An Efficient Analytical Method for Vibration Analysis of a Beam on Elastic Foundation with Elastically Restrained Ends

Abstract

An efficient analytical method for vibration analysis of a Euler-Bernoulli beam on elastic foundation with elastically restrained ends has been reported. A Fourier sine series with Stoke’s transformation is used to obtain the vibration response. The general frequency determinant is developed on the basis of the analytical solution of the governing differential equation for all potential solution cases with rigid or restrained boundary conditions. Numerical analyses are performed to investigate the effects of various parameters, such as the springs at the boundaries to examine how the elastic foundation parameters affect the vibration frequencies.

1. Introduction

Beams resting on elastic foundations have wide application in engineering practice. The vibration analysis of beams is investigated using various elastic foundation models, such as, Vlasov, Pasternak, and Winkler models. A number of studies have been performed to predict the dynamic response of beams on elastic foundations with different boundary conditions.

Numerous works have been performed to explore the static deflection and vibration response of the beams resting on various elastic foundations. Chun [1] has investigated free vibration of hinged beam. Maurizi et al. [2] have considered the vibration frequencies for a beam with different boundary conditions. Vibration of beams on partial elastic foundations has been studied by Doyle and Pavlovic [3]. Laura et al. [4] have investigated beams which carry concentrated masses subject to an axial force. Abbas [5] has investigated vibration of Timoshenko beams with elastically restrained ends. Free vibration and stability behavior of uniform beams and columns with nonlinear elastic end rotational restraints has been considered by Rao and Naidu [6]. Free vibration behaviour of an Euler-Bernoulli beam resting on a variable Winkler foundation has been considered by Kacar et al. [7]. Civalek [8] has implemented differential quadrature and harmonic differential quadrature methods for buckling analysis of thin isotropic plates and elastic columns. H. K. Kim and M. S. Kim [9] have considered vibration of beams with generally restrained boundary conditions. A number of studies have been reported investigating the free vibration of beams on elastic foundation [10–25].

Although vibration analysis of beams on elastic foundation is a widely studied topic, there are only few papers that exist in the literature pertaining to the analysis of beams with elastically restrained ends. In this study, an efficient method is introduced for the analysis of the free vibration behavior of Euler-Bernoulli beams on an elastic foundation with elastic restraints. A Fourier sine series together with Stokes' transformation is used to evaluate the free vibration frequencies. The general frequency determinant is constructed by applying Stokes' transformation to the boundary conditions. Free vibration analyses of elastically supported beam on an elastic foundation are carried out and comparisons are made between with and without elastic foundation.

2. Application of Stokes’ Transformation

Beam on elastic foundation with elastically restrained ends is depicted in Figure 1. Based on the Euler-Bernoulli beam theory, the equation of motion for a beam resting on a Winkler-type elastic foundation is given by 𝜕𝜌𝐴²𝑣(𝑥,𝑡)𝜕𝑡²𝜕+𝐸𝐼⁴𝑣(𝑥,𝑡)𝜕𝑥⁴+𝑘_𝑤𝑣(𝑥,𝑡)=0, (1) where 𝐸𝐼 is the flexural rigidity, 𝜌 is the mass density, 𝐴 is the cross-sectional area of the beam, and 𝑘_𝑤 is the stiffness of the foundation per unit length. Assuming harmonic vibration, the lateral displacement function 𝑣(𝑥,𝑡) can be written in the form 𝑣(𝑥,𝑡)=𝜓(𝑥)cos(𝜔𝑡), (2) where 𝜓(𝑥) is the modal displacement function and 𝜔 is the natural frequency. The function 𝜓(𝑥) is described herein in three separate forms as follows: 𝜓(𝑥)=𝜑₀𝜑𝑥=0_𝐿𝑥=𝐿_∞𝑚=1𝐴_𝑚sin𝑚𝜋𝑥𝐿.0<𝑥<𝐿 (3) The derivatives of 𝜓(𝑥) are based on Stokes’ transformation: 𝑑𝜓(𝑥)=𝜑𝑑𝑥^𝐿−𝜑₀𝐿+^∞_𝑚=1cos𝛼_𝑚𝑥2(−1)^𝑚𝜑_𝐿−𝜑₀𝐿+𝛼_𝑚𝐴_𝑚,𝑑²𝜓(𝑥)𝑑𝑥²=−^∞_𝑚=1𝛼_𝑚sin𝛼_𝑚𝑥2(−1)^𝑚𝜑_𝐿−𝜑₀𝐿+𝛼_𝑚𝐴_𝑚,𝑑³𝜓(𝑥)𝑑𝑥³=𝜑_𝐿^′′−𝜑₀^′′𝐿+^∞_𝑚=1cos𝛼_𝑚𝑥2(−1)^𝑚𝜑_𝐿^′′−𝜑₀^′′𝐿−𝛼²_𝑚2(−1)^𝑚𝜑_𝐿−𝜑₀𝐿+𝛼_𝑚𝐴_𝑚,𝑑⁴𝜓(𝑥)𝑑𝑥⁴=−^∞_𝑚=1𝛼_𝑚sin𝛼_𝑚𝑥×2(−1)^𝑚𝜑_𝐿^′′−𝜑₀^′′𝐿−𝛼²_𝑚2(−1)^𝑚𝜑_𝐿−𝜑₀𝐿+𝛼_𝑚𝐴_𝑚, (4)(5)(6)(7) where 𝛼_𝑚=𝑚𝜋𝐿 (8) and lateral displacement function can be written as a sum of Fourier components; (3) and (7) are substituted into (1) to result in −sin𝑥𝛼_𝑚𝐿𝐴_𝑚𝐴𝜌𝜔²−𝐸𝐼𝛼⁴_𝑚−𝑘_𝑤+2𝐸𝐼𝛼_𝑚𝛼²_𝑚𝜑₀−(−1)^𝑚𝜑_𝐿+(−1)^𝑚𝜑_𝐿^′′−𝜑₀^′′=0. (9) By using above equation, the Fourier coefficient 𝐴_𝑚 can be written as follows: 𝐴_𝑚=2𝐸𝐼𝛼_𝑚𝛼²_𝑚𝜑₀−(−1)^𝑚𝜑_𝐿+(−1)^𝑚𝜑_𝐿^′′−𝜑₀^′′𝐿−𝐴𝜌𝜔²+𝐸𝐼𝛼⁴_𝑚+𝑘_𝑤 (10) and lateral displacement function can be written as a sum of Fourier components: 𝑣(𝑥,𝑡)=^∞_𝑚=12𝐸𝐼𝛼_𝑚𝛼²_𝑚𝜑₀−(−1)^𝑚𝜑_𝐿+(−1)^𝑚𝜑_𝐿^′′−𝜑₀^′′𝐿−𝐴𝜌𝜔²+𝐸𝐼𝛼⁴_𝑚+𝑘_𝑤×sin𝑚𝜋𝑥𝐿cos(𝜔𝑡). (11) When the stiffness of the foundation per unit length (𝑘_𝑤) is taken as zero, the above equation turns to be useful for a beam without elastic foundation [9].

Figure 1 

The beam on elastic foundation with elastically restrained ends.

3. The General Frequency Determinant for Different Boundary Conditions

The beam on elastic foundation is assumed to be elastically restrained by means of translational and rotational springs (see Figure 1). Then the boundary conditions are 𝑆₀𝜑₀𝑑=−𝐸𝐼³𝑣(𝑥,𝑡)𝑑𝑥³𝑆,𝑥=0,_𝐿𝜑_𝐿𝑑=𝐸𝐼³𝑣(𝑥,𝑡)𝑑𝑥³Ω,𝑥=𝐿,₀𝑑𝑣𝑑𝑑𝑥=𝐸𝐼²𝑣(𝑥,𝑡)𝑑𝑥²Ω,𝑥=0,_𝐿𝑑𝑣𝑑𝑑𝑥=−𝐸𝐼²𝑣(𝑥,𝑡)𝑑𝑥²,𝑥=𝐿, (12) where 𝑆₀,𝑆_𝐿 are translational spring constants and Ω₀,Ω_𝐿 are rotational springs constants. The substitution of (4) and (6) into the above boundary conditions leads to four homogeneous equations −𝑆₀−^∞_𝑚=12𝑚²𝜋²−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴𝜑₀𝐿²+^∞_𝑚=12𝑚²(−1)^𝑚𝜋²−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴𝜑_𝐿𝐿²+1+^∞_𝑚=12−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴𝜑₀^′′−1+^∞_𝑚=12(−1)^𝑚−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴𝜑_𝐿^′′=0,_∞𝑚=12𝑚²(−1)^𝑚𝜋²−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴𝜑₀𝐿²−𝑆_𝐿+^∞_𝑚=12𝑚²𝜋²−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴𝜑_𝐿𝐿²−1+^∞_𝑚=12(−1)^𝑚−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴𝜑₀^′′+1+^∞_𝑚=12−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴𝜑_𝐿^′′−=0,Ω₀+2Ω₀^∞_𝑚=1−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴𝜑₀𝐿²+Ω₀+2Ω₀^∞_𝑚=1(−1)^𝑚−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴𝜑_𝐿𝐿²+−1+2Ω₀^∞_𝑚=1𝑚²𝜋²−𝑚⁴−𝛿+𝜆⁴𝜑₀^′′+−2Ω₀^∞_𝑚=1𝑚²(−1)^𝑚𝜋²−𝑚⁴−𝛿+𝜆⁴𝜑_𝐿^′′=0,Ω_𝐿+2Ω_𝐿^∞_𝑚=1(−1)^𝑚−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴𝜑₀𝐿²−Ω_𝐿+2Ω_𝐿^∞_𝑚=1−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴𝜑_𝐿𝐿²+−2Ω_𝐿^∞_𝑚=1𝑚²(−1)^𝑚𝜋²−𝑚⁴−𝛿+𝜆⁴𝜑₀^′′+−1+2Ω_𝐿^∞_𝑚=1𝑚²𝜋²−𝑚⁴−𝛿+𝜆⁴𝜑_𝐿^′′=0, (13) where 𝐿𝛿=⁴𝑘_𝑤𝐸𝐼𝜋⁴,𝜆⁴=𝜌𝐴𝐿⁴𝐸𝐼𝜋⁴𝜔²,𝑆₀=𝑆₀𝐿³,𝐸𝐼𝑆_𝐿=𝑆_𝐿𝐿³,𝐸𝐼Ω₀=Ω₀𝐿,𝐸𝐼Ω_𝐿=Ω_𝐿𝐿,𝐸𝐼 (14) and one can obtain the following system of linear algebraic equations in matrix form to be solved for the constants (𝜑₀/𝐿², 𝜑_𝐿/𝐿², 𝜑₀^′′, 𝜑_𝐿^′′): 𝜑₁₁𝜑₁₂𝜑₁₃𝜑₁₄𝜑₂₁𝜑₂₂𝜑₂₃𝜑₂₄𝜑₃₁𝜑₃₂𝜑₃₃𝜑₃₄𝜑₄₁𝜑₄₂𝜑₄₃𝜑₄₄𝜑₀𝐿²𝜑_𝐿𝐿²𝜑₀^′′𝜑_𝐿^′′=0. (15) The eigen values of the above systems of equations give the free vibration frequencies 𝜑_𝑖𝑗=0;𝑖,𝑗=1,2,3,4, (16) where 𝜑₁₁=−𝑆₀−^∞_𝑚=12𝑚²𝜋²−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴,𝜑₁₂=^∞_𝑚=12𝑚²(−1)^𝑚𝜋²−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴,𝜑₁₃=1+^∞_𝑚=12−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴,𝜑₁₄=−1−^∞_𝑚=12(−1)^𝑚−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴,𝜑₂₁=^∞_𝑚=12𝑚²(−1)^𝑚𝜋²−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴,𝜑₂₂=𝑆_𝐿−^∞_𝑚=12𝑚²𝜋²−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴,𝜑₂₃=−1−^∞_𝑚=12(−1)^𝑚−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴,𝜑₂₄=1+^∞_𝑚=12−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴,𝜑₃₁=−Ω₀−2Ω₀^∞_𝑚=1−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴,𝜑₃₂=Ω₀+2Ω₀^∞_𝑚=1(−1)^𝑚−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴,𝜑₃₃=−1+2Ω₀^∞_𝑚=1𝑚²𝜋²−𝑚⁴−𝛿+𝜆⁴,𝜑₃₄=−2Ω₀^∞_𝑚=1𝑚²(−1)^𝑚𝜋²−𝑚⁴−𝛿+𝜆⁴,𝜑₄₁=Ω_𝐿+2Ω_𝐿^∞_𝑚=1(−1)^𝑚−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴,𝜑₄₂=−Ω_𝐿−2Ω_𝐿^∞_𝑚=1−𝛿+𝜆⁴−𝑚⁴−𝛿+𝜆⁴,𝜑₄₃=−2Ω_𝐿^∞_𝑚=1𝑚²(−1)^𝑚𝜋²−𝑚⁴−𝛿+𝜆⁴,𝜑₄₄=−1+2Ω_𝐿^∞_𝑚=1𝑚²𝜋²−𝑚⁴−𝛿+𝜆⁴. (17)

4. Numerical Results

In this section, the well-known problem of a beam on an elastic foundation is analyzed by the proposed method. To calculate the frequency parameters (𝜆_𝑖), we solve (16). Firstly, in order to verify the accuracy of the proposed formulation, a comparison of the frequency results with the same results available from other numerical methods for the beams with classical supporting conditions is carried out. It is interesting to note that restrained boundary conditions will degenerate into the classical ones, provided that proper values are given to spring parameters in (16).

4.1. Validation of the Proposed Method

In this subsection, it is desired to evaluate the accuracy of the proposed method when applied to some special cases of the model. To the authors’ knowledge, the differential transform method [26, 27] has been used for vibration analysis of beam with rigid boundary conditions. As aforementioned the proposed method can be used to determine the vibration frequencies of a beam in various classical supporting conditions, as well as any desired boundary conditions. It should be noted that, by letting 𝑆₀=∞, 𝑆_𝐿=∞, Ω₀=∞, and Ω_𝐿=∞, (16) will automatically degenerate into the beam clamped at both ends. When the spring parameters are used to represent a cantilever beam, the frequency determinant is written in the same order by letting 𝑆₀=∞, Ω₀=∞, 𝑆_𝐿=0, and Ω_𝐿=0. Frequencies of a simply supported beam can be achieved by using the values 𝑆₀=∞, 𝑆_𝐿=∞, Ω₀=0, and Ω_𝐿=0 in (16).

In the numerical verifications, the frequency parameters are calculated by the present approach using the first 50 terms of the infinite series. Elastic spring parameters are taken as 𝑆₀=𝑆_𝐿=Ω₀=Ω_𝐿=10000 for the beam with clamped-clamped ends and 𝑆₀=Ω₀=10000, 𝑆_𝐿=Ω_𝐿=0 for the cantilever beam. For convenience, the following reference parameter is introduced: 𝐾₀=𝛿𝜋⁴. (18) The elastic foundation parameter is taken as (𝐾₀=1/𝜋⁴) in Tables 1 and 2. In the second validation analysis, a simple supported beam is considered. In Table 3, vibration frequency parameters are given for various values of the (𝐾₀) parameter (𝐾₀ = 10, 50, 100,  500, 1000, 2000).

Table 1 

Verification of the proposed method for a beam with clamped-clamped ends.

Table 2 

Verification of the proposed method for a beam with clamped-free ends.

Table 3 

Verification of the first three frequency parameters for a simple supported beam.

The most important observation from Tables 1, 2, and 3 is due to the fact that all frequency parameters of the beam are calculated by using the first 50 terms of the infinite series. Improvement in accuracy can be gained by increasing the terms of infinite series. However, as seen from Tables 1, 2, and 3, the present results seem to be more acceptable.

After verifying correctness of proposed method, the effects of different spring parameters on the lateral vibration of a beam are discussed. The effect of the spring parameter on the vibration responses of beam is demonstrated in Figure 2. The results in Figure 2 are calculated by using (16) for the values of (𝛿=1,2,3,…,10). One can observe that the first three frequency parameters are increased by considering the effects of the elastic foundation parameter. There is an abrupt change in the first modes when the foundation parameter varies from 0 to 1. On the other hand, with the consideration of the elastic foundation parameter, all vibration frequencies of the beam become dependent on the spring parameters.

(a) 𝑆0=10, 𝑆𝐿=10, Ω0=105, and Ω𝐿=105

(b) 𝑆0=105, 𝑆𝐿=10, Ω0=105, and  Ω𝐿=10

Figure 2 

The effects of different spring parameters on the vibration frequencies.

4.2. Beam on Elastic Foundation with Elastically Restrained Ends

For comparison purposes, the variation in the ratio of vibration frequencies of beam with embedding elastic medium to that without embedding medium with different elastic foundation parameters is plotted for the first three modes. To compare the results of analytical analysis with no elastic foundation case, a 𝜆_𝑖/𝜆₀ ratio is considered. The index (𝑖) denotes the mode number and (0) denotes the case without elastic foundation.

Analyses are performed to investigate the effects of elastic foundation parameter to examine how it affects the vibration frequencies of the system. The results of analysis for both cases are depicted in Figures 3–6. Fixing the spring parameters (𝑆₀=10⁵, 𝑆_𝐿=10⁵, Ω₀=10⁵, and Ω_𝐿=10⁵) and varying the elastic foundation parameter (𝛿) result in a significant change in the vibration frequencies (see Figure 3). For the case in hand, changing the elastic foundation parameter (𝛿) from 0 to 10 results in an increase in the first mode of about 30 percent, whereas second and third modes are not much affected from the existence of the elastic foundation parameter, as can be noted from the figure.

Figure 3 

Effect of elastic foundation parameter on the vibration frequencies for 𝑆0=105, 𝑆𝐿=105, Ω0=105, and Ω𝐿=105.

Figure 4 

Effect of elastic foundation parameter on the vibration frequencies for 𝑆0=105, 𝑆𝐿=105, Ω0=10, and Ω𝐿=10.

Figure 5 

Effect of elastic foundation parameter on the vibration frequencies for 𝑆0=105, 𝑆𝐿=10, Ω0=105, and Ω𝐿=10.

Figure 6 

Effect of elastic foundation parameter on the vibration frequencies for 𝑆0=10, 𝑆𝐿=10, Ω0=105, and Ω𝐿=105.

The enhancement of the first mode is observed for the different boundary conditions as presented in Figures 4 and 5. For a beam with boundary conditions 𝑆₀=10⁵, 𝑆_𝐿=10⁵, Ω₀=10, and Ω_𝐿=10, as the elastic foundation parameter (𝛿) changes from 0 to 10, the first frequency parameter increases by about 45, as can be noted from Figure 4. Finally, for a beam with boundary conditions 𝑆₀=10, 𝑆_𝐿=10, Ω₀=10⁵, and Ω_𝐿=10⁵, this enhancement reaches about 260 percent, as can be noted from Figure 6.

The first three frequency parameters obtained from the analysis and predicted by the suggested formulas are presented as shown in Tables 4 and 5. It can be seen from the tables that, by increasing the the elastic foundation parameter, the first three vibration frequencies increase. It can be noted that the first frequency increase is more than the others. As is obvious from the tables, the proposed analytical method offers acceptable results. This parametric study points out to the possibility of enhancing the frequency parameters of beams with restrained boundary conditions.

Table 4 

The first three frequency parameters for different boundary conditions (𝜆₁, 𝜆₂, and 𝜆₃).

Table 5 

The first three frequency parameters for different boundary conditions (𝜆₁, 𝜆₂, and 𝜆₃).

5. Conclusion

On the basis of Euler-Bernoulli beam theory, the free vibration response of a beam resting on a Winkler-type elastic foundation with restrained boundary conditions has been investigated. A simplified analytical method is developed, which can be used for a beam with any types of boundary conditions. The general frequency determinant is calculated by a combination of Fourier series expansion and Stokes' transformation. The influence of the elastic foundation and spring parameters on the natural frequencies is examined in some numerical examples. The results of the present analytical method demonstrate good agreement with the results of other methods.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.