.7530
187.8207
279.0755
0.4
7.6207
31.6402
71.4122
127.1143
198.8076
19.8231
52.5571
100.9830
165.0931
245.1303
0.6
6.1724
27.0932
60.8831
108.0671
168.7522
17.0641
44.8967
85.9592
140.2721
208.0589
0.8
4.3128
22.1136
49.1297
86.6404
134.9467
14.0097
36.3471
69.2653
113.0049
167.9583
C-P
C-F
Case  1
0
16.4111
54.9276
116.4430
201.0000
308.7099
3.5340
23.7075
68.1642
135.4187
225.7516
0.2
15.3662
49.9039
105.0941
180.9523
277.5884
3.9253
22.6573
62.5140
122.8555
203.8977
0.4
14.2204
44.6316
93.1433
159.8062
244.7549
4.4598
21.5947
56.6601
109.7005
180.9238
0.6
12.9302
38.9644
80.2439
136.9256
209.2062
5.2495
20.5907
50.5440
95.6707
156.2332
0.8
11.3909
32.4806
65.3274
110.2051
167.3687
6.5769
19.9787
44.1206
79.9627
127.8152
Case  2
0
17.3037
53.9917
110.4619
186.8845
283.5206
4.1801
24.4362
66.2609
127.6897
209.1407
0.2
16.2634
49.2161
99.9160
168.4860
255.1649
4.5112
23.3314
60.8734
116.0144
189.0897
0.4
15.1061
44.1433
88.7009
148.8877
224.9241
4.9675
22.1766
55.2115
103.6541
167.7838
0.6
13.7727
38.6134
76.4763
127.5011
191.9048
5.6563
21.0079
49.1851
90.3173
144.6597
0.8
12.1299
32.2583
62.5395
103.3128
154.9786
6.8715
20.0744
42.7624
75.5952
118.9682The results obtained by the numerical examples in this section indicate that the proposed model shows good adaptability to the variation of cross sections, boundary conditions, and material gradient functions in either a polynomial or an exponential gradient function. And for the free vibration of tapered beams with different type of material nonhomogeneity, the results of higher accuracy and lower computational cost are obtained for the proposed SFPM model compared with other reported models. What is more, the natural frequencies of the tapered beams of the same boundary conditions with homogenous and nonhomogeneous materials vary in a similar trend toward the variation of the taper ratios. The material properties, taper ratios, and boundary conditions play an important role in determining the natural transverse frequencies for axially FG tapered Euler-Bernoulli beams.


## 4.1. A Comparison of the Results for Uniform Homogenous Beams

As the first example, we consider the case of a homogeneous beam with uniform cross sections. For this case Young’s modulus(
E
(
x
)
=
E
0
), mass density (
ρ
(
x
)
=
ρ
0
), cross-sectional area (
A
(
x
)
=
A
0
), and second moment of inertia (
I
(
x
)
=
I
0
) are not changed. We calculate the nondimensional natural transverse frequencies, ω
-, of cantilever beams (C-F) by choosing a different spline node number (
N
) in (12) to check the convergence of the presented method (i.e., SFPM). The first four nondimensional natural frequencies, ω
-
i, are tabulated in Table 1 and compared with the exact solutions [35] and other numerical methods [14]. The comparison indicated that the SFPM solution has rapid convergence. As N increases from 5 to 25, the first three nondimensional natural frequencies are identical to the exact solutions. This finding indicates that the present method is efficient except for the case of N
=
5, in which the fourth and fifth nondimensional natural frequencies do not match the exact solution because of the insufficient order of the eigenvalue equation, that is, (27). However, the proposed model does not have any convergence problems. As N increases from 15 to 25, computational time and cost increase, but the results of the natural frequencies via SFPM vary slightly. Thus, we take spline node number N
=
15 in the following examples. Moreover, the first five nondimensional natural transverse frequencies for C-C, C-P, and P-P beams are calculated and tabulated in Table 2 with N
=
15. This finding verifies that SFPM can be applied to analyze the free vibration problem of beams with various boundary conditions.Table 1
First five nondimensional transverse natural frequenciesω
-
i for uniform beam; boundary condition: C-F.
ω
-
i
SFPM
Exact [35]
Reference [14]
N
=
5
N
=
10
N
=
15
N
=
20
N
=
25
ω
-
1
3.516
3.516
3.516
3.516
3.516
3.516
3.516
ω
-
2
22.049
22.035
22.035
22.035
22.035
22.034
22.0345
ω
-
3
62.11
61.716
61.701
61.698
61.698
61.701
61.6972
ω
-
4
124.526
121.062
120.929
120.91
120.905
120.909
120.9019
ω
-
5
205.179
200.702
199.994
199.899
199.875
199.855
—Table 2
First five nondimensional transverse natural frequenciesω
-
i for uniform beams with various boundary conditions.
ω
-
i
C-C
C-P
P-P
SFPM
Exact [35]
SFPM
Exact [35]
SFPM
Exact [35]
ω
-
1
22.373
22.372
15.418
15.413
9.87
9.869
ω
-
2
61.674
61.669
49.965
49.956
39.479
39.478
ω
-
3
120.912
120.912
104.253
104.244
88.828
88.826
ω
-
4
199.898
199.855
178.297
178.275
157.921
157.913
ω
-
5
298.69
298.529
272.132
272.019
246.769
246.739


## 4.2. Effect of the Variable Cross Section

The second example is devoted to the Euler-Bernoulli beam with a rectangular cross section and has a breadth and height that taper linearly, as shown in Figure2. Thus, the cross-sectional area A
(
x
) and the second moment of inertia I
(
x
) that vary along axis are defined as (28), as mentioned earlier.Firstly, the natural transverse frequencies of the tapered homogenous beams with various boundary conditions (C-F, C-C, and P-P) are calculated by using SFPM, for all beams,N
=
15 is selected, and two cases of cross-sectional variation, namely, the case of only the height tapered linearly (
c
h
=
c
0
) and the case of both breadth and height tapered linearly (
c
b
=
c
h
=
c
0
), are both investigated. In order to validate the performance of the proposed method, FEM models (3D), which are implemented in ANSYS (see in Figure 5), are employed to investigate the free vibration of the tapered homogenous beams. In the FEM models, 3D solid element SOLID64 is employed, and all beams are meshed into 1600 elements to ensure the computational accuracy for fundamental and high-order frequencies and to avoid producing the abnormal element due to the increase of taper ratios. The nondimensional natural transverse frequencies for C-F, C-C, and P-P beams are tabulated in Tables 3–5, which indicate that the results of SFPM model agree very well with those of FEM models and DTM model reported by Rajasekaran [4]. The taper ratio (c
0) plays an important role in the tapered homogenous beams. For all beams, the natural frequencies decrease with the increase of the taper ratio (c
0) except for the fundamental frequencies of C-F beams, which increase with c
0; for C-F beams, it is obvious that the frequencies of the case of height tapered linearly (
c
h
=
c
0
) are smaller than those of the case of both height and breadth tapered linearly (
c
b
=
c
h
=
c
0
), while, for C-C and P-P beams, the frequencies of the case of height tapered linearly (
c
h
=
c
0
) are equal to those of the case of both height and breadth tapered linearly (
c
b
=
c
h
=
c
0
). However, in this study, given N
=
15 is employed, the SFPM model shows a considerable decrease of the dimensions of the eigenvalue equations compared with the FEM and DTM models [4], leading to remarkable decrease in computational cost. Furthermore, FEM models implemented in ANSYS are unable to analyze the free vibration of tapered beams with material nonhomogeneity along the longitudinal direction.Table 3
Nondimensional transverse natural frequenciesω
-
i for tapered homogenous beam; boundary condition: C-F.
c
0
c
b
=
0, c
h
=
c
0
c
b
=
c
h
=
c
0
ω
-
1
ω
-
2
ω
-
3
ω
-
1
ω
-
2
ω
-
3
0.2
SFPM
3.6083
20.6212
56.1954
3.8551
21.0569
56.6335
Reference [4]
3.6083
20.621
56.1923
3.8551
21.0567
56.6303
FEM
3.6113
20.5759
55.8008
3.8580
21.0108
56.2356
0.4
SFPM
3.7371
19.114
50.3567
4.31878
20.0501
51.3378
Reference [4]
3.7371
19.1138
50.3537
4.3188
20.05
51.3346
FEM
3.7398
19.0857
50.0932
4.3213
20.0187
51.0670
0.6
SFPM
3.93428
17.488
44.0279
5.00904
19.0651
45.7417
Reference [4]
3.9343
17.4879
44.0248
5.009
19.0649
45.7384
FEM
3.9366
17.4717
43.8706
5.0110
19.0463
45.5747
0.8
SFPM
4.2925
15.7429
36.8881
6.19639
18.3858
39.8376
Reference [4]
4.2925
15.7427
36.8852
6.1964
18.3855
39.8339
FEM
4.2946
15.7366
36.8135
6.1978
18.3751
39.7517Table 4
Nondimensional transverse natural frequenciesω
-
i for tapered homogenous beam; boundary condition: C-C.
c
0
c
b
=
0, c
h
=
c
0
c
b
=
c
h
=
c
0
ω
-
1
ω
-
2
ω
-
3
ω
-
1
ω
-
2
ω
-
3
0.2
SFPM
20.0784
55.3432
108.5080
20.0968
55.3683
108.5357
FEM
20.0525
54.9593
106.9451
20.0708
54.9832
106.9690
0.4
SFPM
17.6357
48.5873
95.2438
17.7203
48.7027
95.3717
FEM
17.6321
48.3680
94.2691
17.7151
48.4791
94.3873
0.6
SFPM
14.9624
41.1721
80.6681
15.1914
41.4862
81.0196
FEM
14.9753
41.0829
80.1634
15.2019
41.3883
80.4955
0.8
SFPM
11.8486
32.5079
63.6182
12.3970
33.2794
64.5189
FEM
11.8684
32.4963
63.4193
12.4054
33.2308
64.2326Table 5
Nondimensional transverse natural frequenciesω
-
i for tapered homogenous beam; boundary condition: P-P.
c
0
c
b
=
0, c
h
=
c
0
c
b
=
c
h
=
c
0
ω
-
1
ω
-
2
ω
-
3
ω
-
1
ω
-
2
ω
-
3
0.2
SFPM
8.8462
35.4461
79.7399
8.8246
35.4665
79.7721
FEM
8.8370
35.2994
78.9997
8.8153
35.3191
79.0321
0.4
SFPM
7.7295
31.1945
70.0980
7.6314
31.2881
70.2453
FEM
7.7234
31.0975
69.6109
7.6251
31.1904
69.7559
0.6
SFPM
6.4667
26.6014
59.6076
6.2087
26.8534
60.0036
FEM
6.4631
26.5453
59.3230
6.2050
26.7957
59.7141
0.8
SFPM
4.9201
21.3485
47.5048
4.3536
21.9447
48.4357
FEM
4.9184
21.3218
47.3703
4.3515
21.9142
48.2878Figure 5
FEM model for homogenous tapered beams by using ANSYS.Then, we study the free vibration of axially FG tapered beam with Young’s modulus, and the material densities along the axis are defined as(47)
E
x
=
E
0
1
+
x
L
,
ρ
x
=
ρ
0
1
+
x
L
+
x
L
2
,where E
0 and ρ
0 are Young’s modulus and mass density, respectively, at x
=
0.The material properties defined by (47) follow the polynomial law distribution, and (47) can be simplified as (31), with n
1
=
1 and n
2
=
2: (48)
E
x
=
∑
i
=
0
1
E
~
i
x
i
,
ρ
x
=
∑
i
=
0
2
ρ
~
i
x
i
,in which(49)
E
~
i
=
E
0
L
i
i
=
0,1
;
ρ
~
i
=
ρ
0
L
i
i
=
0,1
,
2
.Using (48), the structural stiffness and mass matrices can be calculated by (34). In Figure 6, the convergence of natural frequencies for axially FG tapered cantilever beams (
c
b
=
c
h
=
0.8
) (material gradation following the polynomial law) with respect to the spline node number N is discussed; it is noticeable that the natural frequencies achieve a convergence value when N increase from 5 to 15 (about 4-digit precision), and when N increase from 15 to 25, the frequencies vary a little but the computational cost increased remarkably. Therefore, N
=
15 is a suitable choice for the proposed SFPM models to ensure both the accuracy and efficiency for the free vibration of the axially FG tapered beams with different taper ratios.Figure 6
Convergence of natural frequencies for axially FG tapered cantilever beams with respect to spline node numberN
(
c
b
=
c
h
=
0.8
).
(a)
(b)
(c)The natural transverse frequencies for axially FG tapered beams with different breadth taper ratio(
c
b
) and height taper ratio (
c
h
) are calculated by using the proposed method, and the results are tabulated in Tables 6–8. The results of SFPM agree with the results of DTEM reported by Shahba and Rajasekaran [3]. This observation indicates that the proposed method can be used to investigate the free vibration of axially FG tapered beams with high accuracy. The variations of the first two nondimensional natural frequencies of the axially FG beams with various boundary conditions for different values of taper ratios (
c
b
,
c
h
) are illustrated in Figures 7–9.Table 6
Nondimensional transverse natural frequenciesω
-
i for axially FG tapered beam; boundary condition: C-F.
c
h
c
b
0
0.2
0.4
0.6
0.8
ω
-
1
ω
-
2
ω
-
1
ω
-
2
ω
-
1
ω
-
2
ω
-
1
ω
-
2
ω
-
1
ω
-
2
0
SFPM
2.4256
18.6042
2.6054
19.0042
2.8508
19.5304
3.2137
20.2959
3.8310
21.6760
Reference [3]
2.4256
18.6041
2.6054
19.0041
2.8508
19.5303
3.2137
20.2958
3.8310
21.6759
0.2
SFPM
2.5051
17.3802
2.6863
17.7501
2.9336
18.2379
3.2993
18.9501
3.9219
20.2432
Reference [3]
2.5051
17.3802
2.6863
17.7501
2.9336
18.2379
3.2994
18.9501
3.9220
20.2432
0.4
SFPM
2.6155
16.0705
2.7987
16.4092
3.0486
16.8571
3.4181
17.5139
4.0471
18.7164
Reference [3]
2.6155
16.0705
2.7988
16.4092
3.0486
16.8571
3.4181
17.5139
4.0471
18.7164
0.6
SFPM
2.7835
14.6508
2.9699
14.9567
3.2237
15.3628
3.5985
15.9616
4.2355
17.0694
Reference [3]
2.7836
14.6508
2.9699
14.9567
3.2237
15.3627
3.5985
15.9616
4.2355
17.0694
0.8
SFPM
3.0871
13.1142
3.2794
13.3850
3.5401
13.7466
3.9232
14.2848
4.5695
15.2955
Reference [3]
3.0871
13.1142
3.2794
13.3849
3.5401
13.7466
3.9232
14.2848
4.5695
15.2954Table 7
Nondimensional transverse natural frequenciesω
-
i for axially FG tapered beam; boundary condition: C-C.
c
h
c
b
0
0.2
0.4
0.6
0.8
ω
-
1
ω
-
2
ω
-
1
ω
-
2
ω
-
1
ω
-
2
ω
-
1
ω
-
2
ω
-
1
ω
-
2
0
SFPM
20.4721
56.5492
20.4152
56.4716
20.2883
56.2981
20.0186
55.9206
19.3845
54.9713
Reference [3]
20.4721
56.5482
20.4151
56.4706
20.2883
56.2971
20.0186
55.9195
19.3844
54.9699
0.2
SFPM
18.2170
50.4796
18.1996
50.4560
18.1286
50.3593
17.9436
50.1011
17.4565
49.3718
Reference [3]
18.2170
50.4792
18.1995
50.4556
18.1286
50.3589
17.9436
50.1006
17.4564
49.3711
0.4
SFPM
15.8282
44.0246
15.8498
44.0553
15.8350
44.0371
15.7367
43.9027
15.4025
43.4066
Reference [3]
15.8281
44.0236
15.8497
44.0542
15.8349
44.0359
15.7366
43.9013
15.4021
43.4043
0.6
SFPM
13.2294
36.9654
13.2896
37.0509
13.3319
37.1137
13.3238
37.1105
13.1529
36.8680
Reference [3]
13.2291
36.9636
13.2894
37.0490
13.3316
37.1116
13.3234
37.1079
13.1521
36.8635
0.8
SFPM
10.2236
28.7498
10.3231
28.8919
10.4257
29.0416
10.5170
29.1851
10.5343
29.2419
Reference [3]
10.2217
28.7406
10.3211
28.8822
10.4234
29.0311
10.5143
29.1728
10.5301
29.2239Table 8
Nondimensional transverse natural frequenciesω
-
i for axially FG tapered beam; boundary condition: P-P.
c
h
c
b
0
0.2
0.4
0.6
0.8
ω
-
1
ω
-
2
ω
-
1
ω
-
2
ω
-
1
ω
-
2
ω
-
1
ω
-
2
ω
-
1
ω
-
2
0
SFPM
9.0286
36.3718
9.0599
36.3418
9.0867
36.3153
9.0994
36.2969
9.0685
36.2769
Reference [3]
9.0285
36.3715
9.0599
36.3415
9.0867
36.3151
9.0994
36.2966
9.0685
36.2766
0.2
SFPM
8.1341
32.5236
8.1462
32.5123
8.1498
32.5079
8.1336
32.5164
8.0646
32.5326
Reference [3]
8.1341
32.5234
8.1462
32.5121
8.1498
32.5077
8.1336
32.5161
8.0646
32.5323
0.4
SFPM
7.1531
28.4747
7.1455
28.4822
7.1254
28.5003
7.0794
28.5370
6.9703
28.5928
Reference [3]
7.1531
28.4744
7.1455
28.4819
7.1254
28.5001
7.0794
28.5367
6.9703
28.5924
0.6
SFPM
6.0357
24.1101
6.0082
24.1371
5.9638
24.1791
5.8868
24.2469
5.7351
24.3497
Reference [3]
6.0357
24.1098
6.0082
24.1367
5.9637
24.1787
5.8868
24.2465
5.7351
24.3491
0.8
SFPM
4.6520
19.1315
4.6046
19.1804
4.5355
19.2510
4.4264
19.3591
4.2284
19.5302
Reference [3]
4.6520
19.1304
4.6045
19.1792
4.5354
19.2496
4.4263
19.3576
4.2281
19.5281Figure 7
The variation of the first two nondimensional natural frequencies of axially FG beams with respect to taper ratios; boundary condition: C-F.
(a)
(b)Figure 8
The variation of the first two nondimensional natural frequencies of axially FG beams with respect to taper ratios; boundary condition: C-C.
(a)
(b)Figure 9
The variation of the first two nondimensional natural frequencies of axially FG beams with respect to taper ratios; boundary condition: P-P.
(a)
(b)Figures7–9 illustrate that the varying trend of the natural frequencies is distinguished from beams with different boundary conditions. In general, the natural transverse frequencies are considerably affected by the variation of height taper ratio c
h. Figure 7 and Table 6 indicate that the increase of taper ratios c
h and c
b leads to the increase of the fundamental natural frequencies for C-F beams, whose second frequencies decrease with c
h and slightly increase with c
b. For C-C and P-P beams, Figures 8 and 9 and Tables 7 and 8 indicate that all frequencies remarkably decrease with c
h, whereas the variations with the increase of c
b are not monotonic. Thus, predicting the natural frequencies for axially FG tapered beams of different boundary conditions with respect to the taper ratios is impossible.Notice that the proposed model needs considerably less computational cost and modeling time compared with the DTM and DTEM models developed by Rajasekaran [4] and Shahba et al. [2], respectively. In the DTM and DTEM models, 132 simple points and 12 elements are used, and the eigenvalue equation in the dimension of 578
×
578 needs to be solved. However, in the proposed model, as N
=
15 is selected based on the convergence study, the eigenvalue equation is only in the dimension of 18
×
18, which shows that the proposed method has higher remarkable ascending efficiency than the reported methods.Furthermore, the natural transverse frequencies for tapered beams with consideration of material nonhomogeneity and homogeneity are distinguished by comparing the results in Table3 with those in Table 4. The material nonhomogeneity defined by (47) decreases the natural frequencies for tapered C-F beams compared with homogenous beams. Thus, the material properties obviously affect the dynamic characteristics of the tapered beams.


## 4.3. Effect of Axial Gradient Parameter

Finally, the free vibration for tapered rectangular beams with axially exponentially graded material is investigated to verify the adaptability of the proposed method to different gradient functions. Similar works have been conducted by Huang and Li [14]. The material properties are defined as follows: (50)
Y
x
=
Y
1
1
-
e
α
x
/
L
-
1
e
α
-
1
+
Y
2
e
α
x
/
L
-
1
e
α
-
1
,
i
f
α
≠
0
,
Y
x
=
Y
1
1
-
x
L
+
Y
2
x
L
,
i
f
α
=
0
,in which Y
1 and Y
2 correspond to the material properties, such as Young’s modulus E
(
x
) and mass density ρ
(
x
) at x
=
0 and x
=
L, respectively, and α is the gradient parameter that describes the volume fraction change of both constituents involved. The variation of Y
(
x
) against x is shown in Figure 10 for Y
1
=
3
Y
2.Figure 10
Variation of gradation material properties defined by (50) with Y
1
=
3
Y
2.The material properties defined by (50) can be classified into two types of distribution that depends on the value of gradient index α; one is the polynomial type that is defined by (31) (α
=
0) with n
1
=
n
2
=
1 and the other is exponential type that is defined by (32) (α
≠
0), which can be simplified into the following equations:(51)
Y
x
=
Y
~
1
+
Y
~
2
e
α
x
/
L
,
i
f
α
≠
0
,
Y
x
=
Y
~
1
+
Y
~
2
x
,
i
f
α
=
0
,where(52)
Y
~
1
=
Y
1
-
Y
2
-
Y
1
e
α
-
1
,
Y
~
2
=
Y
2
-
Y
1
e
α
-
1
,
i
f
α
≠
0
,
Y
~
1
=
Y
1
,
Y
~
2
=
Y
2
-
Y
1
L
,
i
f
α
=
0
.Using (51), the stiffness and mass matrices can be calculated by (34) and (37) for α
=
0 and α
≠
0, respectively. In the following calculations, the two materials chosen are aluminum (Al) and zirconia (ZrO2). The material properties are presented by [14] (53)
Z
r
O
2
:
E
z
=
2
.
0
×
10
11
N
/
m
2
,
ρ
z
=
7900
k
g
/
m
3
,
A
l
:
E
a
=
7
.
0
×
10
1
0
N
/
m
2
,
ρ
a
=
2702
k
g
/
m
3
.The nondimensional frequencies are defined by (54)
ω
-
=
ω
ρ
a
A
0
L
4
E
a
I
0
.In consideration of the two cases in this paper, one is Al rich near the end x
=
0 and ZrO2 rich near the end at x
=
L (Case 1), whereas the other is Al rich near the end at x
=
L and ZrO2 rich near the end at x
=
0 (Case 2).The free vibration of axially FG beams with uniform cross-sectional(
c
b
=
c
h
=
0
) and material properties defined as (50) is calculated to verify the proposed method (i.e., SFPM). The results for the fundamental nondimensional natural transverse frequencies are tabulated in Table 9, and the high-order frequencies calculated with SFPM are tabulated in Table 10. Figure 11 gives the convergence of the natural frequencies for axially exponentially FG cantilever beams (C-F) (α
=
-
10) with respect to spline node number N; the variations of the first three nondimensional natural frequencies of axially FG beams with respect to gradient parameter α are depicted in Figures 12–15.Table 9
Nondimensional fundamental natural transverse frequenciesω
-
i for axially FG beams with respect to the gradient index.
α
P-P
C-C
C-P
C-F
Case  1
Case  2
Case  1
Case  2
Case  1
Case  2
Case  1
Case  2
−10
Reference [14]
11.4532
9.9358
24.0576
24.7949
16.4775
17.2993
3.5656
4.1800
SFPM
11.4560
9.9366
24.0660
24.8068
16.4111
17.3037
3.5340
4.1801
−3
Reference [14]
11.2443
10.3669
23.9456
24.9375
16.0307
17.8701
3.1421
4.8317
SFPM
11.2443
10.3669
23.9446
24.9366
16.0262
17.8701
3.1410
4.8317
0
Reference [14]
10.8663
10.8663
24.3752
24.3752
15.8734
17.9147
2.9256
5.0156
SFPM
10.8664
10.8664
24.3756
24.3756
15.8735
17.9148
2.9256
5.0156
3
Reference [14]
10.3669
11.2443
24.9375
23.9456
15.7171
17.8867
2.8544
4.8466
SFPM
10.3669
11.2443
24.9366
23.9446
15.7169
17.8874
2.8545
4.8466
10
Reference [14]
9.9358
11.4532
24.7949
23.9456
15.4930
17.9050
3.0431
4.4629
SFPM
9.9366
11.4560
24.8068
24.0660
15.4958
17.9155
3.0431
4.4629Table 10
Nondimensional higher-order transverse natural frequenciesω
-
i for axially FG beams with respect to the gradient index.
α
P-P
C-C
C-P
C-F
Case  1
Case  2
Case  1
Case  2
Case  1
Case  2
Case  1
Case  2
−10
ω
-
2
45.6887
40.0700
68.1328
66.2401
54.9276
53.9917
23.7075
24.4362
ω
-
3
102.6601
90.5367
135.4215
127.6871
116.4430
110.4619
68.1642
66.2609
ω
-
4
182.4345
161.3146
225.7493
209.1345
201.0000
186.8845
135.4187
127.6897
ω
-
5
285.1358
252.4647
339.2851
310.8793
308.7099
283.5206
225.7516
209.1407
−3
ω
-
2
44.8550
41.9691
67.7638
67.1081
54.4731
54.9320
23.0531
25.4924
ω
-
3
100.8244
94.5153
134.4815
130.2588
115.5467
112.8131
67.4843
67.7907
ω
-
4
179.1822
168.0682
223.7508
214.4078
199.1799
191.6794
134.1969
130.8653
ω
-
5
280.0397
262.7506
335.7204
319.7813
305.4568
291.7087
223.4883
214.9894
0
ω
-
2
43.6649
43.6649
67.5918
67.5918
53.9610
55.9731
22.3541
26.4650
ω
-
3
98.1296
98.1296
132.8706
132.8706
113.7904
115.8588
66.1736
70.2707
ω
-
4
174.3787
174.3787
220.0032
220.0032
195.4598
197.5581
131.4575
135.6415
ω
-
5
272.5370
272.5370
329.1761
329.1761
299.1037
301.2218
218.6135
222.8467
3
ω
-
2
41.9691
44.8550
67.1081
67.7638
52.8041
56.9021
21.4951
27.1414
ω
-
3
94.5153
100.8244
130.2588
134.4815
110.6243
118.4554
63.6755
72.7816
ω
-
4
168.0682
179.1822
214.4078
223.7508
189.4610
202.4128
126.6073
140.3662
ω
-
5
262.7506
280.0397
319.7813
335.7204
289.4704
308.9241
210.6519
230.2512
10
ω
-
2
40.0700
45.6887
66.2401
68.1328
50.7956
57.7893
20.7361
27.0569
ω
-
3
90.5367
102.6601
127.6871
135.4215
106.3245
120.4638
60.1871
74.3559
ω
-
4
161.3146
182.4345
209.1345
225.7493
182.1851
205.9533
120.0294
144.0360
ω
-
5
252.4647
285.1358
310.8793
339.2851
278.4567
314.4208
200.4314
236.3716Figure 11
Convergence of natural frequencies for axially exponentially FG cantilever beams with respect to spline node numberN
(
α
=
-
10
).
(a)
(b)
(c)Figure 12
Variation of first three nondimensional natural frequencies of an axially FG beam with respect toα; boundary condition: C-C.Figure 13
Variation of first three nondimensional natural frequencies of an axially FG beam with respect toα; boundary condition: P-P.Figure 14
Variation of first three nondimensional natural frequencies of an axially FG beam with respect toα; boundary condition: C-F.Figure 15
Variation of first three nondimensional natural frequencies of an axially FG beam with respect toα; boundary condition: C-P.Figure11 indicates that convergence solutions of natural frequencies with high accuracy will be obtained for the proposed model of the axially exponentially FG beams at N
=
15. Therefore, for axially FG beams with different gradient index investigated in this section, let N
=
15 be the suitable choice for all models to achieve the demands for both the high accuracy and low computational cost. Table 9 indicates that the results of fundamental frequencies of the proposed model agree with those reported by Huang and Li [14] very well. The comparison of the results for Cases 1 and 2 listed in Tables 9 and 10 indicates that the gradient index α is important for the natural frequencies.As shown in Figures12 and 13, the natural frequencies of P-P beams and C-C beams for Case 1 with index α are identical to those for Case 2 with index −α. However, the varying tendency of the natural frequencies for C-F and C-P beams with respect to gradient index is different because of the asymmetry of the boundary conditions, as shown in Figures 14 and 15. The fundamental natural transverse frequencies for Case 2 are larger than those for Case 1 with similar index α. However, the high-order natural frequencies, which show a complex variation tendency, do not always follow this law. For all beams, the natural frequencies for Case 2 show an ascending tendency with the increase of gradient index, whereas the natural frequencies for Case 1 show an opposite trend.The natural transverse frequencies of the tapered beams (c
b
=
c
h
=
c
0) with material nonhomogeneity defined by (50) (α
=
-
10), which belongs to exponential law, are calculated by the proposed model to investigate the effects of the taper ratio, and the results are tabulated in Table 11, which indicate that all natural transverse frequencies for the beams with different boundary conditions generally decrease with the taper ratio, except for the beams with C-F boundary, whose fundamental natural frequencies (as shown in italic bold fonts in Table 11) increase with the taper ratio. By contrast, the high-order frequencies show an opposite trend, and the same varying trends have been found for the tapered beams with material gradation following the polynomial law (Section 4.2) with respect to the taper ratios.Table 11
Nondimensional natural transverse frequenciesω
-
i for axially FG tapered beam with respect to taper ratio c
b
=
c
h
=
c
0, α
=
-
10.
c
0
P-P
C-C
ω
-
1
ω
-
2
ω
-
3
ω
-
4
ω
-
5
ω
-
1
ω
-
2
ω
-
3
ω
-
4
ω
-
5
Case  1
0
11.4560
45.6887
102.6601
182.4345
285.1358
24.0660
68.1328
135.4215
225.7493
339.2851
0.2
10.2635
41.0849
92.2393
163.8387
256.0119
21.5207
61.0451
121.4017
202.4339
304.2973
0.4
8.8941
36.2851
81.2897
144.1972
225.1632
18.8852
53.6029
106.6116
177.8075
267.3534
0.6
7.2513
31.1822
69.5172
122.9219
191.6076
16.1104
45.6006
90.5999
151.0965
227.2883
0.8
5.0932
25.5025
56.1391
98.3922
152.5382
13.0533
36.4076
71.8833
119.5672
179.6797
Case  2
0
9.9366
40.0700
90.5367
161.3146
252.4647
24.8068
66.2401
127.6871
209.1345
310.8793
0.2
8.8494
35.9365
81.2126
144.6952
226.4368
22.3802
59.6013
114.7530
187.8207
279.0755
0.4
7.6207
31.6402
71.4122
127.1143
198.8076
19.8231
52.5571
100.9830
165.0931
245.1303
0.6
6.1724
27.0932
60.8831
108.0671
168.7522
17.0641
44.8967
85.9592
140.2721
208.0589
0.8
4.3128
22.1136
49.1297
86.6404
134.9467
14.0097
36.3471
69.2653
113.0049
167.9583
C-P
C-F
Case  1
0
16.4111
54.9276
116.4430
201.0000
308.7099
3.5340
23.7075
68.1642
135.4187
225.7516
0.2
15.3662
49.9039
105.0941
180.9523
277.5884
3.9253
22.6573
62.5140
122.8555
203.8977
0.4
14.2204
44.6316
93.1433
159.8062
244.7549
4.4598
21.5947
56.6601
109.7005
180.9238
0.6
12.9302
38.9644
80.2439
136.9256
209.2062
5.2495
20.5907
50.5440
95.6707
156.2332
0.8
11.3909
32.4806
65.3274
110.2051
167.3687
6.5769
19.9787
44.1206
79.9627
127.8152
Case  2
0
17.3037
53.9917
110.4619
186.8845
283.5206
4.1801
24.4362
66.2609
127.6897
209.1407
0.2
16.2634
49.2161
99.9160
168.4860
255.1649
4.5112
23.3314
60.8734
116.0144
189.0897
0.4
15.1061
44.1433
88.7009
148.8877
224.9241
4.9675
22.1766
55.2115
103.6541
167.7838
0.6
13.7727
38.6134
76.4763
127.5011
191.9048
5.6563
21.0079
49.1851
90.3173
144.6597
0.8
12.1299
32.2583
62.5395
103.3128
154.9786
6.8715
20.0744
42.7624
75.5952
118.9682The results obtained by the numerical examples in this section indicate that the proposed model shows good adaptability to the variation of cross sections, boundary conditions, and material gradient functions in either a polynomial or an exponential gradient function. And for the free vibration of tapered beams with different type of material nonhomogeneity, the results of higher accuracy and lower computational cost are obtained for the proposed SFPM model compared with other reported models. What is more, the natural frequencies of the tapered beams of the same boundary conditions with homogenous and nonhomogeneous materials vary in a similar trend toward the variation of the taper ratios. The material properties, taper ratios, and boundary conditions play an important role in determining the natural transverse frequencies for axially FG tapered Euler-Bernoulli beams.


## 5. Conclusion

In this study, a new model for the analysis of the free vibration of axially FG tapered Euler-Bernoulli beams is developed with SFPM. The effects of the variation in cross-sectional and material properties along the longitudinal directions are investigated. In the proposed approach, the beam is discretized with a set of uniformly scattered spline nodes along the beam axis direction instead of meshes, thereby resulting in convenience for modeling and programming. The displacement field of the beam is approximated with cubic B-spline interpolation functions, which shows good adaptability to different boundary conditions. Given the special features of SFPM, the global structural stiffness and mass matrices for axially FG tapered beams are directly generated after spline discretization instead of element meshes, generation, and assembling. The effectiveness of the proposed method is confirmed by comparing our numerical results with those available in the literature, including the case of rectangular beams with linearly varying cross-sectional breadth or height and axially FG tapered beams with polynomial or exponential function gradient. All results show that the proposed method can analyze the free vibration of Euler-Bernoulli beams with arbitrary axial gradient and varying cross sections in various boundary conditions. Moreover, a stable solution with high accuracy can be achieved with a small spline node number.



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*Source: 5891030-2016-04-05.xml*