Abstract
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.
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. 1a) 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 1Nondimensional 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 2Nondimensional 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 3Square 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 4Nondimensional 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 5Nondimensional 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 6Nondimensional 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 7Nondimensional 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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