332 

 

American Academic Scientific Research Journal for Engineering, Technology,  and Sciences   

ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

 

http://asrjetsjournal.org/  

 

Application the Adomian Decomposition Method to the 

Dynamic Response Analysis of Cracked Beam under 

Moving Load 

Naseradin Abujnah
* 

Faculty of Technical Engineering Mesallata, Mesallata, Libya 

Email: naabujnah@gmail.com 

Abstract 

In this article, the Adomian method of decomposition (ADM) is used to examine the vibration of a simply 

supported cracked beam (SSCB) under a moving load based on the Euler Bernoulli hypothesis. the system 

modelled as two segments of the beam are to be connected by a massless elastic rotational spring. Bearing in 

mind each segment of a cracked beam a substructure that can be modeled using ADM. ADM derives for the first 

time on the basis of kinematic conditions and boundary conditions the characteristic / eigenvalue equation and 

mode shape functions of the cracked beam under a moving load. The results obtained from ADM are compared 

to the results obtained from the method of finite elements (FEM). ADM results show remarkable superiority 

compared to FEM. 

Keywords: Adomian decomposition method; Euler-Bernoulli beam; Cracked beam; moving load; mode shape 

functions; eigenvalue equation. 

1. Introduction  

An important topic in the field of mechanical engineering and structural engineering is the analysis of vibrations 

of bridges under moving loads. A moving load will cause larger deflections and higher stresses as compared 

with equivalent static loads. Cracks may occur as a result of stresses on the bridge. The cracks are the main 

reason for the failure of the bridge structure, and when a bridge structure is subjected to cracks, its stiffness will 

decrease, thus reducing the bridge structure's lifetime. The depth and location of a crack is likely to be predicted 

by the changes in vibration parameters. From the earliest starting point of railroad development at the middle of 

the last century, the issue of vibrations of bridges under moving loads has pulled in numerous researchers.  No 

bibliographic account of previous work on this subject will be given here. Here. Some selective new papers [1, 

2,3,4,5,6,7] are quoted, providing additional information on the issue. 

------------------------------------------------------------------------ 

* Corresponding author.  

http://asrjetsjournal.org/


American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 332-348 

333 

 

A relatively new computational method called the Adomian Decomposition Method (ADM) [8,9,10,11,12,13] is 

applied in this paper to examine the problem of vibration for a cracked beam under moving load. The ADM is a 

powerful and useful way to solve linear and nonlinear differential equations. 

The aim of the ADM is to find the linear and nonlinear, ordinary, or partial differential equation solution 

without relying on small parameters such as the example of the perturbation method. By using the ADM as a 

sum of an infinite series and a quick convergence to a particular solution, the solution is considered [9,10]. The 

main advantages of ADM are computational flexibility and do not require any linearization or assumptions of 

smallness. The ADM has been recently applied to the problem of vibration of structural and mechanical 

systems[9,10,11,12,13] . The ADM is a well-known method of systematic solution of linear or non-linear 

operator equations and deterministic or stochastic operators, including normal differential equations, partial 

differential equations, integrated equations, integrated differential equations, etc. 

The ADM offers an effective methodology for computational methods and numerical simulations in the applied 

sciences and engineering, with real-world applications. It allows us, without unphysically restrictive 

assumptions, to solve both nonlinear initial value problems and boundary value problems, including the need for 

linearization, disturbance, ad hoc assumptions, divination of the initial term or set of basic functions, etc 

[14,15,16]. Using the ADM, the governing differential equation for each segment of the cracked beam becomes 

a recursive algebraic equation. The boundary conditions and conditions of continuity at the crack position 

become simple algebraic frequency equations which are appropriate for symbolic computation. In addition, after 

some basic algebraic operations on these frequency equations, we can simultaneously obtain the natural 

frequency and the corresponding mode shape solution of the closed-form series. in literature has been used this 

method for free vibration analysis, only in one research, Bilik and Karaçay [17] applied Modified Adomian 

Decomposition Method in order to analysis the vibration response of intact bridge subjected to a constant load.  

2. Theoretical Model Using ADM 

The ADM is an efficient method for determining the analytical solution of a large class of vibrational systems 

without linearization or weak assumptions of nonlinearity, which can result in a huge numerical estimation. 

Therefore, it is physically more accurate and practical. Although the solution obtained by the ADM is an infinite 

series, a n-term approach usually serves as a practical and suitable solution. In addition, an effective solution 

with very low values of n is often obtained[18] The solution can be improved for numerical purposes by 

considering further terms, and if it exists, the answer will be the exact solution to the problem. As discussed 

earlier, the ADM provides the solution to dynamic problems in a rapidly convergent series with easily 

computable terms. Because the methodology is addressed perfectly in Refs [18,19], the details of this paper will 

not be discussed. The ADM is used to solve the dynamic governing equation of an SSCB beam carrying  a 

moving load based on the theory of Euler – Bernoulli beams with general elastic boundary conditions and 

conditions of kinematics. Consider an SSCB with a moving load shown in Fig.1 with a single-sided transverse 

crack whose depth is 𝑎, The crack is in the 𝑥! position. where 𝑥" and 𝑥# represent end points. The width, depth 

and length of the beam are, respectively, b, h and l. A load F with constant velocity is moving on the beam from 

left edge of the beam to its right end.  



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 332-348 

334 

 

 

Figure 1: A simply supported cracked beam under moving load 

To include the effect of crack, a massless elastic rotational spring is to connect two segments of the Euler-

Bernoulli beam. This spring is the flexibility due to the crack's compliance. Euler-Bernoulli beam theory 

suggests that we can mention the equation of motion for forced vibration of both portions as follows:  

𝐸𝐼
𝜕4𝑤𝑟(𝑥, 𝑡)

𝜕𝑥4
+ 𝐴

𝜕2𝑤𝑟(𝑥, 𝑡)

𝜕𝑡2
= �̅�𝛿(𝑥 − 𝑣𝑡),           𝑥𝑟−1 < 𝑥 < 𝑥𝑟      , 𝑟 = 1,2 (1) 

Where r = 1 and 2 denoted the portion of the beam on the left of the crack and the portion of the beam on the 

right side of the crack respectively, E is Young’s modulus of the beam material, I is the moment of inertia of the 

beam’s cross-section, 𝜌 is the density of material, A is the cross-sectional area of the beam, �̅� is the moving load 

and 𝛿(𝑥 − v𝑡) denoted the Dirac delta function. 

The boundary conditions requirement of a SSCB case are 

𝑤1(0) = 𝑤2(𝐿) = 0   ,
𝑑2𝑤1(0)

𝑑𝑥2
=

𝑑2𝑤2(𝐿)

𝑑𝑥2
= 0   (2) 

While the kinematics conditions are given by 

    𝑤1(𝐿𝑐−) = 𝑤2(𝐿𝑐+),   ,
𝑑2𝑤1(𝐿𝑐−)

𝑑𝑥2
=

𝑑2𝑤2(𝐿𝑐+)

𝑑𝑥2
  𝑎𝑛𝑑 

𝑑3𝑤1(𝐿𝑐−)

𝑑𝑥3
=

𝑑3𝑤2(𝐿𝑐+)

𝑑𝑥3
 (3) 

Where the 𝐿𝑐− and 𝐿𝑐+ symbols have indicated the positions just before and after the crack. 

The crack is Supposed to be completely opened during the vibration phase, and the impact of the crack can only 

be seen at the position of the crack. The crack's existence induces discontinuity or shift in the beam's slope at the 

crack's position. that can be expressed as[20] 

∆Θ = 𝐶𝑚𝑀 (4) 

where 𝑀 and 𝐶𝑚 are the bending moment transmitted by the cracked section and the flexibility constant 

respectively. 

Here 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 332-348 

335 

 

𝐶𝑚 =
ℎ

𝐸𝐼
𝑄(𝛼,cross section geometry ) (5) 

here 𝛼 is the amount of crack depth indicated by 

𝛼 =
𝑎

ℎ
 (6) 

The 𝑄(𝛼) function, which is the configuration function for the opening region of the crack, can be evaluated by 

fracture mechanics as[20][21] 

𝑄(𝛼) = 2 (
𝛼

1 − 𝛼
)

2

[5.93 − 19.69𝛼 + 37.14𝛼2 − 35.84𝛼3 + 13.12𝛼4] (7) 

We rewrite Eq. (4) as 

∆Θ =
𝑑𝑤2(𝐿𝑐+)

𝑑𝑥
−

𝑑𝑤1(𝐿𝑐−)

𝑑𝑥
= ℎ𝑄(𝛼)

𝑑2𝑤2(𝐿𝑐+)

𝑑𝑥2
 (8) 

A series solution of Eq. (1) can be sought in the form 

According to modal analysis approach (for harmonic free vibration), the 𝑤𝑟(𝑥, 𝑡) can be separated in space and 

time: 

𝑤𝑟(𝑥, 𝑡) = 𝜑𝑟(𝑥)𝑒𝑖𝜔𝑡 , 𝑖 = √−1 (9) 

Substituting the Eq.  (9) into the Eq. (1) and the separating time t and space x variables, an ordinary differential 

equation can be obtained for two segments of the cracked beam. 

𝑑4𝜑𝑟(𝑥)

𝑑𝑥4
− Ψ4𝜑𝑟(𝑥) = 0,             𝑥𝑟−1 < 𝑥 < 𝑥𝑟 ,                         𝑟 = 1,2 (10) 

And 

 Ψ4 =
𝐴𝜔2

𝐸𝐼
 (11) 

Where  𝜑(𝑟)(𝑥) is the cracked beam eigenfunctions for the region of the beam before and after the crack and  𝜔 

is the normal frequency. 

To derive the vibration mode shapes of cracked beam the ADM is applied to Eq. (10). First Eq. (10) is rewritten 

as an operator form[14] 

𝐻𝑥[𝜑𝑟(𝑥)] − Ψ4𝜑𝑟(𝑥) = 0             𝑥𝑟−1 < 𝑥 < 𝑥𝑟 , 𝑟 = 1,2 (12) 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 332-348 

336 

 

Where 𝐻𝑥 =
𝑑4

𝑑𝑥4 ,the inverse operator of 𝐻𝑥 is then a 4-fold integral operator defined by  

𝐻𝑥
−1 = ∫ ∫ ∫ ∫ (⋅)𝑑𝑥𝑑𝑥𝑑𝑥𝑑𝑥

𝑥

0

𝑥

0

𝑥

0

𝑥

0

 (13) 

Appling 𝐻𝑥
−1 on both sides of Eq. (12) yields  

𝜑𝑟(𝑥) = 𝜑𝑟(0) +
𝑥

1!
𝜑𝑟

(1)(0) +
𝑥2

2!
𝜑𝑟

(2)(0) +
𝑥3

3!
𝜑𝑟

(3)(0) + 𝛹4𝐻𝑥
−1[𝜑𝑟(𝑥)], 𝑥𝑟−1 < 𝑥 < 𝑥𝑟 (14) 

where 𝜑𝑟
(𝑖)

  denotes the ith-order derivative with respect to x and sets 𝜑𝑟(𝑥) =  𝜑𝑟
(0)(𝑥)                                                                                                                        

 Based on the ADM[22][2] , the solution of the vibration mode function can be expressed as a series:  

𝜑𝑟(𝑥) = ∑ 𝜑𝑟
[𝑗](𝑥)

∞

𝑗=0

 ,       𝑥𝑟−1 < 𝑥 < 𝑥𝑟 , 𝑟 = 1,2  (15) 

Substituting Eq. (15) into Eq. (14) by assuming  𝜑𝑟
[0](𝑥) = 𝜑𝑟(0) + 𝜑𝑟

(1)(0)𝑥 +
𝑥2

2!
𝜑𝑟

(2)(0) +
𝑥3

3!
𝜑𝑟

(3)(0) Gives 

∑𝜑𝑟
[𝑗](𝑥) = 𝜑𝑟

[0]
+ 𝐻𝑥

−1 [𝛹4 ∑ 𝜑𝑟
[𝑗](𝑥)

∞

𝑗=0

]

∞

𝑗=0

 ,       𝑥𝑟−1 < 𝑥 < 𝑥𝑟 , 𝑟 = 1,2  (16) 

Consequently, for each term of the series, it can be written as: 

𝜑𝑟
[0](𝑥) = ∑

𝑥𝑢

𝑢!
𝜑𝑟

(𝑢)(0)

3

𝑢=0

, (17) 

as the initial term of the decomposition, and 

𝜑𝑟
[𝑗](𝑥) = Ψ4𝐻𝑥

−1[𝜑𝑟
[𝑗−1](𝑥)],   𝑗 ≥ 1 ,       𝑥𝑟−1 < 𝑥 < 𝑥𝑟 , 𝑟 = 1,2       (18) 

As the recurrence relation of the decomposition. Consequently, all components of the decomposition can be 

identified and evaluated from Eq. (18). This is, by substituting Eq. (17) into Eq. (18), one can obtain 

𝜑𝑟
[1](𝑥) = 𝛹4𝐻𝑥

−1[𝜑𝑟
[0](𝑥)] =  𝛹4 ∑

𝑥4+𝑢

(4 + 𝑢)!
𝜑𝑟

(𝑢)(0),

3

𝑢=0

 

 

(19) 

𝜑𝑟
[2](𝑥) = 𝛹4𝐻𝑥

−1[𝜑𝑟
[1](𝑥)] =  𝛹8 ∑

𝑥8+𝑢

(8 + 𝑢)!
𝜑𝑟

(𝑢)(0),

3

𝑢=0

 (20) 

 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 332-348 

337 

 

And 

𝜑𝑟
[𝑗](𝑥) = 𝛹4𝑗𝐻𝑥

−1[𝜑𝑟
[𝑗−1](𝑥)] =  𝛹4𝑗 ∑

𝑥4𝑗+𝑢

(4𝑗 + 𝑢)!
𝜑𝑟

(𝑢)(0),

3

𝑢=0

𝑗 ≥ 1 (21) 

In practice, the solution will be the n-term approximation  

𝜑𝑟
𝑛(𝑥) = ∑ 𝜑𝑟

[𝑗](𝑥) =

𝑛−1

𝑗=0

∑𝛹4𝑗

𝐽−1

𝑗=0

∑
𝑥4𝑗+𝑛

(4𝑗 + 𝑛)!

3

𝑢=0

𝜑𝑟
(𝑢)(0) (22) 

and from the convergence of ADM[22] , 𝜑𝑟
𝑛 approaches 𝜑𝑟

[𝑛](𝑥) as 𝑛 → ∞ 

𝜑𝑟(𝑥) = lim
𝑛→∞

𝜑𝑟
𝑛(𝑥) = ∑𝜑𝑟

[𝑗](𝑥)

∞

𝑗=0

 (23) 

For simplicity, the constants are set as 𝑎𝑟 = 𝜑𝑟(0), 𝑏𝑟 = 𝜑𝑟
(1)(0), 𝑐𝑟 = 𝜑𝑟

(2)(0), 𝑑𝑟 = 𝜑𝑟
(3)(0) 

 here 𝑎𝑟 , 𝑏𝑟, 𝑐𝑟 and 𝑑𝑟 are constants associated with the r-th segments (r =1,2). 

The final vibration mode shape function has been shown in Eq. (23). One can consider as many terms as needed 

to obtain a more accurate deflection function. If the series converges, the j-term partial sum will be the 

approximate closed-form solution of the differential equation[22] 

By substituting the final vibration mode shape function into the four boundary conditions and kinematics 

conditions of the cracked beam, a system of eight equations with unknown constants is given as 

𝑇𝑠1
𝑛 (Ψ𝑛)𝑎𝑟 + 𝑇𝑠2

𝑛 (Ψ𝑛)𝑏𝑟 + 𝑇𝑠3
𝑛 (Ψ𝑛)𝑐𝑟 + 𝑇𝑠4

𝑛 (Ψ𝑛)𝑑𝑟 = 0 (24) 

Where 𝑇𝑠1
𝑛 (Ψ𝑛),  𝑇𝑠2

𝑛 (Ψ𝑛),  𝑇𝑠3
𝑛 (Ψ𝑛) and 𝑇𝑠4

𝑛 (Ψ𝑛) are the coefficients of unknown constants 𝑎𝑟 , 𝑏𝑟, 𝑐𝑟, and 𝑑𝑟 

corresponding to the n-term solution series as a function of 𝛹𝑛. Also, s changes from one to four with respect to 

each boundary condition.  

These constants in the second segment (𝑎2, 𝑏2, 𝑐2 and 𝑑2) are related to those in the first segment (𝑎1, 𝑏1, 𝑐1 and 

𝑑1)  through the compatibility conditions in Eqs. (3) and (4) and cab be expressed as  

{

𝑎2

𝑏2

𝑐2

𝑑2

} = [
𝑧11

𝑛 (Ψ𝑛) ⋯ 𝑧14
𝑛 (Ψ𝑛)

⋮ ⋱ ⋮
𝑧41

𝑛 (Ψ𝑛) ⋯ 𝑧44
𝑛 (Ψ𝑛)

] {

𝑎1

𝑏1

𝑐1

𝑑1

} = 𝑍4×4 {

𝑎1

𝑏1

𝑐1

𝑑1

} (25𝑎) 

Where 𝑍4×4 is a 4 × 4 transfer matrix which depends on eigenvalue Ψ𝑗 

After applying the boundary conditions, the following equation can be obtained as: 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 332-348 

338 

 

{
0
0
} = 𝑇2×4 {

𝑎2

𝑏2

𝑐2

𝑑2

} = 𝑇2×4 𝑍4×4 {

𝑎1

𝑏1

𝑐1

𝑑1

} = 𝑄2×4 {

𝑎1

𝑏1

𝑐1

𝑑1

} (25𝑏) 

Where 

where    𝑇2×4 = [
𝑡11
𝑛 (Ψ𝑛)

𝑡21
𝑛 (Ψ𝑛)

    
𝑡12
𝑛 (Ψ𝑛)

𝑇22
𝑛 (Ψ𝑛)

    
𝑡13
𝑛 (Ψ𝑛)

𝑡23
𝑛 (Ψ𝑛)

    
𝑡14
𝑛 (Ψ𝑛)

𝑡24
𝑛 (Ψ𝑛)

] (25𝑐) 

 

𝑄2×4 = 𝑇2×4 𝑍4×4 = [
𝑞11

𝑛 (Ψ𝑛)

𝑞21
𝑛 (Ψ𝑛)

    
𝑞12

𝑛 (Ψ𝑛)

𝑞22
𝑛 (Ψ𝑛)

    
𝑞13

𝑛 (Ψ𝑛)

𝑞23
𝑛 (Ψ𝑛)

    
𝑞14

𝑛 (Ψ𝑛)

𝑞24
𝑛 (Ψ𝑛)

] (25𝑑) 

Thus, the existence of nontrivial solution requires 

|
𝑞11

𝑛 (𝛹𝑛) 𝑞13
𝑛 (𝛹𝑛)

𝑞21
𝑛 (𝛹𝑛) 𝑞23

𝑛 (𝛹𝑛)
| = 0 (26) 

Equation (26) leads to the eigenvalues equation solving 𝛹𝑛, which is the n-th eigenvalue related with the n-th 

natural frequency of the beam. Substituting 𝛹𝑛 into Eq. (26) and choosing an arbitrary constant, other three 

unknown constants can be evaluated. Then Eq. (23) gives the vibration mode shapes function for both span 

segments (the region before and after crack).  

2. Forced responses 

The original equation of motion (Eq. (1)) can be expressed as  

𝜕4𝑤𝑟(𝑥, 𝑡)

𝜕𝑥4
+
𝐴

𝐸𝐼

𝜕2𝑤𝑟(𝑥, 𝑡)

𝜕𝑡2
=

�̅�

𝐸𝐼
𝛿(𝑥 − v𝑡),         𝑟 = 1,2 (27) 

Using the modal expansion theory, the forced response 𝑤𝑟(𝑥, 𝑡) can be expressed as 

𝑤𝑟(𝑥, 𝑡) = ∑𝜑𝑟
𝑛(𝑥) 

𝑛

𝑗=1

𝜂𝑟
𝑛(𝑡),                𝑟 = 1,2 (28) 

where 𝜑𝑟
𝑛(𝑥) are eigenfunctions of the cracked beam and which are  obtained from the above section (section 

2),𝜂𝑟
𝑛(𝑡) are generalized coordinates and n is  the number of terms used to approximate the solution. Substitute 

Eq. (28) into Eq. (27), multiplying by 𝜑𝑟(𝑥), and integrating from 0 to l leads to 

𝑑2𝜂𝑟(𝑡)

𝑑𝑡2
+ 𝜔2𝜂𝑟(𝑡) = 𝐹𝑠𝜑𝑟(v𝑡),           𝑟 = 1,2   (29) 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 332-348 

339 

 

where             𝐹𝑠 =
�̅�

𝜌𝐴
 

To derive the generalized coordinates of cracked beam the ADM is applied to Eq. (29). First Eq. (29) is 

rewritten as an operator form. 

𝐺𝑡[𝜂𝑟(𝑡)] + 𝜔2𝜂𝑟(𝑡) = 𝐹𝑠𝜑𝑟(v𝑡),        𝑟 = 1,2             (30) 

Where 𝐺𝑡 =
𝑑4

𝑑𝑡4 ,the inverse operator of 𝐺𝑡
−1 is then a 2-fold integral operator defined by  

𝐺𝑡
−1 = ∫ ∫ (⋅)𝑑𝑡𝑑𝑡 

𝑡

0

𝑡

0

 (31) 

Appling 𝐺𝑡
−1 on both sides of Eq. (30) yields  

𝜂𝑟(𝑡) = 𝜂𝑟(0) + 𝑡𝜂𝑟
(1)(0) − 𝐺𝑡

−1[𝜔𝑗
2𝜂𝑟(𝑡) − 𝐹𝑠𝜑𝑟(v𝑡)] (32) 

Based on the ADM, the solution of the vibration mode function can be expressed as a series: 

𝜂𝑟
𝑛(𝑡) = ∑ 𝜂𝑟

[𝑗](𝑡)

∞

𝑗=0

   (33) 

Consequently, for each term of the series, it can be written as: 

𝜂𝑟
[0](𝑡) = ∑ [

𝑡𝑢

𝑢!
𝜂𝑟

(𝑢)(0)]

1

𝑢=0

+ 𝐺𝑡
−1[𝐹𝑠𝜑𝑟(v𝑡)] (34) 

Substituting Eq. (22) into Eq. (34) gives 

𝜂𝑟
[0](𝑡) = ∑ [

𝑡𝑢

𝑢!
𝜂𝑟

(𝑢)(0)]

1

𝑢=0

+ 𝐹𝑠𝐺𝑡
−1 [∑𝛹4𝑗 ∑

(v𝑡)4𝑗+𝑛

(4𝑗 + 𝑛)!

3

𝑢=0

𝜑𝑟
(𝑢)(0)

𝐽−1

𝑗=0

] 

= ∑ [
𝑡𝑢

𝑢!
𝜂𝑟

(𝑢)(0)]

1

𝑢=0

+ 𝐹𝑠 [∑ 𝛹4𝑗

𝐽−1

𝑗=0

∑
v4𝑗+𝑛. 𝑡4𝑗+𝑛+2

(4𝑗 + 𝑛 + 2)!

3

𝑢=0

𝜑𝑟
(𝑢)(0)] (35) 

as the initial term of the decomposition, and 

𝜂𝑟
[𝑗](𝑡) = −𝜔2𝑗𝐺𝑡

−1[𝜂𝑟
[𝑗−1](𝑡)],   𝑗 ≥ 1 , 𝑟 = 1,2       (36) 

As the recurrence relation of the decomposition. Consequently, all components of the decomposition can be 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 332-348 

340 

 

identified and evaluated from Eq. (36). This is, by substituting Eq. (35) into Eq. (36), one can obtain 

𝜂𝑟
[1](𝑡) = −𝜔2𝐺𝑡

−1[𝜂𝑟
[0](𝑡)] 

= −𝜔2 [∑ [
𝑡𝑢+2

(𝑢 + 2)!
𝜂𝑟

(𝑢)(0)]

1

𝑢=0

+ 𝐹𝑠 [∑ 𝛹4𝑗

𝐽−1

𝑗=0

∑
v4𝑗+𝑛. 𝑡4𝑗+𝑛+4

(4𝑗 + 𝑛 + 4)!

3

𝑢=0

𝜑𝑟
(𝑢)(0)]], 

 

 

(37) 

 

𝜂𝑟
[2](𝑡) = −𝜔4𝐺𝑡

−1[𝜂𝑟
[1](𝑡)] 

= 𝜔4 [∑ [
𝑡𝑢+4

(𝑢 + 4)!
𝜂𝑟

(𝑢)(0)]

1

𝑢=0

+ 𝐹𝑠 [∑ 𝛹4𝑗

𝐽−1

𝑗=0

∑
v4𝑗+𝑛. 𝑡4𝑗+𝑛+6

(4𝑗 + 𝑛 + 6)!

3

𝑢=0

𝜑𝑟
(𝑢)(0)]], 

 

 

 

(38) 

𝜂𝑟
[3](𝑡) = −𝜔4𝐺𝑡

−1[𝜂𝑟
[2](𝑡)] 

= −𝜔6 [∑ [
𝑡𝑢+6

(𝑢 + 6)!
𝜂𝑟

(𝑢)(0)]

1

𝑢=0

+ 𝐹𝑠 [∑ 𝛹4𝑗

𝐽−1

𝑗=0

∑
v4𝑗+𝑛. 𝑡4𝑗+𝑛+8

(4𝑗 + 𝑛 + 8)!

3

𝑢=0

𝜑𝑟
(𝑢)(0)]], 

 

 

(39) 

and 

𝜂𝑟
[𝑚](𝑡) = −𝜔2𝑚𝐺𝑡

−1[𝜂𝑟
[𝑚−1](𝑡)]= 

−𝜔2𝑚 [∑ [
𝑡𝑢+2𝑚

(𝑢 + 2𝑚)!
𝜂𝑟

(𝑢)(0)]

1

𝑢=0

+ 𝐹𝑠 [∑ 𝛹4𝑗

𝐽−1

𝑗=0

∑
v4𝑗+𝑛 . 𝑡4𝑗+𝑛+2𝑚+2

(4𝑗 + 𝑛 + 2𝑚 + 2)!

3

𝑢=0

𝜑𝑟
(𝑢)(0)]] ,𝑚 ≥ 1 (40) 

In practice, the solution will be the n-term approximation  

𝜂𝑟
𝑛 = ∑ 𝜂𝑟

[𝑚](𝑡)

𝑛−1

𝑚=0

 

= ∑

[
 
 
 
 

−𝜔2𝑚 [∑ [
𝑡𝑢+2𝑚

(𝑢 + 2𝑚)!
𝜂𝑟

(𝑢)(0)]

1

𝑢=0

+ 𝐹𝑠 [∑𝛹4𝑗

𝐽−1

𝑗=0

∑
v4𝑗+𝑛. 𝑡4𝑗+𝑛+2𝑚+2

(4𝑗 + 𝑛 + 2𝑚 + 2)!

3

𝑢=0

𝜑𝑟
(𝑢)(0)]]

]
 
 
 
 𝑛−1

𝑚=0

 , 𝑚 ≥ 1 

 

 

 

 

(41) 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 332-348 

341 

 

and from the convergence of ADM [ ], 𝜂𝑟
𝑛 approaches 𝜂𝑟

[𝑚](𝑡) as 𝑛 → ∞ 

𝜂𝑟(𝑡) = lim
𝑛→∞

𝜂𝑟
𝑛(𝑡) = ∑ 𝜂𝑟

[𝑚](𝑡)

∞

𝑚=0

,    𝑟 = 1,2 (42) 

The final generalized coordinates function for two segments (the region before and after the crack) has been 

shown in Eq. (42) One can consider as many terms as needed to obtain a more accurate deflection function. If 

the series converges, the j-term partial sum will be the approximate closed-form solution of the differential 

equation. The constants have been found by applying the four boundary conditions and kinematics conditions of 

the cracked beam. 

The dynamic response of cracked beam under moving load can be found from substituting Eq. (23) and Eq. (42) 

into Eq. (28). 

4. Numerical results and discussion 

4.1. Validation 

In this subsection, the accuracy and efficiency of ADM solutions are investigated through a comparison with 

those obtained from FEM. The ADM model developed in this article is applied to a SSCB under a moving load. 

The numerical values of geometrical and material parameters of the Euler Bernoulli beam are: length L = 5(m), 

Young’s modulus E = 2.1×1011(Gpa), cross-sectional second moment of inertia I = 
𝑏×ℎ3

12
(𝑚4),width b = 

0.5(m), height h = 1(m), mass density ρ = 7860(kg/𝑚3) (Mahmoud, 2001). To obtain the dynamic response of 

cracked beams under a moving load, Equations (23), (28) and (42) is used for an ADM analysis, the 

corresponding numerical results are denoted by ADM. On the other hand, for an FEM analysis, the total number 

of finite elements used in the analysis is varied in order to improve the numerical accuracy; and the 

corresponding results are denoted by FEM. The dynamic transverse deflections against time response are 

normalized relative to the value, 
𝐹𝐿3

48𝐸𝐼
, which is the related static deflection was applied to the mid-span beam 

due to load amplitude F. 

4.2 Moving load with constant velocity 

For example, a moveable load with a constant load amplitude F = 2000(N), a constant velocity v = 80(m/s) and 

initial zero conditions 0 are taken into consideration. Fig.2, indicates the normalized deflection for mid-point of 

the beam. In this figure, 
v𝑡

𝐿
  indicate to the normalized position of moving load once over the beam. Fig. 2 shows 

that the results predicted by FEM model would converge to those predicted by the present ADM method 

provided that the total number of elements used in FEM could increase beyond 100. In Fig. 2, it is seen that the 

results of ADM are in good agreement with those of exact solution and closed to FEM solution. This could 

verify the higher accuracy of the present ADM method. The percent proportional error (PPE) is defined in Eq. 

(43) and shows the numerical error of ADM and FEM methods in comparison with exact solution and Table 1 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 332-348 

342 

 

shows this ratio of the error. 

 

PPE(%) =
Exact solution − ADM or FEM response 

Exact solution 
× 100 

(43) 

Although the numerical error would be diminished by increasing the number of elements, it could be seen that 

the efficiency of using ADM is much bigger than that using numerous elements in FEM. Section 4 assumes that 

the initial conditions for numerical examples are zero, whereas the moving load crosses the beam for once and it 

could be a pseudo initial condition for a Euler Bernoulli beam. Pseudo initial speed situation means a pulse 

function that limits Dirac delta function at a not so small-time interval That might mimic the function of 

theoretical impulses but is very close to it. Fig.3 shows a normalized dynamic response for different load speeds 

with ADM and FEM (100 elements) at the mid-span of a cracked beam. As predicted, an increase in load 

velocity will cause the maximum dynamic deflection to correspondingly increase. Fig.4 shows the effect of 

moving load amplitude on the dynamic deflection response at the midpoint of a cracked beam at constant 

velocity v = 70 (m/s) under moving load. It can be seen that by increasing the load amplitude from 1000 (N) to 

2000 (N), the mid-point deflection of cracked beam is increased. 

Fig.5 shows the difference between the mid-point deflection of the cracked beam and the same one for an intact 

beam (uncracked beam). It is seen that the maximum mid-span deflection of the beam with open crack is greater 

than that of the healthy beam. In addition, the time response of mid-span deflection causes a horizontal shift-in-

time. Such vertical and horizontal shifts in the response time signals will occur because the stiffness of the 

cracked beam is lower than that of the healthy beam. The good agreement between the results from ADM and 

from those of transfer matrix method (Gopalakrishnan and his colleagues 1992) , lead to the utilization of ADM 

for this case study. Maximum normalized relative error deflection (MNRED) is defined in Eq. (44) and shows 

the numerical error of ADM and FEM methods in comparison with the transfer matrix method. These numerical 

error ratios are given in Table 2. The efficiency of using ADM in comparison with FEM could be seen again in 

this table. 

MNRED(%) =
Transfer matrix response − ADM or FEM response 

response Transfer matrix response 
× 100 

(44) 

The effect on the mid-point deflection of the beam is shown in Fig. 6. In this case, both the vertical and the 

horizontal changes are extended to the higher values of magnitude and time for the beam's midpoint deflection, 

respectively. The horizontal shift-in-time phenomenon demonstrates the cracked beam's fundamental period-

lengthening event corresponding to the deficiency in stiffness. This is more pronounced with higher levels of 

crack depth. In Eq. 45, The factor of crack magnification (FCM) displays the numerical ratio of the average 

deflection between cracked and intact beams subjected to moving load with constant velocity, v =70 (m/s). 

Table 3 shows these ADM numerical results.  



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 332-348 

343 

 

FCM =
maximum mid − point cracked beam deflection

maximum mid − point intact beam deflection
× 100 (45) 

The effect of an open crack location on the dynamic response of a SSCB under moving load with constant 

velocity, v =70 (m/s), is investigated in Fig. 7 by ADM. It seems that getting the crack location away from the 

beam's midpoint would decrease the midpoint deflection of the beam until it becomes nil at the beam's two ends. 

 

(a) 

 

(b) 

Figure 2:  (a) Comparison between the normalized deflection at mid-span of a SSCB subjected to moving load 

with constant velocity v=80 (m/ s), by using ADM, FEM and exact solution (80 and 100 elements) (b) zoom-in 

subfigure (a) shows the comparison between these three methods 

Table 1: Difference between maximum normalized defection at the mid-point of a simply- supported beam for 

ADM, FEM, and exact solution, v = 80 (m/s) 

Method Maximum normalized deflection of mid-span PPE (%) 

Exact solution -1.0361 - 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 332-348 

344 

 

ADM 

FEM (80 elements) 

FEM (100 elements) 

 

-1.0860 

-1.0836 

-1.0843 

 

0.0092 

0.2302 

0.1657 

 

 

(a) 

 

(b) 

Figure 3:  (a) moving load velocity (70, 80, 90(m/s)) effects on the normalized deflection at mid-span of a 

SSCB by using ADM and FEM (100 elements) (b) zoom-in subfigure (a) shows the comparison between these 

two method 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 332-348 

345 

 

 

Figure 4:  The effect of the amplitude of the moving load on the mid-span deflection of an SSCB undergoing a 

moving load with constant velocity v=70 (m/s) and α=0.25 using ADM 

 

(a) 

 

(b) 

Figure 5:  (a) Comparison between the normalized deflection at mid-span of a simply supported cracked and 

intact beam subjected to moving load with constant velocity v=70 (m/ s), by using Transfer matrix, ADM and 

FEM (60 and 100 elements) (b) zoom-in subfigure (a) shows the comparison between these three methods 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 332-348 

346 

 

 

Figure 6:  Crack depth ratio effects on the deflection at mid-span of a SSCB subjected to moving load with 

constant velocity(v=70m/s) by using ADM 

Table 2: Difference between maximum normalized defection in mid-point of simply supported beam in ADM, 

FEM and transfer matrix method, v=70 (m/s) and 𝛼 = 0.25  

 

Method Maximum normalized deflection of mid-span MNRED (%) 

Transfer matrix 

ADM 

FEM (30 elements) 

FEM (60 elements) 

FEM (90 elements) 

 

-1.4269 

-1.4268 

-1.4128 

-1.4139 

-1.4166 

 

1 

0.007 

0.9882 

0.9111 

0.7218 

 

 

Figure 7:  Crack location effects on the normalized deflection at mid-span of a SSCB subjected to moving load 

with constant velocity(v=70m/s) and α=0.25, by using ADM 

Table 3: Effect of crack size on the ratio between maximum normalized deflections of cracked and healthy 

beams under moving load with constant velocity, v=70 (m/s), by using ADM 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 88, No1, pp 332-348 

347 

 

𝛼 Maximum normalized deflection of mid-span FCM 

0 

0.15 

0.25 

0.35 

0.45 

0.55 

-1.061 

-1.1726 

-1.4268 

-1.8977 

-2.537 

-3.6196 

1 

1.105 

1.345 

1.789 

2.391 

3.412 

4. Conclusions 

In this article, the vibrations of cracked Euler Bernoulli beams under moving load are analyzed using ADM. The 

open crack was modeled by a massless rotational spring. The main characteristics of ADM were its higher 

accuracy and its superiority comparison with other methods, such as FEM. It was shown that the present ADM 

could provide extremely accurate solutions, as compared with those obtained from an FEM having even one 

hundred elements, the following results were observed: Vertical and horizontal shifts of the (both healthy and 

cracked) beam mid-point deflection to the higher values were a result of 

• increase of load movement speed. (Move load quicker) 

• Increase the moving load amplitude. (Greater moving load) 

• Increased the height ratio of the crack depth to a beam. (Cracks deeper) 

The reduction of the maximum deflection at the midpoint of the cracked Euler Bernoulli beam could be the 

result of shifting crack location away from the center. 

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