An analytical solution technique that provides fast, efficient, and accurate results is developed for fourth-order nonlinear differential equations describing the transverse oscillations of a beam on an elastic foundation. 77 Table 4.3 Eigenfrequencies of a clamped-clamped beam on an elastic foundation with and without a nonlinear member.
Introduction
General
Methodology and techniques
Although the AMDM has been used to solve problems related to beam response, it has not been applied to finite beams resting on foundations subjected to harmonic loading. This is the originality of the current work, which will be analyzed in the following paragraphs.
Literature review
- Beams and Foundations
- General methods for beam vibrations analysis
- Adomian Decomposition Method
The model is based on the assumption of shear interaction between the springs that create the shear effect of the foundation. By applying the shear layer, continuity within the soil medium is thus achieved.
Theory development
Beam and foundation theory
- Euler-Bernoulli beam resting on a Winkler foundation
- Timoshenko beam resting on Pasternak foundation
- Boundary Conditions
Despite the effectiveness of Euler-Bernoulli beam theory in obtaining accurate results of vibration frequencies of thin beams, Timoshenko beam theory predicts more accurate results in thick non-thin beams where shear or torsional effects must be considered. Here is the length of the beam, the cross-sectional area, the moment of inertia of the cross-section, the modulus of rigidity, the modulus.
Adomian Decomposition Method
- Adomian Modified Decomposition Method
- Adomian Polynomials for nonlinear term
To get an overview of the ADM, consider the differential equation. where is a general nonlinear ordinary or partial differential operator consisting of both linear and nonlinear terms. Linear components are decomposed into where is reversible, which is considered the highest order derivative to overcome integrations with complicated Greenian functions, and the remainder of the liner operator is defined by. where all non-linear components are described. is defined as the twofold integration operator. 2.24). The nonlinear component is decomposed as. and are the terms that result from the integration of the source term and from the initial or boundary conditions. where are the Adomain polynomials determined for each nonlinearity.
It must be tested for the number of terms to achieve the convergence. There have been several modifications of the Adomian decomposition method made by various researchers to increase the speed of convergence or to change the application of the original method. To use AMDM is decomposed into the infinite summation of a convergent series. 2.36), and the non-linear component is decomposed using Adomian polynomials,.
In fact, the coefficients cannot be calculated exactly and the truncated set ∑. will be used to approximate the solutions. The accuracy of the solution depends on the number of terms at which convergence is observed. To be able to create and calculate a series that controls the number of links for accurate results, computer software is required. 2.29) describing Adomain polynomials is coded in MATLAB software.
Validation of Adomian Decomposition Method
- Linear Boundary Value Problem
- Non-Linear Boundary Value Problem
Therefore, approximations such as the truncated Taylor expansions will be considered for the exponential term in Eq. The second subscript here refers to the number of terms used in the truncated Taylor expansions for . As it can be seen, the difference between the ADM values and actual values decreases as the number of iterations, , increases.
The results show that the results of ADM become more accurate and approach the actual values as the number of terms increases. As can be seen, the difference between the ADM values and the actual values decreases as the number of terms, , increases. The results show that ADM works well even with the unlined boundary value problem.
Moreover, the ADM results with non-linear BVP provide more accurate results than linear BVP. This shows that ADM can be used to obtain reasonably accurate results with linear and especially non-linear equations.
Adomian Modified Decomposition Method Applied to Various Foundation Types
AMDM applied to Euler-Bernoulli beam and Winkler foundation with no loading 43
- Results
3.4) contains a nonlinear term called the nonlinear Fredholm integral coefficient. Thus, the initial term ( ) can be represented as a function and and from Eq. This general theory can now be used to analyze the oscillation frequencies of a beam with different boundary conditions.
-Clamped Uniform Beam boundary conditions were chosen to investigate the vibration of the Euler-Bernoulli beam resting on a Winker foundation without any load. - Tensioned boundary condition was chosen due to its better physical representation of the railway track beam compared to other boundary conditions. When the first two boundary conditions for Eq. 2.20) state the connections shown in Eq. into the last two boundary conditions of Eq. 2.20) the following two algebraic equations involving and can be obtained. 3.21), and additional numerical results are calculated.
Natural frequencies ( ) were calculated for the first three modes using four different foundation stiffness values. From the Table 3.1 it can be seen that frequencies increase as the stiffness value of the foundation increases. From the ratio of the results, it can be noted that the presence of the non-linear term increases the natural frequencies quite significantly and also the ratio grows with the increase of the number of modes.
AMDM applied to Euler-Bernoulli beam and Winkler foundation with harmonic
- Mathematical formulation
- Discontinuous load expansion
- Application of the AMDM
- Results
The method gives fast convergence especially for the first mode which gives an exact result only after. This can be applied to the concentrated load by assuming that the load is distributed over an infinitesimal interval δ. 3.38) and making the use non-dimensional. 3.36) becomes non-dimensional throughout by substituting Eq. AMDM can now be applied for equalization. 3.37) following the same procedure as described in section 3.1.
By ( ) ∑ and its second derivatives in Eq. Recurrence ratio for the first 4 terms of coefficients can be written as in Eq. Algebraic equation resulting from boundary conditions in Eq. 3.42) will be used to find the coefficients for solving Eq. It can be seen that with beam subjected to concentrated loading shows similar pattern as unloaded beam.
By substituting the obtained ( ) back into Eqs. 3.24), the polynomial that describes the shape function of the first mode is derived. However, at large values, the shape changes are significant, and we can see that the amplitude of the third mode shape changes greatly. We can see that the perfect convergence for the first mode is only after 10.
AMDM applied to Euler-Bernoulli beam and Pasternak foundation with harmonic
- Mathematical formulation
- Distributed harmonic high-speed moving loading
- Application of the AMDM
- Results
Vibration response of the Euler-Bernoulli beam resting on the Pasternak foundation is now calculated using Clamp-free boundary conditions where the spring constants become, as per Eq. Algebraic equation resulting from boundary conditions in Eq. 3.53) will be used to find the coefficients for solving Eq. The effect of the stiffness parameter on beam vibration frequency in the presence of the constant shear parameter was also analyzed and the numerical results are provided in Table 3.5.
Although the comparison is only performed up to and, it can be observed that the overall ratio of each mode in Figure 3.8, where the shear parameter is varied while keeping the stiffness constant, is greater than in Figure 3.9, where the stiffness is varied while keeping the shear constant . Furthermore, both figures show that the first mode is the most affected by each parameter than the other modes. Again a good convergence of the obtained results of vibration frequencies of the beams on the Pasternak foundation can be seen from Figure 3.10 where the first three modes converge successfully after about 16 iterations.
The time is determined by , where is the length of the beam, is the speed of the moving charge. The effect of the shear parameter on the deflection at the middle of the beam was also analyzed. The result is shown in figure 3.12 which shows that the shear parameter significantly affects the deflection of the beam under the moving load.
Extension of AMDM application to Special Case
Application of AMDM to a Euler-Bernoulli Beam with Axial Loading
- Boundary conditions
- Application of the Adomian Modified Decomposition Method
- Results
The first analyzed case is the unclamped uniform beam resting on the Winkler foundation and experiencing an axial compressive force. First, calculations were made for a clamped beam resting on an elastic foundation for ̅ and with and without the nonlinear term in Eq. Results are obtained for the first four modes and compared with results from the literature [30], as shown in Table 4.1.
It can be seen that natural frequencies are higher with the presence of the nonlinear term and its effect increases as with the increase of modes. In can be seen that the first mode is mostly affected by both axial force and stiffness increase than higher modes. As expected, the negative axial force creates tension in the beam and reduces the transverse deflection while positive axial force creates compression and induces the vibration for the first mode, but has less effect on the third mode.
Natural frequencies of clamped beam were calculated for two different foundation stiffness values while keeping axial force zero, and compared with other results from literature. In Table 4.3 results are shown for the first four modes with and without the non-linear term. Calculated natural frequencies are used to develop a polynomial for the first three mode shape functions using Eq.
Concluding Remarks
After the analysis of the beam resting on the two-parameter Pasternak foundation, it was found that the displacement parameter of the foundation had a greater effect on the frequency than the stiffness parameter. Furthermore, investigation of beam subjected to harmonic moving load showed that the amplitude of the beam deflection decreases as the shear coefficient of the foundation increases. The influence of concentrated masses and Pasternak soils on the free vibrations of Euler beams - exact solution.
Younesian, Response of beams on nonlinear viscoelastic foundations to harmonic moving loads, Computers and Structures. Dynamic response of a finite-length Euler-Bernoulli beam on linear and nonlinear viscoelastic foundations to a concentrated moving force. Analysis of the vibration of an elastic beam supported on elastic ground using the differential transformation method.
The Homotopy Perturbation Method for Free Vibration Analysis of Beam on Elastic Foundation, Structural Engineering and Mechanics. 33] C.-N Chen: Vibration of prismatic beam on an elastic foundation by the differential quadrature element method.
