Modification of the Large Parameter Approach for the Periodic Solutions of Nonlinear Dynamical Systems

Abstract

This paper focuses on the modification of the large parameter approach (LPA), a novelty procedure, for estimating the periodic solutions of two degrees-of-freedom (DOF) autonomous quasi-linear systems with a first integral. This strategy is crucial because it provides an effective approach to recognizing approximate solutions to problems for which it is impossible to obtain exact solutions. These problems arise in the fields of physics, engineering, aerospace, and astronomy. They can be solved analytically using several perturbation approaches that depend on a small parameter that can be recognized according to the initial conditions and the body parameters of each problem. Therefore, we propose a large parameter instead of a small one to solve the aforementioned 2DOF systems, as well as provide a comparison between the suggested procedure and the previous approaches.

1. Introduction

In this section, we will take a closer look at some innovative perturbation approaches used by physicists and mathematicians to obtain the required approximate solutions for problems that cannot be fully solved. These methods are used in a wide range of areas, including classical mechanics, elastic body mechanics, oscillators, gyroscopic systems, and rotational dynamics [1,2,3,4,5]. These problems have received a lot of research attention from many distinguished researchers throughout the past two centuries due to their importance in many applications in both civilian and military life, such as in devices that measure direction or maintain stability in aircraft, spacecraft, and submarines.

The governing system (GS) of several problems of rigid body (RB) rotational motion has been solved using the approach of small parameter (ASP) [6,7], such as in [8,9,10,11]. In Ref. [8], the rotatory motion of the RB under the influence of the Newtonian field of force (NFF) was examined using the aforementioned approach to obtain its approximate periodic solutions. It was found that these solutions have singular points for the integer values of the body’s natural frequency. To address these singularities, the impact of the gyrostatic moment (GM) on the body’s principal axes was taken into account to generalize this problem; as in [9], when the body was acted upon by the third component of this moment. Moreover, the full action of this moment was applied in [10] and [11], respectively, for the Euler–Poinsot and Kovalevskaya cases when the influence of the gravitational field was taken into account. The lack of singular points in the obtained solutions adds to the significance and influence of the GM on the body’s motion because it addresses all of these points, which have previously been identified in [9,12,13]. Additionally, ref. [14] used the same technique to obtain the solutions of a symmetric RB near the Lagrange gyroscope when it is influenced by a field of gravity. It is assumed that the body’s center of mass is situated in very close proximity to the axis of dynamic symmetry. This problem was broadened in [15] when the effects of all projections of the GM on the body’s main axis were taken into account. In addition to the action of just two projections of the GM in [16,17], the influence of the NFF was also considered. To demonstrate the accuracy of the analytical results, a comparison between the obtained analytical outcomes and the numerical ones was provided in [17].

In Refs. [18,19], it was discussed how to deal with the turbulent rotational motion of an RB close to a uniform precession motion with a variable recovery moment under some applied limiting circumstances relating to the body’s angular velocity and turbulent moments. Solutions of the GS were achieved up to approximations of the first and second orders. This issue was studied in [20], where the non-zero third component of the GM was applied and generalized in [21]. More extensions of this problem were found in [22,23] for the body’s motion in an NFF besides the influence of the GM. Some related applications to dissipative torque, projections of angular velocity, and scenarios of atmospheric symmetric RB and tiny torque were presented in [24] for the motion of a charged body in the presence of external torques and forces.

Other perturbation approaches [25] that depend on the existence of the small parameter, such as Krylov–Bogoliubov–Mitropolski, averaging, multiple scales, and others, which are used to find approximate solutions of the GS of several dynamical systems, were used in [26,27,28,29,30,31,32]. In most of the relevant earlier works, the ASP was used to construct orders of analytic periodic solutions, where the small parameter is inserted based on the premise that it is inversely proportional to one component of the angular velocity, which is assumed to be sufficiently large around the minor or major axes of the body’s inertia ellipsoid. Nobody considered looking for an appropriate tiny value for the component of the body’s angular velocity. Such a presumption permits the problem to have a new domain. The ASP was used in [27] to investigate the periodic solutions to RB rotatory motion. It was assumed that the center of the body’s mass is significantly different from the dynamic axis of symmetry, and it was considered the fixed point of rotation. These solutions were evaluated in [28] where the NFF was applied to the body’s motion. In [29], the averaging method was used to obtain the approximate solutions of the GS of motion when it was acted upon by NFF and an electromagnetic field. In [30], the rotatory problem of a dynamically spherical RB coupled with a viscoelastic element was described. This element was modeled as a moving mass connected to a point along the major axis of inertia by a spring and damper. The motion was examined in the framework of a nutation angle-dependent restoring moment. Finally, the numerical outcomes of this problem were looked at in [33,34] in the context of the governing averaging scheme when the body is affected by restoring and perturbing moments. According to these studies, the body has a high initial angular velocity around the dynamic principal axis of inertia, and the desired solutions are obtained using the averaging method.

This study modifies the LPA for estimating the periodic solutions of 2DOF autonomous quasi-linear systems, which represents its novelty. This approach is essential because it offers a practical way to obtain approximate analytic solutions for problems whose exact solutions are elusive. The idea of this modification and its applications to time-independent systems can be illustrated below.

2. A System with 2DOF

In [33], the author used the small parameter theory method for constructing $2\pi$-periodic solutions of a 2DOF system which do not depend on time explicitly. In this work, we suppose that the conditions of the motion cannot achieve the small parameter. In this case, we must use a large one $\epsilon$ and qualify the LPA for solving the following assumed 2DOF system [34]

$$\begin{array}{l}{a}_{11}{\ddot{x}}_1+a_{12}{\ddot{x}}_2+c_{11}{x}_1+c_{12}{x}_2={\epsilon }^{-1}{F}_1(x_1,x_2,{\dot{x}}_1,{\dot{x}}_2,{\epsilon }^{-1}),\\ a_{21}{\ddot{x}}_1+a_{22}{\ddot{x}}_2+c_{21}{x}_1+c_{22}{x}_2={\epsilon }^{-1}{\varphi }_1(x_1,x_2,{\dot{x}}_1,{\dot{x}}_2,{\epsilon }^{-1}),\end{array}$$

(1)

where $a_{kl},c_{kl}(k,l=1,2)$ are constants and $F_1,{\varphi }_1$ represent analytic functions in their arguments. At $\epsilon \to \infty$, one obtains a generating system that is linearly conservative, with coefficients that are constants, where

$$a_{12}=a_{21}, c_{12}=c_{21}.$$

(2)

The correct periodic solutions of the generating system of (1) may be written in the form:

$$x_k=A_k{e}^{i\omega t}, (k=1,2),$$

(3)

where $\omega$ is the frequency. Substituting (3) into the generating system of (1) then leads to the following frequency equation,

$$\left|\begin{array}{cc}{c}_{11}-{\omega }^2{a}_{11}& c_{12}-{\omega }^2{a}_{12}\\ c_{21}-{\omega }^2{a}_{21}& c_{22}-{\omega }^2{a}_{22}\end{array}\right|=0.$$

(4)

The solution of Equation (4) yields the roots ${\omega }_1^2$ and ${\omega }_2^2$ of the frequency equation, and accordingly, we find periodic solutions in three different cases, as follows:

1

The two frequencies are commensurate.

2

The two frequencies are equal.

3

The two frequencies are non-commensurate.

System (1) can be rewritten in the form of a special case as follows:

$$\begin{array}{l}{\ddot{x}}_1+{\omega }^2{x}_1={\epsilon }^{-2}{F}_1(x_1,{\dot{x}}_1,x_2,{\dot{x}}_2,{\epsilon }^{-1}),\\ {\ddot{x}}_2+x_2={\epsilon }^{-2}{\varphi }_1(x_1,{\dot{x}}_1,x_2,{\dot{x}}_2,{\epsilon }^{-1}),\end{array}$$

(5)

where

$$F={\displaystyle \sum _{k=2}^{\infty }{\epsilon }^{2-k}}{F}_k, \varphi ={\displaystyle \sum _{k=2}^{\infty }{\epsilon }^{2-k}}{\varphi }_k, \dot{x}\equiv \frac{dx}{d\tau },$$

(6)

This system has the first integral [35]

$$x_2^2+{\dot{x}}_2^2+2{\epsilon }^{-1}(\nu x_1{x}_2+{\nu }_1{\dot{x}}_1{\dot{x}}_2+s_{21})+{\epsilon }^{-2}(\dots )=L,$$

(7)

where $\nu ,{\nu }_1,$ and $L$ are known constants, and $s_{21}=s_{21}(\tau ,{\epsilon }^{-1})$ satisfies the condition $s_{21}(0,{\epsilon }^{-1})=0$.

At $\epsilon \to \infty$, one can obtain the generated system of (5), which has the frequencies of $\omega$ and 1. Therefore, the $2\pi$-periodic required solutions of system (5) are classified in three different cases; $\omega$ is a rational number, $\omega =1$, and $\omega$ is an irrational number.

3. Commensurate Case

In this section, we present the independent periodicity conditions for the required general solutions with a period correction from their generating ones. Also, we determine the initial basic amplitudes of the generating solutions and their deviations from the general solutions. Therefore, we will consider the first case in which the frequency $\omega =m/n$ is such that $m$ and $n$ are integer numbers and $n\ne 0$. In this case, the generated system (5) produces periodic solutions with a period $T_0=2\pi n$ that can be written as follows:

$$\begin{array}{l}{x}_1(\tau ,0)=x_1^{(0)}=M_1\cos \omega \tau +M_2\sin \omega \tau ,\\ x_2(\tau ,0)=x_2^{(0)}=M_3\cos \tau ,\end{array}$$

(8)

where $M_j(j=1,2,3)$ are independent constants on $\epsilon$. Then, the general required solutions of a system (5) that fulfill the below initial circumstances are searched for, see Figure 1 and Figure 2

$$\begin{array}{l}{x}_1(0,{\epsilon }^{-1})=M_1+{\beta }_1, {\dot{x}}_1(0,{\epsilon }^{-1})=\omega (M_2+{\beta }_2),\\ x_2(0,{\epsilon }^{-1})=M_3+{\beta }_3, {\dot{x}}_2(0,{\epsilon }^{-1})=0,\end{array}$$

(9)

where ${\beta }_j$ are the dependent analytic functions on $\epsilon$ that equal zero at $\epsilon \to \infty$.

Since the elementary system (5) does not explicitly depend on time, the conditions (9) do not preclude the generality of solutions.

Based on the generated periodic solution (8), we can assume that the general periodic solutions of system (5) are of the period $(T_0+\alpha )$, where $\alpha$ is a dependent analytical function of $\epsilon$ that can be vanished at $\epsilon \to \infty$, and has the following form:

$$\begin{array}{l}{x}_1(\tau ,{\epsilon }^{-1})=(M_1+{\beta }_1)\cos \omega \tau +(M_2+{\beta }_2)\sin \omega \tau +{\displaystyle \sum _{k=2}^{\infty }{\epsilon }^{-k}}{G}_k(\tau ),\\ x_2(\tau ,{\epsilon }^{-1})=(M_3+{\beta }_3)\cos \tau +{\displaystyle \sum _{k=2}^{\infty }{\epsilon }^{-k}}{H}_k(\tau ),\end{array}$$

(10)

in which it can be reduced to the generated solutions (8) at $\epsilon \to \infty$. The functions $G_k(\tau )$ and $H_k(\tau )$ are expressed in Maclaurin’s series as follows:

$$G_k(\tau )=g_k(\tau )+\frac{\partial g_k}{\partial M_1}{\beta }_1+\frac{\partial g_k}{\partial M_2}{\beta }_2+\frac{\partial g_k}{\partial M_3}{\beta }_3+\frac{1}{2}\frac{{\partial }^2{g}_k}{\partial M_1^2}{\beta }_1^2+\cdots ,$$

(11)

$$H_k(\tau )=h_k(\tau )+\frac{\partial h_k}{\partial M_1}{\beta }_1+\frac{\partial h_k}{\partial M_2}{\beta }_2+\frac{\partial h_k}{\partial M_3}{\beta }_3+\frac{1}{2}\frac{{\partial }^2{h}_k}{\partial M_1^2}{\beta }_1^2+\cdots .$$

(12)

Also, by decomposing the functions $F_k$ and ${\varphi }_k$ in a neighborhood to the ${\epsilon }^{-1}={\beta }_j=0$, on the right-hand side of a system (5), we obtain:

$$\begin{array}{ll}{F}_k(x_1,{\dot{x}}_1,x_2,{\dot{x}}_2)& =F_k(x_1^{(0)},{\dot{x}}_1^{(0)},x_2^{(0)},{\dot{x}}_2^{(0)})+{(\frac{\partial F_k}{\partial x_1})}_0{\beta }_1+{(\frac{\partial F_k}{\partial {\dot{x}}_1})}_0{\beta }_2\\ & +{(\frac{\partial F_k}{\partial x_2})}_0{\beta }_3+\frac{1}{2}{(\frac{{\partial }^2{F}_k}{\partial {x_1}^2})}_0{\beta }_1^2+\cdots .\end{array}$$

(13)

Substituting Equations (10)–(13) into the elementary system (5) and then equating the terms with similar powers of $\epsilon$ on both sides obtains the differential equations below that define the independent functions $g_k(\tau )$ and $h_k(\tau )$ on $\epsilon$

$$\begin{array}{ll}{\ddot{g}}_k+{\omega }^2{g}_k& =F_k(x_1^{(0)},{\dot{x}}_1^{(0)},x_2^{(0)},{\dot{x}}_2^{(0)})\equiv F_k^{(0)},\\ {\ddot{h}}_k+h_k& ={\varphi }_k(x_1^{(0)},{\dot{x}}_1^{(0)},x_2^{(0)},{\dot{x}}_2^{(0)})\equiv {\varphi }_k^{(0)}.\end{array}$$

(14)

The integration of these equations yields:

$$\begin{array}{l}{g}_k(\tau )=\frac{1}{\omega }{\displaystyle \underset{0}{\overset{\tau }{\int }}{F}_k^{(0)}}(t_1)\sin \omega (\tau -t_1)d{t}_1,\\ h_k(\tau )={\displaystyle \underset{0}{\overset{\tau }{\int }}{\varphi }_k^{(0)}}(t_1)\sin \omega (\tau -t_1)d{t}_1.\end{array}$$

(15)

The quantities $M_1, \omega M_2,$ and $M_3$ represent the initial values for the generated solutions given by (8), while the deviations ${\beta }_1({\epsilon }^{-1}), \omega {\beta }_2({\epsilon }^{-1}),$ and ${\beta }_3({\epsilon }^{-1})$, with the correction of the period $\alpha$ can be determined from the following periodicity conditions of the solutions $x_1,x_2$ and their first derivatives:

$$\begin{array}{l}{\psi }_1=x_1(T_0+\alpha ,{\epsilon }^{-1})-x_1(0,{\epsilon }^{-1})=0, {\psi }_2={\dot{x}}_1(T_0+\alpha ,{\epsilon }^{-1})-{\dot{x}}_1(0,{\epsilon }^{-1})=0,\\ {\psi }_3=x_2(T_0+\alpha ,{\epsilon }^{-1})-x_2(0,{\epsilon }^{-1})=0, {\psi }_4={\dot{x}}_2(T_0+\alpha ,{\epsilon }^{-1})-{\dot{x}}_2(0,{\epsilon }^{-1})=0.\end{array}$$

(16)

As a result of the existence of integral (7) for system (5), we observe that one of the periodicity conditions (16) is not independent, namely, ${\psi }_3=0$, because it is automatically realized if all the other three conditions are met, i.e., $({\psi }_1={\psi }_2={\psi }_4=0)$.

Based on the integral (7) and the conditions (9) at $\tau =0$, we obtain the equations that are fulfilled by $M_3$ and ${\beta }_3$, as follow

$$0<M_3=\sqrt{L}<\infty , {\beta }_3=-{\epsilon }^{-1}\nu (M_1+{\beta }_1)+\cdots .$$

(17)

Expanding the periodicity requirements (16) regarding the powers of $\alpha$ and keeping only the first-order terms to obtain the following solutions’ periodicity requirements leads to

$$\begin{array}{l}{x}_1(T_0,{\epsilon }^{-1})-M_1-{\beta }_1+\alpha \omega (M_2+{\beta }_2)=0,\\ {\dot{x}}_1(T_0,{\epsilon }^{-1})-\omega (M_2+{\beta }_2)-\alpha {\omega }^2(M_1+{\beta }_1)=0,\\ {\dot{x}}_2(T_0,{\epsilon }^{-1})-\alpha (M_3+{\beta }_3)=0.\end{array}$$

(18)

Using the last equations in (18) and (10), we can achieve the following correction of the period:

$$\alpha =\frac{{\dot{x}}_2(T_0,{\epsilon }^{-1})}{(M_3+{\beta }_3)}={\epsilon }^{-2}[{\dot{H}}_2(T_0)+{\epsilon }^{-1}{\dot{H}}_3(T_0)+\dots ]/(M_3+{\beta }_3).$$

(19)

With the substitution of the above period’s correction (19) and the solution (10) into the first two equations in (18), we can obtain the next system that determines $M_1, M_2, {\beta }_1,$ and ${\beta }_2$

$$\begin{array}{l}{G}_2(T_0)+{\epsilon }^{-1}{G}_3(T_0)+\omega (M_2+{\beta }_2)[{\dot{H}}_2(T_0)+{\epsilon }^{-1}{\dot{H}}_3(T_0)+\cdots ]/(M_3+{\beta }_3)+\cdots =0,\\ {\dot{G}}_2(T_0)+{\epsilon }^{-1}{\dot{G}}_3(T_0)-{\omega }^2(M_1+{\beta }_1)[{\dot{H}}_2(T_0)+{\epsilon }^{-1}{\dot{H}}_3(T_0)+\cdots ]/(M_3+{\beta }_3)+\cdots =0.\end{array}$$

(20)

We observe that the case $\omega =1$ is distinguished from the current comparable situation as a special case, in which the period for the generating system is $T_0=2\pi$.

4. Non-Commensurate Case

In this case, $\omega$ is considered to be an irrational number, and the period of the generating solutions is $T_0=2\pi$. According to the second equation of the generating system from (5), one can write the generating solutions in the forms:

$$x_1=0, x_2=M_3\cos \tau .$$

(21)

Therefore, the general periodic solutions for this case take the form [36]:

$$\begin{array}{l}{x}_1(\tau ,{\epsilon }^{-1})={\eta }_1({\beta }_3,{\epsilon }^{-1})\cos \omega \tau +{\eta }_2({\beta }_3,{\epsilon }^{-1})\sin \omega \tau +{\displaystyle \sum _{k=2}^{\infty }{\epsilon }^{-k}}{G}_k(\tau ),\\ x_2(\tau ,{\epsilon }^{-1})=(M_3+{\beta }_3)\cos \tau +{\displaystyle \sum _{k=2}^{\infty }{\epsilon }^{-k}}{H}_k(\tau ),\end{array}$$

(22)

where ${\beta }_3={\beta }_3({\epsilon }^{-1})$ vanishes when $\epsilon \to \infty$. The functions ${\eta }_q({\beta }_3,{\epsilon }^{-1}), (q=1,2)$ are then determined since ${\eta }_q({\beta }_3,0)=0$. Therefore, we can write functions such as those in the following forms of series:

$${\eta }_q({\beta }_3,{\epsilon }^{-1})={\displaystyle \sum _k^{\infty }(Q_k^{(q)}}+\frac{\partial Q_k^{(q)}}{\partial M_3}{\beta }_3+\frac{1}{2}\frac{{\partial }^2{Q}_k^{(q)}}{\partial {M_3}^2}{{\beta }_3}^2+\cdots ){\epsilon }^{-k}.$$

(23)

Replacing $(M_1+{\beta }_1)$ and $(M_2+{\beta }_2)$, respectively, by ${\eta }_1({\beta }_3,{\epsilon }^{-1})$ and ${\eta }_2({\beta }_3,{\epsilon }^{-1})$, due to the periodicity criteria (18) and the series (23) gives us an infinite number of equations that satisfy the coefficients $Q_k^{(q)}$.

5. Discussion of the Obtained Results

In this section, we present the significance of the obtained results. From Equation (13), we can find the analytical functions $F_k(x_1,{\dot{x}}_1,x_2,{\dot{x}}_2)\to F_k({x_1}^{(0)},{{\dot{x}}_1}^{(0)},{x_2}^{(0)},{{\dot{x}}_2}^{(0)})$ when ${\epsilon }^{-1}={\beta }_j=0 (j=1,2,3)$, which are the required solutions, and they tend to be the ones generated in the neighborhood of ${\epsilon }^{-1}$. In this case, the analytical function $\alpha$ tends to zero when the general period $\tau$ tends to $T_0=2\pi$. Therefore, the analytical functions ${\beta }_j$ in $\epsilon$ vanish at $\epsilon \to \infty$. That is, the functions ${\eta }_q({\beta }_3,{\epsilon }^{-1})$ in (23) are reduced to ${\displaystyle \sum _k^{\infty }{Q}_k^{(q)}}{\epsilon }^{-k}$ in the neighborhood of ${\epsilon }^{-1}$. According to the periodicity conditions (18), the obtained solutions (22) are independent periodic ones, in which they are dependent on the defined analytical functions $G_k(\tau )$ and $H_k(\tau )$.

6. Conclusions

In this paper, the LPA has been modified and adapted to solve an autonomous quasi-linear system consisting of 2DOF nonlinear second-order differential equations. A sufficiently large parameter $\epsilon \to \infty$ has been defined separately for each problem according to the inverse proportional to one of the projections of the angular velocity on the RB’s principal axes (assumed to be sufficiently small) for the minor or major axis of the body’s inertia ellipsoid, in addition to suitable conditions on the mass center position. This technique produces a new domain for the solutions of the RB problem to be examined according to new initial circumstances. This technique can be applied to discover problems’ solutions in aerospace engineering, astronomy, vibrating systems, nonlinear slow-spin dynamical systems, and others. These solutions are in agreement with the numerical ones that can be gained by numerical methods. Moreover, it solves such problems in a complement domain of ASP; that is, if this approach solves the problems in the domain $f(t,r_0\to \infty ,\epsilon \to 0)$, the LPA solves them in the complement domain $f(t,r_0\to 0,\epsilon \to \infty )$, which is the angular velocity component. It saves a lot of the body’s kinetic energy, which initially moves the body to start the spin motion. The solutions and the geometric interpretation of the body motion can be obtained in terms of the large parameter, slow velocity, and time. Thus, many problems with the pendulum’s motion can be solved in view of the new definition of the large parameter strategy in this article. It is important to realize that ASP failed to solve these problems in our new domain because the discovery of the small parameter in this domain is impossible.