Analytical and numerical investigation of jet engine vibration equation using least square homotopy perturbation method

Abstract

This research focuses on solving the nonlinear second-order jet engine vibration equation utilizing a hybrid analytical technique named the least square homotopy perturbation method (LSHPM). The numerical and graphical comparison of the solutions obtained using the homotopy perturbation method (HPM), LSHPM, and the MATLAB built-in solver bvp5c is presented across four distinct cases. Additionally, a comparative analysis between the solutions derived from LSHPM and those reported in previous literature is also presented. The tabular and graphical representation of the solutions, along with the numerical validation through residual error analysis, are given. Furthermore, the convergence analysis of the LSHPM for its stability and solution reliability is provided. The graphical and numerical representation of the residual error analysis reveals that LSHPM exhibits superiority over HPM in terms of rapid convergence and accuracy. The strong agreement between the results obtained from HPM and bvp5c with those of LSHPM demonstrates that LSHPM offers a more efficient, reliable, and fast convergent solution of the initial and boundary value problem.

Keywords

nonlinear oscillations jet engine vibration system LSHPM bvp5c Numerical Results

Introduction

In recent years, the study of nonlinear oscillation has become a critically important and widely studied area in the field of science and engineering due to its complex behavior and wide range of real-world applications. One key application is the formulation and analysis of the jet engine vibration equation. It is a second-order nonlinear ordinary differential equation that describes the vibrational behavior of engine components such as rotors, shafts, and blades under different conditions. It captures the effect of aerodynamic forces, mechanical imbalance, thermal stresses, etc. It enables engineers to predict and analyze critical dynamic phenomena such as resonance. The mathematical modeling of these vibrations provides a valuable insight during the design phase of a jet engine to enhance the structural integrity. Applications of the jet engine vibration equation include prediction of natural frequencies, simulations of blade vibrations, and development of vibration control strategies. By solving these equations, researchers and engineers can understand how design parameters such as mass, length, damping, stiffness, etc. improve engine efficiency, reliability, durability, and safety while minimizing the risk of failure. Kawser et al.¹ used He’s frequency amplitude method to analyze the nonlinear oscillations in the jet engine vibration system. In another study, Kawser et al.² utilized the HPM to determine the approximate solution of the jet engine vibration equation. Kim Q. et al.³ developed a new thermoelastic model to analyze the vibrations and thermal behavior of the system using the higher-order shear deformation theory with the incorporation of isogeometric analysis. Zhou et al.⁴ explored the bandgap characteristics of periodic structures utilizing the finite element method. Qin, S. et al.⁵ conducted in-situ vibration measurements to evaluate the structural condition of an arch bridge constructed with steel tubes filled with concrete. In another study, Qin, S. et al.⁶ explored the inappropriate frequency band division in different civil structures under the high noise and modulation effect using the empirical wavelet transform. Atta et al.⁷ investigated the analytical solution of the time-fractional Newell–Whitehead–Segel equation utilizing the spectral collocation technique. Sedighi et al.⁸ studied the beam vibrations under the preload discontinuity and yielded the accurate function to tackle the nonlinearity utilizing He’s parameter expanding approach. In another study, Sedighi et al.⁹ used the homotopy analysis method to determine the profile of the disc cam by optimizing the dissipated energy.

There are many physical phenomena in mathematical physics, applied mathematics, and biological mathematics modeled in the form of linear and nonlinear differential equations, but various scientific problems show nonlinearity. Nonlinear problems are a crucial element in applied mathematics. In most cases, we cannot find the exact solution of nonlinear problems. The fast advancement in nonlinear science has allowed scientists to come up with a variety of methods, including the Adomian’s Decomposition Method,^10–12 Laplace Adomian Decomposition Method,^13–15 modified homotopy perturbation method (MHPM),^16–18 He’s Frequency Amplitude Method (HFAM),^19,20 Variational Iteration Method,^21–23 power series approach,^24,25 and Linstedt–Poincare Method^26,27 for solving various nonlinear initial and boundary value problems (IBVPs). These above-mentioned techniques lack the capability to compute accurate and precise approximate solutions. In most cases, these methods exhibit divergence behavior when analyzing the residual errors of obtained solutions. Researchers have developed various alternative techniques to overcome these challenges. The LSHPM is a distinctive technique that offers an effective, accurate, and fast-converging solution for solving IBVPs over a finite and semi-infinite domain. Some researchers have tested this technique to solve IBVPs over finite and semi-infinite domains. Qayyum and Oscar²⁸ explored the different linear and nonlinear problems of the second- to seventh-order differential equations using the LSHPM. Pasca et al.²⁹ examined the motion of laminar fluid flow in the blood vessels under the influence of a magnetic field. Rafiq et al.³⁰ used the LSHPM to investigate the free vibrations in the nonlinear oscillation based on the dynamic system that can be fully described by a single independent variable. Oyedepo et al.³¹ solved the fractional integro-differential equation utilizing the LSHPM. The LSHPM was used by Bota and Caruntu³² to determine the solution of the nonlinear Jeffery–Hamel problem under the MHD effects. Qureshi et al.³³ explored the incompressible flow of Casson fluid between the two parallel plates. Tahir et al.³⁴ utilized the LSHPM to study the boundary layer flow of different nanofluids over a horizontally moving flat surface plate. The LSHPM is a hybrid approach that combines the HPM and least square optimizer. This method is used to increase the accuracy and convergence of the solution by reducing the residual error on the specified domain. This property of reducing the residual error of the problems, increases the effectiveness of this technique for solving initial and boundary value problems defined on both finite and semi-infinite intervals.

This research investigates the solution of the jet engine vibration equation. It has been solved utilizing the recently developed technique named the least square homotopy perturbation method. LSHPM is an upgraded version of HPM that provides an iterative framework to determine an approximate analytical solution of the differential equations. The reason why researchers prefer LSHPM is that it does not only deal with linear and nonlinear differential equations but also deals with the fractional differential equations and, in conjunction, a combination of fractional differential equations and integro-differential equations. The LSHPM provides more precise and effective outcomes with minimum iterations. The weaknesses or limitations of the proposed technique are: The LSHPM relies on the HPM solution and the finding of unknown constants. If an incorrect initial guess approximation is used in the HPM solution, then the LSHPM may exhibit divergence behavior.

This paper is organized as follows. First, the mathematical formulation of the jet engine vibration equation along with its initial conditions is given. Next, the basic idea as a summary of the LSHPM and bvp5c methodologies along with their block figures is presented. The convergence analysis of the LSHPM is then discussed. Subsequentlly, the analytical approximation of the proposed problem along with the graphical and numerical comparison and analysis of the solution obtained using the HPM, LSHPM, and MATLAB’s built-in solver bvp5c are provided. Finally, the concluding remarks of the proposed problem solution are presented.

Governing equations

An aircraft’s jet engine system is illustrated in Figure 1. To simulate the horizontal movement of a jet engine as Kawser described in Ref. 2, a rigid body is supported by an elastic beam. The engine has a mass m and its moment of inertia about its center of gravity point C is J. It is also possible to model the elastic beam as a massless rod, hinged at point A, with rotating spring constant k that provides restoring torque kθ. As shown in Figure 1, θ represents the angle between the rod and vertical line. For small angular displacement, that is, |θ|≪ 1 the motion of the jet engine can be described by a jet engine motion equation in terms of θ to determine the system’s natural frequency.

Figure 1.

Physical model of jet engine vibration.

Since the hinge rotates about point A. The moment of inertia of jet engine about its center of gravity axis C is denoted by J. The moment of inertia of jet engine about its axis at point A can be calculated by using the parallel axis theorem. It is represented as follows

J_{A} = J + m L^{2}

(1)

The jet engine experiences a gravitational force of magnitude mg. To analyze the forces at point A, elbow is replaced by its horizontal and vertical reaction force components R_Ax and R_Ay. Given that angular displacement θ and angular acceleration $\ddot{θ}$ are in counterclockwise direction, rotating spring generates a clockwise restoring torque kθ and moment of inertia $J_{A} \ddot{θ}$ . According to the D’Alembert’s principle, the free body is shown in Figure 1 is considered to be in dynamic equilibrium under these forces and torques. It is given as follows

J_{A} \ddot{θ} + k θ + m g L s i n θ = 0

(2)

Equation (2) can be represented as follows

\frac{J + m L^{2}}{L} \ddot{θ} + \frac{k}{L} θ + m g \sin θ = 0

(3)

The most common type of damping is viscous damping, where damping force is directly proportional to speed. To account for this effect, the viscous damping term $- 2 δ \dot{θ}$ is added to equation (3), where δ represents damping coefficient. Incorporating this term, the jet engine vibration equation becomes

\frac{J + m L^{2}}{L} \ddot{θ} + \frac{k}{L} θ + m g \sin θ = - 2 δ \dot{θ}

(4)

It can also be represented as

\frac{J + m L^{2}}{L} \ddot{θ} + 2 δ \dot{θ} + \frac{k}{L} θ + m g \sin θ = 0

(5)

By incorporating the periodic external force Fcosωt in equation (5), the model is modified as follows

\frac{J + m L^{2}}{L} \ddot{θ} + 2 δ \dot{θ} + \frac{k}{L} θ + m g \sin θ - F c o s ω t = 0

(6)

Expanding the Taylor series of sinθ and neglecting terms of order higher than θ³ due to the smallness of θ. The Taylor series of sinθ is truncated after the cubic term θ³. Assuming that θ is very small, that is, |θ|≪ 1. This small angle approximation linearizes the governing equation to some extent while preserving the nonlinearity through θ³. In contrast, this small angle approximation becomes less accurate for larger angular displacement. Therefore, the jet engine vibration equation remains valid for small to moderate oscillations, where angular motion is limited. Hence Sinθ ≈ θ − θ³/6 is a reasonable representation. We obtain the equation for nonlinear jet engine vibration as follows

\frac{J + m L^{2}}{L} \ddot{θ} + 2 δ \dot{θ} + (\frac{k}{L} + m g) θ - \frac{1}{6} m g θ^{3} - F c o s ω t = 0

(7)

with appropriate initial conditions

θ (0) = a, \dot{θ} (0) = 0

Methodology

We apply the recently formulated technique, LSHPM along with bvp5c, to solve the linear and non-linear differential equations.

LSHPM

Consider a non-linear differential equation

M (w) - g (ξ) = 0, ξ \in Ω,

(8)

subject to the boundary conditions

B (w, \frac{\partial w}{\partial n}) = 0, ξ \in Γ .

(9)

In equations (8) and (9), M represents a differential operator, w is the unknown function to be calculated, B denotes a boundary operator, g represents the source term, n is the normal vector to the given boundary, and Γ refers to the boundary of the domain Ω. The equation (8) can also be written as follows

L (w) + N (w) - h (ξ) = 0, ξ \in Ω .

(10)

In equation (10), L and N represent linear and nonlinear operators. We first of all intend to solve equation (10) with the help of the traditional HPM technique as detailed in references 35, 36. In order to accomplish our objective, we construct a homotopic function $H : R^{n} \times [0,1] \to R^{m}$ , which follows the homotopic property

H (w, h) = (1 - h) [L (w) - L (w_{0})] + h [L (w) + N (w) - g (ξ)] = 0, ξ \in Ω, h \in [0,1] .

(11)

Here, h represents the embedding parameter, while initial guess w₀ applies to equation (8), to meet the boundary conditions (9). Clearly, from the equation (11), it is shown that

H (w, 0) = L (w) - L (w_{0}) = 0,

(12)

and

H (w, 1) = L (w) + N (w) - g (ξ) = M (w) - g (ξ) = 0, ξ \in Ω .

(13)

As the unknown parameter h varies from 0 to 1, u₀ gradually coincides u(ξ). The solution of the previously mentioned equation can be presented as a power series in h. It is also demonstrated in previous works.^37,38

w = w_{0} + h w_{1} + h^{2} w_{2} + h^{3} w_{3} + \dots .

(14)

Taking h → 1, then we have

\tilde{w} = \lim_{h \to 1} w = w_{0} + w_{1} + w_{2} + w_{3} + \dots,

(15)

For solving most of the nonlinear differential equations, we find the approximate solution by truncating the series after a few terms. However, the truncated series solution may not give the exact solution or may diverge on extended domains. The least square method is incorporated. Let u_N be the truncated series solution. We find the residual function of equation of (10) by replacing u with u_N, we get

R (ξ) = L (w_{N}) (ξ) + N (w_{N}) (ξ) - g (ξ), ξ \in Ω .

The least square method is incorporated in the HPM truncated series solution w_N. Let $\tilde{U}$ be the LSHPM series solution given by

\tilde{U} (ξ) = \sum_{i = 0}^{N} c_{i} ϕ_{i},

where ϕ₀, ϕ₁, ϕ₂, ⋯ , ϕ_N are linearly independent functions, which are obtained from HPM series solutions and the

c_{i}^{'} s

are the unknown constants. The unknown coefficients

c_{i}^{'} s

of ϕ_i’s are used to manage convergence of solution

\tilde{U}

. we put the unidentified function

\tilde{U}

in the place of approximate solution v in equation (3) to construct the residual function as

\hat{R} (ξ, c_{i}) = L (\tilde{U}) + N (\tilde{U}) - g (ξ), ξ \in Ω .

(16)

Obtaining the residual sum of squares as follows

J (c_{i}) = \int_{Ω} {\hat{R}}^{2} (ξ, c_{i}) d ξ, i = 0, 1, 2, \dots .

(17)

Now we will determine the unknown constants $c_{i}^{'} s$ by minimizing ∂J/∂c_i = 0 leads to a system of algebraic equations. Since the basis functions $ϕ_{i}^{'} s$ are linearly independent. The resulting system is non-degenerate and guarantees a unique system of coefficients $c_{i}^{'} s$ . After that, by putting these optimal values of $c_{i}^{'} s$ into $\tilde{U}$ , we get the required solution by using LSHPM.

Figure 2 illustrates the block diagram representing the key steps contained in the LSHPM for solving the boundary layer flow problems.

Figure 2.

Block diagram of LSHPM methodology.

bvp5c

The bvp5c code is a finite difference method that uses the four-stage Lobatto IIIa formula, as mentioned in Ref. 39. This collocation-based formula gives a C¹ continuous solution with uniform fifth-order reliability defined on the interval [x₁, x₂], where $x_{1}, x_{2} \in R$ . The application of the Lobatto IIIa formula using an implicit Runge–Kutta method. The bvp5c code is a MATLAB built-in function formulated to solve the nonlinear ordinary differential equations, and it provides an efficient and smooth solution process.

In the bvp5c solver, mesh size is taken with a uniform grid having 100 equispaced points in the interval ranging from 0 to 20. The relative and absolute tolerances are established to be equal to 10⁻⁵ to make sure that the computed results are accurate. The initial approximation is represented in vector form as

[0.01 \cos (1.41 t), - 0.01 \sin (1.41 t)]

Figure 3 represents the procedure for finding the solution of the given model using MATLAB’s built-in solver bvp5c, highlighting the step-by-step process starting with the variable declaration and ending with obtaining numerical results.

Figure 3.

Block diagram of bvp5c solver.

Convergence analysis

In this section, we present the convergence analysis of semi-analytical technique, LSHPM when applied to nonlinear problems. This theorem is a special case of Banach’s fixed point theorem.

Theorem

Suppose u₀(ξ), u₁(ξ), u₂(ξ), …, u_n(ξ) be defined as given in (15). The series solution (15) converges if there exist 0 < r < 1, such that

‖u_k+1‖ ≤ r‖u_k‖, $\forall k \geq k_{0}, for some k_{0} \in N .$

Proof

Consider a nonlinear differential equation

\tilde{L} (v) + \tilde{N} (v) - g (ξ) = 0, ξ \geq 0 .

(18)

Suppose a series solution of (18), obtained using the HPM in the form such that

v (ξ) = \sum_{n = 0}^{\infty} u_{n} (ξ) .

(19)

Consider {α_n} be the sequence of partial sums of series solution

\{\begin{cases} α_{0} & = u_{0} (ξ) \\ α_{1} & = u_{0} (ξ) + u_{1} (ξ) \\ α_{2} & = u_{0} (ξ) + u_{1} (ξ) + u_{2} (ξ) \\ ⋮ \\ α_{n} & = u_{0} (ξ) + u_{1} (ξ) + u_{2} (ξ) + \dots + u_{n} (ξ) \end{cases}

Now, we prove that ${α_{n}}_{n = 0}^{\infty}$ is a Cauchy sequence in Hilbert space H. To establish this, let us consider the following

‖ α_{n + 1} - α_{n} ‖ = ‖ u_{n + 1} (ξ) ‖ \leq r ‖ u_{n} (ξ) ‖ \leq r^{2} ‖ u_{n - 1} (ξ) ‖ \leq r^{3} ‖ u_{n - 2} (ξ) ‖ \leq \dots \leq r^{n + 1} ‖ u_{0} (ξ) ‖ .

(20)

For each $n, i \in N and n \geq i$

\begin{align} ‖ α_{n} - α_{i} ‖ & = ‖ (α_{n} - α_{n - 1}) + (α_{n - 1} - α_{n - 2}) + \dots + (α_{i + 1} - α_{i}) ‖ \\ \leq ‖ α_{n} - α_{n - 1} ‖ + ‖ α_{n - 1} - α_{n - 2} ‖ + \dots + ‖ α_{i + 1} - α_{i} ‖ \\ \leq r^{n} ‖ u_{0} (ξ) ‖ + r^{n - 1} ‖ u_{0} (ξ) ‖ + \dots + r^{i + 1} ‖ u_{0} (ξ) ‖ \\ = (r^{i + 1} + r^{i + 2} + \dots + r^{n}) ‖ u_{0} (ξ) ‖ \\ = r^{i + 1} (1 + r + r^{2} + \dots + r^{n - i - 1}) ‖ u_{0} (ξ) ‖ \\ = r^{i + 1} \cdot \frac{1 - r^{n - i}}{1 - r} ‖ u_{0} (ξ) ‖ . \end{align}

(21)

Since 0 < r < 1, we have

\lim_{n, i \to \infty} ‖ α_{n} - α_{i} ‖ = 0 .

Hence, {α_n} is a Cauchy sequence in the Hilbert space H. Therefore, the series solution u(t) expressed in (10) converges in H.

Now, we again suppose the modified series for the trial solution of LSHPM as follows

\tilde{U} (ξ) = \sum_{i = 0}^{N} c_{i} ϕ_{i} (ξ),

(22)

where $c_{i}^{'} s$ are unidentified constants introduced to control the convergence of the series solution, and $ϕ_{i}^{'} s$ are the components of the HPM truncated series solution. The convergence of HPM is highly dependent on the initial approximation derived from the linear part of the differential equation. In contrast, the LSHPM convergence depends on the HPM solution and unknown constant c_i’s, resulting in greater reliability and faster convergence in the solution. For this purpose, a trial solution (13) is substituted into the original equation to get and a residual function can be represented as follows

\hat{R} (ξ, c_{i}) = \tilde{L} (\tilde{U}) + \tilde{N} (\tilde{U}) - g (ξ), ξ \geq 0 .

(23)

By finding the residual sum of squares, we have

J (c_{i}) = \int_{a}^{b} {\hat{R}}^{2} (ξ, c_{i}) d ξ, i = 0, 1, 2, \dots, a, b \in R .

(24)

From equation (24), the sum of the square residual function is minimized over the same function space. Therefore, the residual error of the LSHPM is always less as compared to the HPM residual error given in (25) in the L²-norm

‖ \hat{R} ‖ \leq ‖ R ‖

(25)

Here, $\hat{R}$ is the residual function obtained from the LSHPM and R represents the residual function obtained from the HPM. Hence, upon comparing the approximate solutions obtained using the LSHPM and HPM with the exact solution $\hat{u} (ξ)$ of the given problem, we observe the following

‖ \hat{u} (ξ) - \tilde{U} (ξ) ‖ \leq ‖ \hat{u} (ξ) - v (ξ) ‖

(26)

Therefore, the LSHPM converges faster than the HPM. These results are validated in the subsequent problem.

Now, we will solve the problems (7) by taking different parameter values.

Solution by LSHPM

This section utilizes LSHPM to determine an analytical approximate solution for the jet engine vibration equation, which shows promising numerical results.

Case 1:

For F = 0 N, δ = 0 kgms⁻¹, j = 240 kgm², m = 40 kg, L = 2.7 m, k = 4 kgm²s⁻², g = 9.81 ms⁻², a = 0.1 m, using the HPM, the resulting homotopy equations for (7) can be represented as follows

\begin{align} H (θ, p) & = \frac{j + m L^{2}}{L} \cdot \frac{d^{2} θ (x)}{d x^{2}} + 2 δ \cdot \frac{d θ (x)}{d x} + (\frac{k}{L} + m g) θ (x) - p (\frac{1}{6} m g θ {(x)}^{3} + F \cos (ω x)) . \end{align}

(27)

Now we assume the series solution of (27) as follows

θ = θ_{0} + p θ_{1} + p^{2} θ_{2} + p^{3} θ_{3} + \dots .

(28)

After substituting (28) into equation (27) and the resulting equations based on the powers of p are expressed as follows

p^{0} : \{\begin{cases} \frac{j + m L^{2}}{L} \cdot \frac{d^{2} θ_{0} (x)}{d x^{2}} + 2 δ \cdot \frac{d θ_{0} (x)}{d x} + (\frac{k}{L} + m g) θ_{0} (x), \\ θ_{0} (0) = a, θ_{0}^{'} (0) = 0, \end{cases}

(29)

p^{1} : \{\begin{cases} \frac{j + m L^{2}}{L} \cdot \frac{d^{2} θ_{1} (x)}{d x^{2}} + 2 δ \cdot \frac{d θ_{1} (x)}{d x} + (\frac{k}{L} + m g) θ_{1} (x) - \frac{1}{6} m g θ_{0} {(x)}^{3} - F \cos (ω x) . \\ θ_{1} (0) = 0, θ_{1}^{'} (0) = 0, \end{cases}

(30)

By finding the solution of equations (29) and (30), the solution of these equations will provide the values of parameters θ₀, θ₁. By combining the solutions, we achieve the approximate solution as follows

\begin{aligned} θ (x) = θ_{0} + θ_{1} . \end{aligned}

(31)

This implies

θ (x) = 0.09998443376932335351863693 \cos (1.414399771 x) - 0.000005188743558882160454357390 \cos (4.243199313 x) .

(32)

Furthermore, we can achieve an improved accuracy level to a greater extent by adding more iterations in (31). In equation (32), θ(x) consists of cos (1.41x), cos (4.24x).

With the given information, we suppose that the LSHPM assumed solution of equation (7) as follows

\hat{θ} = c_{0} \cos (1.414399771 x) + c_{1} \cos (4.243199313 x)

(33)

where c₀, c₁, are unidentified constants. By using the initial conditions on (33) we get the values of c₀ given as follows

\begin{aligned} c_{0} = 0.1 - c_{1} . \end{aligned}

(34)

and incorporating the values of c₀ into the proposed solution (33), we get

\hat{θ} = (0.1 - 1.0 c_{1}) \cos (1.414399771 x) + c_{1} \cos (4.243199313 x)

(35)

To construct the residual function, put values of $\tilde{f}$ in the place of f into the equation (7), we have

\begin{aligned} \hat{R} (c_{1}, c_{2}, x) = \frac{J + m L^{2}}{L} \ddot{\tilde{θ}} + 2 δ \dot{\tilde{θ}} + (\frac{κ}{L} + m g) \tilde{θ} - \frac{1}{6} m g {\tilde{θ}}^{3} - F \cos (ω t) \end{aligned}

(36)

The squared residual function can be represented as follows

\begin{aligned} J (c_{1}, c_{2}) = \int_{0}^{50} {(\hat{R} (c_{1}, c_{2}, x))}^{2} d x . \end{aligned}

(37)

The optimal values of $c_{i}^{'} s$ , where i ranges from 1 to 2, can be estimated by evaluating the partial derivatives ∂J/∂c_i using MAPLE software, we have

c_{1} = - 0.0000051830723398496351053,

substitute them into equation (34) and then subsequently into the proposed solution as described in (35), we have

\hat{θ} = 0.10000518307233984964 \cos (1.414399771 x) - 0.0000051830723398496351053 \cos (4.243199313 x) .

(38)

Case 2:

Without periodic force F = 0 and choosing δ = 18 kgms⁻¹, j = 250 kgm², m = 50 kg, L = 2.5 m, k = 5 kgm²s⁻², g = 9.81 ms⁻², and a = 0.5 m, with initial conditions: $θ (0) = a, \dot{θ} (0) = 0$ .

The HPM required solution is given as follows

\begin{align} θ (x) & = 0.002616 e^{- 0.08 x} [((\cos (2.954649 x) + 0.245092 \cos (5.909299 x) + 0.054467 \sin (2.954649 x) \\ + 0.046975 \sin (5.909299 x) + 1.495677) \cos (1.477324 x) \\ + (- 0.268560 \cos (2.954649 x) + 0.245092 \sin (5.909299 x) + 1.973863 \sin (2.954649 x) \\ - 0.046975 \cos (5.909299 x) - 27.620029) \sin (1.477324 x)) e^{- 0.16 x} \\ + 38.284267 \sin (1.477324 x) + 188.364184 \cos (1.477324 x)] \end{align}

(39)

Using the same procedure from equations (24)–(29), we obtain our required LSHPM solution as follows

\begin{align} \hat{θ} (x) & = 0.00261636 e^{- 0.08 x} [((- 677.08284 \cos (2.95464982 x) - 5.12205 \cos (5.90929964 x) \\ + 49.46774 \sin (2.95464982 x) + 17.92467 \sin (5.90929964 x) + 686.10517) \cos (1.47732491 x) \\ + (- 85.49971 \cos (2.95464982 x) - 5.11951 \sin (5.90929964 x) - 686.86245 \sin (2.95464982 x) \\ - 17.92583 \cos (5.90929964 x) - 94.27857) \sin (1.47732491 x)) e^{- 0.16 x} \\ + 37.84109 \sin (1.47732491 x) + 187.20468 \cos (1.47732491 x)] \end{align}

(40)

Case 3:

Without damping force, that is, δ = 0 kgms⁻¹ and choosing F = 5 N, j = 220 kgm², m = 45 kg, L = 2 m, k = 3.5 kgm²s⁻², g = 9.81 ms⁻², ω = 30^o, and a = 0.03 m, with initial conditions: $θ (0) = a, \dot{θ} (0) = 0$ .

The HPM solution can be expressed as follows

\begin{align} θ (x) & = 0.0301 \cos (1.488623525274271621135631 x) \\ - 0.0000001400697343 \cos (4.465870575822814863406893 x) \\ - 0.000027846341659018204824323 \cos (30.0 x) . \end{align}

(41)

By using the same procedure applied in equations (24)–(29), the desired LSHPM solution can be expressed as follows

\begin{align} \hat{θ} & = 0.030027985063914917296 \cos (1.488623525274271621135631 x) - 0.0000001387444966 \cos (4.465870575822814863406893 x) - 0.000027846319418293960575 \cos (30.0 x) . \end{align}

(42)

Results and discussions

This section is devoted to a detailed discussion of the effects of variation of different parameters for the computed results obtained by solving equation (7) subject to the initial conditions using the LSHPM.

• In Table 1, we compare the solutions obtained from HPM, LSHPM, and bvp5c for case 1. The results obtained from LSHPM offer a strong agreement compared to bvp5c and HPM results up to three decimal places, whereas the HPM and bvp5c provide slightly different values at some specific time level. It is important to emphasize that residual errors obtained through the LSHPM solution (38) show higher precision than the HPM solution (32). It is highlighted that the LSHPM solution is based on merely two terms of the series solution of HPM; the precision of both methods may be enhanced by adding additional terms in the series solution (31).

• Table 2 presents a comparison of the solutions obtained from HPM, LSHPM, and bvp5c for case 2. The LSHPM results exhibit strong agreement with bvp5c and HPM results, matching up to two decimal places. In contrast, slight discrepancies between HPM and bvp5c results are observed at some specific time levels. Notably, the residual errors obtained from the LSHPM solution (40) demonstrate higher precision than the HPM solution (39).

• In Table 3, a comparison of the solutions derived from the LSHPM, HPM, and bvp5c for case 3 is discussed. The LSHPM results are closely aligned with bvp5c and HPM results, maintaining consistency up to three decimal places. The HPM and bvp5c demonstrate a minor deviation in their results at a certain time level. Importantly, the residual errors associated with the LSHPM solution (42) give higher precision than those from the HPM solution (41).

• Table 4 provides a comparative analysis of the solutions evaluated using the LSHPM, HPM, and bvp5c for case 4. The HPM and bvp5c results are closely aligned with the LSHPM results up to three decimal places. However, the analysis of residual errors confirms that the LSHPM solution achieves greater accuracy than the HPM solution.

• In Table 5, we compare the results obtained from LSHPM with those reported in the previous study,¹ which utilized the Homotopy Frequency Amplitude Method (HFAM) across the four different cases of the jet engine vibration equation. It is observed that the LSHPM results show a strong agreement with the HFAM outcomes for case 1 and case 2. However, notable discrepancies arise in case 3 and case 4, because the solution that is provided in Ref. 1 is not accurate for these specific cases. The residual error analysis across all four cases supports the reliability and efficiency of the LSHPM approach as a robust semi-analytical approximate technique.

• In Figure 4, subfigure (4a) represents the graphical comparison of the jet engine vibration equation outcomes derived from LSHPM, bvp5c, and HPM for case 1. In the absence of damping and periodic external force, the vibration profile follows the sinusoidal pattern with the same wavelength and amplitude over the time interval x = 0 to x = 50. It is clearly shown that all three solution curves are overlapping each other, which indicates the strong agreement of HPM and bvp5c results with the LSHPM outcomes and supports the correctness of LSHPM results. Additionally, subfigure (4b) provides a graphical view of a complete period of the vibration system within a specific region.

• In Figure 5, subfigure (5a) presents a graphical representation of the jet engine vibration solution obtained using LSHPM, HPM, and bvp5c for case 2. In the scenario where periodic external force is absent and damping force is present, the wavelength and amplitude of the vibration system rapidly decrease across the time interval x = 0 to x = 50. The close alignment of all three solution curves highlights the consistency of HPM and bvp5c outputs with those derived from LSHPM, thereby validating the accuracy of the LSHPM approach. Subfigure (5b) further illustrates a complete cycle of the vibration system within a specific region, providing a complete view of the system’s vibration behavior.

• In Figure 6, subfigure (6a) displays the graphical illustration of jet engine vibration equation outputs obtained using LSHPM, HPM, and bvp5c for case 3. In this case, where the system is subjected to periodic external force but not influenced by a damping force, the vibration system follows a sinusoidal pattern with uniform wavelength and amplitude over the interval x = 0 to x = 50. All three solution curves overlap, which indicates a strong agreement of HPM and bvp5c results with those derived from LSHPM, thereby affirming the reliability and accuracy of the LSHPM approach. Additionally, subfigure (6b) illustrates a complete cycle within a particular region, offering a detailed view of the system’s dynamic behavior.

• In Figure 7, subfigure (7a) illustrates the jet engine vibration equation solution derived from LSHPM, HPM, and bvp5c for case 4. In the presence of both damping and periodic external force, the amplitude and wavelength of the vibration profile gradually decrease over the time interval x = 0 to x = 50. It is observed that all three solution curves demonstrate a strong agreement with each other that indicates the reliability and correctness of LSHPM results. Furthermore, subfigure (7b) presents a complete view of one vibration cycle within a specific section, providing a more detailed view of vibration behavior.

• Figure 8 illustrates the residual errors of both HPM and LSHPM techniques for case 1. It is observed that HPM residual error shows a significant deviation behavior; the deviation of HPM residual error graphs indicates instability of the method. In contrast, LSHPM residual error does not show deviation and remains consistent throughout, ensuring that it gives a reliable, rapid-convergent, and efficient solution compared to the HPM method. When we analyze the numerical solution of both methods, they provide a strong agreement with each other up to three decimal places, but a major difference is shown in their residual error graphs. This is due to the coefficients of the basis function in the HPM solution (32) and the LSHPM solution (38). The coefficients obtained using LSHPM are optimized to minimize the residual error, thereby ensuring a more reliable and fast convergent solution. While HPM has a lack of consistency beyond three decimal places. In contrast, LSHPM demonstrates both precision and robust convergence, making it a superior method in this context.

• Figure 9, displays a graphical comparison between HPM and LSHPM residual errors for case 2. It is clearly shown that HPM residual error is not consistent and shows fluctuations in its graph, whereas the LSHPM residual error remains consistent throughout. Therefore, the LSHPM solution (40) is more accurate than the HPM solution (39).

• In Figure 10, we compare the residual errors of both HPM and LSHPM solutions for case 3. The residual error associated with HPM exhibits noticeable fluctuations, indicating the instability and inconsistency in the method. In contrast, LSHPM residual error remains smooth and uniform throughout the domain. As a result, the LSHPM solution (42) is considered more precise than the HPM solution (41).

• In Figure 11, the residual errors of HPM and LSHPM solutions for case 4 are compared. The residual error corresponding to HPM shows significant deviations throughout the domain that reveal the instability in the method. Whereas LSHPM residual error remains consistent over the entire domain. Therefore, the LSHPM solution is fast convergent and accurate compared to HPM solution.

Table 1.

Comparisons between the solutions outcomes obtained from the LSHPM, bvp5c, and HPM for the parameters F = 0 N, δ = 0 kgms⁻¹, j = 240 kgm², m = 40 kg, L = 2.7 m, k = 4 kgm²s⁻², g = 9.81 ms⁻², and a = 0.1 m.

x	HPM	LSHPM	bvp5c	HPM residual error	LSHPM residual error
0	0.099979245025764	0.100000000000000	0.100000000000000	−0.049009294894243	−0.000249216084698
5	0.070461499090785	0.070476116816715	0.070787957641960	−0.034557857665439	−0.000175527094981
10	−0.000683071565997	−0.000683213182379	0.000197959307115	0.000335046659078	0.000001699844495
15	−0.071423901501409	−0.071438719147761	−0.070507811170475	0.035029567455274	0.000177928575197
20	−0.099969919100153	−0.099990672133357	−0.099999216890910	0.049004736668217	0.000249192799309
25	−0.069485940585598	−0.069500355668978	−0.071066994457483	0.034079676695666	0.000173092898150
30	0.002049087054404	0.002049511877293	−0.000593874645859	−0.001005077445753	−0.000005099220591
35	0.072372968426533	0.072387983233620	0.070226559723277	−0.035494718043202	−0.000180296884451
40	0.099941943057359	0.099962690267834	0.099996867545305	−0.048991062798355	−0.000249122947450
45	0.068497407841460	0.068511617598221	0.071344916465233	−0.033595113835385	−0.000170626445203
50	−0.003414719636136	−0.003415427586767	0.000989778799102	0.001674920648494	0.000008497658050

Table 2.

Comparisons between the solutions outcomes obtained from the LSHPM, bvp5c, and HPM for the parameters F = 0 N, δ = 18 kgms⁻¹, j = 250 kgm², m = 50 kg, L = 2.5 m, k = 5 kgm²s⁻², g = 9.81 ms⁻², and a = 0.5 m.

x	HPM	LSHPM	bvp5c	HPM residual error	LSHPM residual error
0	0.500000000000000	0.500000000000000	0.500000000000000	0	−0.000947773433406
5	0.190713171354532	0.189973488676661	0.190137046075549	−0.184927818896807	−0.000435262084137
10	−0.101064467015189	−0.100709088317072	−0.100332671381916	−0.070255183703054	−0.000164447859280
15	−0.151258982913555	−0.150388111679863	−0.149903376440056	0.007953558287374	0.000012414374309
20	−0.048061042476595	−0.047678240885455	−0.047582988967185	0.005592918644887	0.000017608768552
25	0.038809525865401	0.038624917026643	0.038408588209037	0.003297276078491	0.000008845099534
30	0.045182853480802	0.044890846317814	0.044703999819018	−0.000758505967222	−0.000002088649256
35	0.009906282059899	0.009812955970064	0.009799202184148	−0.000065335712903	−0.000000204680443
40	−0.014317979938302	−0.014242416657173	−0.014165101690267	−0.000109668317846	−0.000000304620304
45	−0.013103740341691	−0.013015951223706	−0.012962747299079	0.000031537350720	0.000000095826318
50	−0.001482468579035	−0.001463413928814	−0.001466099705233	0.000000266119818	0.000000000782338

Table 3.

Comparisons between the solutions outcomes obtained from the LSHPM, bvp5c, and HPM for the parameters F = 5 N, δ = 0 kgms⁻¹, j = 220 kgm², m = 45 kg, L = 2 m, k = 3.5 kgm²s⁻², g = 9.81 ms⁻², ω = 30^o, and a = 0.03 m.

x	HPM	LSHPM	bvp5c	HPM residual error	LSHPM residual error
0	0.030072013588606	0.030000000000000	0.030000000000000	−0.00150423379700	−0.000007492929930
5	0.012002648070056	0.011973883956296	0.011985387123051	−0.00059571986223	−0.000002965836255
10	−0.020496292342910	−0.020447252252526	−0.020428872447373	0.00102079989511	0.000005063437468
15	−0.028374488760575	−0.028306554043445	−0.028319020111786	0.00141859785086	0.000007059753619
20	−0.002157344325377	−0.002152116080055	−0.002202165611179	0.00010813871834	0.000000555533968
25	0.026667959750502	0.026604200589270	0.026574981704988	−0.00133583911696	−0.000006618771947
30	0.023470978098918	0.023414818398582	0.023461868371957	−0.00117109323037	−0.000005812330044
35	−0.007920541094667	−0.007901640903500	−0.007816864840529	0.00039165375333	0.000001924997286
40	−0.029810347730634	−0.029739093032459	−0.029724394061350	0.00149890851769	0.000007426233812
45	−0.015909033967121	−0.015871012441100	−0.015966797341899	0.00079051255514	0.000003905938559
50	0.017091957922338	0.017051071405684	0.016947646775515	−0.00084971529038	−0.000004210406174

Table 4.

Comparisons between the solutions outcomes obtained from the LSHPM, bvp5c, and HPM for the parameters F = 12 N, δ = 6 kgms⁻¹, j = 200 kgm², m = 56 kg, L = 2.25 m, k = 3 kgm²s⁻², g = 9.81 ms⁻², ω = 60^o, and a = 0.15 m.

x	HPM	LSHPM	bvp5c	HPM residual error	LSHPM residual error
0	0.150000000000000	0.150000000000000	0.150000000000000	−0.000000133967677	0.000000000857197
5	−0.015844174126153	−0.015844173481092	−0.015841941246094	−0.000094879153854	−0.000000441592872
10	−0.109550742815248	−0.109550743125384	−0.109534363275361	0.001685038387500	0.000007840071953
15	0.039044134506804	0.039044133157414	0.039032790507705	0.000907315578254	0.000004222435829
20	0.073100503108582	0.073100504032790	0.073073726602393	−0.001789385651593	−0.000008325764414
25	−0.047937225809564	−0.047937224403563	−0.047914738408670	−0.001115514518886	−0.000005191375332
30	−0.043140844893964	−0.043140846375977	−0.043117957061455	0.000790717994876	0.000003678485692
35	0.047240375669996	0.047240374626469	0.047211959099600	0.000647487003395	0.000003013421354
40	0.020558721172122	0.020558722974776	0.020545688998141	−0.000172437483132	−0.000000080134493
45	−0.040998967361849	−0.040998966871135	−0.040970795297762	−0.000190161010981	−0.000000885251839
50	−0.005023851507877	−0.005023853353377	−0.005020468838706	0.000007202869744	0.000000032465947

Table 5.

Comparisons of numerical values obtained from LSHPM and previously reported results for four different cases.

x	Case 1		Case 2
x	HFAM results¹	Present study	HFAM results¹	Present study
0	0.300000	0.300000000000000	0.250000	0.250000000000000
5	0.202265	0.202477197027064	0.164911	0.164957559983220
10	−0.027259	−0.027299587669016	0.108767	0.108880562576927
15	−0.239023	−0.239177222937421	0.0717265	0.071816148315340
20	−0.295046	−0.295068326053252	0.0472931	0.047315877703740
25	−0.158826	−0.159066512571981	0.0311783	0.031134099602578
30	0.080879	0.080988204037559	0.0205514	0.020458940850720
35	0.267887	0.267971385605156	0.0135446	0.013425686113704
40	0.280348	0.280430192640134	0.00892541	0.008798090462414
45	0.110143	0.110367905306151	0.00588064	0.005757455044709
50	−0.131828	−0.131982039205807	0.00387397	0.003762298283855
	Case 3		Case 4
0	0.200000	0.200000000000000	0.250000	0.250000000000000
5	0.101762	0.194979696117314	−0.0598265	0.166422840860755
10	−0.0948027	0.180102880996934	0.182523	0.110537559132072
15	−0.193393	0.156049917774317	−0.0961173	0.073129569691505
20	−0.0958415	0.124096805352553	0.146364	0.048096795267010
25	0.099737	0.085913655233214	−0.10484	0.031374975190779
30	0.197091	0.043359859978895	0.1222	0.020283192589470
35	0.0981166	−0.001462492122138	−0.0994338	0.013039189281823
40	−0.0990003	−0.046200464165333	0.101589	0.008403184912217
45	−0.19793	−0.088526021931920	−0.0861919	0.005463877647592
50	−0.100489	−0.126368595534437	0.0810748	0.003555140005406

Figure 4.

(a) Graphical illustration of jet engine vibration system for case 1 (b) Complete period without damping and periodic external force.

Figure 5.

(a) Graphical illustration of jet engine vibration system for case 2 (b) Complete period with damping and without periodic external force.

Figure 6.

(a) Graphical illustration of jet engine vibration system for case 3 (b) Complete period without damping and with periodic external force.

Figure 7.

(a) Graphical illustration of jet engine vibration system for case 4 (b) Complete period with both damping and periodic external force.

Figure 8.

Graphical illustration of the LSHPM and HPM residual errors for case 1.

Figure 9.

Graphical illustration of the LSHPM and HPM residual errors for case 2.

Figure 10.

Graphical illustration of the LSHPM and HPM residual errors for case 3.

Figure 11.

Graphical illustration of the LSHPM and HPM residual errors for case 4.

Case 1:

Case 2:

Case 3:

Case 4:

Conclusion

This research investigates the effectiveness of LSHPM for solving nonlinear jet engine vibration equation across different operational conditions. The problem has been addressed utilizing LSHPM, HPM, and the bvp5c solver across four distinct cases. A comparative analysis of the solution derived using the LSHPM, bvp5c solver, and HPM is presented, including a detailed evaluation of residual errors from Tables 1–4. The numerical results obtained from HPM and bvp5c exhibit a strong agreement with those derived from LSHPM, confirming the accuracy and rapid convergence of the LSHPM approach. A graphical comparison of the solution profiles and residual errors further supports LSHPM across all four cases. These findings have potential real-world application in the aerospace industry, where they could contribute to enhancing jet engine design and performance.

This study investigates the implementation of the LSHPM for solving the jet engine vibration equation. This method shows a strong alignment with the bvp5c solver and HPM in terms of numerical results. However, HPM demonstrates divergence behavior in residual error analysis, indicating it is not a reliable converging tool for solving this problem. While bvp5c and LSHPM provide closely matching results. LSHPM exhibits clear advantages. Unlike bvp5c, which yields only numerical data, LSHPM provides a semi-analytical closed-form solution that is reliable and efficient. A notable advantage of the LSHPM is to determine numerical validation through residual error analysis, which is not possible for the bvp5c solver. The main characteristics of this technique are its rapid convergence, reliability, and computational efficiency, making it a valuable tool for solving complex boundary layer flow problems. We conclude that LSHPM provides a new channel for scientists and engineers by providing them with an efficient and powerful methodology for finding precise solutions in closed form, especially for problems defined over semi-infinite domains. In addition, this technique can be modified for finding the solution of fractional differential equations.

Footnotes

Acknowledgments

The Researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2026).

ORCID iDs

Muhammad Rafiq

Hijaz Ahmad

Taha Radwan

Author contributions

Investigation: M.T. ; Formal Analysis: M.R. ; Conceptualization: M.R. ; Methodology: M.R. ; Analysis of data: H.A. ; Writing-review editing: M.T. ; Software: M.B. ; Writing-Original draft: M.R.; Re modelling design: M.B. ; Re-Validation: M.T. and T. R. In addition, the contribution of all authors is almost equal.

Funding

The Researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2026).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

All relevant data generated or analyzed during this study are fully incorporated within this published article.

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