Free vibration analysis of tapered columns under selfweight using pseudospectral method
Gopinathan Sudheer^{1} , Pillutla Sri Harikrishna^{2} , Yerikalapudy Vasudeva Rao^{3}
^{1}GVP College of Engineering for Women, Visakhapatnam, India
^{2}GITAM University, Visakhapatnam, India
^{3}IIT Bhubaneswar, Bhubaneswar, India
^{1}Corresponding author
Journal of Vibroengineering, Vol. 18, Issue 7, 2016, p. 45834591.
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
JVE Conferences
This paper deals with the vibration of tapered column which is affected by gravity using a pseudospectral formulation. The formulation is simple and easytoimplement 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, nonuniform, 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 nonlinearity while the geometric nonlinearity arises from nonuniform crosssectional areas [2]. In conventional column vibration and buckling problems, the selfweight 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 freestanding 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 selfweight on the natural frequencies of a hanging cantilever beam using the RayleighRitz 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 selfweight. 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 selfweight, 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 nonrotating 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 RungeKutta 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 nonuniform 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 EulerBernoulli beam theory [21]. Assuming the effects of rotational inertia and transverse shear deformation to be negligible, the equation for small vibrations of a nonuniform Euler Bernoulli column subjected to an axial force $F$ is given by [13]:
where $\left(x,y\right)$ are the longitudinal and transverse coordinates of the column with the origin at the base, $EI$ is the flexural rigidity, $\rho $ is the mass per length and $t$ is the time. For a standing column of height $L$, axial force is given by:
where $g$ is the acceleration due to gravity. Introducing $y\left(x,t\right)=w\left(x\right){e}^{ikt}$, $EI\left(x\right)=E{I}_{o}l\left(x\right)$, $\rho \left(x\right)={\rho}_{o}r\left(x\right)$, where $E{I}_{o}$ and ${\rho}_{o}$ 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:
where $\omega =k{L}^{2}\sqrt{{\rho}_{o}/E{I}_{o}}$ and $\beta =g{\rho}_{o}{L}^{3}/E{I}_{o}$.
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 $l\left(x\right)={\left(1cx\right)}^{m}$, $r\left(x\right)={\left(1cx\right)}^{n}$ where $m$, $n$ are positive constants and $0\le c\le 1$ representing the degree of taper, Eq. (3) becomes:
where:
${b}_{2}\left(x\right)=\frac{m\left(m1\right){c}^{2}}{{\left(1cx\right)}^{2}}+\frac{\beta \left\{{\left(1cx\right)}^{n+1}{\left(1c\right)}^{n+1}\right\}}{c\left(n+1\right){\left(1cx\right)}^{m}}=\beta \left(1x\right),\left(c=0\right),$
${b}_{3}\left(x\right)=\beta {\left(1cx\right)}^{nm},$
${b}_{4}\left(x\right)={\omega}^{2}{\left(1cx\right)}^{nm}.$
For clampedfree (CF) columns with clamped end $x=0$ and free end $x=$ 1, the boundary conditions can be written as:
If $c=$ 0, the column is uniform and for $c=$ 1, the column has a pointy tip.
For $c\ne $ 1, Eq. (6) reduces to:
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 AA which is perpendicular to the thickness direction $m=$1, $n=$1 and if the column vibrates about the axis BB 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)
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 GaussChebyshevLobatto points are employed as the collocation points at which the residuals are minimized. The physical domain $0\le x\le 1$ is transformed into $1\le X\le 1$ by the transformation $X=2x1$. With this transformation Eq. (4) reduces to:
where ${B}_{i}\left(X\right)={b}_{i}\left(x\right)/{2}^{i}$, $\mathrm{}i=$1, 2, 3, 4.
We assume:
where ${a}_{k}$, $\left(k=0,\dots ,N\right)$ are unknown constants and ${T}_{k}\left(X\right)$$\left(k=\mathrm{0,1},\dots ,N\right)$ are Chebyshev polynomials defined by [22] ${T}_{k}\left(X\right)=\mathrm{c}\mathrm{o}\mathrm{s}\left(k\mathrm{c}\mathrm{o}{\mathrm{s}}^{1}\left(X\right)\right)=\mathrm{c}\mathrm{o}\mathrm{s}\left(k\theta \right)$ for $k=$ 0, 1, 2,…, where $\theta =\mathrm{c}\mathrm{o}{\mathrm{s}}^{1}\left(X\right)$.
The transformation $X=\mathrm{c}\mathrm{o}\mathrm{s}\theta $ converts the Chebyshev series into a Fourier cosine series. In the proposed methodology, we compute the basis functions and their derivatives using trigonometric functions:
The elementary identity:
is repeatedly applied to convert Eq. (8) into an equivalent differential equation on $\theta \in \left[0\pi \right]$. As $\theta \to 0$, $\pi $ 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:
yielding a system of $\left(N3\right)$ equations in $N+1$ unknowns ${a}_{k}\text{.}$ Imposing the boundary conditions given by Eq. (5), Eq. (6) we get a system of 4 equations in $\left(N+1\right)$ unknowns. The resulting $\left(N+1\right)$ by $\left(N+1\right)$ 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 selfweight. The third example considers the free vibration of nonuniform column with no selfweight. 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 nondimensional frequency parameter $\left(\omega \right)$ for the first five modes of a nonuniform beam whose cross section is an open web or tower type [20] is considered. To highlight this, we consider the clampedfree boundary condition with a taper ratio of $c=$0.1 in the absence of gravity. Taking the exact values of $\left(\omega \right)$ given in [20] as base values, we compute the values of $\left(\omega \right)$ for $N$ varying from 20 to 28. The results obtained to sixdigit 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. Nondimensional vibration frequencies for $m=$4, $n=$0
$\left(\omega \right)$

$N$


20

22

24

26

27

28


${\omega}_{1}$

3.376722

3.376722

3.376722

3.376722

3.376722

3.376722

${\omega}_{2}$

20.248149

20.248149

20.248149

20.248149

20.248149

20.248149

${\omega}_{3}$

55.966597

55.966595

55.966595

55.966595

55.966595

55.966595

${\omega}_{4}$

109.302542

109.301970

109.301978

109.301978

109.301978

109.301978

${\omega}_{5}$

180.403245

180.432576

180.430075

180.430210

180.430205

180.430205

3.2. Vibration of a class of Nonuniform beams in the absence of gravity
In [20], new exact solutions were presented for a class of nonuniform 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 nonuniform beams presented in [20]. Ignoring rotational inertia and shear deformation, the EulerBernoulli small deflection beam equation is obtained by taking $F\equiv 0$ in Eq. (1). To test the validity of the method for different end conditions we consider ClampedFree (CF), PinnedPinned (PP) and FreeFree (FF) end conditions. At clamped and free ends the boundary conditions can be used from Eqs. (56). For a pinned end the conditions are $w=0$, ${d}^{2}w/d{x}^{2}=0$.
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. Nondimensional 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

CF

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


FF

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


PP

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 selfweight
The equation for the vibration of a uniform column under selfweight 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 $\beta $ denotes a hanging column. The first two square of nondimensional frequency values $\left({\omega}^{2}\right)$ 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 Nondimensional frequency values of a column under selfweight
${\omega}^{2}$

$\beta $


0

–20

–50

–100


Wang [4]

PSM

Wang [4]

PSM

Wang [4]

PSM

Wang [4]

PSM


${\omega}_{1}^{2}$

12.362

12.362363

43.53

43.530595

89.65

89.653010

165.6

165.604518

${\omega}_{2}^{2}$

485.50

485.518818

657.7

657.648489

913.1

913.078445

1333

1332.919282

3.4. Free Vibration of nonuniform column
The governing equation of a nonuniform column varying with bending stiffness given by $1+\alpha x$ as in [11] under no selfweight is obtained by taking $n=$ 0, $m=$ 1, $\beta =$ 0 and $c=\alpha $ in Eqn. (4). In [11], the free vibration of nonuniform 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 simplesimple (SS) supports/pinnedpinned (PP) supports and clampedsimple (CS) supports at bottom and top were treated using MWCM technique while the clampedclamped (CC) supports and clampedfree (CF) supports were treated using SBCGE technique. However, the present Pseudospectral formulation is capable of treating the different boundary conditions with ease to compute the nondimensional frequencies $\left(\omega \right)$ of nonuniform columns. The computed first three frequency values for $\alpha =$ –0.5, 0.5 in the case of SS and CC 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 nonuniform columns with good accuracy.
Table 4. Nondimensional frequency values of nonuniform column
Nondimensional
frequency values

$\alpha =$ –0.5

$\alpha =$ 0.5

$\alpha =$ –0.5

$\alpha =$ 0.5


MWCM
[11]

PSM

MWCM
[11]

PSM

SBCGE
[11]

PSM

SBCGE
[11]

PSM


(SS)

(SS)

(CC)

(CC)


${\omega}_{1}$

8.479

8.479450

11.003

11.003523

19.098

19.098602

24.888

24.888283

${\omega}_{2}$

33.834

33.834311

43.976

43.976308

52.709

52.709121

68.633

68.633917

${\omega}_{3}$

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 RungeKutta 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 crosssection shape parameters $\left(m,n\right)$, we consider the values (3, 1), (1, 1) and (4, 2) for a fair comparison. The nondimensional vibration frequencies are computed for the taper values of $c=$ 0.1, 0.3 and 0.7 with the gravity parameter $\left(\beta \right)$ taking the values 0, 2.5 and 7.5. The computations in the PSM are carried out using$N=$ 25. The results obtained are presented in Tables 57 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, $\beta =$ 2.5 and $c=$ 0.3, $\beta =$ 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 $\left(\beta \right)$ 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. Nondimensional frequencies for $m=$ 3, $n=$ 1
$\beta $

$c$


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. Nondimensional frequencies for $m=$ 1, $n=$ 1
$\beta $

$c$


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. Nondimensional frequencies for $m=$ 4, $n=$ 2
$\beta $

$c$


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 nonuniform gravity loaded EulerBernoulli column/beam is obtained numerically as the governing fourth order differential equation with variable coefficients does not yield any closed form solutions. The numerical techniqueChebyshev 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 nonuniform beams in the absence of gravity, solutions obtained using two variants of DQM for a nonuniform column under different end conditions and also against the solutions obtained using an initial value method in the case of a uniform column under selfweight. 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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