Abstract
In this article, the free vibrations of EulerBernoulli and Timoshenko beams with arbitrary varying crosssection are investigated analytically using the perturbation technique. The governing equations are linear differential equations with variable coefficients and the Wentzel, Kramers, Brillouin approximation is adopted for solving these eigenvalue equations and determining the natural frequencies and mode shapes. This method relates the solution of equations with the solving of some successive algebraic equations. A parametric study is performed and the effects of different profiles and different combinations of boundary conditions on the natural frequencies are investigated. To confirm the reliability of the present method, the analytical results are checked with those obtained from the finite elements method and other literatures which are found to be in a good agreement. The calculations show that the presented procedure is very effective to find the modal characteristics of the varying crosssections beams.
1. Introduction
A beam is an important element which is used as a part of some structures. To achieve a better distribution of rigidity and reducing the weight, the beams with variable crosssections are used. The dynamic analysis of beams is frequently encountered in engineering practices. This analysis becomes more complicated in the cases of beams with the variable crosssection. The wind turbine blades are a typical example of the beams with variable thickness subjected to dynamic loads.
The natural frequency is a design parameter associated with engineering vibrations. During the past decades, the extensive research efforts have been presented concerning the linear dynamic analysis of beams. Jategaonkar and Chehil [1] determined the natural frequencies of a linear EulerBernoulli (EB) beam with varying section properties by evaluating the effective inertia, area, and mass. An approximated finite elements (FE) method was introduced by Eisenberger and Reich [2] to analyze the nonuniform beams. They used the displacement functions of a constant crosssection beam to find the approximated stiffness and consistent mass matrices. Rossi and Laura [3] determined the natural frequencies and dynamic behavior of linearly tapered Timoshenko beams subjected to different combinations of edge supports by the FE procedures. De Rosa and Auciello [4] studied the dynamic behavior of beams with a linearly varying crosssection. The equation of motion was solved in terms of Bessel functions. Abrate [5] presented simple formulas for predicting the fundamental natural frequency of nonuniform beams with the general shape and arbitrary boundary conditions by RayleighRitz method. Zhou and Cheung [6] studied the vibrational characteristics of tapered EB beams with a continuously varying rectangular crosssection by the RayleighRitz method. Byoung et al. [7] studied the free vibrations of tapered EB beams with general boundary conditions. The natural frequencies were calculated by combining the RungeKutta and the determinant search methods. Kukla and Zamojska [8] applied Green’s function method for frequency analysis of an EB beam with the varying crosssection. Rensburgand and Merweb [9] presented a systematic approach to solve the eigenvalue problems associated with the uniform Timoshenko beam model by the FE method. Ece et al. [10] studied the vibrations of an isotropic EB beam which has a variable crosssection. The governing equation was reduced to an ordinary differential equation in spatial coordinate for a family of crosssection geometries with exponentially varying width. De Rosa and Lippiello [11] studied the natural frequencies of tapered beams by using the EB theory in the presence of an arbitrary number of rotationally, axially and elastically flexible constraints. The dynamic analysis was performed by means of the cell discretization method. Zamorska [12] used Green’s function and power series methods for the free vibrations problem of nonuniform EB beams. Mahmoud et al. [13] applied the differential transformation method for the free vibrations analysis of EB beams with uniform and nonuniform crosssections. Boiangiu et al. [14] solved the differential equations for free bending vibrations of straight EB beams with variable crosssection using Bessel’s functions. In order to improve the analytical accuracy, Leiping et al. [15] found the stiffness matrix of Timoshenko beam element with the arbitrary section. According to the relationship between geometrical deformation and internal force, by integral of sectional curvature, the shearing strain, and axial strain, the stiffness matrix of the Timoshenko beam element was derived. Yuan et al. [16] proposed a novel method to simplify the governing equations for the free vibrations of Timoshenko beams with both geometrical nonuniformity and material inhomogeneity along the beam axis. They converted the governing equations to uncoupled forms. Each of obtained equations was in the form of Sturm–Liouville type with variable coefficients. The authors solved these equations for polynomial and exponential variations of parameters which have the exact solutions in terms of Bessel’s and hypergeometric functions. Korabathina and Koppanat [17] developed the “coupled displacement field method” for calculating the fundamental frequency of Timoshenko beam which reduces the computational efforts compared with respect to the other methods. Zhao et al. [18] introduced the Chebyshev polynomials to analyze the free vibrations of axially functionally graded EB and Timoshenko beams with a nonuniform crosssection. Chen et al. [19] reexamined the free vibrations of rotating tapered Timoshenko beams using the technique of variational iteration. Nourifar et al [20] utilized the differential transform method for free vibrations analysis of rotating EB beam with an exponentially varying crosssection.
The review of literatures demonstrates the FE and analytical methods to find the natural frequencies in the beams. Although the FE is straight forward but building a FE model for a beam with arbitrary varying cross section, is difficult especially when the optimization with trial and error is the final purpose. In analytical filed, the most authors used the known functions for the solution to find the eigenvalue of the system. In this text, an analytical solution based on the perturbation technique is presented for the flexural vibrations of EB and Timoshenko beams, with variable crosssection and the natural frequencies and mode shapes are determined. We do not restrict our solution for special boundary conditions, thickness profiles or using special functions for solution. The numerical examples are presented to demonstrate the accuracy and the efficiency of the presented method. The essential features and novel aspects of the presented formulation are summarized as follows:
The crosssection has an arbitrary symmetric shape and leads to the equations with variable coefficients.
An analytical procedure is demonstrated to find the eigenvalues of one or two coupled differential equations with variable coefficient.
The formulation has a short running time without any restrictions such as the mesh pattern in FE.
It is possible to use this method for different boundary conditions.
2. Euler Bernoulli beam
Consider an isotropic, homogenous nonuniform symmetric EB beam with the length $l$, density $\rho $ and crosssectional area $A\left(x\right)$.
Fig. 1Schematic of beam
The governing equation of the beam with varying crosssection and small deflection is as [22]:
In Eq. (1), $w\left(x,t\right)$ is the beam transverse displacement, $x$, $t$ are the spatial and time parameter, $E$ is the Young modulus, $I\left(x\right)$, $A\left(x\right)$ are the area moment of inertia and area cross section of the beam. The boundary conditions for EB beam are as the following:
$\text{Free}\text{}\text{edge}:\mathrm{}\mathrm{}{w}_{,xx}=0,\mathrm{}\mathrm{}\mathrm{}{w}_{,xxx}=0.$
In this paper, the perturbation technique is used for solving the governing equations. We start by converting the governing equation to the dimensionless form, using the following parameters:
${x}^{*}$, ${t}^{*}$are dimensionless location and time, respectively, ${w}^{*}$ is dimensionless transverse displacement, ${w}_{0}$, ${t}_{0}$ are characteristic deflection and time, ${k}_{0}$ is gyration radius of the crosssection and $\epsilon $ is a small quantity which is considered as the perturbation parameter. By using Eq. (3), the dimensionless form of Eq. (1) (in terms of displacement) is as the following:
Eq. (4) is a linear partial differential equation with variable coefficients. We consider ${\lambda}_{1}={\epsilon}^{1}$ as a large parameters and use the Wentzel, Kramers, Brillouin )WKB( approximation [21] for determining the natural frequencies and mode shapes. We assume the solutions in the following form:
where $\omega $ is nondimensional natural frequency. By substituting Eq. (5) into Eq. (4) we have:
We seek an expansion for ${W}^{*}$ in the following form:
By substituting Eq. (7) into Eq. (6) and considering a straightforward expansion for $G$ in the following form, we have:
We substitute Eqs. (7), (8) into Eq. (6), and separate different orders of $\epsilon $ as the following:
From Eq. (9), four roots for ${G}_{0}^{\mathrm{\text{'}}}$ are determined. Then ${G}_{1}^{\mathrm{\text{'}}}$, ${G}_{2}^{\mathrm{\text{'}}}$ can obtain From Eqs. (10), (11), so we have ${G}^{\mathrm{\text{'}}}={G}_{0}^{\mathrm{\text{'}}}+\epsilon {G}_{1}^{\mathrm{\text{'}}}+{\epsilon}^{2}{G}_{2}^{\mathrm{\text{'}}}$. The solutions of Eq. (7) is in the following form:
By applying the boundary conditions into Eqs. (12), a system of homogenous algebraic equations is constructed. The nontrivial solution of this system results eigenvalues $\omega $ and mode shapes.
3. Timoshenko beam
We consider the free vibrations of homogeneous Timoshenko beam with variable cross section as follows [22]:
where $\psi \left(x,t\right)$ denotes the rotation angle due to bending, $G$ is the shear modulus, ${A}_{s}=$$A$/$k$, and $k$ is called the Timoshenko shear correction factor, depends on the geometry. The shear correction is introduced to take care of the nonuniformity in the shear force across the section. For a rectangular section $k\approx $ 1.20, for a circular section $k\approx $ 1.11 [22]. The boundary conditions for Timoshenko beam are as follows:
$\text{Free}\text{}\text{edge}\text{:}\text{}{\psi}_{,x}=0,\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\left({w}_{,x}\psi \right)=0.$
Combining two equations gives a forth order differential equation with variable coefficient as follows:
To convert the equation as dimensionless we use Eqs. (3). Substitution Eqs (3) into Eq. (15) gives a dimensionless equation of motion represented by:
We seek an expansion for ${\psi}^{*}$ in Eq. (16) in the following form:
Substituting Eq. (17) into Eq. (16) and applying a straightforward expansion for $G$ as Eq. (8) and, separating the terms with the same order of $\epsilon $ results:
From Eq. (18) four roots for ${{G}_{0}}^{\mathrm{\text{'}}}$ are determined, subsequently substituting Eq. (19), (20) gives ${{G}_{1}}^{\mathrm{\text{'}}}$, ${{G}_{2}}^{\mathrm{\text{'}}}$ so we have ${G}^{\mathrm{\text{'}}}={G}_{0}^{\mathrm{\text{'}}}+\epsilon {G}_{1}^{\mathrm{\text{'}}}+\epsilon {G}_{2}^{\mathrm{\text{'}}}$ and the solutions is in the following form:
By applying Eq. (5) into the first Eqs. (13), and using Eq. (3) to convert the resultant equation to dimensionless form and substituting Eq. (21) in the obtained equation, it is possible to calculate the transverse displacement ${W}^{*}$. Consequently, ${w}^{*}\left({x}^{*},{t}^{*}\right)$ is determined from Eq. (5). By applying, the boundary conditions (Eqs. 14), a system of homogenous algebraic equations is constructed. For a nontrivial solution, the coefficient matrix has to equate to zero. The solutions of this equation which are determined by the bisection method are the natural frequencies of the system.
4. FE analysis
ANSYS Workbench 16.2 FE package has been used for the modal analysis of the variable section beam. All the reported results (except that mentioned cases) are related to the rectangular crosssection with constant width and variable thickness with the characteristics listed in Table 1.
Table 1Beam properties
$L=$ Length (mm)  500 
$b=$ Width (mm)  32 
${d}_{0}={h}_{0}$ (mm)  25 
${d}_{1}={h}_{1}$ (mm)  45 
Elastic modulus (GPa)  200 
$\rho $ (kg/m^{3})  7800 
${d}_{0}$, ${h}_{0}$ are the section diameter and thickness of the circular and rectangular crosssections at ${x}^{*}=$ 0 respectively and ${d}_{1}$, ${h}_{1}$ related to ${x}^{*}=$ 1 
5. Parametric study
The analytical calculations have been performed on Maple mathematical environment. As case studies, we consider the beams with rectangular cross section and constant, linear, parabolic, trigonometric thickness distributions. All the reported frequencies are dimensionless and by multiplying in ${t}_{0}$ (Eq. (3)), one can find them in terms of rad/s. To check the formulation, we investigate at first the beam with constant and linear thicknesses variations. Table 2 shows the frequencies of the EB and Timoshenko beams with constant thicknesses and simply supported (SS) boundary conditions. The results have been reported for two cases: EB(HG) and Timoshenko (HG) which are related to the exact solution of equations [22] and WKB which is for the presented formulation. It is seen that the difference between the results is small, so the program works correctly. Also, for the higher modes, the difference between the approximated and exact methods increases. From Table 2, by decreasing the ration of the length to thickness ($L/h$), the natural frequencies and the difference between EB and Timoshenko theories increase especially in the higher modes. The difference percentage of results for HG has been defined as:
In Table 3, a comparison of EB and Timoshenko beams with the results of Shukla [23] has been reported for the linear thickness profile (Fig. 2) and simply supported boundary conditions. Shukla [23] used the EB theory and the FE method for determining the natural frequencies. The results show that in the higher modes, the Timoshenko results are closer to the FE solution with respect to the EB.
Table 2Comparison of EB and Timoshenko beams with constant thickness (SS)
$L/h$  Theory  1st mode  2nd mode  3rd mode 
25  EB (HG)  0.11397  0.45584  1.02570 
EB (WKB)  0.11396  0.45586  1.02568  
Timoshenko (HG)  0.11368  0.45131  1.00307  
Timoshenko (WKB)  0.11396  0.45588  1.02590  
Diff (%)  0.25  1.00  2.26  
15.6  EB(HG)  0.18262  0.73053  1.64365 
EB(WKB)  0.18264  0.73054  1.64372  
Timoshenko (HG)  0.18146  0.71225  1.55617  
Timoshenko (WKB)  0.18264  0.73075  1.64610  
Diff (%)  0.64  2.57  5.62  
10  EB (HG)  0.28489  1.13969  2.56422 
EB (WKB)  0.28491  1.13964  2.56420  
Timoshenko (HG)  0.28048  1.07442  2.02705  
Timoshenko (WKB)  0.28494  1.14158  2.58680  
Diff (%)  1.57  6.07  26.50 
Table 3Comparison of EB and Timoshenko beams with results of Shukla (2013)
Theory  1st mode  2nd mode  3rd mode 
Shukla (2013)  0.17145  0.70560  1.58791 
EB  0.19390  0.78484  1.76270 
Diff (%)  1.35  3.33  3.47 
Timoshenko  0.17400  0.68034  1.52548 
Diff (%)  1.49  3.58  3.93 
Fig. 2Simply supported beam with linearly variable thickness (E= 2.109 GPa, ρ= 7995.74 Kg/m3) (Shukla, 2013)
The various functions with equal volume for the beam area cross section $\text{(}{\int}_{0}^{1}{A}^{\mathrm{*}}\left({x}^{\mathrm{*}}\right)d{x}^{\mathrm{*}}=\mathrm{}$ constant) has been considered. For the rectangular cross section, ${A}^{*}\left({x}^{*}\right)=b{h}^{*}\left({x}^{*}\right)$ and the various thickness functions has been listed in Table 4 and comparison of the frequencies of EB and Timoshenko beam have been reported in Table 5. The differences percentage has been defined similar Table 2.
Table 4Thickness profile with different functions
Type 1  ${h}^{*}={a}_{1}{x}^{*}+{b}_{1}$  ${a}_{1}=$ 0.8, ${b}_{1}=$ 1 
Type 2  ${h}^{*}={a}_{1}{x}^{{*}^{3}}+{b}_{1}{x}^{{*}^{2}}+{c}_{1}$  ${a}_{1}=$ –1.6, ${b}_{1}=2.4$, ${c}_{1}=$ 1 
Type 3  ${h}^{*}={a}_{1}{x}^{*{}^{4}}+{b}_{1}{x}^{*{}^{2}}+{c}_{1}$  ${a}_{1}=$ –1, ${b}_{1}=1.8$, ${c}_{1}=$ 1 
Type 4  ${h}^{*}={b}_{1}\mathrm{c}\mathrm{o}\mathrm{s}\left(\pi {x}^{*}\right)+{c}_{1}$  ${b}_{1}=$ –0.4, ${c}_{1}=$ 1.4 
Type 5  ${h}^{*}={a}_{1}\mathrm{s}\mathrm{i}\mathrm{n}h\left({x}^{*}\right)+{b}_{1}\mathrm{c}\mathrm{o}\mathrm{s}h\left({x}^{*}\right)+{c}_{1}$  ${a}_{1}=$ 0.866, ${b}_{1}=\text{0.4}$, ${c}_{1}=$ 1.4 
Table 5Comparison of EB and Timoshenko beams with different thickness functions (SS)
Thickness function  Theory  1st mode  2nd mode  3rd mode 
Type 1  EB  0.19390  0.78484  1.76270 
Timoshenko  0.19367  0.78517  1.76637  
Diff (%)  0.12  0.04  0.21  
Type 2  EB  0.20686  0.78304  1.74937 
Timoshenko  0.20218  0.78190  1.75233  
Diff (%)  2.32  0.15  0.17  
Type 3  EB  0.23235  0.79136  1.75057 
Timoshenko  0.18676  0.77510  1.74662  
FE  0.18606  0.75727  1.63868  
Diff (%)  24.41  2.10  0.23  
Type 4  EB  0.20195  0.78161  1.74759 
Timoshenko  0.20198  0.78201  1.75122  
Diff (%)  0.02  0.05  0.21  
Type 5  EB  0.19337  0.78493  1.76358 
Timoshenko  0.19343  0.78528  1.76726  
Diff (%)  0.03  0.04  0.21 
According to Table 5, the different profiles with the same weight can change the frequencies or distribution of density is important. For profile Type 3, there is a discrepancy between Timoshenko and EB frequencies. For more investigation, we analyzed this case using the FE procedure. It is seen the obtained results for the FE and Timoshenko are nearly the same, but the EB result is not confidence.
Tables 6 and 7 comprise three first frequencies of EB and Timoshenko beams with thickness variation (Type 2) for some different boundary conditions. The notations C and S stand for the clamped and simply supported boundary conditions respectively. In SC, SS, CC, … the former letter indicates boundary condition at ${x}^{*}=$ 0 and the latter one indicates the boundary condition at ${x}^{*}=$ 1 e.g. SC stands for the simply supported at ${x}^{*}=$ 0 and clamped at ${x}^{*}=$ 1. Tables 6 demonstrate that the highest frequency corresponds to the CC boundary conditions while the lowest frequency occurs for SS which was expected. Also the results for two ($G={G}_{0}\left({x}^{\mathrm{*}}\right)+\epsilon {G}_{1}\left({x}^{\mathrm{*}}\right)$ (2)), and three terms ($G={G}_{0}\left({x}^{\mathrm{*}}\right)+\epsilon {G}_{1}\left({x}^{\mathrm{*}}\right)+{\epsilon}^{2}{G}_{2}\left({x}^{\mathrm{*}}\right)$ (3)) in the expansion have been reported in these tables. Although the differences of results in the most cases are small but for the nonsimilar boundaries (i.e. SC, CS) in EB theory, the obtained results from two terms expansion are in doubt because it reports the same frequencies for the SC and CS. So, we reported the results for three terms expansion.
Table 6EB and Timoshenko results with two and three terms (Type 2)
–  Mode  SS (2)  SS (3)  Diff (%)  CC (2)  CC (3)  Diff (%) 
EB  1  0.19337  0.20686  6.52  0.43835  0.44665  1.86 
2  0.77349  0.78304  1.22  1.20833  1.21696  0.71  
3  1.74035  1.74937  0.52  2.36882  2.37744  0.36  
Timoshenko  1  0.19360  0.20218  4.24  0.43984  0.44824  1.87 
2  0.77265  0.78190  1.18  1.21641  1.22533  0.73  
3  1.74317  1.75233  0.52  2.39684  2.40613  0.39 
Table 7EB and Timoshenko results with two and three terms (Type 2)
–  Mode  CS (2)  CS (3)  Diff (%)  SC (2)  SC (3)  Diff (%) 
EB  1  0.30208  0.28920  4.26  0.30208  0.33338  9.39 
2  0.97894  0.97321  0.59  0.97894  1.00235  2.34  
3  2.04249  2.04058  0.09  2.04249  2.06190  0.94  
Timoshenko  1  0.23490  0.27123  13.4  0.34988  0.34404  1.70 
2  0.95222  0.97010  1.84  1.00968  1.01050  0.08  
3  2.03441  2.04911  0.72  2.07621  2.08000  0.18 
In Table 8 comparison of three first frequency of EB and Timoshenko beams with various crosssections, SS boundary conditions and the thickness variation Type 2 have been listed. The beam with different sections and the same volume have different frequencies. Also, the Timoshenko theory does not have advantage with respect to the EB for calculating the first frequency of the beam with the circular section in this case study.
Table 8Comparison of EB and Timoshenko beams with various crosssections profile (Type 2SS)
Beam section  Theory  1st mode  2nd mode  3rd mode 
Rectangular  EB  0.20686  0.78304  1.74937 
Timoshenko  0.20218  0.78190  1.75233  
Circular  EB  0.35916  1.33839  2.97013 
Timoshenko  0.32000  0.96993  2.14803  
–  FE  0.38061  1.04338  1.97235 
In Fig. 3 three first mode shapes of the EB beam with CC conditions have been shown. In despite of the beams with constant thickness, for a beam with variable thickness the intersection point of mode shape curve with the horizontal axis is not exactly in the middle of the beam.
Fig. 3Three mode shapes for thickness EB beam (Type 2, CC)
a)
b)
c)
In Fig. 4 the effect of thickness function on the first mode shape of the EB beam with SS boundary conditions can be seen. As shown, the sinusoidal function is not suitable for the mode shape in beams with variable thickness and the peak of the graph is not necessarily at the midpoint of the nonuniform beam.
Fig. 4First mode shape for five type’s thickness functions of EB beam (SS)
6. Conclusions
A mathematical approach has been applied to investigate the free vibration of EB and Timoshenko beams with arbitrary symmetric variable crosssection. The governing equations were solved analytically by the perturbation technique. A sensitivity analysis was performed to investigate the influences of boundary conditions, crosssection shape and various thickness functions on the transverse natural frequency of the beam. In summary:
1) An arbitrary crosssection led to differential motion equation with variable coefficients. Using WKB approximation, new form of equations can obtain which, has a closedform solution in each order of $\epsilon $.
2) The calculations can be performed using a simple code in a mathematical environment such as Maple and it is not necessary to construct a FE model. In the other word, the calculations can be performed faster than the FE and this study provides a unified and systematic procedure which is seemingly simpler and more straightforward than the other methods.
3) By using the proposed method, any natural frequency and mode shape function can be obtained one at a time.
4) The presented method can use for different boundary conditions.
5) By increasing the thickness, the natural frequency increases and the difference between EB beam and Timoshenko beam increases.
6) The boundary conditions have less effect for higher modes.
7) It can be seen that the variation of crosssection has a significant effect on the mode shapes. The known trial functions that satisfy the geometric boundary conditions such as sinusoidal function are not suitable as admissible functions for mode shapes in a beam with variable thickness in the methods such as Galerking.
8) It is also possible to use this method for the other crosssection shapes and it can also be useful for designing nonuniform EB and Timoshenko beams which may be required to vibrate with a particular frequency.
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