The vibration analysis of the elastically restrained functionally graded Timoshenko beam with arbitrary cross sections

Abstract

In this study, an investigation on the free vibration of the beam with material properties and cross section varying arbitrarily along the axis direction is studied based on the so-called Spectro-Geometric Method. The cross-section area and second moment of area of the beam are both expanded into Fourier cosine series, which are mathematically capable of representing any variable cross section. The Young’s modulus, the mass density and the shear modulus varying along the lengthwise direction of the beam, are also expanded into Fourier cosine series. The translational displacement and rotation of cross section are expressed into the Fourier series by adding some polynomial functions which are used to handle the elastic boundary conditions with more accuracy and high convergence rate. According to Hamilton’s principle, the eigenvalues and the coefficients of the Fourier series can be obtained. Some examples are presented to validate the accuracy of this method and study the influence of the parameters on the vibration of the beam. The results show that the first four natural frequencies gradually decrease as the coefficient of the radius $β$ increases, and decreases as the gradient parameter n increases under clamped–clamped end supports. The stiffness of the functionally Timoshenko beam with arbitrary cross sections is variable compared with the uniform beam, which makes the vibration amplitude of the beam have different changes.

Keywords

Functionally graded material Timoshenko beam vibration variable cross section Reyleigh-Ritz method

Introduction

Functionally graded material (FGM), a new type of composite materials which has continuous gradient changing in compositions and structures, was first proposed by materials scientists^1–3 to solve the problem of materials under harsh conditions in technical fields such as aeronautics and astronautics. The mechanical performance of plates, shells and beams made from FGM can be improved obviously. The material properties of FGM have shown different features in any desired spatial orientation by different theories such as exponential law,⁴ power law,⁵ and sigmoid law.⁶ FGM have great potential in many fields due to its superior performance than ordinary materials. The numerical method is used to study the free vibration of FGM beam in previous studies. In this paper, the exact solution is obtained to analyze the free vibration of FGM Timoshenko beam with arbitrary cross sections. Great efforts have been made for the vibration of the FGM beam by many engineers and researchers. Moreover, most of the researchers studied the type of the beam with material properties varying thicknesswise^7–16 and only a few researchers studied the FGM beam with material properties varying lengthwise.^17–20 Wu et al.¹⁷ studied the vibration of axially functionally graded (AFG) beams through semi-inverse method. Huang²¹ studied the free vibration of AFG beams with variable cross sections by the finite element method.

For the beam with non-uniform cross section, it is widely investigated by many researchers due to its better mechanical performance. A closed-form solution about non-uniform deflection beams was obtained by Romano.²² Fertis and Zobel²³ studied this problem by the equivalent system method. Ece et al.²⁴ investigated the vibration of the non-uniform cross-section beam with exponentially varying width. Numerical methods such as Frobenius method,^25,26 Rayleigh-Ritz method,²⁷ differential transform method,²⁸ the differential quadrature method,²⁹ semi-inverse method³⁰ and finite element method^31–33 have also been used to study the free vibration of the non-uniform cross-section beam.

It is well known that the influence of transverse shear deformations and rotary inertia in Euler-Bernoulli beams^34,35 is ignored while it is not neglected by Timoshenko beams. The methods of differential transform and differential quadrature are adopted to analyze the free vibration of the FGM beam.^36,37 The power series method and dynamic stiffness method are used in Leung et al.³⁸ for analyzing the free vibration of non-uniform Timoshenko beams. The method that transforms complex couple functions into a single government function via introducing an additional function is adopted by Huang et al.³⁹ to discuss the free vibration of AFG Timoshenko beam with variable cross section. Recently, many researchers are focusing on solving the vibration of plates, shells and beams with different end supports. A unified solution for the vibration of plates with holes under elastic boundary conditions is given in Wang et al.⁴⁰ Wang et al.⁴¹ made a prediction of break-out sound from a rectangular cavity via an elastically mounted panel. The free vibration of FGM plates and doubly curved shells with elastic ends supports are studied in literature.^42–44 The vibration problems of a FGM beam with thermo-elastic environment have been investigated by Chen et al.⁴⁵ according to a higher-order shear deformation beam theory. A numerical method of viewing a shifting as a multi-curved beam has been addressed for solving the vibration of the curved shifting in Wang et al.⁴⁶

As far as the authors know, there is no unified method which was proposed to study the vibration of the functional graded Timoshenko beam with arbitrary cross sections. In this paper, an unified procedure is proposed to study the elastically restrained FGMT beam with cross sections varying arbitrarily based on the spectro-geometric method (SGM)⁴⁷ in which the translational displacement and the rotation of cross section are both expressed into the Fourier series with some polynomial functions. The cross-sectional area, the second moment of area, the Young’s modulus, the mass density and the shear modulus have been expanded into Fourier cosine series. Several examples are introduced to validate the accuracy and reliability of this method. This method is generally applicable to any kind of beams with material properties and cross-sectional area varying arbitrarily along the axis direction under any elastic boundary conditions.

Theoretical formulation

Structural model

Figure 1 shows the FGMT beam with variable cross section. The beam is elastically restrained at the ends by translational and rotational springs, k₀, k₁, K₀ and K₁.

Figure 1.

The non-uniform cross section FGMT beam with elastically boundary conditions.

Figure 2.

A FGMT beam with variable circular cross section.

The coupled governing equations of the FGMT beam are expressed as⁴⁸

E (x) I (x) \frac{\partial^{2} θ}{\partial x^{2}} + k G (x) A (x) (\frac{\partial w}{\partial x} - θ) - ρ (x) I (x) \frac{\partial^{2} θ}{\partial t^{2}} = 0

(1)

ρ (x) A (x) \frac{\partial^{2} w}{\partial t^{2}} - k G (x) A (x) (\frac{\partial^{2} w}{\partial x^{2}} - \frac{\partial θ}{\partial x}) = 0

(2)

where

w

θ

I (x)

A (x)

E (x)

G (x)

k

and

ρ (x)

represent the transverse displacement of the beam, the rotation of cross section, second moment of area, cross-sectional area, Young’s modulus, shear modulus, shear coefficient and mass density, respectively.

The end supports of the FGMT beams with elastic constraints are given as

k_{0} w = k A (0) G (0) (\frac{d w}{d x} - θ) K_{0} θ = E (0) I (0) \frac{d θ}{d x} x = 0

(3)

k_{1} w = k A (L) G (L) (\frac{d w}{d x} - θ) K_{1} θ = - E (L) I (L) \frac{d θ}{d x} x = L

(4)

where

k_{0}

and

k_{1}

represent the linear spring constants, and

K_{0}

and

K_{1}

denote the rotational spring constants at x = 0 and x = L, respectively. Equations (3) and (4) represent a group of boundary conditions, and one can obtain different end supports by setting spring constants to be of different values.

The translational displacement and rotation of cross section of the beam can be expressed as

w (x) = \sum_{m = 0}^{N} A_{m} \cos λ_{m} x + \sum_{i = 1}^{4} C_{i} ξ_{i} (x)

(5)

θ (x) = \sum_{m = 0}^{N} B_{m} \cos λ_{m} x + \sum_{i = 1}^{2} D_{i} ξ_{i} (x)

(6)

where

λ_{m} = m π / L

. The additional polynomials,

ξ_{i} (x)

, are introduced to overcome the discontinuity of the initial displacements and its derivatives when they are extended periodically to the end of the beam. In the Timoshenko beam, the four additional polynomials of displacement w(x) are introduced to calculate the vibration of Timoshenko beam considering the second moment of area I(x) and shear modulus G(x). However, I(x) and G(x) are neglected for Euler beam.

The selected additional polynomials should satisfy any subsequent differential operations and should be sufficiently smooth over [0, L]. Mathematically, the term “sufficiently smooth” means that the first and third derivative of supplementary functions exist and keep continuous on the entire beam. In this paper, the selected additional polynomials are as follows

ξ_{1} (x) = \frac{9 L}{4 π} \sin (\frac{π x}{2 L}) - \frac{L}{12 π} \sin (\frac{3 π x}{2 L})

(7)

ξ_{2} (x) = - \frac{9 L}{4 π} \cos (\frac{π x}{2 L}) - \frac{L}{12 π} \cos (\frac{3 π x}{2 L})

(8)

ξ_{3} (x) = \frac{L^{3}}{π^{3}} \sin (\frac{π x}{2 L}) - \frac{L^{3}}{3 π^{3}} \sin (\frac{3 π x}{2 L})

(9)

ξ_{4} (x) = - \frac{L^{3}}{π^{3}} \cos (\frac{π x}{2 L}) - \frac{L^{3}}{3 π^{3}} \cos (\frac{3 π x}{2 L})

(10)

It is obvious from equations (7) to (10) that

ξ_{1}^{'} (0) = ξ_{3}^{″'} (0) = ξ_{2}^{'} (L) = ξ_{4}^{'''} (L) = 1

(11)

Solution for the system

For solving the vibration of the FGMT beam, the Rayleigh-Ritz method will be introduced in the following. The Lagrange function of the free vibration of the beam is

L = T - U

(12)

where

T

and

U

denote the total kinetic energy and the total potential energy of the beam.

U

and T can be expressed as

\begin{matrix} U = \frac{}{12} \int_{0}^{L} [E (x) I (x) {(\frac{d θ}{d x})}^{2} + k A (x) G (x) {(\frac{d w}{d x} - θ)}^{2}] d x \\ {+ {\frac{}{12} (k_{0} w^{2} + K_{0} θ^{2}) |}_{x = 0} + \frac{}{12} (k_{1} w^{2} + K_{1} θ^{2}) |}_{x = L} \end{matrix}

(13)

T = \frac{}{12} \int_{0}^{L} [ρ (x) I (x) {(\frac{d θ}{d t})}^{2} + ρ (x) A (x) {(\frac{d w}{d t})}^{2}] d x

(14)

It should be noted that the potential energy of the springs at boundaries are included in the total potential energy, and the stiffness of these elastic springs can be any value satisfying one’s requirements.

It is not easy to solve the integral in equations (13) and (14) with $A (x)$ , $I (x)$ , $E (x)$ , $G (x)$ and $ρ (x)$ varying arbitrarily along the beam. In order to overcome this problem, $A (x)$ , $I (x)$ , $E (x)$ , $G (x)$ and $ρ (x)$ are all expanded into the Fourier cosine series, as shown

A (x) = \sum_{m = 0}^{\infty} S_{m} \cos λ_{m} x

(15)

I (x) = \sum_{m = 0}^{\infty} I_{m} \cos λ_{m} x

(16)

E (x) = \sum_{m = 0}^{\infty} H_{m} \cos λ_{m} x

(17)

ρ (x) = \sum_{m = 0}^{\infty} Q_{m} \cos λ_{m} x

(18)

G (x) = \sum_{m = 0}^{\infty} G_{m} \cos λ_{m} x

(19)

where

λ_{m} = m π / L

S_{m}

I_{m}

H_{m}

Q_{m}

and

G_{m}

are Fourier coefficients which are actually obtained by Discrete Cosine Transform (DCT).⁴⁹

Substituting equations (5), (6), (13), (14) into equation (12) and minimizing all the unknown coefficients, the governing equations of the FGMT beam can be obtained as

(K - ω^{2} M) X = 0

(20)

where

K

M

and

X

are the stiffness matrix, the mass matrix and the column vector containing all the unknown coefficients in equations (5) and (6), respectively. By solving equation (20), one can derive the natural frequencies and the mode shapes of the beam.

Results and discussion

To validate the accuracy of the proposed method, several numerical examples will be first considered. The influence of the parameters of the beam on the vibration behaviors is also analyzed.

Validation of the present method

First, the free vibration of a uniform beam is considered in this part. In this case, the parameters of the beam are set as the same as those in literature^38,39 in which the parameters are $E_{0}$ = 200 GPa, $ρ_{0} = 5700 kg / m^{3}$ , $I_{0} / A {}_{0}$ = 0.01, $κ$ = 5/6 and $Ω = ω L^{2} \sqrt{ρ_{0} A_{0} / E_{0} I_{0}}$ . In literature,³⁸³⁹ the vibration of the beam is derived based on the power series method and the transforms complex couple functions method, respectively. The influence of the truncation coefficient N on the dimensionless natural frequency which is introduced to study the convergence of the present method is presented in Table 1. It is observed that the natural frequencies gradually converge by increasing N and it is almost unchanged when N is larger than 9. So N = 15 is efficient to study the vibration characteristics of the beam in the following parts. The dimensionless natural frequencies of the beam with different boundary conditions are shown in Table 2. It is observed that the results derived by the present method match well with those in literature.^38,39

Table 1.

The first four dimensionless natural frequencies with different N.

$Ω_{i}$	Present results
$Ω_{i}$	N = 7	N = 9	N = 11	N = 13	N = 15
1	13.8347	13.8347	13.8347	13.8347	13.8347
2	28.5179	28.5179	28.5179	28.5179	28.5179
3	45.6659	45.6659	45.6659	45.6659	45.6659
4	61.8621	61.8620	61.8620	61.8620	61.8620

Table 2.

The first five dimensionless natural frequencies of the uniform beam.

Boundary conditions		$Ω_{1}$	$Ω_{2}$	$Ω_{3}$	$Ω_{4}$	$Ω_{5}$
C-F	Present	3.2271	14.4689	31.5025	47.9090	62.3470
	Ref. [40]	3.227128	14.468930	31.502970	48.082740	62.626921
	Ref. [39]	3.23	14.47	31.50	47.91	62.35
C-C	Present	13.8347	28.5179	45.6659	61.8620	68.2836
	Ref. [40]	13.834758	28.517926	45.667237	61.867699	68.292529
	Ref. [39]	13.84	28.54	45.67	61.86	68.28
F-F	Present	16.7920	33.8149	51.5214	58.9920	73.7397
	Ref. [40]	16.791957	33.814869	51.526943	58.993336	73.963612
	Ref. [39]	16.79	33.82	51.52	58.99	73.74

To further validate the accuracy of the proposed method, the free vibration of a beam with variable cross section varies are studied. The parameters of the beam are set as the same with those in Huang et al.,³⁹ as $E_{0}$ =200 GPa, $ρ_{0} = 5700 kg / m^{3}$ , $I_{0} / A {}_{0}$ = 0.01, respectively. The cross-sectional area $A (x)$ and the second moment of area $I (x)$ are denoted as

A (x) = A_{0} (1 - \frac{c x}{L})

(21)

I (x) = I_{0} {(1 - \frac{c x}{L})}^{3}

(22)

where c is the geometrical parameter. The first five dimensionless natural frequencies of the beam with different boundary conditions are listed as in Table 3 and compared with those in Huang et al.³⁹ It could be seen that these two results agree with each other very well.

Table 3.

The first five dimensionless natural frequencies of a non-uniform beam.

Boundary conditions		C-C		C-F		S-S	S-F	C-S	F-F
Boundary conditions		Present	Ref. [40]	Present	Ref. [40]	S-S	S-F	C-S	F-F
c = 0.2	$Ω_{1}$	13.2218	13.222267	3.3345	3.330651	7.1255	5.1029	4.4932	14.5230
	$Ω_{2}$	27.7776	27.778218	14.2939	14.289208	22.5708	19.0679	17.0313	31.2850
	$Ω_{3}$	44.6964	44.698311	30.7127	30.710797	40.7504	36.6908	33.1620	49.1846
	$Ω_{4}$	61.8060	61.818070	47.7532	47.753286	59.6109	55.1533	50.8415	63.7143
	$Ω_{5}$	72.5534	72.565691	65.0067	65.012364	78.4804	73.1453	69.0169	76.2644

In this part, the vibration of a FGMT beam with non-uniform cross section will be taken into consideration to further validate the accuracy of the present method. The cross-sectional area $A (x)$ and the second moment of area $I (x)$ vary according to equation (21) and (22). The Young’s modulus $E (x)$ and the mass density $ρ (x)$ of the beam vary as

E (x) = (E_{1} - E_{0}) {(\frac{x}{L})}^{n} + E_{0}

(23)

ρ (x) = (ρ_{1} - ρ_{0}) {(\frac{x}{L})}^{n} + ρ_{0}

(24)

where

n

is the material gradient parameter. The material properties of the beam are set as those in literature.^33,39

E_{0}

= 200 GPa,

ρ_{0} = 5700 kg / m^{3}

E_{1} = 70 GPa

ρ_{1} = 2702 kg / m^{3}

I_{0} / A {}_{0}

= 0.01. The gradient parameter n is 2 and c is 0.1 in this section. The first four dimensionless natural frequencies of this beam with different boundary conditions are shown in Table 4. It is observed that the results derived by the present method match very well with those in literature.³³,.39

Table 4.

The first four dimensionless natural frequencies of a FGMT beam with non-uniform cross section.

Boundary conditions	$Ω_{i}$	n = 1		n = 2
Boundary conditions	$Ω_{i}$	Present	Ref. [40]	Present	Ref. [40]	Ref. [34]
C-F	1	3.9443	3.944636	3.9370	3.935789	3.9359
	2	14.9224	14.936404	15.1481	15.153330	15.1577
	3	30.5315	30.572737	31.2010	31.223895	31.2638
	4	46.3363	46.408879	47.5432	47.585724	47.7164
C-C	1	12.6572	12.68157575	12.4421	12.463286	12.4689
	2	26.4415	26.4910078	26.3401	26.380442	26.4153
	3	42.5640	42.64203444	42.9015	42.961075	43.0904
	4	58.5490	58.668491249	59.3178	59.3402335	59.6829

Free vibration of FGMT beams with variable circular cross section

In this part, the free vibration of a FGMT beam with circular cross section, shown in Figure 2, is studied. The parameters of the beam are $ρ_{0} = 7800 kg / m^{3}$ , $E_{0} = 202 GPa$ , $ρ_{1} = 2700 kg / m^{3}$ , $E_{1} = 70 GPa$ and $ν = 0.3$ . The material properties and cross sections are

E (x) = (E_{1} - E_{0}) {(\frac{x}{L})}^{n} + E_{0}

(25)

I (x) = π R {(x)}^{4} / 4

(26)

A (x) = π R {(x)}^{2}

(27)

ρ (x) = (ρ_{1} - ρ_{0}) {(\frac{x}{L})}^{n} + ρ_{0}

(28)

R (x) = β {(x - \frac{L}{2})}^{2} + R_{center}

(29)

G (x) = E (x) / (1 + ν)

(30)

where R(x) is the radius of the beam,

E_{1}

ρ_{1}

and

E_{0}

ρ_{0}

are Young’s modulus and the mass density at

x = L

and

x = 0

, respectively,

n

is the material gradient parameter,

β

is the radius coefficient,

R_{center}

is the radius at the center of the beam.

The dimensionless natural frequency parameter $Ω$ is introduced to effectively study the influence of the parameters on the vibrational behaviors.

Ω = ω L^{2} \sqrt{\frac{ρ_{0} A_{0}}{E_{0} I_{0}}}

(31)

The radius of the beam at the center is set as 0.05 m and the length is 1 m. The first four dimensionless natural frequencies of the beam with different $β$ and $n$ under clamped–clamped boundary conditions are given in Table 5. It could be seen that the natural frequencies gradually decrease as $β$ increases and decreases as n increases. The first four dimensionless natural frequencies with $β$ = 0.1 are listed in Table 6. It could be observed that the natural frequencies with clamped-clamped end support decrease as the increasing of n.

Table 5.

The first four dimensionless natural frequencies with different $β$ and different n under clamped-clamped boundary conditions.

n	$Ω_{i}$	$β = - 0.15$	$β$ = −0.1	$β =0$	$β =0 .1$	$β = 0.2$	$β = 0.3$
0	1	36.5495	25.7483	21.3771	20.4000	19.9452	19.5802
	2	115.4109	78.6468	55.8060	46.4450	40.9382	37.0959
	3	263.60	160.5796	102.6743	79.9726	67.0003	58.2666
	4	447.6878	261.5574	158.4153	118.7607	96.6086	82.0775
1	1	35.4838	25.1709	20.9458	19.9815	19.5271	19.1641
	2	114.4444	77.9257	55.1937	45.8655	40.3882	36.5768
	3	263.7313	160.0268	101.9908	79.3142	66.3864	57.6986
	4	448.3317	261.0738	157.6949	118.0639	95.9651	81.4883
2	1	34.6395	24.4125	20.0854	19.1230	18.7170	18.4140
	2	111.6361	76.2441	53.9473	44.7408	39.3446	35.6037
	3	262.6696	158.5227	100.5897	78.0625	65.2543	56.6693
	4	448.1775	259.7040	156.2466	116.7557	94.7961	80.4410

Table 6.

The first four dimensionless natural frequencies with $β =0 .1$ .

Boundary conditions	$Ω_{i}$	n = 0	n = 1	n = 2	n = 3
C-C	1	20.4000	19.9815	19.1230	18.7765
	2	46.4450	45.8655	44.7408	44.2552
	3	79.9726	79.3142	78.0625	77.4790
	4	118.7607	118.0639	116.7557	116.1060
C-F	1	2.5968	3.5198	3.5163	3.4201
	2	16.1178	17.8649	17.7166	17.6181
	3	42.5372	44.1997	44.1692	44.2973
	4	77.2404	78.8305	78.8983	79.1572
C-S	1	12.6203	13.6815	13.2787	13.0464
	2	37.1641	37.9636	37.4569	37.2956
	3	70.3177	70.9606	70.4022	70.2475
	4	109.6585	110.1772	109.5892	109.4255
S-S	1	6.7763	6.7239	6.7656	6.7903
	2	27.6868	27.8181	27.8990	27.8943
	3	60.1835	60.2387	60.2037	60.1517
	4	99.9577	99.9512	99.8401	99.7485
S-F	1	9.5858	10.2955	10.5825	10.6658
	2	32.7368	33.7635	34.2769	34.5051
	3	67.0539	68.1010	68.6724	69.0217
	4	108.1090	109.1420	109.7502	110.1819
F-F	1	12.8119	13.1952	13.7336	13.9382
	2	38.1784	38.8016	39.6385	39.9865
	3	74.2521	74.8806	75.8030	76.2443
	4	116.4693	117.0830	118.0544	118.5572

Figures 3 and 4 show the first four transverse displacement and the rotation of cross section mode shapes of the uniform beam with $β$ = 0.2 and n = 2 compared with the uniform Timoshenko beam with radius 0.05 m under clamped–clamped boundary conditions. It is observed from Figures 3 and 4 that the maximum amplitude of the mode shapes appears at the right side of the beam for the transverse displacement mode shapes and the rotation of cross section mode shapes. For the higher order mode shapes, the amplitude of the right side of the beam is slightly larger than that of the left side. It could be explained that the average stiffness of the left side of the beam is larger than that of the right side while the uniform beam has same stiffness on both sides.

Figure 3.

The first four transverse displacement mode shape compared with uniform Timoshenko beam. (a): first order; (b) second order; (c) third order; (d) fourth order.

Figure 4.

The first four rotations of cross section mode shapes compared with uniform Timoshenko beam. (a): first order; (b) second order; (c) third order; (d) fourth order.

Effects of material parameters and spring stiffness on the vibration of beams

Effects of the coefficient of the radius $β$

In this part, the effects of $β$ on the natural frequencies and mode shapes with n = 2 with clamped–clamped boundary condition of the beam are discussed. The ratio of the non-dimensional natural frequencies $\frac{u_{i}}{u_{0}}$ with different $β$ are plotted in Figure 5, in which $u_{i}$ is the non-dimensional natural frequency corresponding to different $β$ , $u_{0}$ is the non-dimensional natural frequency at $β$ = 0. Figure 5 implies that the natural frequencies decrease with the increasing of $β$ , and this trend is more obvious on higher order natural frequency.

Figure 5.

Non-dimensional natural frequencies compared with a FGM beam with uniform cross section (n = 2).

The first four mode shapes of the transverse displacement and the rotation of cross section by different $β$ are shown in Figures 6 and 7, respectively. It can be seen that the amplitude of the first-order transverse displacement mode shape reaches the maximum on the middle right of the beam and the transverse displacement decreases with the increasing of $β$ . The vibration amplitudes of the position around the center of the beam increase but the first-order transverse displacement and the second-order rotation of cross-section mode shape show decreasing trend with increasing $β$ .

Figure 6.

Transverse displacement mode shape with different $β$ (n = 2). (a): first order; (b) second order; (c) third order; (d) fourth order.

Figure 7.

Rotational mode shapes with different $β$ (n = 2). (a): first order; (b) second order; (c) third order; (d) fourth order.

Effects of the gradient parameter n

Furthermore, the influence of the gradient parameter n on the dimensionless natural frequencies and mode shapes is investigated in this part. Here $R_{center}$ is set as 0.05 m, and $β$ = 0.2, and the boundary condition is set as clamped-clamped end support. The ratio of the non-dimensional natural frequencies $\frac{u_{i}}{u_{0}}$ with different n is displayed in Figure 8. It implies an important information that the higher-order natural frequencies decrease when n is less than 20. However, the natural frequencies almost keep unchanged when n is more than 20.

Figure 8.

The radio of the non-dimensional natural frequencies with different n.

Figures 9 and 10 plot the first four mode shapes of the transverse displacement and the rotation of cross section with different n, respectively. It could be seen that the rotational mode shapes have one more vibration extremum compared with the transverse displacement ones for the same order mode shape.

Figure 9.

Transverse displacement mode shape with different n. (a): first order; (b) second order; (c) third order; (d) fourth order.

Figure 10.

Rotational mode shapes with different n. (a): first order; (b) second order; (c) third order; (d) fourth order.

Effects of elastic supports

The influence of elastic boundary conditions of the vibration of the beam is studied. The coefficient of radius $β$ is 0.2, and the gradient parameter n is 2 in this part. The first four frequency parameters with one side of the beam is set as clamped end support and the other side is fixed by an infinite translational spring k₁ and a variable rotational spring K₁ from 0 to infinite is displayed in Figure 11. The frequency parameters with one end of the beam clamped and the other end restrained by a variable translational spring k₁ is shown in Figure 12. From Figures 11 and 12, it takes a conclusion that different spring values have different effects on the natural frequencies and the higher order natural frequencies change with lager spring values. And then, the first four dimensionless natural frequencies which are influenced by a variable translational spring k₀ and a variable rotational spring K₀ are displayed in Figure 13. Supposing translational spring k₀ and the rotational spring K₀ is changing from 0 to 10¹³, the other end of the beam is set as Clamped. One can see from Figure 13 that the dimensionless natural frequencies are increasing gradually with k₀ and K₀ becoming bigger, and the dimensionless natural frequency is almost unchanged when the spring value is big enough. And the natural frequencies of the beam are more sensitive to the translational springs than the rotational springs.

Figure 11.

First four non-dimensional natural frequencies with variable K₁( $β$ = 0.2, n = 2).

Figure 12.

First four non-dimensional natural frequencies with variable k₁( $β$ = 0.2, n = 2).

Figure 13.

First four non-dimensional natural frequencies with variable k₀ and K₀( $β$ =0.2, n = 2). (a): first order; (b) second order; (c) third order; (d) fourth order.

Effects of a variable translational spring k₀ and different $β$

In this part, the effects of a variable translational spring k₀ and different $β$ on the first four non-dimensional natural frequencies and mode shapes with n = 2 with clamped-clamped boundary condition of the beam in Figure 14 are discussed. Supposing translational spring k₀ is changing from 0 to 10¹³, the other end of the beam is set as clamped, and $β$ is continuously changing from 0 to 0.3. One can see from Figure 14 that the non-dimensional natural frequency keeps unchanged when k₀ is relatively small, then it rises gradually when k₀ is bigger than 10⁶, and finally it almost unchanged when k₀ is close to infinity. On the other hand, the non-dimensional natural frequency decreases gradually by increasing $β$ and setting k₀ as a constant value. Additionally, the efforts of k₀ and $β$ on the higher order non-dimensional natural frequency are much higher than the first order one.

Figure 14.

First four non-dimensional natural frequencies with variable k₀ and different $β$ (n = 2). (a): first order; (b) second order; (c) third order; (d) fourth order.

Effects of the gradient parameter n and coefficient of the radius $β$

The influence of the gradient parameter n and coefficient of the radius $β$ on the dimensionless natural frequencies is considered in this part. Supposing n is changed from 0 to 30 and $β$ is continuously changed from 0 to 0.3, the end support is C-C. Figure 15 shows the first four dimensionless natural frequencies with different n and variable $β$ . As shown in Figure 15, there is a steady decrease with increasing $β$ . The first-order non-dimensional natural frequency jumps when n is relatively small, and then it rises slowly when n is increasing gradually. The second, third and forth order non-dimensional natural frequencies show the same trend as the first order.

Figure 15.

First order non-dimensional natural frequency with different n and $β$ . (a): first order; (b) second order; (c) third order; (d) fourth order.

Conclusion

This paper proposed a unified procedure to study the free vibration of an elastically restrained FGMT beam with any variable cross section. The translational displacement and the rotation of cross section are both expressed into the Fourier series with some polynomial functions based on the so-called spectro-geometric method. The cross-sectional area, the second moment of area, the Young’s modulus, the mass density and the shear modulus have also been expanded into Fourier cosine series so that this method could be adopted to analyze any kind of FGMT beams with any variable cross sections. Several examples are introduced to validate the accuracy and reliability of the method. The natural frequencies and the mode shapes of the beams with different ends supports are studied. This method is generally applicable to any kinds of FGM beams with smoothly variable cross section under elastic boundary conditions.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China (grant no. 51805341), the Natural Science Foundation of Jiangsu Province (grant no. BK20180843), China Postdoctoral Science Foundation (grant no. 2020M671412) and Jiangsu Planned Projects for Postdoctoral Research Funds (grant no. 2020Z055).

ORCID iD

Guofang Li

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The vibration analysis of the elastically restrained functionally graded Timoshenko beam with arbitrary cross sections

Abstract

Keywords

Introduction

Theoretical formulation

Structural model

Solution for the system

Results and discussion

Validation of the present method

Free vibration of FGMT beams with variable circular cross section

Effects of material parameters and spring stiffness on the vibration of beams

Effects of the coefficient of the radius β

Effects of the gradient parameter n

Effects of elastic supports

Effects of a variable translational spring k0 and different β

Effects of the gradient parameter n and coefficient of the radius β

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Effects of the coefficient of the radius $β$

Effects of a variable translational spring k₀ and different $β$

Effects of the gradient parameter n and coefficient of the radius $β$