Modified Krylov-Bogoliubov-Mitropolskii method for solving damped nonlinear oscillators

Abstract

In the recent past, a simplified solution was obtained by using a modified Krylov-Bogoliubov-Mitropolskii method for a cubic Duffing oscillator in the presence of a linear damped force. However, a similar solution is not always possible for another class of nonlinear oscillators where the inertia type force is involved in the nonlinear function along with the restoring forces and in the presence of linear damping. In this paper, an alternative modification of the Krylov-Bogoliubov-Mitropolskii method is introduced to overcome this limitation. The approximated solutions are achieved and comparison with the fourth-order Runge-Kutta method which are represented graphically. The comparison reveals excellent consistency between them.

Keywords

Lindstedt-Poincare method KBM method nonlinear damped oscillation

Introduction

Considerable attention has been focused on studying nonlinear damped oscillators in the fields of nonlinear sciences and engineering which are appearing mathematically in the form of nonlinear differential equations. The nonlinear differential equations play a vital role in modelling the inherent nonlinear properties of cantilever beam models, like, helicopter rotor blades, large-span bridges, aircraft wings and so on. The study of nonlinear damped oscillatory systems also has great importance in both theoretical and practical applications. For the damped nonlinear oscillators, several methods have been developed to determine approximate solutions. The classical perturbation techniques^1–4 has been randomly used to solve nonlinear problems to obtain approximate solutions. However, for strong nonlinear damped oscillatory problems, the classical perturbation techniques are almost failed to obtain desire results. A semi-analytical method^5,6 has been applied to solve nonlinear damped oscillatory systems. The modification of the Lindstedt-Poincare (LP) method gives desired solutions as well as frequencies close to the resonance for strongly nonlinear oscillatory problems.^7–13 However, this modification is failed if it is considered the damping effect in the nonlinear function. The Krylov-Bogoliubov-Mitropolskii (KBM) method provides a significant framework for analyzing nonlinear damped oscillatory problems^14–16 with small nonlinearities. Popov et al.¹⁷ and Mendelson¹⁸ extended this method to solve strongly nonlinear damped oscillatory problems. However, they have considered only the first-order approximation in the amplitude and phase variables to avoid complexity of mathematical calculation. The trial approximate solution of the KBM method has considered as an expanding series involving a small parameter.^19–27 Amer et al.^28–30 has implemented the KBM method to analyze the 3-D motion for rigid body. Recently, a modified Krylov-Bogoliubov-Mitropolskii method³¹ has been presented for solving damped nonlinear oscillators with large oscillation. Nevertheless, this modification is not always fruitful for another sort of nonlinear oscillator where the inertia type force is involved in the nonlinear function along with the restoring forces and in the presence of linear damping. In this paper, an alternative modification has been introduced to remove this limitation. This is the prime significance in this paper.

The rest of this paper is organized as follows: Firstly, we present the details discussion of the existing and modified KBM method. Then, we implement the modified KBM method to the nonlinear oscillators where the inertia type force is involved in the nonlinear function along with the restoring forces and in the presence of linear damping force. Afterwards, results and discussions are presented in detail. And finally, concluding remarks are given.

Methodology

The KBM method^14,15 was originally developed to solve the following nonlinear equation

\ddot{x} + ω_{0}^{2} x + ε f (x, \dot{x}) = 0, x (0) = a_{0}, \dot{x} (0) = 0

(1)

where

ε

is a small parameter,

ω_{0} > 0

is unperturbed frequency, over dots denote differentiation with respect to

t

and

f

is a nonlinear function such as

f (- x, - \dot{x}) = - f (x, \dot{x})

. According to this method, a solution is expanded in ascending powers of

ε

x = b (t) \cos θ (t) + ε u_{1} (b (t), θ (t)) + ε^{2} u_{2} (b (t), θ (t)) + \dots

(2)

where the amplitude and phase variables are denoted by

b (t)

and

θ (t)

separately which satisfy the following two first-order differential equations¹⁶ as

\dot{b} = ε B_{1} (b) + ε^{2} B_{2} (b) + \dots

(3)

\dot{θ} = ω_{0} + ε C_{1} (b) + ε^{2} C_{2} (b) + \dots

(4)

Later, Popov et al.¹⁷ (also Mendelson¹⁸) extended the KBM method for solving another type of nonlinear equation as

\ddot{x} + μ \dot{x} + ω_{0}^{2} x + ε f (x) = 0, x (0) = a_{0}, \dot{x} (0) = 0

(5)

where

\frac{μ}{2} < ω_{0}

μ

is a damped constant and

f (- x) = - f (x)

. It was considered the same solution equation (2), whereas the amplitude and phase variables satisfying the different form of the first-order differential equations as

\dot{b} = - μ b / 2 + ε B_{1} (b) + ε^{2} B_{2} (b) + \dots

(6)

\dot{θ} = ω + ε C_{1} (b) + ε^{2} C_{2} (b) + \dots, ω = \sqrt{ω_{0}^{2} - \frac{μ^{2}}{4}}

(7)

where

ω

is reduced frequency.

Recently, Alam et al.³¹ modified the KBM method for solving cubic Duffing oscillator in the presence of a linear damped force where the amplitude and phase variables of equations (6) and (7) were rearranged as

\dot{b} = - \frac{μ a}{2} / \sqrt{1 + ε B_{1} (b) + ε^{2} B_{2} (b) + \dots}

(8)

\dot{θ} = ω \sqrt{1 + ε C_{1} (b) + ε^{2} C_{2} (b) + \dots}

(9)

In this paper, it is remarkably important that the nonlinear function $f$ contains $\ddot{x}$ , such that $f (- x, - \dot{x}, - \ddot{x}) = - f (x, \dot{x}, \ddot{x})$ . In this case, the phase variable equation (9) diverges. Therefore, it is considered equation (9) into another form as

\dot{θ} = ω / \sqrt{1 + ε C_{1} (b) + ε^{2} C_{2} (b) + \dots}

(10)

Now differentiating $x$ (equation (2)) twice with respect to $t$ and substituting it together with equations (8) and (9) into equation (5) and equating the coefficients of various powers of $ε$ , the following equations are obtained

(- \frac{μ^{2}}{8} b^{2} \frac{d B_{1}}{d b} + ω^{2} b C_{1}) \cos θ - \frac{μ ω}{4} (2 b B_{1} + b^{2} \frac{d C_{1}}{d b}) \sin θ + \frac{μ^{2}}{4} (b^{2} \frac{\partial^{2} u_{1}}{\partial b^{2}} - b \frac{\partial u_{1}}{\partial b})

+ μ ω (\frac{\partial u_{1}}{\partial θ} - b \frac{\partial^{2} u_{1}}{\partial b \partial θ}) + ω^{2} \frac{\partial^{2} u_{1}}{\partial θ^{2}} + ω_{0}^{2} u_{1}

+ f (b \cos θ, - \frac{μ}{2} b \cos θ - ω b \sin θ, (\frac{μ^{2}}{4} - ω^{2}) b \cos θ + μ ω b \sin θ) = 0

(11)

(- \frac{μ^{2}}{8} b^{2} \frac{d B_{2}}{d b} + ω^{2} b C_{2} + \frac{μ^{2}}{16} b B_{1}^{2} - ω^{2} b C_{1}^{2} + \frac{μ^{2}}{4} b^{2} B_{1} \frac{d B_{1}}{d b}) \cos θ

- \frac{μ ω}{4} (2 b B_{2} + b^{2} \frac{d C_{2}}{d b} - \frac{3}{2} b B_{1}^{2} - \frac{μ}{2} ω b B_{2} + \frac{μ}{4} ω a B_{1} C_{1} \dots) \sin θ

+ \frac{μ^{2}}{4} (b^{2} \frac{\partial^{2} u_{2}}{\partial b^{2}} - b \frac{\partial u_{2}}{\partial b}) + μ ω (\frac{\partial u_{2}}{\partial θ} - b \frac{\partial^{2} u_{2}}{\partial b \partial θ}) + ω^{2} \frac{\partial^{2} u_{2}}{\partial θ^{2}} + ω_{0}^{2} u_{2}

- \frac{μ^{2}}{4} b^{2} (2 \frac{d B_{1}}{d b} \frac{\partial u_{1}}{\partial b} + B_{1} \frac{\partial^{2} u_{1}}{\partial b^{2}}) - \frac{μ}{4} ω (2 B_{1} \frac{\partial u_{1}}{\partial θ} + b \frac{d B_{1}}{d b} \frac{\partial u_{1}}{\partial θ} - 2 b C_{1} \frac{\partial^{2} u_{1}}{\partial b \partial θ})

+ ω^{2} C_{1} \frac{\partial^{2} u_{2}}{\partial θ^{2}} + u_{1} f_{x} (\dots) = 0

(12)

The above equations are linear and easy to solve. Now solving equation (11), $B_{1}$ , $C_{1}$ and $u_{1}$ are obtained by assuming that $u_{1}, u_{2}, \dots$ do not contain the first harmonic terms. Then substituting the values of $B_{1}$ , $C_{1}$ and $u_{1}$ into equation (12) and solving it, $B_{2}$ , $C_{2}$ and $u_{2}$ are obtained. In a similar procedure, other functions are obtained; but it is a tremendously difficult task to obtain more than a sixth approximation. As a result, an alternate approach is required to remove the algebraic complicity. Since, the limiting values of $B_{1}, B_{2}, \dots$ ; $C_{1}, C_{2}, \dots$ and $u_{1}, u_{2}, \dots$ , at $μ \to 0$ are used (details descriptions are given³¹), equation (5) is rewritten as

\ddot{x} + ω_{0}^{2} x + ε μ \dot{x} + ε f (x, \dot{x}, \ddot{x}) = 0, x (0) = a, \dot{x} (0) = 0

(13)

Herein similar equations to equations (11) and (12) $\dots$ take the following form by substituting equations (8) and (10) into equation (13) as

ω_{0}^{2} b C_{1} \cos θ + ω_{0}^{2} (\frac{\partial^{2} u_{1}}{\partial θ^{2}} + u_{1}) + f (b \cos θ, - ω_{0} b \sin θ, - ω_{0}^{2} b \cos θ) = 0

(14)

(ω_{0}^{2} b C_{2} + (ω_{0}^{2} C_{1}^{2} - \frac{μ^{2}}{4}) b) \cos θ - \frac{μ ω_{0}}{4} (2 b B_{1} + b^{2} \frac{d C_{1}}{d b}) \sin θ

+ ω_{0}^{2} (\frac{\partial^{2} u_{2}}{\partial θ^{2}} + u_{2}) + u_{1} f_{x} (\dots) + (- \frac{μ}{2} b \cos θ + \frac{ω_{0}}{2} C_{1} \sin θ + ω_{0} \frac{\partial u_{1}}{\partial θ}) f_{\dot{x}} (\dots)

+ \dots = 0, \dots

(15)

Comparing the above two sets of equations, it is clear that equations (14) and (15), $\dots$ are much simpler than equations (11) and (12), $\dots$ . However, another problem arises in the later approach. Solving equation (14), $C_{1}$ and $u_{1}$ are obtained only. Then solving equation (15), $B_{1}$ , $C_{2}$ and $u_{2}$ are obtained. Therefore, $B_{1}, B_{2}, \dots, B_{n}$ are obtained from $n + 1$ -th approximation. Still now, the latter approach comparatively easy task and saves a lot of calculations.

Example

Let us consider a nonlinear oscillator¹⁹ in the presence of the linear damped force as

\ddot{x} + x + ε μ \dot{x} + ε {\dot{x}}^{2} x + ε x^{2} \ddot{x} = 0

(16)

For this oscillator first and second approximate solutions were obtained in ref 17,18. In ref 31, the third approximation has been determined. In this paper, the tenth approximation is obtained. However, some lower-order functions are shown as the following

B_{1} = C_{1} = \frac{b^{2}}{2}, B_{2} = \frac{9 b^{4}}{64}, C_{2} = - \frac{3 b^{4}}{32} + \frac{μ^{2}}{4}

B_{3} = - \frac{161 b^{6}}{1024} + \frac{μ^{2} b^{2}}{4}, C_{3} = \frac{37 b^{6}}{512} - \frac{μ^{2} b^{2}}{8}

B_{4} = \frac{925 b^{8}}{4096} - \frac{{457 μ}^{2} b^{4}}{1024}, C_{4} = - \frac{549 b^{8}}{8192} + \frac{{77 μ}^{2} b^{4}}{512} + \frac{μ^{2}}{16}, \dots

(17)

and

u_{1} = - \frac{b^{3} \cos 3 θ}{16}, u_{2} = \frac{{9 b}^{5} \cos 3 θ}{256} + \frac{{3 b}^{5} \cos 5 θ}{256} + \frac{μ b^{3} \sin 3 θ}{64}

u_{3} = - \frac{{μ^{2} b}^{5} \cos 3 θ}{256} - \frac{{105 b}^{7} \cos 3 θ}{4096} + \frac{μ b^{5} \sin 3 θ}{1024} - \frac{167 b^{7} \cos 5 θ}{12288}

- \frac{5 μ b^{5} \sin 5 θ}{1024} - \frac{37 b^{7} \cos 7 θ}{12288}

(18)

As a limit $μ \to 0$ , the $ε$ -series of amplitude equation (8) and phase equation (10) become

1 + ε B_{1} + \dots + ε^{8} B_{8} = 1 + \frac{ε b^{2}}{2} + \frac{9 ε^{2} b^{4}}{64} - \frac{161 ε^{3} b^{6}}{1024} + \frac{925 ε^{4} b^{8}}{4096} + \dots + \frac{252832093 ε^{8} b^{16}}{268435456}

(19)

1 + ε C_{1} + \dots + ε^{8} C_{8} = 1 + \frac{ε b^{2}}{2} - \frac{3 ε^{2} b^{4}}{32} + \frac{37 ε^{3} b^{6}}{512} - \frac{549 ε^{4} b^{8}}{8192} + \dots - \frac{166499429 ε^{8} b^{16}}{1610612736}

(20)

and at this limit, the solution equation (2) becomes

x = b \cos θ - \frac{{ε b}^{3}}{16} \cos 3 θ + \frac{{3 ε^{2} b}^{5}}{256} (3 \cos 3 θ + \cos 5 θ)

- \frac{{ε^{3} b}^{7}}{12288} (315 \cos 3 θ + 167 \cos 5 θ + 37 \cos 7 θ)

(21)

Applying the initial condition $t \to 0, θ \to 0$ and then $x \to a$ , equation (21) becomes

a = b - \frac{{ε b}^{3}}{16} + \frac{{3 ε^{2} b}^{5}}{64} - \frac{{173 ε^{3} b}^{7}}{4096} + \frac{{2785 ε^{4} b}^{9}}{65536} - \frac{{48149 ε^{5} b}^{11}}{1048576} + \dots + \frac{{477088111 ε^{8} b}^{17}}{6442450944}

(22)

From equation (22), $b$ is transferred in a series of $a$ ²⁶ as

b = a + \frac{{ε a}^{3}}{16} - \frac{{9 ε^{2} a}^{5}}{256} + \frac{{89 ε^{3} a}^{7}}{4096} - \frac{{235 ε^{4} a}^{9}}{16384} + \frac{{653 ε^{5} b}^{11}}{65536} - \dots - \frac{{53698019 ε^{8} b}^{17}}{12884901888}

(23)

After utilizing of equation (23), $ε$ -series of amplitude, phase and solution are transformed to $a$ as

1 + ε B_{1} + \dots + ε^{8} B_{8} = 1 + \frac{ε a^{2}}{2} + \frac{13 ε^{2} a^{4}}{64} - \frac{159 ε^{3} a^{6}}{1024} + \frac{87 ε^{4} a^{8}}{512} + \dots + \frac{8961987 ε^{8} a^{16}}{33554432}

(24)

1 + ε C_{1} + \dots + ε^{8} C_{8} = 1 + \frac{ε a^{2}}{2} - \frac{ε^{2} a^{4}}{32} + \frac{ε^{3} a^{6}}{64} - \frac{77 ε^{4} a^{8}}{8192} + \dots - \frac{1419349 ε^{8} a^{16}}{536870912}

(25)

and

x = a \cos θ - \frac{{ε a}^{3}}{16} (\cos 3 θ - \cos θ) + \frac{{3 ε^{2} a}^{5}}{256} (\cos 5 θ + 2 \cos 3 θ - 3 \cos θ)

- \frac{{ε^{3} a}^{7}}{12288} (37 \cos 5 θ + 122 \cos 5 θ + 108 \cos 3 θ - 267 \cos θ) + \dots

(26)

According to the modified KBM method,³¹ equations (24) and (25) are expressed in $α$ -series as

1 + ε B_{1} + \dots + ε^{8} B_{8} = 1 + α + \frac{29 α^{2}}{16} + \frac{177 α^{3}}{128} + \frac{311 α^{4}}{128} + \frac{13011 α^{5}}{8192} + \dots + \frac{5887799 α^{8}}{1572864}

(27)

1 + ε C_{1} + \dots + ε^{8} C_{8} = 1 + α + \frac{7 α^{2}}{8} + \frac{7 α^{3}}{8} + \frac{435 α^{4}}{512} + \frac{435 α^{5}}{512} + \dots + \frac{1745835 α^{8}}{2097152}

(28)

In the case of large oscillations, $α \to 1$ . In this situation, both series are divergent. However, multiplying these series by $1 - α$ , the coefficients of them are much reduced. So that the amplitude and phase equations become

\dot{a} = - \frac{μ a}{2} / \sqrt{ρ_{a} (1 + \frac{a^{2}}{2})}

(29)

where

ρ_{a} = 1 + \frac{13 α^{2}}{16} - \frac{55 α^{3}}{128} + \frac{67 α}{64} - \frac{6893 α^{5}}{8192} + \frac{143827 α^{6}}{98304} - \frac{2205131 α^{7}}{1572864} + \frac{548931 α^{8}}{262144}

and

\dot{θ} = 1 / \sqrt{ρ_{θ} (1 + \frac{a^{2}}{2})}

(30)

where

ρ_{θ} = 1 - \frac{α^{2}}{8} - \frac{13 α^{4}}{512} - \frac{45 α^{6}}{4096} - \frac{12885 α^{8}}{2097152}

Equation (29) is analytically solved to obtain the amplitude $a$ with the initial condition $a (0) = a_{0}$ . Substitute this value into equation (30), and the phase value $θ$ is numerically solved.

Results and discussion

The KBM method was originally used to solve some weak nonlinear problems with small damping effects. Then its extended versions^17,18 were implemented to solve strong nonlinear problems with significant damping effects. In a recent published paper,³¹ the method was modified to solve the cubic Duffing oscillator in the presence of a linear damped force with large oscillations. However, this technique is not directly used for handling some nonlinear oscillators when the nonlinear function $f$ is also dependent on the inertia force $\ddot{x}$ . In this paper, equation (16) ¹⁹ is considered and an approximate solution is obtained for different initial conditions and system parameters which are shown in Figures 1–8.

The comparison between approximated solutions with the corresponding numerical ones (Runge-Kutta fourth-order method) of equation (16) for different initial amplitudes $a = 0.25$ , $a = 0.5$ , $a = 1.0$ and the system parameters $ε = 1$ , $μ = 0.05$ have been shown in Figures 1–3. Also, in Figures 4–6, the obtained solutions have been compared with the corresponding numerical ones for different initial amplitudes $a = 0.25$ , $a = 0.5$ , $a = 1.0$ and the system parameters $ε = 1$ , $μ = 0.1$ . The values of $ρ_{a}$ and $ρ_{θ}$ are almost equal to $1$ when $a \leq 1$ . Under these circumstances, the amplitude decreases exponentially, and the presented results show excellent agreement with the numerical results.

In Figures 7 and 8, the comparison has been shown between approximate solutions of equation (16) for initial amplitude $a = 2.0$ and $a = 3.0$ and the system parameters $ε = 1$ , $μ = 0.1$ together with corresponding numerical ones. When the initial amplitude $a > 1$ , the values of $ρ_{a}$ and $ρ_{θ}$ are changed swiftly and the amplitude decreases linearly up to the amplitude $a = 1$ . After that, it decreases exponentially and slightly deviates from the numerical results.

Conclusion

In this paper, an alternative modified KBM method has been implemented to solve nonlinear oscillators where the inertia type force is involved in the nonlinear function along with the restoring forces and in the presence of linear damping. The solutions obtained for various system parameters as well as various initial values of $x (0)$ have been compared to numerical solutions. It indicates that the solutions are excellent for $a \leq 1$ when $ε = 1$ and different values of damping coefficient. The approximated solutions show a good coincidence as compared to numerical ones for different initial amplitudes and system parameters. The proposed alternative modification of the KMB method has removed the limitation of the existing one and reveals the novelty, reliability and wider applicability in nonlinear science and engineering.

Footnotes

Acknowledgments

The authors are really grateful to the honorable editor and reviewers for their constructive suggestions/comments to enrich the quality of this paper.

Declaration of conflicting interests

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

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Md. Mohaiminul Islam

Md. Zahangir Alam

Md. Alal Hosen

References

Jones

. Remarks on the perturbation process for certain conservative systems. Int J Non Lin Mech 1978; 13(2): 125–128.

Azad

Hosen

Rahman

, et al. A perturbation technique to compute initial amplitude and phase for the Krylov-Bogoliubov-Mitropolskii method. Tamkang J Math 2012; 43(4): 563–575.

Elliott

Tehrani

Langley

. Nonlinear damping and quasi-linear modelling. Phil Trans R Soc A 2015; 373(2051): 20140402.

Mitropolskii

Van Dao

. Applied asymptotic methods in nonlinear oscillations. Springer Science & Business Media, 2013, vol 55.

Mohammadian

Shariati

. Application of AG method and its improvement to nonlinear damped oscillators. Sci Iran 2020; 27(1): 203–214.

Mohammadian

. Approximate analytical solutions to nonlinear damped oscillatory systems using a modified algebraic method. J Appl Mech Tech Phys 2021; 62: 70–78.

Cheung

Chen

Lau

. A modified Lindstedt-Poincaré method for certain strongly non-linear oscillators. Int J Non Lin Mech 1991; 26(3-4): 367–378.

. Modified Lindstedt-Poincare methods for some strongly non-linear oscillations: part I: expansion of a constant. Int J Non Lin Mech 2002; 37(2): 309–314.

Amore

Aranda

. Improved Lindstedt–Poincaré method for the solution of nonlinear problems. J Sound Vib 2005; 283(3-5): 1115–1136.

10.

Ramos

. An artificial parameter–Linstedt–Poincaré method for oscillators with smooth odd nonlinearities. Chaos Solitons Frac 2009; 41(1): 380–393.

11.

Alam

Yeasmin

Ahamed

. Generalization of the modified Lindstedt-Poincare method for solving some strong nonlinear oscillators. Ain Shams Eng J 2019; 10(1): 195–201.

12.

Yeasmin

Sharif

Rahman

, et al. Analytical technique for solving the quadratic nonlinear oscillator. Results Phys 2020; 18: 103–303.

13.

Yeasmin

Rahman

Alam

. The modified Lindstedt-Poincare method for solving quadratic nonlinear oscillators. J Low Freq Noise Vib Act Control 2021; 40(3): 1351–1368.

14.

Krylov

Bogoliubov

. Introduction to non-linear mechanics. Princeton University Press, 1950.

15.

Bogoliubov

Mitropolskii

. Asymptotic methods in the theory of non-linear oscillations. CRC Press, 1961, vol 10.

16.

Nayfeh

. Perturbation methods. Wiley, 1973.

17.

Popov

. A generalization of the bogolubov asymptotic method in the theory of non-linear oscillations. Dokl Akad Nauk SSSR 1956; 111(2): 308–311.

18.

Mendelson

. Perturbation theory for damped nonlinear oscillations. J Math Phys 1970; 11(12): 3413–3415.

19.

Liao

. An approximate solution technique not depending on small parameters: a special example. Int J Non Lin Mech 1995; 30(3): 371–380.

20.

Van Horssen

Ter Brake

. On the multiple scales perturbation method for difference equations. Nonlinear Dyn 2009; 55: 401–418.

21.

Oliveira

. History of krylov-bogoliubov-mitropolsky methods of nonlinear oscillations. Adv Hist Stud 2017; 6(1): 40–55.

22.

Sattar

. An asymptotic method for second-order critically damped nonlinear equations. J Franklin Inst 1986; 321(2): 109–113.

23.

Vázquez

. The nonlinearly damped oscillator. ESAIM Control, Optim Calc Var 2003; 9: 231–246.

24.

Shamsul Alam

Roy

Rahman

, et al. An analytical technique to find approximate solutions of nonlinear damped oscillatory systems. J Franklin Inst 2011; 348(5): 899–916.

25.

Sugie

Ishihara

. Uniform global asymptotic stability for oscillators with nonlinear damping and nonlinear restoring terms. Commun Nonlinear Sci Numer Simul 2021; 103: 105969.

26.

Jian

Kuksin

. Krylov-Bogolyubov averaging. Russ Math Surv 2020; 75(3): 427–444.

27.

Alhejaili

Salas

El-Tantawy

. The Krylov-Bogoliubov-Mitropolsky method for modeling a forced damped quadratic pendulum oscillator. AIP Adv 2023; 13(8): 085029.

28.

Amer

Abady

. On the application of KBM method for the 3-D motion of asymmetric rigid body. Nonlinear Dyn 2017; 89: 1591–1609.

29.

Amer

Elkafly

Galal

. The 3D motion of a charged solid body using the asymptotic technique of KBM. Alex Eng J 2021; 60(6): 5655–5673.

30.

Amer

Farag

Amer

. The dynamical motion of a rigid body for the case of ellipsoid inertia close to ellipsoid of rotation. Mech Res Commun 2020; 108: 103583.

31.

Alam

Yeasmin

, et al. A modified Krylov-Bogoliubov-Mitropolskii method for solving damped nonlinear oscillators with large oscillation. Int J Non Lin Mech 2023; 155: 104459.