# Free Vibrations with Large Amplitude of Axially Loaded Beams on an Elastic Foundation Using the Adomian Modified Decomposition Method

Authors: Desmond Adair; Askar Ibrayev; Alima Tazabekova; Jong R. Kim
Journal: Shock and Vibration
 (2019)

---

## Abstract

Analytical solutions describing free transverse vibrations with large amplitude of axially loaded Euler–Bernoulli beams for various end restrains resting on a Winkler one-parameter foundation are obtained using the Adomian modified decomposition method (AMDM). The AMDM allows the governing equation to become a recursive algebraic equation, and, after some additional simple mathematical operations, the equations can be cast as an eigenvector problem whose solution results in the calculation of natural frequencies and corresponding closed-form series solution of the mode shapes. Important to the use of the Adomian modified decomposition method is the treatment of the nonlinear Fredholm integral coefficient, which forms part of the governing equation. In addition to the calculation of natural frequencies and mode shapes, investigations are made of the effects on the free vibrations of the Winkler parameter and of increasing the axial loading.


---

## Body

## 1. Introduction

Uniform slender beams resting on an elastic foundation, while subjected to axial loading, are common in structural systems undergoing actual operating conditions. Analysis of such systems, both linear and nonlinear, has been of interest to civil and railway engineering. For example, when ambient temperatures increase, rails and concrete slabs, often used in urban transport systems, tend to expand, so causing compressive in-plane forces, leading to changes in natural frequencies and eventually to buckling. In-plane compressive forces are also found in prestressed beams. If the amplitudes of the vibrations remain small, the governing equation is usually in the form of a linear differential equation which is relatively simple to solve. However for large amplitude vibration, nonlinear terms are introduced into the governing equation which needs to be treated. Rails and concrete slabs often rest on foundations generally classified as elastic, viscoelastic, Winkler, and Pasternak.Boundary value problems (BVPs) have been the subject of several analytical methods, recently developed, to calculate beams with relatively simple configurations. The variational iteration method (VIM) often attributed to He [1] is a modification of a general Lagrange multiplier method and has been used as a powerful tool for solving ordinary differential equations [2, 3]. Another recent method developed is the homotopy perturbation method (HPM) [4, 5] which has been used for problems involving nonlinear differential equations. Less recent methods used to investigate the vibration problem for nonuniform Euler–Bernoulli beams have been the Rayleigh–Ritz method [6], closed-form solutions [7], and Green’s function method [8]. Several methods have used the Frobenius series [9] and also by discretizing the beam into beam elements [10]. There have been some early studies of vibrating beams under axial loading [11–13], where the effect of increasing the axial loading on the mode shapes and natural frequencies of the beam was investigated. There has been some work already done for large amplitude vibration. Bhashyam and Prathap [14] used the Galerkin finite element method to study nonlinear vibration, and Özkaya [15] calculated the response of a beam mass system with clamped ends by applying a method known as the method of multiple scales.For beams resting on foundations, an understanding of the beam-foundation interaction is needed. The foundation increases resistance to movement and can significantly change the modal characteristics of the beam. Many practical cases in engineering related to foundation-beam interaction can be modelled by assuming the beam resting on an elastic foundation with the Winkler elastic foundation model [16] used extensively. This model assumes the foundation to be made up of an infinitely many closed-spaced linear springs and is a one-parameter model. A limitation of this model is the assumption that there is no interaction between the springs. To overcome this defect, several two-parameter models have been suggested, such as Filonenko–Borodich, Pasternak, and Vlasov and Leontiev foundation models [17]. Studies using a constant Winkler foundation can be found in the literature [18–20].In the present work, the adomian modified decomposition method [21, 22] is utilized to calculate free transverse vibration characteristics of axially loaded Euler–Bernoulli beams with various end restrains, resting on a Winkler one-parameter foundation. The method is chosen as it has proved efficient and accurate [23, 24] for solving linear and nonlinear differential equations, and it has the advantage of computational simplicity. In addition, it does not involve linearization, discretization, perturbation, or a priori assumptions, which may alter the physics of the problem considered [21]. For the AMDM, the solution is considered to be the sum of an infinite series with rapid convergence [25]. Using the AMDM, the governing differential equation becomes a recursive algebraic equation and the boundary conditions become simple algebraic frequency equations, which are suitable for symbolic computation. After some simple algebraic operations on the frequency equations for any ith natural frequency, the closed-form series solution of any ith mode shape can be obtained. Calculations are made for clamped-free and clamped-clamped boundary conditions together with an investigation of the effects of increasing the axial loading and Winkler parameter on the natural vibrations.


## 2. Principle of Adomian Modified Decomposition Method (AMDM)

The basic theory of AMDM is briefly stated here. Consider the equation(1)Fyx=gx,where F represents a general nonlinear ordinary differential operator involving both linear and nonlinear parts and gx is a given function. The linear terms in Fy are decomposed into Ly+Ry, where L is an invertible operator, which for AMDM is taken as the highest-order derivative, and R is the remainder of the linear operator. Equation (1) can now be written as(2)Ly+Ry+Ny=g,where Ny represents the nonlinear terms of Fy and equation (2) represents an initial value or boundary value problem.On solving forLy, equation (2) can be transformed to(3)y=Φ+L−1g−L−1Ry−L−1Ny,where Φ is the integration constant and LΦ=0 is satisfied. To use the AMDM, y is decomposed into the infinite sum of a convergent series:(4)y=∑m=0∞Cmxm,and the nonlinear term is decomposed using Adomian polynomials, Am:(5)Ny=∑m=0∞xmAmC0,C1,…,Cm.The functiongx can also be decomposed as(6)gx=∑m=0∞gmxm.Putting equations (4)–(6) into equation (3) gives(7)y=∑m=0∞Cmxm=Φ+L−1∑m=0∞gmxm−L−1R∑m=0∞Cmxm−L−1∑m=0∞xmAmC0,C1,…,Cm.The coefficientCm can be calculated using a recurrence relation and the power series solutions of linear homogeneous differential equations in initial value problems yield simple recurrence relations for the coefficient Cm. In practice, the coefficients cannot be determined exactly, and the solutions can only be approximated by a truncated series ∑m=0n−1Cmxm.


## 3. Mathematical Formulation

In this work, a uniform beam under axial load while resting on a Winkler foundation is considered. As shown in Figure1, the beam has a length, l, a uniform rectangular cross section, A, a cross-sectional moment of inertia, I, and the beam is considered as made of isotropic material with a modulus of elasticity, E, and density, ρ.Figure 1
Beam under axial loading while resting on an elastic foundation.The model for the foundation is the relatively simple Winkler model whose stiffness changes along the beam length and is a function of the spatial coordinate along the beam in thex direction.According to the theory of structural vibrations [26, 27], on using the Euler–Bernoulli beam model, the strain energy induced by a large displacement amplitude is given by(8)U=12∫0lEI∂2wx,t∂x22dx+12∫0lEA∂ux,t∂x+12∂wx,t∂x22dx+12∫0lkwxwx,t2dx.The large amplitude of the vibrations necessitates the inclusion of the nonlinear term shown in equation (8).Here,u and w are the axial and transverse displacements, respectively, and kwx is the foundation stiffness coefficient. The kinetic energy is given by(9)T=12∫0lρA∂wx,t∂t2dx.The external work done by the axial load is(10)W=P2∫0l∂wx,t∂x2dx.By invoking Hamilton’s principle and using the Lagrangian of the system,(11)δ∫t1t2T−U+Wdt=0.On substituting equations (8)–(10) into equation (11), the following governing equation can be obtained after eliminating axial displacement:(12)EI∂4wx,t∂x4+ρA∂2wx,t∂t2+P∂2wx,t∂x2+kwxwx,t−EA2l∂2wx,t∂x2∫0l∂wx,t∂x2dx=0,where k0 is a constant and gx is a function of the spatial coordinate along the beam length.According to modal analysis for harmonic-free vibration,wx,t can be separable in space and time as(13)wx,t=ϕxht,where ϕx is the modal deflection and ht is a harmonic function of time t. If ω denoted the circular frequency of ht, then ∂2wx,t/∂t2=−ω2ϕxht and the eigenvalue problem of equation (12) reduces to(14)EId4ϕxdx4+Pd2ϕxdx2−EA2ld2ϕxdx2∫0ldϕxdx2dx+kwϕx−ρAω2ϕx=0.Equation (14) is now made nondimensional using(15)X=xl,ϕX=ϕxl,P¯=Pl2EI,K0=kwl4EI,λ=ρAω2l4EI,and becomes(16)d4ϕXdX4+P¯d2ϕXdX2−12d2ϕXdX2∫01dϕXdX2dX+K0−λϕX=0.


## 4. Boundary Conditions

Two cases are considered in this work, namely, beams which are clamped-clamped and clamped-free, respectively:For the clamped-clamped case, the boundary conditions atX=0 and X=1 are(17)ϕX=dϕXdX=0.For the clamped-free case, the boundary conditions atX=0 and X=1 are(18)ϕ0=dϕ0dX=0,d2ϕ1dX2=0,d3ϕ1dX3+P¯dϕ1dX=0.It is convenient for the AMDM to describe boundary conditions in terms of rotational and translational flexible ends as shown in Figure2.Figure 2
Boundary condition described by rotational and translational flexible ends.The boundary conditions can be written in the dimensionless form as(19)d2ϕ0dX2−κL0dϕ0dX=0,d3ϕ0dX3+κL1ϕ0=0,d2ϕ1dX2+κR0dϕ1dX=0,d3ϕ1dX3+P¯dϕ1dX−κR1ϕ1=0,where the coefficients are nondimensionalized as(20)κL1=kL1l3EI,κR1=kR1l3EI,κL0=kL0lEI,κR0=kR0lEI.


## 5. Application of the Adomian Modified Decomposition Method (AMDM)

According to the AMDM,ϕX in equation (16) can be expressed as an infinite series, i.e.,(21)ϕX=∑m=0∞CmXm,where the unknown coefficients, Cm, are determined recurrently. If a linear operator G≡d4/dX4 is used, then the inverse operator of G is a four fold operator defined as(22)G−1=∫0x∫0x∫0x∫0x⋯dXdXdXdX.Equation (16) now can be written as(23)ϕX=ΦX−G−1P¯d2ϕXdX2−12d2ϕXdX2∫01dϕXdX2dX+K0−λϕX,where ΦX is a polynomial depending on the boundary conditions.


### 5.1. Nonlinear Fredholm Integral Coefficient

The terms in equation (23) are linear except for the nonlinear Fredholm integral coefficient shown in equation (24). Before continuing with the main solution method, the nonlinear term will be treated first through the use of appropriate Cauchy products. Consider(24)12∫01dϕXdX2dXd2ϕXdX2.Use is now made of the term(25)ϕX=∑m=0∞CmX−X0m.Differentiating(26)dϕXdX=∑m=0∞m+1Cm+1X−X0m=∑m=0∞bmX−X0m,d2ϕXdX2=∑m=0∞m+1m+2Cm+2X−X0m.On settingX0=0,(27)d2ϕXdX2=∑m=0∞m+1m+2Cm+2Xm=∑m=0∞bmXm.So(28)dϕXdX2=∑m=0∞bmX−X0m∑l=0∞blX−X0l=∑m=0∞BmX−X0m,where Bm=∑l=0mblbm−l.So(29)12∫01dϕXdX2dX=12∫01∑m=0∞BmX−X0mdX=12∑m=0∞∫01BmX−X0mdX=12∑m=0∞BmX−X0m+1m+1X=0X=1.AsX0=0 in the current analysis, then(30)12∫01dϕXdX2dX=∑m=0∞Bm2m+1=∑m=0∞αm.Combining the two strands of analysis gives(31)12∫01dϕXdX2dXd2ϕXdX2=∑m=0∞αm∑l=0∞βlXm=∑m=0∞Xm∑l=0mαlβm−l=∑m=0∞ζmXm,where(32)ζm=∑l=0mαlβm−l.


### 5.2. Linear and Nonlinear Terms Combined

Using the analysis of the last subsection andϕX=∑m=0∞CmXm, equation (23) now becomes(33)ϕX=ΦX+G−1−P¯∑m=0∞m+1m+2Cm+2Xm+∑m=0∞ζmXm−K0−λ∑m=0∞CmXm,where ΦX=∑m=03CmXm=ϕ0+ϕ′0X+ϕ″0X2/2+ϕ‴0X3/6 is the initial term. The recurrence relation for the coefficients Cm can now be stated as(34)C0=ϕ0,C1=ϕ′0,C2=ϕ″02,C3=ϕ‴06,and for m≥4 as(35)Cm=1mm−1m−2m−3∑j=0m−4−P¯j+1j+2Cj+2+ζj−K0−λCj.The coefficientsCm can be found from the recurrence equations (34) and (35), and the solution for ϕX is calculated using equation (33). The series solution is ϕX=∑m=0∞CmXm, although all of the coefficients Cm cannot be determined, and thus, the solutions must be approximated by the truncated series ∑m=0n−1CmXm and successive approximations are ϕnX=∑m=0n−1CmXm, as n increases and the boundary conditions are met.Thus,ϕ1X=C0, ϕ2X=ϕ1X+C1X, ϕ3X=ϕ2X+C2X2, and ϕ4X=ϕ3X+C3X3 serve as approximate solutions with increasing accuracy as n⟶∞. The four coefficients Cjj=0,1,2,3 depend on the boundary conditions used (either equation (17) or (18)). For example, for the clamped-free boundary conditions at X=0, the two coefficients C0 and C1 can be chosen as arbitrary constants, and the other two coefficients C2 and C3 can be expressed as functions of C0 and C1. Thus, from equations (19) and (34), the following is obtained:(36)C2=κL02C1,C3=−κR06C0.Thus the initial termΦX is only a function of C0 and C1, and from the recurrence relation of equation (36), the coefficients Cmm≥4 are functions of C0, C1, and λ. By substituting ϕnX into the boundary conditions of equation (19) when X=1, we have(37)fr0nλC0+fr1nλC1=0,r=1,2.For nontrivial solutions ofC0 and C1, the frequency equation is given as(38)f10nλf11nλf20nλf21nλ=0.Theith estimated eigenvalue λin corresponding to m is obtained from equation (38), i.e., the ith estimated dimensionless natural frequency Ωnin=λin is also obtained and n is determined by(39)Ωnin−Ωnin−1≤ε,where Ωnin−1 is the ith estimated dimensionless natural frequency corresponding to n−1 and ε is a preset sufficiently small value. If equation (39) is satisfied, then Ωnin is the ith dimensionless natural frequency Ωni. By substituting Ωnin into equation (37),(40)C1=−fr0nΩninfr1nΩninC0,r=1,2,and all of the other coefficients Cm can be obtained from equations (34) and (35). Furthermore, the ith mode shape ϕin corresponding to the ith eigenvalue Ωnin is obtained by(41)ϕinX=∑m=0n−1CmiXm,where CmiX is CmX in which λ is substituted by λi and ϕin is the ith eigenfunction corresponding to the ith eigenvalue λi. By normalizing equation (41), the ith normalised eigenfunction is defined as(42)ϕ¯inX=ϕinX∫01ϕinX2dX,where ϕ¯inX is the ith mode shape function of the beam corresponding to the ith natural frequency ωin=λinEI/ρAl4=ΩninEI/ρAl4.This general theory is now applied to a uniform Euler–Bernoulli beam under different boundary conditions.


## 5.1. Nonlinear Fredholm Integral Coefficient

The terms in equation (23) are linear except for the nonlinear Fredholm integral coefficient shown in equation (24). Before continuing with the main solution method, the nonlinear term will be treated first through the use of appropriate Cauchy products. Consider(24)12∫01dϕXdX2dXd2ϕXdX2.Use is now made of the term(25)ϕX=∑m=0∞CmX−X0m.Differentiating(26)dϕXdX=∑m=0∞m+1Cm+1X−X0m=∑m=0∞bmX−X0m,d2ϕXdX2=∑m=0∞m+1m+2Cm+2X−X0m.On settingX0=0,(27)d2ϕXdX2=∑m=0∞m+1m+2Cm+2Xm=∑m=0∞bmXm.So(28)dϕXdX2=∑m=0∞bmX−X0m∑l=0∞blX−X0l=∑m=0∞BmX−X0m,where Bm=∑l=0mblbm−l.So(29)12∫01dϕXdX2dX=12∫01∑m=0∞BmX−X0mdX=12∑m=0∞∫01BmX−X0mdX=12∑m=0∞BmX−X0m+1m+1X=0X=1.AsX0=0 in the current analysis, then(30)12∫01dϕXdX2dX=∑m=0∞Bm2m+1=∑m=0∞αm.Combining the two strands of analysis gives(31)12∫01dϕXdX2dXd2ϕXdX2=∑m=0∞αm∑l=0∞βlXm=∑m=0∞Xm∑l=0mαlβm−l=∑m=0∞ζmXm,where(32)ζm=∑l=0mαlβm−l.


## 5.2. Linear and Nonlinear Terms Combined

Using the analysis of the last subsection andϕX=∑m=0∞CmXm, equation (23) now becomes(33)ϕX=ΦX+G−1−P¯∑m=0∞m+1m+2Cm+2Xm+∑m=0∞ζmXm−K0−λ∑m=0∞CmXm,where ΦX=∑m=03CmXm=ϕ0+ϕ′0X+ϕ″0X2/2+ϕ‴0X3/6 is the initial term. The recurrence relation for the coefficients Cm can now be stated as(34)C0=ϕ0,C1=ϕ′0,C2=ϕ″02,C3=ϕ‴06,and for m≥4 as(35)Cm=1mm−1m−2m−3∑j=0m−4−P¯j+1j+2Cj+2+ζj−K0−λCj.The coefficientsCm can be found from the recurrence equations (34) and (35), and the solution for ϕX is calculated using equation (33). The series solution is ϕX=∑m=0∞CmXm, although all of the coefficients Cm cannot be determined, and thus, the solutions must be approximated by the truncated series ∑m=0n−1CmXm and successive approximations are ϕnX=∑m=0n−1CmXm, as n increases and the boundary conditions are met.Thus,ϕ1X=C0, ϕ2X=ϕ1X+C1X, ϕ3X=ϕ2X+C2X2, and ϕ4X=ϕ3X+C3X3 serve as approximate solutions with increasing accuracy as n⟶∞. The four coefficients Cjj=0,1,2,3 depend on the boundary conditions used (either equation (17) or (18)). For example, for the clamped-free boundary conditions at X=0, the two coefficients C0 and C1 can be chosen as arbitrary constants, and the other two coefficients C2 and C3 can be expressed as functions of C0 and C1. Thus, from equations (19) and (34), the following is obtained:(36)C2=κL02C1,C3=−κR06C0.Thus the initial termΦX is only a function of C0 and C1, and from the recurrence relation of equation (36), the coefficients Cmm≥4 are functions of C0, C1, and λ. By substituting ϕnX into the boundary conditions of equation (19) when X=1, we have(37)fr0nλC0+fr1nλC1=0,r=1,2.For nontrivial solutions ofC0 and C1, the frequency equation is given as(38)f10nλf11nλf20nλf21nλ=0.Theith estimated eigenvalue λin corresponding to m is obtained from equation (38), i.e., the ith estimated dimensionless natural frequency Ωnin=λin is also obtained and n is determined by(39)Ωnin−Ωnin−1≤ε,where Ωnin−1 is the ith estimated dimensionless natural frequency corresponding to n−1 and ε is a preset sufficiently small value. If equation (39) is satisfied, then Ωnin is the ith dimensionless natural frequency Ωni. By substituting Ωnin into equation (37),(40)C1=−fr0nΩninfr1nΩninC0,r=1,2,and all of the other coefficients Cm can be obtained from equations (34) and (35). Furthermore, the ith mode shape ϕin corresponding to the ith eigenvalue Ωnin is obtained by(41)ϕinX=∑m=0n−1CmiXm,where CmiX is CmX in which λ is substituted by λi and ϕin is the ith eigenfunction corresponding to the ith eigenvalue λi. By normalizing equation (41), the ith normalised eigenfunction is defined as(42)ϕ¯inX=ϕinX∫01ϕinX2dX,where ϕ¯inX is the ith mode shape function of the beam corresponding to the ith natural frequency ωin=λinEI/ρAl4=ΩninEI/ρAl4.This general theory is now applied to a uniform Euler–Bernoulli beam under different boundary conditions.


## 6. Numerical Results

### 6.1. Clamped-Free Uniform Beam

The first case considered is the clamped-free uniform beam resting on an elastic foundation and experiencing axial compressive force The case was chosen to test (validate) the accuracy of the AMDM.The boundary conditions are as given in equations (18) and (19) with the spring constants becoming κL0⟶∞,κR0⟶0,κL1=∞,κR1=0.WhenX=0, the first two boundary conditions of equation (19) yield the relationships shown in equation (36), and when X=1 by substituting ϕnX=∑m=0n−1CmXm into the last two boundary conditions of equation (19), the following two algebraic equations (written in full) involving C0 and C1 are obtained:(43)∑m=0n−3m+1m+2Cm+2+κR0∑m=0n−2m+1Cm+1=f11nλC0+f12nλC1=0,∑m=0n−4m+1m+2m+3Cm+3+P¯∑m=0n−2m+1Cm+1−κR1∑m=0n−1Cm=f21nλC2+f22nλC3=0.The case of the clamped-free uniform beam without an elastic foundation support or without any axial force and with small vibration amplitude was first chosen to test (validate) the accuracy of the AMDM as comparisons can be made with what is already given in the literature. For this, the values ofP¯, K0, and nonlinear term in equation (16) were set to zero. The first five natural frequencies, Ωni,i=1,…,5 are shown in Table 1 and compared with those obtained by Reference [9]. Excellent agreement was found.Table 1
First five natural frequenciesΩni,i=1,…,5 for the small vibration amplitude.
Method
Ω
n
1
Ω
n
2
Ω
n
3
Ω
n
4
Ω
n
5
Present
3.516010
22.034484
61.697213
120.901920
199.859536
Reference [9]
3.5160
22.0345
61.6972
120.902
199.860The results are shown in Table2, with, and without, the nonlinear term given in equation (16). Here, Ωnoni are the results with the nonlinear term used and Ωni are the results when the nonlinear term was not used.Table 2
Natural frequencies of the cantilever beam on an elastic foundation withP¯=0 and K0=1.
Modei
Reference [27] Ωni
PresentΩni
PresentΩnoni
PresentΩnoni/Ωni
1
3.66
3.6712
3.9869
1.089
2
22.22
22.2343
24.4049
1.098
3
62.69
62.7421
69.1014
1.101
4
124.29
124.3811
138.0632
1.111It can be seen from Table2 that the results agree with those of Reference [27] fairly well where the nonlinear term is absent although generally the present results are slightly higher than those calculated by Chen [27]. Also shown in Table 2 are the results calculated for large amplitudes. This was effected through the inclusion of the nonlinear term of equation (16). The ratio of results calculated with and without the inclusion of the nonlinear term is also given in Table 2. The presence of the nonlinear terms increases quite substantially the natural frequencies, and it can be seen that this increase grows in line with increasing the mode number.Figure3 presents the variation of natural frequency modes with increase in axial force and the foundation stiffness. During the variation of the axial force the elastic stiffness, K0, was held constant and for the variation of the elastic stiffness, the axial force, P¯, was held constant. It is noticeable that the first mode is significantly affected by both variations in comparison with the higher modes, and in particular, the first mode natural frequency is greatly affected by increasing in axial force.Figure 3
Variation of frequency modes with axial force and foundation stiffness.
(a)
(b)By substituting the convergedΩn1n into equations (21) and (35) and normalizing the result using equation (42), a polynomial can be obtained to describe the first mode shape function. The same procedure can be employed for other natural frequencies to find the mode shapes for higher mode numbers. The variation of the first and third mode shapes for various Winkler parameters is illustrated on Figure 4. Here, the axial force was set to zero. The changes in the mode shape were not too significant until the value of K0 became reasonably large. It can be seen from the results for the third mode that the increase in elastic stiffness affects both the amplitude and the phase of the shape function.Figure 4
(a) First and (b) third mode shapes for various stiffness parameters withP¯=0.
(a)
(b)A similar exercise was carried out to ascertain the trends for the mode shapes when the axial force was increased or when the beam was tensioned as opposed to being compressed. The effects on the first and third modes are shown in Figure5. Here, the elastic stiffness was held constant at one. It can be seen that the greatest effect was on the first mode and, except at the extrema, increasing the axial load had very little effect on higher modes.Figure 5
Effect of axial load on mode shapes withK0=1. (a) 1st mode. (b) 3rd mode.
(a)
(b)Important to this study was the speed of obtaining accurate converged solutions. An example of the rate of convergence is given in Figure6 for the case of clamped-free boundary conditions with P¯=0 and K0=1. As can be seen, the AMDM method was fast converging with results for the first mode obtained after a very few iterations. Convergence for higher modes took longer, although not prohibitively.Figure 6
Convergence plots for natural frequencies whenP¯=0 and K0=1.


### 6.2. Clamped-Clamped Uniform Beam

For this case, the boundary conditions are as given in equations (17) and (19) with the spring constants becoming κL0⟶∞,κR0⟶∞,κL1⟶∞,andκR1⟶∞:(44)C2=κL02C1,C3=−κL16C0,∑m=0n−1Cm=f11nλC0+f12nλC1=0,∑m=0n−2m+1Cm+1=f21nλC0+f22nλC1=0.Calculations were made for the clamped-clamped beam of the natural frequencies with no axial force and for two values of foundation stiffness as shown in Table3. The results for the present method compare well with those reported in the literature. These results were obtained with the nonlinear term of equation (16) not used.Table 3
Natural frequencies of the clamped-clamped beam on an elastic foundation withP¯=0 and without the nonlinear term.
K
0
=
0
K
0
=
100
Present
Reference [28]
Reference [29]
Present
Reference [28]
Reference [29]
Ω
n
1
4.73004
4.7314
4.73
4.95246
4.9515
4.95
Ω
n
2
7.85320
7.8533
7.854
7.91103
7.9044
7.904
Ω
n
3
10.9955
10.9908
10.996
11.0121
11.0096
11.014Table4 shows results for the first four modes with and without the nonlinear term present. Again Ωnoni is the result when the nonlinear term is used, and Ωni is the result when the nonlinear term is not used. It can be seen that the present calculated values of Ωni are in good agreement although again most of the results are slightly higher than those calculated by Chen [27]. When the nonlinear term is included, the results for the natural frequencies increase with the ratio of increase found to be similar in magnitude to those reported by Mei [30].Table 4
Natural frequencies of the clamped-clamped beam on an elastic foundation withP¯=0 and K0=1.
Modei
Reference [27] Ωni
PresentΩni
PresentΩnoni
PresentΩnoni/Ωni
1
22.435
22.614
23.971
1.06
2
62.094
62.051
74.461
1.19
3
122.659
123.54
149.772
1.21
4
204.935
205.332
256.665
1.25Again polynomials were obtained to describe the first three mode shape functions for the clamped-clamped beam. The first three mode shapes for the clamped-clamped beam are shown in Figure7 after normalization and compared well with those reported in the literature.Figure 7
First three mode shapes for a clamped-clamped beam withK0=1 and P¯=1.Finally, the present method of calculation was validated using a backbone curve for the clamped-clamped beam as shown in Figure8. The first mode results reported by Gupta et al. [31] were chosen with reasonable agreement found with a similar trend found between the two result sets, as illustrated in Figure 8. Here, the value of the foundation stiffness was set at zero.Figure 8
Comparison of backbone curves for the 1st mode of the clamped-clamped beam.


## 6.1. Clamped-Free Uniform Beam

The first case considered is the clamped-free uniform beam resting on an elastic foundation and experiencing axial compressive force The case was chosen to test (validate) the accuracy of the AMDM.The boundary conditions are as given in equations (18) and (19) with the spring constants becoming κL0⟶∞,κR0⟶0,κL1=∞,κR1=0.WhenX=0, the first two boundary conditions of equation (19) yield the relationships shown in equation (36), and when X=1 by substituting ϕnX=∑m=0n−1CmXm into the last two boundary conditions of equation (19), the following two algebraic equations (written in full) involving C0 and C1 are obtained:(43)∑m=0n−3m+1m+2Cm+2+κR0∑m=0n−2m+1Cm+1=f11nλC0+f12nλC1=0,∑m=0n−4m+1m+2m+3Cm+3+P¯∑m=0n−2m+1Cm+1−κR1∑m=0n−1Cm=f21nλC2+f22nλC3=0.The case of the clamped-free uniform beam without an elastic foundation support or without any axial force and with small vibration amplitude was first chosen to test (validate) the accuracy of the AMDM as comparisons can be made with what is already given in the literature. For this, the values ofP¯, K0, and nonlinear term in equation (16) were set to zero. The first five natural frequencies, Ωni,i=1,…,5 are shown in Table 1 and compared with those obtained by Reference [9]. Excellent agreement was found.Table 1
First five natural frequenciesΩni,i=1,…,5 for the small vibration amplitude.
Method
Ω
n
1
Ω
n
2
Ω
n
3
Ω
n
4
Ω
n
5
Present
3.516010
22.034484
61.697213
120.901920
199.859536
Reference [9]
3.5160
22.0345
61.6972
120.902
199.860The results are shown in Table2, with, and without, the nonlinear term given in equation (16). Here, Ωnoni are the results with the nonlinear term used and Ωni are the results when the nonlinear term was not used.Table 2
Natural frequencies of the cantilever beam on an elastic foundation withP¯=0 and K0=1.
Modei
Reference [27] Ωni
PresentΩni
PresentΩnoni
PresentΩnoni/Ωni
1
3.66
3.6712
3.9869
1.089
2
22.22
22.2343
24.4049
1.098
3
62.69
62.7421
69.1014
1.101
4
124.29
124.3811
138.0632
1.111It can be seen from Table2 that the results agree with those of Reference [27] fairly well where the nonlinear term is absent although generally the present results are slightly higher than those calculated by Chen [27]. Also shown in Table 2 are the results calculated for large amplitudes. This was effected through the inclusion of the nonlinear term of equation (16). The ratio of results calculated with and without the inclusion of the nonlinear term is also given in Table 2. The presence of the nonlinear terms increases quite substantially the natural frequencies, and it can be seen that this increase grows in line with increasing the mode number.Figure3 presents the variation of natural frequency modes with increase in axial force and the foundation stiffness. During the variation of the axial force the elastic stiffness, K0, was held constant and for the variation of the elastic stiffness, the axial force, P¯, was held constant. It is noticeable that the first mode is significantly affected by both variations in comparison with the higher modes, and in particular, the first mode natural frequency is greatly affected by increasing in axial force.Figure 3
Variation of frequency modes with axial force and foundation stiffness.
(a)
(b)By substituting the convergedΩn1n into equations (21) and (35) and normalizing the result using equation (42), a polynomial can be obtained to describe the first mode shape function. The same procedure can be employed for other natural frequencies to find the mode shapes for higher mode numbers. The variation of the first and third mode shapes for various Winkler parameters is illustrated on Figure 4. Here, the axial force was set to zero. The changes in the mode shape were not too significant until the value of K0 became reasonably large. It can be seen from the results for the third mode that the increase in elastic stiffness affects both the amplitude and the phase of the shape function.Figure 4
(a) First and (b) third mode shapes for various stiffness parameters withP¯=0.
(a)
(b)A similar exercise was carried out to ascertain the trends for the mode shapes when the axial force was increased or when the beam was tensioned as opposed to being compressed. The effects on the first and third modes are shown in Figure5. Here, the elastic stiffness was held constant at one. It can be seen that the greatest effect was on the first mode and, except at the extrema, increasing the axial load had very little effect on higher modes.Figure 5
Effect of axial load on mode shapes withK0=1. (a) 1st mode. (b) 3rd mode.
(a)
(b)Important to this study was the speed of obtaining accurate converged solutions. An example of the rate of convergence is given in Figure6 for the case of clamped-free boundary conditions with P¯=0 and K0=1. As can be seen, the AMDM method was fast converging with results for the first mode obtained after a very few iterations. Convergence for higher modes took longer, although not prohibitively.Figure 6
Convergence plots for natural frequencies whenP¯=0 and K0=1.


## 6.2. Clamped-Clamped Uniform Beam

For this case, the boundary conditions are as given in equations (17) and (19) with the spring constants becoming κL0⟶∞,κR0⟶∞,κL1⟶∞,andκR1⟶∞:(44)C2=κL02C1,C3=−κL16C0,∑m=0n−1Cm=f11nλC0+f12nλC1=0,∑m=0n−2m+1Cm+1=f21nλC0+f22nλC1=0.Calculations were made for the clamped-clamped beam of the natural frequencies with no axial force and for two values of foundation stiffness as shown in Table3. The results for the present method compare well with those reported in the literature. These results were obtained with the nonlinear term of equation (16) not used.Table 3
Natural frequencies of the clamped-clamped beam on an elastic foundation withP¯=0 and without the nonlinear term.
K
0
=
0
K
0
=
100
Present
Reference [28]
Reference [29]
Present
Reference [28]
Reference [29]
Ω
n
1
4.73004
4.7314
4.73
4.95246
4.9515
4.95
Ω
n
2
7.85320
7.8533
7.854
7.91103
7.9044
7.904
Ω
n
3
10.9955
10.9908
10.996
11.0121
11.0096
11.014Table4 shows results for the first four modes with and without the nonlinear term present. Again Ωnoni is the result when the nonlinear term is used, and Ωni is the result when the nonlinear term is not used. It can be seen that the present calculated values of Ωni are in good agreement although again most of the results are slightly higher than those calculated by Chen [27]. When the nonlinear term is included, the results for the natural frequencies increase with the ratio of increase found to be similar in magnitude to those reported by Mei [30].Table 4
Natural frequencies of the clamped-clamped beam on an elastic foundation withP¯=0 and K0=1.
Modei
Reference [27] Ωni
PresentΩni
PresentΩnoni
PresentΩnoni/Ωni
1
22.435
22.614
23.971
1.06
2
62.094
62.051
74.461
1.19
3
122.659
123.54
149.772
1.21
4
204.935
205.332
256.665
1.25Again polynomials were obtained to describe the first three mode shape functions for the clamped-clamped beam. The first three mode shapes for the clamped-clamped beam are shown in Figure7 after normalization and compared well with those reported in the literature.Figure 7
First three mode shapes for a clamped-clamped beam withK0=1 and P¯=1.Finally, the present method of calculation was validated using a backbone curve for the clamped-clamped beam as shown in Figure8. The first mode results reported by Gupta et al. [31] were chosen with reasonable agreement found with a similar trend found between the two result sets, as illustrated in Figure 8. Here, the value of the foundation stiffness was set at zero.Figure 8
Comparison of backbone curves for the 1st mode of the clamped-clamped beam.


## 7. Conclusions

A fast, efficient, and accurate method of solution, namely, the Adomian modified decomposition method (AMDM) was developed to calculate natural vibrations of an Euler–Bernoulli beam with large amplitude resting on a Winkler foundation. The method is free of linearization, discretization, perturbation, ora priori assumptions and nonlinear terms are relatively easily treated. A practical advantage of the AMDM is the ease of applying the boundary conditions where the vibrational analysis for different boundary conditions simply involves changing the values of the corresponding parameters with no need to change the solution procedures or the algorithms employed.The numerical comparisons for both boundary conditions used here indicate that the current numerical results are in satisfactory agreement with those found by other methods, with perhaps some of the results found here better than some reported.



---

*Source: 3405075-2019-02-26.xml*