Free vibration analysis of tapered columns under self-weight using pseudospectral method

Gopinathan Sudheer1 , Pillutla Sri Harikrishna2 , Yerikalapudy Vasudeva Rao3

1GVP College of Engineering for Women, Visakhapatnam, India

2GITAM University, Visakhapatnam, India

3IIT Bhubaneswar, Bhubaneswar, India

1Corresponding author

Journal of Vibroengineering, Vol. 18, Issue 7, 2016, p. 4583-4591. https://doi.org/10.21595/jve.2016.17089
Received 18 April 2016; received in revised form 10 September 2016; accepted 27 October 2016; published 15 November 2016

Copyright © 2016 JVE International Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Abstract.

This paper deals with the vibration of tapered column which is affected by gravity using a pseudospectral formulation. The formulation is simple and easy-to-implement and is capable of dealing with different end conditions. Numerical examples of the effects of taper, cross section shapes and gravity on the vibration of columns are illustrated. The effectiveness of the pseudospectral method for vibration analysis of tapered heavy columns is validated by comparing the results with numerical techniques such as the numerical initial value method and differential quadrature method.

Keywords: pseudospectral, non-uniform, columns, taper, gravity.

1. Introduction

Elastic columns are a class of important structural components that find wide applications in civil, mechanical and aerospace engineering fields [1]. The strength of an elastic column basically depends on its material and geometrical properties. The material selection and Young’s modulus determine whether a column has material non-linearity while the geometric non-linearity arises from non-uniform cross-sectional areas [2]. In conventional column vibration and buckling problems, the self-weight is often neglected and when taken into consideration, the column is referred to as a heavy column [3]. The standing heavy column is fundamental in mechanics and models tall structures and free-standing antennas [4].

Greenhill [5], using Bessel’s functions, first investigated the stability of a uniform column due to its own weight. Schafer [6] studied the effect of self-weight on the natural frequencies of a hanging cantilever beam using the Rayleigh-Ritz method. The finite element method was used by Yokoyama [7] to investigate the vibration characteristics of uniform hanging beams under gravity. Virgin et al. [8] performed analytical and experimental studies on the effect of gravity on the vibration of vertical cantilevers. Duan and Wang [3] presented analytical solutions for the buckling of columns including self-weight. Okay et al. [9] applied the variational iteration method to determine the buckling loads and mode shapes of heavy columns under its own weight. An analytic method involving the Fredholm integration method was used by Huan and Li [10] to analyze the buckling behavior of axially nonuniform graded columns. The differential quadrature (DQ) method was used by Mahmoud et al [11] to investigate the effect of column geometry on the natural frequencies and mode shapes. Taha and Essam [12] used the DQ method to study the stability behavior and free vibration of axially loaded tapered columns with elastic end restraints. Recently, Wang [13] used a numerical initial value method to study the influence of gravity as well as taper on the vibration of a standing column.

Although many methods have been presented to analyze problems concerned with taper and self-weight, most of them apply to specific cases determined by the form of the equations. This study presents a simple numerical technique that is capable of handling different cases using the pseudospectral formulation. In terms of effective mathematical techniques, Pesudospectral (PS) methods have been used in recent years for structural engineering analysis. Lee and Schultz [14] applied the Chebyshev PS method to solve the vibration of Timoshenko beams and Mindlin plates. Yagci et al. [15] used a spectral Chebyshev technique for solving linear and nonlinear beam equations. Sari and Butcher [16] used the PS method for the free vibration analyses of non-rotating and rotating Timoshenko beams with damaged boundaries. In [17], the PS method is used to investigate the dynamic response of Timoshenko beams made of functionally graded materials. A Chebyshev PS method is presented in [18] for the static analysis of the geometrically exact beams undergoing large deflections. However, to the author’s knowledge no analytical solution exists for the important problem of the vibration of a standing heavy tapered column. A numerical initial value method proposed in [13] that combines the initial value method [19] with Runge-Kutta method and bisection method is the only available technique present in literature. The important problem demands techniques that are easy to implement and computationally inexpensive.

The intention of this work is to explore the application of a novel formulation of the Chebyshev Pseudospectral method to the vibration analysis of heavy and tapered columns. The method is first validated by computing the frequencies of vibration of non-uniform beams where their density and the flexural rigidity vary along the longitudinal axis. The influence of gravity, taper and gravity and taper on the vibrations of columns is then analyzed and the results are compared with those obtained using exact solutions [20], a Differential Quadrature Method [11] and a numerical initial value method [13].

2. Equations

The vibration of long and slender beams/columns is an important problem in applied mechanics and are generally modeled by the Euler-Bernoulli beam theory [21]. Assuming the effects of rotational inertia and transverse shear deformation to be negligible, the equation for small vibrations of a non-uniform Euler Bernoulli column subjected to an axial force F is given by [13]:

(1)
2 x 2 E I x 2 y x 2 + x F x y x + ρ x 2 y t 2 = 0 ,

where x,y are the longitudinal and transverse coordinates of the column with the origin at the base, EI is the flexural rigidity, ρ is the mass per length and t is the time. For a standing column of height L, axial force is given by:

(2)
F = g x L ρ x d x ,

where g is the acceleration due to gravity. Introducing yx,t=wxeikt, EIx=EIolx, ρx=ρorx, where EIo and ρo are the maximum values of flexural rigidity and mass per length occurring at the base x=0 and k is the frequency of vibration. Normalizing all length by L Eq. (1), becomes:

(3)
d 2 d x 2 l x d 2 w d x 2 + β d d x x 1 r x d w d x = ω 2 r x w ,

where ω=kL2ρo/EIo and β=gρoL3/EIo.

Eq. (3) does not have a closed form solution even for a uniform beam/column [21]. Assuming that the column has linear taper with the rigidity and density varying as lx=1-cxm, rx=1-cxn where m, n are positive constants and 0c1 representing the degree of taper, Eq. (3) becomes:

(4)
d 4 w d x 4 + b 1 x d 3 w d x 3 + b 2 x d 2 w d x 2 + b 3 x d w d x + b 4 x w = 0 ,

where:

b 1 x = - 2 m c 1 - c x ,           c 0 ,
b 2 x = m m - 1 c 2 1 - c x 2 + β 1 - c x n + 1 - 1 - c n + 1 c n + 1   1 - c x m = β 1 - x ,           c = 0 ,
b 3 x = - β 1 - c x n - m ,
b 4 x = - ω 2 1 - c x n - m .

For clamped-free (C-F) columns with clamped end x=0 and free end x= 1, the boundary conditions can be written as:

(5)
w = d w d x = 0 ,           x = 0 ,
(6)
d 2 w d x 2 = 0 ,           d d x l x d 2 w d x 2 = 0 ,           x = 1 .

If c= 0, the column is uniform and for c= 1, the column has a pointy tip.

For c 1, Eq. (6) reduces to:

(7)
d 2 w d x 2 = 0 ,           d 3 w d x 3 = 0 ,           x = 1 .

Eq. (4) is solved subject to the boundary conditions given by Eq. (5) and Eq. (6) for different values of m, n. Though the equations can be solved for general values of m, n, we consider some special cases which correspond to those shown in Fig. 1. In the case of a solid tapered column of circular cross section, m= 4, n= 2, while for a solid column of constant thickness and tapered sides m= 1, n= 1 or m= 3, n= 1. If the column vibrates about the axis A-A which is perpendicular to the thickness direction m= 1, n= 1 and if the column vibrates about the axis B-B which is parallel to the thickness direction m= 3, n= 1 [13].

Fig. 1. a) Tapered column of circular cross section, b) solid column of constant thickness and tapered sides

 a) Tapered column of circular cross section, b) solid column of constant thickness and tapered sides

a)

 a) Tapered column of circular cross section, b) solid column of constant thickness and tapered sides

b)

2.1. Solution procedure

In Chebyshev PS method, the Chebyshev polynomials are employed as the trial functions for the discretization of the unknown function namely w and the Gauss-Chebyshev-Lobatto points are employed as the collocation points at which the residuals are minimized. The physical domain 0x1 is transformed into -1X1 by the transformation X=2x-1. With this transformation Eq. (4) reduces to:

(8)
d 4 w d X 4 + B 1 X d 3 w d X 3 + B 2 X d 2 w d X 2 + B 3 X d w d X + B 4 X w = 0 ,

where BiX=bix/2i,  i=1, 2, 3, 4.

We assume:

(9)
w = k = 0 N a k T k X ,

where ak, k=0,,N are unknown constants and TkXk=0,1,,N are Chebyshev polynomials defined by [22] TkX=coskcos-1X=coskθ for k= 0, 1, 2,…, where θ=cos-1X.

The transformation X=cosθ converts the Chebyshev series into a Fourier cosine series. In the proposed methodology, we compute the basis functions and their derivatives using trigonometric functions:

(10)
i . e .   X = c o s θ θ = c o s - 1 X ,         X   - 1       1   ,           t   0       π   .

The elementary identity:

(11)
d d X - 1 s i n θ d d θ ,

is repeatedly applied to convert Eq. (8) into an equivalent differential equation on θ0  π. As θ0, π the derivatives are evaluated using the Taylor expansions of both numerator and denominator about their common zero. Substituting the value of w given by Eq. (9) into Eq. (8) and using Eq. (10), Eq. (11) the equivalent differential equation is collocated at:

(12)
θ i = N - i   π N ,           i = 1 , , N - 3 ,           0 , π ,

yielding a system of N-3 equations in N+1 unknowns ak. Imposing the boundary conditions given by Eq. (5), Eq. (6) we get a system of 4 equations in N+1 unknowns. The resulting N+1 by N+1 system of equations is expressed as a matrix eigenvalue problem and solved using a standard eigensolver.

3. Numerical results and discussion

In this section, we study the convergence behavior of the PS method first and then consider some numerical examples to validate the efficiency of the Pseudospectral method. The first numerical example is devoted to a beam with constant thickness and linearly tapered width with clamped base and no tip mass. The second is concerned with the vibration of a hanging uniform column under self-weight. The third example considers the free vibration of non-uniform column with no self-weight. The last example concerns the influence of gravity and taper on the vibration of a standing column.

3.1. Convergence behavior of PS method

As a case study, the convergence behavior of the non-dimensional frequency parameter (ω) for the first five modes of a non-uniform beam whose cross section is an open web or tower type [20] is considered. To highlight this, we consider the clamped-free boundary condition with a taper ratio of c= 0.1 in the absence of gravity. Taking the exact values of (ω) given in [20] as base values, we compute the values of (ω) for N varying from 20 to 28. The results obtained to six-digit precision are presented in Table 1. We observe from the table that the converged results have good accuracy in comparison with those of [20].

Table 1. Non-dimensional vibration frequencies for m= 4, n= 0

ω
N
20
22
24
26
27
28
ω 1
3.376722
3.376722
3.376722
3.376722
3.376722
3.376722
ω 2
20.248149
20.248149
20.248149
20.248149
20.248149
20.248149
ω 3
55.966597
55.966595
55.966595
55.966595
55.966595
55.966595
ω 4
109.302542
109.301970
109.301978
109.301978
109.301978
109.301978
ω 5
180.403245
180.432576
180.430075
180.430210
180.430205
180.430205

3.2. Vibration of a class of Non-uniform beams in the absence of gravity

In [20], new exact solutions were presented for a class of non-uniform beams whose density and flexural rigidity vary along the longitudinal axis. To assess the numerical accuracy of the proposed PS method, we obtain the numerical solutions for a class of non-uniform beams presented in [20]. Ignoring rotational inertia and shear deformation, the Euler-Bernoulli small deflection beam equation is obtained by taking F0 in Eq. (1). To test the validity of the method for different end conditions we consider Clamped-Free (C-F), Pinned-Pinned (P-P) and Free-Free (F-F) end conditions. At clamped and free ends the boundary conditions can be used from Eqs. (5-6). For a pinned end the conditions are w=0, d2w/dx2=0.

The computations in the PS method (PSM) are carried out using N= 30 and the results obtained are presented in Table 2 along with the exact values [20]. The exact values given in [20] are obtained through power function solutions. The results are presented in the form of tables to highlight the numerical accuracy and for easy comparison.

Table 2. Non-dimensional vibration frequencies for m= 4, n=0

Taper ratio c
0.1
0.3
0.7
Boundary conditions
Wang [20]
PSM
Wang [20]
PSM
Wang [20]
PSM
C-F
3.3767
3.376722
3.0751
3.075097
2.3151
2.315075
20.248
20.248149
16.680
16.680409
9.3906
9.390641
55.367
55.966595
44.733
44.733218
22.898
22.897800
109.30
109.301978
86.631
86.630457
42.975
42.974975
180.43
180.430205
142.50
142.494970
69.698
69.698967
F-F
20.162
20.161464
15.890
15.889909
7.9004
7.900440
55.566
55.566239
43.715
43.714562
21.303
21.302894
108.92
108.923974
86.625
85.625417
41.369
41.369409
180.05
180.050691
141.49
141.487640
68.084
68.085117
268.96
268.959692
211.31
211.313335
101.46
101.465340
P-P
8.8895
8.889481
6.9698
6.969833
3.2686
3.268554
35.570
35.569613
27.985
27.984517
13.647
13.646465
80.026
80.026031
62.915
62.914773
30.430
30.430199
142.26
142.263141
111.80
111.800265
53.836
53.836215
222.28
222.281555
174.65
174.646486
83.894
83.894116

3.3. Vibration of a uniform column under self-weight

The equation for the vibration of a uniform column under self-weight is obtained by taking c= 0 in Eq. (4). The present scheme is applied to solve the resulting equation using N= 25. The results obtained are compared with the values given in [4]. It is to be noted that negative values of β denotes a hanging column. The first two square of non-dimensional frequency values ω2 obtained using the present method with the base fixed and the top experiencing zero moment and shear is given in Table 3. The closeness of the results obtained using PSM with that of the results obtained using the method of [19] is seen in the table.

Table 3. Square of Non-dimensional frequency values of a column under self-weight

ω 2
β
0
–20
–50
–100
Wang [4]
PSM
Wang [4]
PSM
Wang [4]
PSM
Wang [4]
PSM
ω 1 2
12.362
12.362363
43.53
43.530595
89.65
89.653010
165.6
165.604518
ω 2 2
485.50
485.518818
657.7
657.648489
913.1
913.078445
1333
1332.919282

3.4. Free Vibration of non-uniform column

The governing equation of a non-uniform column varying with bending stiffness given by 1+αx as in [11] under no self-weight is obtained by taking n= 0, m= 1, β= 0 and c=-α in Eqn. (4). In [11], the free vibration of non-uniform column was considered efficiently using the Differential Quadrature Method(DQM). Two variants of DQM namely the modifying weighting coefficient matrices (MWCM) method and substituting boundary conditions into governing equations (SBCGE) techniques were used to treat different types of boundary conditions in [11]. The simple-simple (S-S) supports/pinned-pinned (P-P) supports and clamped-simple (C-S) supports at bottom and top were treated using MWCM technique while the clamped-clamped (C-C) supports and clamped-free (C-F) supports were treated using SBCGE technique. However, the present Pseudospectral formulation is capable of treating the different boundary conditions with ease to compute the non-dimensional frequencies ω of non-uniform columns. The computed first three frequency values for α= –0.5, 0.5 in the case of S-S and C-C supports are presented in Table 4 along with the corresponding values of [11]. It is to be noted that the computations were carried out using N= 15 as in [11] for a fair comparison. The results show that the present formulation of PS method is an efficient method in solving the free vibration of non-uniform columns with good accuracy.

Table 4. Non-dimensional frequency values of non-uniform column

Non-dimensional
frequency values
α = –0.5
α = 0.5
α = –0.5
α = 0.5
MWCM
[11]
PSM
MWCM
[11]
PSM
SBCGE
[11]
PSM
SBCGE
[11]
PSM
(S-S)
(S-S)
(C-C)
(C-C)
ω 1
8.479
8.479450
11.003
11.003523
19.098
19.098602
24.888
24.888283
ω 2
33.834
33.834311
43.976
43.976308
52.709
52.709121
68.633
68.633917
ω 3
76.065
76.065006
98.919
98.919180
103.38
103.383220
134.57
134.575354

3.5. Vibration of a standing tapered heavy column

In [13], the stability and natural vibration of a standing tapered vertical column under its own weight is studied. The method consists in using a simple initial value method combined with interpolation using the Runge-Kutta method to obtain the frequencies of vibration. The present Pseudospectral method is much simpler to implement for computer usage. Though the method is suitable for general values of cross-section shape parameters m, n, we consider the values (3, 1), (1, 1) and (4, 2) for a fair comparison. The non-dimensional vibration frequencies are computed for the taper values of c= 0.1, 0.3 and 0.7 with the gravity parameter β taking the values 0, 2.5 and 7.5. The computations in the PSM are carried out usingN= 25. The results obtained are presented in Tables 5-7 and are compared with the corresponding values given in [13]. In [13], there is a misprint in the 2nd frequency values corresponding to c= 0.7, β= 2.5 and c= 0.3, β= 7.5 for m= 3, n= 1 and the correct values obtained using PSM are given in Table 5. It is observed that as the gravity effect (β) increases, the frequencies decrease until the fundamental frequency is almost zero, at which stage the column buckles. In addition, the closeness of the values with those of [13] brings out the simplicity and accuracy of the present method.

Table 5. Non-dimensional frequencies for m= 3, n= 1

β
c
0
0.3
0.7
Wang [13]
PSM
Wang [13]
PSM
Wang [13]
PSM
0
3.5587
3.558702
3.667
3.666749
4.0817
4.081714
21.338
21.338102
19.889
19.880606
16.625
16.625269
58.980
58.979904
53.322
53.322198
40.588
40.587991
2.5
2.9485
2.948465
3.0621
3.062087
3.4971
3.497114
20.837
20.836806
19.369
19.368556
14.085
16.085256
58.470
58.469500
52.806
52.805953
40.051
40.051452
7.5
0.8466
0.846644
1.0938
1.093815
1.8155
1.815472
19.794
19.794401
19.300
18.300150
14.947
14.946583
57.433
57.432830
51.756
51.755963
38.955
38.954952

Table 6. Non-dimensional frequencies for m= 1, n= 1

β
c
0.1
0.3
0.7
Wang [13]
PSM
Wang [13]
PSM
Wang [13]
PSM
0
3.6310
3.631027
3.9160
3.916033
4.9317
4.931642
22.254
22.254029
22.786
22.785958
24.687
24.687279
61.910
61.909628
62.463
62.436120
64.527
64.526628
2.5
3.0376
3.037632
3.3638
3.363823
4.4793
4.479281
21.771
21.770841
22.333
22.332727
24.315
24.315225
61.416
61.416302
61.976
61.975560
64.153
64.152600
7.5
1.1326
1.132558
1.8024
1.802415
3.3960
3.395995
20.769
20.769043
21.396
21.396124
23.553
23.552729
60.415
60.415453
61.042
61.042332
63.397
63.397027

Table 7. Non-dimensional frequencies for m= 4, n= 2

β
c
0.1
0.3
0.7
Wang [13]
PSM
Wang [13]
PSM
Wang [13]
PSM
0
3.6737
3.673701
4.0669
4.066932
5.5093
5.509268
21.550
21.550253
20.556
20.555506
18.641
18.641218
58.189
59.188637
54.015
54.015186
42.810
42.810666
2.5
3.0821
3.082123
3.5181
3.518063
5.0536
5.053637
21.062
21.062033
20.085
20.084902
18.212
18.212036
58.693
58.692572
53.543
53.543037
42.385
42.385379
7.5
1.2137
1.213739
2.0036
2.003562
3.9836
3.983635
20.048
20.048376
19.108
19.107981
17.322
17.321630
57.686
57.685602
57.584
57.584453
41.521
41.521269

4. Conclusions

Typically, the free vibration frequencies of a non-uniform gravity loaded Euler-Bernoulli column/beam is obtained numerically as the governing fourth order differential equation with variable coefficients does not yield any closed form solutions. The numerical technique-Chebyshev Pseudospectral method explored in this paper introduces a novel formulation of the method in which the basis functions and their derivatives are computed using trigonometric functions. The stability of the method is first studied by obtaining the vibration frequencies of a linearly tapered beam in the absence of gravity. The accuracy of the method is further tested against the exact solutions of a class of non-uniform beams in the absence of gravity, solutions obtained using two variants of DQM for a non-uniform column under different end conditions and also against the solutions obtained using an initial value method in the case of a uniform column under self-weight. Finally, the proposed method is used to find the vibration frequencies of a standing linearly tapered heavy column. A comparison of the results obtained with those of the numerical initial value method shows that the proposed technique is an efficient and reliable method in handling vibration columns of elastic columns/beams. It is also possible to extend the technique to other tapers and consider inclusion of shear effects as in a Timoshenko column.

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Cited By

Journal of Sound and Vibration
Nirmal Ramachandran, Ranjan Ganguli
2018