# Approximate Solutions of the Jet Engine Vibration Equation by the Homotopy Perturbation Method

Authors: M. Abul Kawser; Md Abdul Alim
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, we consider the jet engine vibration system. With the help of the homotopy perturbation method (HPM), we solve the jet engine vibration (JEV) equation, which is a second-order nonlinear differential equation in the presence of damping and external forces. We compare the results with the corresponding numerical method to show the efficiency and reliability of this method. The findings demonstrate how well HPM works as a solution to these concerns, and it is anticipated that HPM will be used in a variety of new problems.


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## Body

## 1. Introduction

Nonlinearity characterizes the majority of scientific issues and physical events. Finding exact analytical answers for these kinds of problems can be challenging, with the exception of a small number of cases. As a result, efforts have been undertaken to create novel methods for getting analytical answers that are reasonably close to the precise solutions [1, 2]. The previous years, several modern methods have been used to find approximate solutions of nonlinear problems, that is the inverse scattering method (ISM) [3], homogeneous balance method (HBM) [4], Hirota’s bilinear method (HBM) [5, 6], the natural decomposition method (NDM) [7], Li-He’s modified homotopy perturbation method (MHPM) [8], the reduced differential transform method (RDTM) [9, 10], the enhanced homotopy perturbation method (EHPM) [11], Adomian decomposition method (ADM) [12], the variational iteration method (VIM) [13, 14], the homotopy perturbation method (HPM) [15, 16], etc. The HPM provides highly accurate solutions to nonlinear problems, which can easily be understood in comparison to the numerical techniques.He et al. developed the homotopy perturbation method (HPM). It does not require linearization process and is a powerful, effective, and efficient method for finding solutions to linear and nonlinear differential equations. The fundamental weakness of the classic perturbation method which is a small parameter assumption has been overcome by the HPM. The embedding parameterp∈0,1 is changed from 0 to 1 and utilized as an expanding parameter for the tiny parameter in the perturbation method in the homotopy perturbation approach, which results in a homotopy equation. The produced homotopy issue is transformed into an easy-to-solve linear equation when p=0, and when p=1 then back into the original equation. As a result, the solution process gradually transforms a linear equation into a nonlinear equation, and when p=1, it converges to the precise answer. Due to the fact that it is independent of an extremely small equation parameter, the solution holds true for both weak and strong nonlinear instances. He applied successfully this method to solve a variety of nonlinear equations in science and engineering [17–19]. This HPM has already been effectively employed by numerous researchers to resolve numerous linear and nonlinear issues such as solving heat radiation equations [20], KdV PDE studied by Ganji and Rafei [21], nonconservative oscillators [22], Laplace equation [23], time fractional partial integro-differential equations [24], wave equations [25], heat conduction and convection equations [26], integral equations [27], couple spring mass system [28] and is also applicable in other areas [29–31]. He [32] applied HPM for solving Blasius differential equation [33] and nonlinear boundary value problems. Also, Anjum et al. [34, 35] summarized and perfected HPM in a series of papers, and demonstrated its effectiveness through nonlinear differential equations. In majority of cases, the HPM can present a series of solutions very rapidly. It usually takes a few numbers of iterations to get a very accurate solution. The purpose of this article is to develop the HPM and use it to solve nonlinear ordinary differential equations for jet engine vibration [36].


## 2. Homotopy Perturbation Method (HPM)

To describe the homotopy perturbation method, we take into consideration the following form of equation:(1)Gvx−fr=0,r∈Ω.As for the boundary circumstances,(2)Bv,dvdx=0,r∈Γ,where G,L,andΓ are the general differential operator, boundary operator, and the boundary of the domain Ω, respectively; and fr is a known analytical function. Now, we divide the differential operator G into two parts of linear L and nonlinear N, and then (1) provides(3)Nvx+Lvx−fr=0.The following homotopy is created by the HPM,ur,p:Ω×0,1⟶R, which satisfies(4)Hu,p=1−pLu−Lv0+pGu−fr=0.(4) can be written as follows:(5)Hu,p=Lu−Lv0+pLv0+pNu−fr=0.Here,v0 and p∈0,1 are an initial approximation and embedding parameters of (1) and satisfies the boundary conditions. We can write the following from (4):(6)Hu,0=Lu−Lv0=0,andHu,1=Gu−fr=0.Here, the changes inp from 0 to 1 are those of ur,p from v0r to vr. This system is referred to as the deformation in topological space, and Lu−Lv0 and Gu−fr are called homotopy. It can be assumed that the first embedded parameter p can be utilized as a tiny parameter and that the solution to (4) can be characterized as a power series in p in accordance with the theory of homotopy perturbations:(7)u=u0+pu1+p2u2+p3u3+⋯⋯.Assumingp=1 in (7), the (1) has the following solution:(8)v=limp⟶1u=u0+u1+u2+u3+⋯⋯.The nonlinear operatorGu is operating the convergent rate. The (8) is convergence in most cases.


## 3. The Jet Engine Vibration Model

The jet engine system shown in Figure1(a) is hidden in the wings of an aircraft. In order to simulate how a jet engine moves horizontally, an elastic beam is used to support a rigid body. The moment of inertia around the center of gravity axis C is J, and the mass of the engine is m. The elastic beam is also modeled by considering it as a massless rod that is hinged at A, and the rotating spring constant κ provides exactly a restoring torque κθ, Figure 1(b) shows the angle θ created by the rod and the vertical line. For small rotations, that is θ≪1, set the jet engine motion equation with θ to find the frequency of natural vibration.Figure 1
The jet engine horizontal vibration model.
(a)(b)In the system, the hinge revolves atA. The jet engine’s moment of inertia around its axis of mass C is J. Using the parallel axis theorem, the jet engine’s moment of inertia around axis A is as follows:(9)JA=J+mL2.The jet engine carries gravitymg. We remove the elbow at A and replace it with two parts RAx and RAy, which are components of the reaction force. Since the angle θ and the angular acceleration θ¨ of the system are anticlockwise, the rotating spring provides a clockwise reset torque κθ and the moment of inertia JAθ¨, respectively. Using D’Alembert’s principle, from the Figure 1(b) the free body is in dynamic equilibrium.Hence,JAθ¨+κθ+mgLsinθ=0,(10)i.e.J+mL2Lθ¨+κLθ+mgsinθ=0.The most typical kind of dampening encountered throughout the introductory course is the viscous damping type. The damping force is proportional to the speed. We add viscous damping term−2δθ˙ to the (10), where δ is the damping coefficient. Then, the jet engine vibration equation is as follows:(11)J+mL2Lθ¨+κLθ+mgsinθ=−2δθ˙,i.e.J+mL2Lθ¨+2δθ˙+κLθ+mgsinθ=0.Adding periodic external forceFcosωt to the model, the model takes the following form:(12)J+mL2Lθ¨+2δθ˙+κLθ+mgsinθ−Fcosωt=0.Expandingsinθ in terms of θ and ignoring the terms higher than θ3 as θ is very small, we attain the nonlinear jet engine vibration equation of the system as follows:(13)J+mL2Lθ¨+2δθ˙+κL+mgθ−16mgθ3−Fcosωt=0.


## 4. Solution to the Problem

According to (4), the homotopy next to (13) is considered to be given by the following equation:(14)J+mL2Lθ¨+2δθ˙+κL+mgθ−p16mgθ3+Fcosωt=0.As a power series inp, consider the solution of (13)(15)θ=θ0+pθ1+p2θ2+⋯⋯.Therefore, substituting (15) into (14) and setting the coefficient of each power of p to zero, the following set of differential equations can be obtained:(16)p0:J+mL2Lθ¨0++2δθ˙0+κL+mgθ0=0,θ00=a,θ˙00=0,p1:J+mL2Lθ¨1+2δθ˙1+κL+mgθ1−16mgθ03−Fcostω=0,θ10=0,θ˙10=0.Case 1.
In the absence of the periodic external force and damping, that isF=0N,δ=0kgms−1 and choosing J=240kgm2,m=40kg,L=2.7m,κ=4kgm2s−2,andg=9.81ms−2.
With the initial conditionsθ00=a=0.1m and θ˙00=0ms−1.
For these values, the above two equations give the following:(17)θ0=0.01cos1.41t,θ1=0.000005cos1.41t−cos4.24t+0.000088tsin1.41t.
So, the approximate solution of (13) is(18)θ=0.100005cos1.41t−0.000005cos4.24t+0.000088tsin1.41t.Case 2.
In the absence of the periodic external force, that isF=0N, and choosing J=250kgm2,m=50kg,L=2.5m,δ=18kgms−1,κ=5kgm2s−2,andg=9.81ms−2.
With the initial conditions:θ00=a=0.5m and θ˙00=0ms−1.
Consequently, the initial two solutions of (13) take the following structures:(19)θ0=e−0.08t0.5cos1.48t+0.027076sin1.48t,θ1=−e−0.08t0.007171cos1.48t−0.07309sin1.48t+e−0.24t0.007804cos1.48t−0.071842sin1.48t−0.000633cos4.43t−0.000157sin4.43t.
Hence, the solution of (13) becomes(20)θ=e−0.08t+0.492829cos1.48t+0.100166sin1.48t+e−0.24t0.007804cos1.48t−0.071842sin1.48t−0.000633cos4.43t−0.000157sin4.43t.Case 3.
In the absence of the damping, that isδ=0kgms−1 and choosing J=220kgm2,m=45kg,L=2m,κ=3.5kgm2s−2,g=9.81ms−2,F=5N,andω=30°.
With the initial conditions,θ00=a=0.03m and θ˙00=0ms−1.
For the above values, the following are the first two components for (13):(21)θ0=0.03cos1.49t,θ1=0.012874cos0.52t−cos1.49t+0.000003tsin1.49t.
Hence, the solution of the (13) takes the following form:(22)θ=0.017126cos1.49t+0.012874cos0.52t+0.000003tsin1.49t.Case 4.
In the presence of the periodic external force and damping, we chooseJ=200kgm2,m=56kg,L=2.25m,δ=6kgms−1,κ=3kgm2s−2,g=9.81ms−2,F=12N,andω=60°.
With the initial conditions,θ00=a=0.15m and θ˙00=0ms−1.
Therefore, the first two solutions of (13) by the homotopy perturbation method are(23)θ0=e−0.03t0.15cos1.6t+0.002617sin1.6t,θ1=0.03803cos1.05t+0.001517sin1.05t−e−0.03t0.038223cos1.6t−0.004386sin1.6t+e−0.08t0.00021cos1.6t−0.006031sin1.6t−0.000017cos4.8t−0.000001sin4.8t.
Thus, the solution of (13) given by the homotopy perturbation method is as follows:(24)θ=0.03803cos1.05t+0.001517sin1.05t+e−0.03t0.111777cos1.6t+0.007003sin1.6t+e−0.08t0.00021cos1.6t−0.006031sin1.6t−0.000017cos4.8t−0.000001sin4.8t.
All the above solutions are shown in Figures2–5 by comparing with the numerical solutions and also to show the accuracy of the results, we have magnified a complete period of each figure. Figures are given as follows:Figure 2
(a) The graphical representation of the nonlinear jet engine vibration system without damping and external driving force and (b) a complete period for Case1.
(a)(b)Figure 3
(a) The graphical representation of the nonlinear jet engine vibration system with damping and without external driving force and (b) a complete period for Case2.
(a)(b)Figure 4
(a) The graphical representation of the nonlinear jet engine vibration system with external driving force and without damping and (b) a complete period for Case3.
(a)(b)Figure 5
(a) The graphical representation of the nonlinear jet engine vibration system with both damping and external driving force and (b) a complete period for Case4.
(a)(b)


## 5. Results and Discussion

The general solution has been found by successfully utilizing the HPM for undamped and damped cases in the absence and presence of external forces of the nonlinear model equation. To test the accuracy of the general solutions, we have graphically compared the solutions for all the cases with the numerical results for different sets of initial values. In order to show how strong the accurateness of the obtained solutions by both methods is, we have shown graphically a complete cycle of a special part of each figure. We have used the realistic values for damping constant, forcing constant, gravitational force, etc., to compare the solution curves with the curves of numerical results for the time period,t = 0 to t = 50. All these comparative graphical representations have been displayed in Figures 2–5, which show nice agreement with the numerical results and strongly support the correctness of our outcomes.


## 6. Conclusion

The jet engine vibration (JEV) system’s nonlinear ordinary differential equation has been effectively solved explicitly and approximately for a number of examples in this study using He’s homotopy perturbation technique (HPM). For this system the HPM gives expected results corresponding to the numerical solutions obtained by “NDSolve” in Mathematica. This method requires less calculation, easier to use, accelerates convergence to the solution and all in all is very simple. According to this research, we can emphatically say that the HPM is actually a powerful analytical method for solving any type of nonlinear problems. We ensure that this method is appropriate and gives very accurate results.



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