A slow rotary dynamic motion of a disc under the influence of torque with the concept of a large parameter

Abstract

A rotary motion is studied, which is used as a paradigm in both mathematics and dynamics. In this motion, the body moves around its axis. Important examples of this motion are the earth, wheels for cars or bicycles, cooling fans, airplanes, radars, and motion of the discs. In this article, we consider one of these motions such as the slow rotational motion of a disc of high natural frequency around a fixed point different from its center of mass. Suppose a constant gyrostatic couple acts on the body about the axis of symmetry. Let us consider the axis of symmetry coincide with one of the main axes of inertia. The controlled nonlinear independent system of equations of motion is obtained in the form of six nonlinear differential equations in the presence of three independent first integrals. This system is reduced to an alternative independent quasi-linear system consisting of two differential equations with only one integral. At first, when the body rotates slowly with a small angular speed about the axis of the symmetry, we achieve a large parameter for the motion which is proportional to the inverse of the angular velocity component. Then, we use the large parameter technique to obtain approximate and high-frequency analytic solutions for this motion. This technique enables us to introduce new conditions of the motion, save a lot of energy required to move the disc, and obtain new approximated solutions in a new domain of the disc parameters. We present a digital example of the problem to clarify the inferences of the motion depending on the parameters of the body. On the other hand, we use the Runge–Kutta algorithmic method to obtain the numerical solutions corresponding to the approximate solutions of the considered independent system. We investigate the inaccuracies and errors of analytic and numerical solutions by comparing them. This problem offers many applications in gyroscope dynamics, industrial engineering, astrophysics, navigation, satellites, and space engineering.

Keywords

Rotational bodies motion euler’s equations gyro’s dynamics mathematical methods numerical methods

Introduction

The problem of the rotational motion of a rigid body around its fixed point attracts the attention of many outstanding researchers in the last years due to its important applications in both civil and military life. In Ref. 1, the rotary motion of a disc around one fixed point acted upon by gravitational and Newtonian force fields is studied. The author has used the third component of the angular speed to achieve a large parameter for solving this problem. Using this technique, the equations of motion are obtained and solved. The analytic and digital solutions are obtained and compared. The geometric investigations of the motion are searched. In Refs. 2–5, the problems of asymmetric or symmetric rigid body when the body rotates in a gravitational or Newtonian or Electromagnetic fields are studied. The KBM technique is applied to construct and solve the system of equations of motion. In Ref. 6, the existence of important integrable cases for bodies rotating about a fixed point in the Newtonian force field is considered. The solutions for this problem are obtained by deriving the Euler–Poisson system of equations of motion. The periodic solutions for the problem are attained by using the small parameter method. On the other hand, the possibility of obtaining a fourth first integral for the body’s motion, in a simple way, is examined in Ref. 7. In Ref. 8, the evolution of rotating motions of a rigid body around its mass center is studied. The author derived the Euler–Poison motion equations for different problems. They presented the analytic solutions for these problems by many outstanding methods. They presented the geometrical illustrations for these motions. In Ref. 9, the authors studied the problem of rigid bodies in the presence of new initial conditions. They derived the equations of motion for these bodies and presented the averaging methods for solving the nonlinear systems to different dynamical models. In Ref. 10, the rigid body problems of different dynamical systems are studied. They presented many conditions and procedures for solving the integrable cases of the motions. They used Euler’s angles to study the behaviors of the bodies at every moment of the time. Nayfeh presented in Refs. 11and 12 many mathematical techniques for solving various problems of vibrations and rotational motions in engineering, physics, and many other fields of technology. Such methods are multiple scales, Poincare’s small parameter, averaging, large parameters, and other numeric ones. In Ref. 13, the authors studied a new type for solving techniques of Euler–Poisson systems for the rigid body rotating problems around a fixed point. Different analytic solutions for the rigid motions are obtained in Refs. 14–19 using the aforementioned methods when the influence of a gyrostatic moment is considered. In Ref. 20, the authors introduced a case like Lagrange’s one for evaluating the motions of a body acted upon by unsteady moments. The controlled equations of motions for this case are considered in presence of new restrictions. New procedures of the solutions are given and analyzed. In Ref. 21, the authors admit the rotary motions around a point that is fixed in space by applying hypergeometric functions. They considered new analytic solutions (which are non-complex). They integrated the herpolhode problem. The authors in Ref. 22 formulated new conditions for important integrable problems of gyrostatic body dynamics. They solved the conditional cases and investigated the description of their motions. In Refs. 23–25, Elmandouh presented new problems for integrating dynamics of rigid bodies and particles with quadratic and cubic integrals in velocities. The authors in Ref. 26 studied the Hamiltonian techniques for integrability cases and chaotic problems in the dynamic of rigid bodies. In Ref. 27, the authors presented some of the mathematical procedures for solving many interesting problems of theoretical mechanics and celestial ones. Ismail in Ref. 28 solved the heavy solid problem for rotatory motions by applying a method of a large parameter. He found the system for the motion equations and their first integrals applying the method of a large parameter, in which the graphical representations for the obtained numerical and analytical solutions with their stabilities are presented. In Ref. 19, Ershkov found the solutions of a Riccati-type for equations of Euler–Poisson concerning rigid body rotational motions over its fixed point. The author in Ref. 30 investigated the torques that applied a gyro dynamical motion. The numerical results of the equations of motion for the considered case are obtained, in which the corresponding first integrals of these equations are obtained. Sahli studied in Ref. 31 the gyrostatic problem in cases of zero and non-zero torques for the rigid body motions. The case of slight displacement of the center of mass of the rigid body under the action of the gyrostatic moment vector is investigated in Refs. 32 and 33, in which the approximate solutions are obtained when the ratio of the body’s frequency becomes irrational. Some generalizations of modern problems of the rigid body classical systems are considered in Ref. 34. Kovalevskaya and Lagrange’s cases are considered in problems of rigid body dynamical motions.^35,36 Leimanis presented in Ref. 37 some of the torqued general problems for the rigid body’s motions over its fixed points.

There are more relevant very important works focusing on different methods of approximate solutions of physical systems and mechanical ones. For example, the coupled and harmonic restoring force nonlinear oscillations and their stabilities analysis are studied in Refs. 38–42. The authors used some perturbed techniques like the rank upgrading one and presented approximated solutions and asymptotic investigations with stabilities properties for different problems.

Some latest methods of oscillations are studied in Refs. 43–45. The authors investigated the approximated solutions for some nonlinear oscillator problems using some perturbation techniques and numeric ones. An example of these techniques is the homotopy perturbation method and its modifications. Also, a variational principle is applied for some nonlinear oscillatory problems that come from micro-electromechanics models. In Refs. 46–49, the homotopy perturbation method and its modification are used to solve some problems for axial vibration for strings, the fractal Toda oscillator, and others. This method solved also the doubly clamped electrical actuated micro-beam-based micro-electromechanical systems.

This work presents the application of the large parameter technique^50–52 for considering the disc problem motion under new initial conditions, in a new domain, and in presence of the gyro torque around one of the principal axes of the body inertia ellipsoid. Initially, we give the body a sufficiently low angular speed component about one of these axes (weak oscillation of the disc). In this case, we cannot use Poincare’s method for solving this problem as known previously because this technique depends on a small parameter, which depends on a high angular speed, that cannot be achieved here. The only road to exiting from this complicated problem is by achieving a large parameter and using it through the large parameter technique to solve the problem.^1,28,50,52 The motion equations system is reduced, applying the first integrals, to a quasilinear independent one of 2 freedom degrees and 1 first integral. The analytic periodic solutions of the problem system are integrated and analyzed dynamically in presence of the first integrals of the motion. The resulted solutions are presented as new domain ones and considered as a generalization of the previous works. These solutions are drowned graphically to show the motion geometrical effect due to the body’s parameters. On the other side, the force order Runge–Kutta method is applied through a software program to find digital solutions for the autonomous system. The solutions are compared together. This problem has important applications in the aerospace engineering industry, and astronomy, for example, the radars, ship and airplanes stabilizers, satellites, racing cars, and pointing devices for the computer.

The problem considerations

Consider the rotary motion of a disc that is fixed at a point $O$ , which is defined as the origin of our dynamical system, acted upon by a gyrostatic torque component $ℓ_{z}$ around the z-axis of the inertia ellipsoid, see Figure 1. We take two coordinate systems of frames named; a fixed one $O X$ YZ and a moving one $O xyz$ , fixed in the body. Consider the moving axes taken to be in the direction of the ellipsoid of inertia axes.

Figure 1.

Describing the disc motion.

Taking into consideration that the disk rotates with a sufficiently low component $r_{0}$ of the angular speed vector around the z-axis. Let the z-axis rotate about Z with a nutation angle $θ_{0}$ which is nearer to $π / 2$ . Thus, the governing equations of the motion are reduced to the following system when putting the motion frequency = 1 in Ref. 37

\begin{array}{l} {\dot{p}}_{1} + q_{1} r_{1} = - {(A r_{0})}^{- 1} q_{1} ℓ_{z} + ε^{- 1} a^{- 1} y_{0}^{'} γ_{1}^{″}, {\dot{q}}_{1} - p_{1} r_{1} = {(B r_{0})}^{- 1} p_{1} ℓ_{z} - ε^{- 1} b^{- 1} x_{0}^{'} γ_{1}^{″}, \\ {\dot{r}}_{1} = ε^{- 2} (x_{0}^{'} γ_{1}^{'} - y_{0}^{'} γ_{1} - C_{1} p_{1} q_{1}), {\dot{γ}}_{1} - r_{1} γ_{1}^{'} = - ε^{- 1} q_{1} γ_{1}^{″}, {\dot{γ}}_{1}^{'} + r_{1} γ_{1} = ε^{- 1} p_{1} γ_{1}^{″}, {\dot{γ}}_{1}^{″} = ε^{- 1} (q_{1} γ_{1} - p_{1} γ_{1}^{'}) \end{array}

(1)

such that

\frac{p}{p_{1}} = \frac{q}{q_{1}} = c \sqrt{γ_{0}^{″}}, r = r_{0} r_{1}, (. \equiv d / d τ), \frac{γ}{γ_{1}} = \frac{γ^{'}}{γ_{1}^{'}} = \frac{γ^{″}}{γ_{1}^{″}} = γ_{0}^{″}, τ = r_{0} t, γ_{0} > 0, 0 < γ_{0}^{″} < 1;

(2)

\begin{array}{l} A + B = C, C C_{1} = (B - A), a C = A, b C = B, a + b = 1, ε = \frac{c \sqrt{γ_{0}^{″}}}{r_{0}}, c^{2} = M g l / C \\ x_{0} = l x_{0}^{'}, y_{0} = l y_{0}^{'}, l^{2} = x_{0}^{2} + y_{0}^{2} \end{array}

(3)

Here, $(A, B, C)$ are the principal inertia moments that generate $(a, b, C_{1})$ ; $(x_{0}, y_{0})$ are the mass center coordinates; $(γ, γ^{'}, γ^{″})$ are the $\hat{\bar{Z}}$ unit vector components in the fixed space direction Z that generate $(γ_{1}, γ_{1}^{'}, γ_{1}^{″})$ ; $(p, q, r)$ are the components of the angular speed that generate $(p_{1}, q_{1}, r_{1})$ ; $ℓ_{z}$ represents the gyrostatic torque vector component around the principal inertia z-axis; $M$ gives the body mass; $g$ refers to the acceleration of gravity; t is the time of the motion that generates the small time-motion τ; and $(r_{0}, γ_{0}, γ_{0}^{″})$ represent the corresponding quantities initial values.

This problem admits the following first independent integrals

r_{1}^{2} - 1 = ε^{- 2} S_{1}, r_{1} γ_{1}^{″} - 1 = ε^{- 1} S_{2}, γ_{1}^{2} + γ_{1}^{' 2} + γ_{01}^{' 2'' - 2}

(4)

where

\begin{array}{l} S_{1} = a (p_{10}^{2} - p_{1}^{2}) + b (q_{10}^{2} - q_{1}^{2}) - 2 [x_{0}^{'} (γ_{10} - γ_{1}) + y_{0}^{'} (γ_{10}^{'} - γ_{1}^{'})], \\ S_{2} = a (p_{10} γ_{10} - p_{1} γ_{1}) + b (q_{10} γ_{10}^{'} - q_{1} γ_{1}^{'}) + {(c C \sqrt{γ_{0}^{″}})}^{- 1} [ℓ_{z} (1 - γ_{1}^{″})] \end{array}

(5)

where

(p_{10}, q_{10}, γ_{10}, γ_{10}^{'})

are the initial values of the corresponding quantities.

Making use of system (1) and equation (4), the following variables are obtained

r_{1} - 1 = 0.5 ε^{- 2} S_{1} + \dots, γ_{1}^{″} - 1 = ε^{- 1} S_{2} - 0.5 ε^{- 2} S_{1} + \dots

(6)

Differentiating the equation number first and number four of a system (1), making use of equation (6), and considering $r_{0}$ is sufficiently small, therefore the quantities $r_{0}^{2}, r_{0}^{3}, \dots$ are neglected. Thus, the remaining equations from system (1) are reduced to

\begin{array}{l} {\ddot{p}}_{1} + ω^{' 2} p_{1} = ε^{- 1} b^{- 1} [x_{0}^{'} + {(A r_{0})}^{- 1} x_{0}^{'} ℓ_{z}] + ε^{- 2} {[- ω^{2} S_{1} p_{1} + b^{- 1} x_{0}^{'} S_{2} + C_{1} p_{1} q_{1}^{2} \\ - q_{1} (x_{0}^{'} γ_{1}^{'} - y_{0}^{'} γ_{1}) + a^{- 1} y_{0}^{'} (q_{1} γ_{1} - p_{1} γ_{1}^{'})] - 0.5 r_{0} -^{1} ℓ_{z} p_{1} {(A b)}^{- 1} S_{1} + {(A b r_{0})}^{- 1} x_{0}^{'} S_{2} ℓ_{z}} - 0.5 ε^{- 3} {(A b r_{0})}^{- 1} x_{0}^{'} S_{1} ℓ_{z} + \dots \end{array}

(7)

{\ddot{γ}}_{1} + γ_{1} = - ε^{- 1} {(B r_{0})}^{- 1} ℓ_{z} p_{1} + ε^{- 2} {x_{0}^{'} (γ_{1}^{' 2} + b^{- 1}) - γ_{1} [(y_{0}^{'} γ_{1}^{'} + q_{1}^{2}) + S_{1}]} + ε^{- 3} (2 b^{- 1} x_{0}^{'}) S_{2} + \dots,

(8)

where

ω^{' 2} = 1 + ℓ_{z} / A b r_{0}

Solving the 1st and the 4th equations of system (1), and then making use of equation (6), we get

r_{1} q_{1} = [1 - {(A r_{0} r_{1})}^{- 1} ℓ_{z} + \dots] (- {\dot{p}}_{1} + ε^{- 1} a^{- 1} y_{0}^{'} γ_{1}^{″}), r_{1} γ_{1}^{'} = ({\dot{γ}}_{1} - ε^{- 1} q_{1} γ_{1}^{″})

(9)

Replacing the variables

p_{1} = p_{2} + ε^{- 1} χ, γ_{1} = γ_{2} + ε^{- 1} a p_{2},

(10)

where

χ = x_{0}^{'} (1 + \frac{ℓ_{z}}{A r_{0}}) / b ω^{' 2}

Using the relations (10), we get

\begin{array}{l} q_{1} - X (a^{- 1} y_{3} - {\dot{p}}_{2}) = ε^{- 1} X a^{- 1} y_{0}^{'} + 0.5 ε^{- 2} [(2 X - 1) S_{11} {\dot{p}}_{2} + 2 X a^{- 1} y_{0}^{'} S_{21}] + \dots, \\ γ_{1}^{'} - {\dot{γ}}_{2} = ε^{- 1} (a - X) {\dot{p}}_{2} + 0.5 ε^{- 2} [2 X (a^{- 1} y_{0}^{'} - S_{21} {\dot{p}}_{2}) - S_{11} {\dot{γ}}_{2}] + \dots \end{array}

(11)

where

X = 1 - \frac{ℓ_{z}}{A r_{0}}, y_{3} = ℓ_{z} / c C \sqrt{γ_{0}^{″}}

We determine $S_{1}$ and $S_{2}$ by Substituting equations (11) and (10) into (5), to get

S_{i} = S_{i 1} + 2^{2 - i} ε^{- 1} S_{i 2} + \dots, (i = 1,2)

(12)

where

\begin{array}{l} S_{11} = a (p_{20}^{2} - p_{2}^{2}) + b X^{2} ({\dot{p}}_{20}^{2} - {\dot{p}}_{2}^{2}) - 2 [x_{0}^{'} (γ_{20} - γ_{2}) + y_{0}^{'} ({\dot{γ}}_{20} - {\dot{γ}}_{2})], \\ S_{12} = a (χ - x_{0}^{'}) (p_{20} - p_{2}) - y_{0}^{'} [a + X (b a^{- 1} X - 1)] ({\dot{p}}_{20} - {\dot{p}}_{2}), \\ S_{21} = a (p_{20} γ_{20} - p_{2} γ_{2}) - b X [({\dot{p}}_{20} {\dot{γ}}_{20} - {\dot{p}}_{2} {\dot{γ}}_{2})], \\ S_{22} = a [a (p_{20}^{2} - p_{2}^{2}) + χ (γ_{20} - γ_{2})] - b X [(a - X) ({\dot{p}}_{20}^{2} - {\dot{p}}_{2}^{2}) - a^{- 1} y_{0}^{'} ({\dot{γ}}_{20} - {\dot{γ}}_{2})] - y_{3} S_{21} \end{array}

(13)

Using the equations (12) and (13) into (6), we get

r_{1} - 1 = 0.5 ε^{- 2} S_{11} + ε^{- 3} S_{12} + \dots, γ_{1}^{″} - 1 = ε^{- 1} S_{21} + ε^{- 2} (S_{22} - 0.5 S_{11}) - ε^{- 3} S_{12} + \dots

(14)

Substituting the equations (10)–(14) into (4), (7), and (8), we get

{\ddot{p}}_{2} + ω^{' 2} p_{2} = ε^{- 2} F (p_{2}, {\dot{p}}_{2}, γ_{2}, {\dot{γ}}_{2}, ε^{- 1}), {\ddot{γ}}_{2} + γ_{2} = ε^{- 2} Φ (p_{2}, {\dot{p}}_{2}, γ_{2}, {\dot{γ}}_{2}, ε^{- 1}),

(15)

\begin{array}{l} γ_{2}^{2} + {\dot{γ}}_{2}^{2} + 2 ε^{- 1} [a γ_{2} p_{2} + (a - X) {\dot{p}}_{2} {\dot{γ}}_{2} + S_{21}] + ε^{- 2} [a^{2} p_{2}^{2} + {(a - X)}^{2} {\dot{p}}_{2}^{2} \\ + 2 X {\dot{γ}}_{2} (a^{- 1} y_{0}^{'} - S_{21} {\dot{p}}_{2}) + S_{21}^{2} + 2 (S_{22} - 0.5 S_{11})] = {(γ_{0}^{″})}^{- 2} - 1 \end{array}

(16)

where

\begin{array}{l} F = f_{1} + ε^{- 1} f_{2} + \dots, Φ = \frac{A^{2} r_{0}}{B ℓ_{z}} χ + ϕ_{1} + ε^{- 1} (ϕ_{2} - a f_{2}) + \dots, \\ f_{1} = x_{0}^{'} b^{- 1} S_{21} + C_{1} X^{2} p_{2} ({\dot{p}}_{2}^{2} - 2 a^{- 1} y_{0}^{'} {\dot{p}}_{2}) - [y_{0}^{'} X (1 + a^{- 1}) γ_{2} - x_{0}^{'} X {\dot{γ}}_{2}] {\dot{p}}_{2} - a^{- 1} p_{2} y_{0}^{'} {\dot{γ}}_{2} + 0.5 S_{11} p_{2} [ℓ_{z} {(A b r_{0})}^{- 1} - 2], \\ - x_{0}^{'} X {\dot{γ}}_{2}] {\dot{p}}_{2} - a^{- 1} p_{2} y_{0}^{'} {\dot{γ}}_{2} + 0.5 S_{11} p_{2} [ℓ_{z} {(A b r_{0})}^{- 1} - 2], ϕ_{1} = - γ_{2} S_{11} + x_{0}^{'} (b^{- 1} + {\dot{γ}}_{2}^{2}) - γ_{2} (y_{0}^{'} {\dot{γ}}_{2} + X^{2} {\dot{p}}_{2}^{2}) + 2 S_{21} x_{0}^{'} b^{- 1}, \\ f_{2} = b^{- 1} x_{0}^{'} S_{22} - (2 p_{2} S_{12} + S_{11} χ) + C_{1} X^{2} χ {\dot{p}}_{2} ({\dot{p}}_{2} - 2 a^{- 1} y_{0}^{'}) - a^{- 1} X x_{0}^{'} y_{0}^{'} {\dot{γ}}_{2} - a X {\dot{p}}_{2} [p_{2} y_{0}^{'} (1 + a^{- 1}) - x_{0}^{'} {\dot{p}}_{2}] - a^{- 1} y_{0}^{'} p_{2} (q_{1} + a {\dot{p}}_{2}) \\ - a^{- 1} y_{0}^{'} χ {\dot{γ}}_{2} + 0.5 ℓ_{z} {(A b r_{0})}^{- 1} (S_{11} χ + 2 p_{2} S_{12} - S_{11} x_{0}^{'}), ϕ_{2} = 2 x_{0}^{'} [(a - X) {\dot{p}}_{2} {\dot{γ}}_{2} + b^{- 1} S_{21}] - y_{0}^{'} [(a - X) γ_{2} {\dot{p}}_{2} + a p_{2} {\dot{γ}}_{2}] \\ - (a p_{2} S_{11} + 2 γ_{2} S_{12}) + X^{2} {\dot{p}}_{2} (2 a^{- 1} y_{0}^{'} γ_{2} - a p_{2} {\dot{p}}_{2}) \end{array}

(17)

Constructing the analytic periodic solutions

We aim to construct the analytic periodic solutions of the system (15). The following initial conditions are admitted since the system (15) is an independent one

p_{2} (0,0) = 0, {\dot{p}}_{2} (0,0) = 0, {\dot{γ}}_{2} (0, ε^{- 1}) = 0

(18)

Based on Ref. 53, we note that the conditions (18) do not affect the required general solutions. The generated system (15) gives the analytic periodic solutions of the period $T_{0} / n = 2 π$ as

p_{2}^{(0)} = M_{1} \cos ω^{'} τ + M_{2} \sin ω^{'} τ, γ_{2}^{(0)} = M_{3} \cos τ

(19)

The constants $(M_{i}; i = 1,2,3)$ will be determined later. Due to the generated solutions (19), we assume the required analytical periodic general solutions in the forms

\begin{array}{l} p_{2} (τ, ε^{- 1}) = (M_{1} + β_{1}) \cos ω^{'} τ + (M_{2} + β_{2}) \sin ω^{'} τ + \sum_{k = 1}^{\infty} ε^{- k} G_{k} (τ), \\ γ_{2} (τ, ε^{- 1}) = (M_{3} + β_{3}) \cos τ + \sum_{k = 1}^{\infty} ε^{- k} H_{k} (τ) \end{array}

(20)

with the generalized period

T (ε^{- 1}) = T_{0} + α (ε^{- 1})

It is worthwhile to mention that $β_{1}, ω^{'} β_{2}$ , and $β_{3}$ represent the deviating between the solutions $p_{2}, {\dot{p}}_{2}$ and $γ_{2}$ from their corresponding generating ones. These deviations give functions of $ε^{- 1}$ and equal zero when $ε \to \infty$ . The solutions (20) have the following initial restrictions

\begin{array}{l} p_{2} (0, ε^{- 1}) = M_{1} + β_{1}, {\dot{p}}_{2} (0, ε^{- 1}) = ω^{'} (M_{2} + β_{2}), \\ γ_{2} (0, ε^{- 1}) = M_{3} + β_{3}, {\dot{γ}}_{2} (0, ε^{- 1}) = 0 \end{array}

(21)

Let us define the functions $(G_{k} (τ), H_{k} (τ); k = 1,2,3, \dots)$ in the following form⁵⁴

U = u + \frac{\partial u}{\partial M_{1}} β_{1} + \frac{\partial u}{\partial M_{2}} β_{2} + \frac{\partial u}{\partial M_{3}} β_{3} + \frac{1}{2} \frac{\partial^{2} u}{\partial M_{1}^{2}} β_{1}^{2} \dots, (\begin{matrix} U = G_{k}, H_{k} \\ u = g_{k}, h_{k} \end{matrix})

(22)

where

ω^{'} g_{k} (τ) = \int_{0}^{τ} f_{k}^{(0)} (t_{1}) \sin ω^{'} (τ - t_{1}) d t_{1}, h_{k} (τ) = \int_{0}^{τ} ϕ_{k}^{(0)} (t_{1}) \sin ω^{'} (τ - t_{1}) d t_{1}

(23)

To determine the initiating functions $f_{1}^{(0)}$ and $ϕ_{1}^{(0)}$ , we put the generating solutions (19) as follows

p_{2}^{(0)} = E \cos (ω^{'} τ - η), γ_{2}^{(0)} = M_{3} \cos τ

(24)

where

E = {(M_{1}^{2} + M_{2}^{2})}^{1 / 2}

and

\tan η = M_{2} / M_{1}

Using the equations (24) and (13), we get

\begin{array}{l} 2 S_{11}^{(0)} = E^{2} [a (2 \cos^{2} η - 1) + b X^{2} ω^{' 2} (2 \sin^{2} η - 1) + (b X^{2} ω^{' 2} - a) \cos 2 (ω^{'} τ - η)] - 4 M_{3} [x_{0}^{'} (1 - \cos τ) + y_{0}^{'} \sin τ], \\ 2 S_{21}^{(0)} = M_{3} E [2 a \cos η + (b ω^{'} X - a) \cos [(ω^{'} - 1) τ - η] - (b ω^{'} X + a) \cos [(ω^{'} + 1) τ - η]], \\ S_{12}^{(0)} = E {a X [\cos η - \cos (ω^{'} τ - η)] - ω^{'} {a^{- 1} b X^{2} y_{0}^{'} [\sin η + \sin (ω^{'} τ - η)] - [a x_{0}^{'} + (a - X) y_{0}^{'}] [\sin η + \sin (ω^{'} τ - η)]}}, \\ S_{22}^{(0)} = a {a E^{2} [\cos^{2} η - \cos^{2} (ω^{'} τ - η)] + χ M_{3} (1 - \cos τ)} + b X {a^{- 1} y_{0}^{'} M_{3} \sin τ - E^{2} ω^{' 2} (a - X) [\sin^{2} η - \sin^{2} (ω^{'} τ - η)]} - y_{3} S_{21}^{(0)} \end{array}

(25)

The substitution of equations (24) and (25) into (17), yields

f_{1}^{(0)} = L (ω^{'}) (M_{1} \cos ω^{'} τ + M_{2} \sin ω^{'} τ) + \dots, ϕ_{1}^{(0)} = M_{3} N (ω^{'}) \cos ω^{'} τ + \dots

(26)

such that

\begin{array}{l} 4 L (ω^{'}) = - [{(a B r_{0})}^{- 1} ℓ_{z} + 2 ω^{2}] {a (2 M_{1}^{2} - 1) + 2 [b ω^{' 2} X^{2} (2 M_{2}^{2} - 1) - M_{3} x_{0}^{'}] - (b ω^{' 2} X^{2} - a) (M_{1}^{2} + M_{2}^{2})} + \dots, \\ N (ω^{'}) = X^{2} ω^{' 2} [(1 + b) (M_{1}^{2} + M_{2}^{2}) - b M_{2}^{2}] + 2 x_{0}^{'} M_{3} - a M_{1}^{2} + \dots \end{array}

(27)

The substitution of equations (26) and (27) into (23), yields

ω^{'} g_{1} (T_{0}) = - π n M_{2} L (ω^{'}), {\dot{g}}_{1} (T_{0}) = π n M_{1} L (ω^{'}), h_{1} (T_{0}) = 0, {\dot{h}}_{1} (T_{0}) = π n M_{3} N (ω^{'})

(28)

The integral (16) according to the conditions (21) when $τ = 0$ , yields

{(M_{3} + β_{3})}^{2} + 2 a ε^{- 1} (M_{1} + β_{1}) (M_{3} + β_{3}) + \dots = {(γ_{0}^{″})}^{- 2} - 1

Let $γ_{0}^{″}$ be independent of $ε^{- 1},$ we get $M_{3}$ and $β_{3}$ as follows

0 < M_{3} = {(1 - γ_{0}^{″ 2})}^{\frac{1}{2}} {(γ_{0}^{″})}^{- 1} < \infty, β_{3} = - ε^{- 1} a (M_{1} + β_{1}) + \dots

(29)

The solutions²⁹ give the following independent periodicity conditions

\begin{array}{l} - π n β_{2} {(ω^{'})}^{- 1} [L_{1} (ω^{'}) - ω^{' 2} N_{1} (ω^{'})] + ε^{- 1} [G_{2} (T_{0}) + \dots] = 0, π n β_{1} [L_{1} (ω^{'}) + N_{1} (ω^{'}) (b^{- 1} y_{1} - ω^{'} β_{1})] + ε^{- 1} [{\dot{G}}_{2} (T_{0}) + \dots] = 0, \\ ε^{- 1} {(M_{3} + β_{3})}^{- 1} [{\dot{H}}_{1} (T_{0}) + ε^{- 1} {\dot{H}}_{2} (T_{0}) + ε^{- 2} {\dot{H}}_{3} (T_{0}) + \dots] = α (ε^{- 1}) \end{array}

(30)

Making use of the equation (28), changing the constants $M_{1}, M_{2}$ and $M_{3}$ by the deviations and constants $β_{1}, β_{2}$ and $M_{3} + β_{3}$ , we get

L_{1} (ω^{'}) - ω^{' 2} N_{1} (ω^{'}) = (β_{1}^{2} + β_{2}^{2}) W_{1} (ω^{'}),

(31)

such that

4 W_{1} (ω^{'}) = [b {(ω^{'} X)}^{2} - a] [2 + {(A B r_{0})}^{- 1} C ℓ_{z}] - 4 (1 + b) {(ω^{' 2} X)}^{2}

Making use of the equations (10), (11), (14), (20), (22), (24), (29), and (30), we get the required analytic periodic solutions and the period correction $α (ε^{- 1})$ as follows

\begin{array}{l} p_{1} = ε^{- 1} x_{0}^{'} {(b ω^{' 2})}^{- 1} [1 + {(A r_{0})}^{- 1} ℓ_{z}] + \dots, q_{1} = ε^{- 1} a^{- 1} y_{0}^{'} X + \dots, r_{1} = 1 - ε^{- 2} M_{3} [x_{0}^{'} (1 - \cos τ) + y_{0}^{'} \sin τ] + \dots, \\ γ_{1} = M_{3} \cos τ + \dots, γ_{1}^{'} = - M_{3} \sin τ + ε^{- 2} X [a^{- 1} y_{0}^{'} - M_{3}^{2} \sin τ [x_{0}^{'} (1 - \cos τ) + y_{0}^{'} \sin τ]] + \dots, \\ γ_{1}^{″} = 1 + ε^{- 2} M_{3} [(a χ + x_{0}^{'}) (1 - \cos τ) + (X b a^{- 1} + 1) y_{0}^{'} \sin τ] + \dots, α (ε^{- 1}) = - 2 ε^{- 1} π n x_{0}^{'} M_{3} + \dots \end{array}

(32)

From these solutions, we note that the large parameter technique gives us the chance to study the problem of the rotational disc under the new condition when the initial rotation is slow while the frequency of the motion is high. The above solutions are obtained in the new domain for the angular speed and the parameter ε.

The numeric solutions

In this section, the numerical solutions of a system (1) are presented to check the validity of the obtained analytic solutions and to study the affection for the parameters of the body on the motion. Thus, to obtain the graphical representations and the behavior of both the analytic solutions and the corresponding numeric ones, we give a digital example as follows

\begin{array}{l} A = 0.853 kg . m^{2}, B = 2.149 kg . m^{2}, r_{0} = 0.001 m, γ_{0}^{″} = 0.463, \\ M = 30 kg, x_{0} = 0.5 m, y_{0} = 1 m, T = 12.566371, \\ ℓ_{z} = (1000, 25000, 3500, 4500, 55000, 65000) kg . m^{2} . s^{- 1} \end{array}

We use the above data through a software program to obtain the analytical solutions and numerical ones in Tables 1–4. Tables 1 and 2 represent the analytical and numerical solutions for

p_{1}

and

γ_{1}

with their derivatives against the time

τ

when

ℓ_{z} = 1000 kg . m^{2} . s^{- 1}

, respectively, whereas Tables 3 and 4 represent the analytical and numerical solutions for

p_{1}

and

γ_{1}

with their derivatives against the time

τ

when

ℓ_{z} = 2000 kg . m^{2} . s^{- 1}

, respectively. From these tables, it is easy to see that the amplitudes of the analytic and numeric waves decrease when the gyro torque

\underline{ℓ}

increases and vice versa. It clears also from equation (7) that the frequency of the motion is sufficiently high since the initial angular velocity is sufficiently small. That is the only suitable technique that solves the problem is the large parameter instead of the small one which used in the previous papers. The advantage of this technique is that it can solve the problem at a low speed and high frequency. The stability of the solutions is investigated through the phase plane diagrams which are given in Figures 2–5. It is obvious that the obtained analytic and numerical solutions are stable, and they are free from chaos. We note also that both analytical and numerical solutions are nearly coinciding.

Table 1.

The analytical solutions $p_{1}$ and $γ_{1}$ with them derivatives against the time $τ$ when $ℓ_{z}$ = 1000.

$τ$	$p_{1}$	$γ_{1}$	${\dot{p}}_{1}$	${\dot{γ}}_{1}$
0	1,154,700	3,179,793	666 × 10⁻¹	212 × 10⁻¹¹
10	1,544,112	2,576,352	572 × 10⁻¹	−186,373
20	1,866,479	995 × 10⁻¹	453 × 10⁻¹	−302,009
30	2,107,803	−964 × 10⁻¹	315 × 10⁻¹	−303,018
40	2,257,608	−255,702	162 × 10⁻¹	−189,017
50	2,309,387	−317,963	264 × 10⁻³	−328 × 10⁻²
60	2,260,894	−259,541	−157 × 10⁻¹	1,837,096
70	2,114,234	−102,612	−310 × 10⁻¹	3,009,679
80	1,875,774	933 × 10⁻¹	−449 × 10⁻¹	3,039,946
90	1,555,868	2,537,413	−569 × 10⁻¹	1,916,409
100	1,168,407	3,179,119	−664 × 10⁻¹	655 × 10⁻²
110	730 × 10⁻¹	2,614,198	−730 × 10⁻¹	−181,027
120	260 × 10⁻¹	1,057,064	−765 × 10⁻¹	−299,895
130	−221 × 10⁻¹	−901 × 10⁻¹	−766 × 10⁻¹	−304,939
140	−692 × 10⁻¹	−251,754	−734 × 10⁻¹	−194,245
150	−113,404	−317,828	−671 × 10⁻¹	−982 × 10⁻²
160	−152,634	−263,271	−578 × 10⁻¹	1,783,242
170	−185,237	−10,879	−460 × 10⁻¹	2,987,903
180	−209,797	870 × 10⁻¹	−322 × 10⁻¹	3,058,513
190	−225,248	2,497,397	−170 × 10⁻¹	1,968,273
200	−230,918	3,177,095	−106 × 10⁻²	131 × 10⁻¹
210	−226,562	2,650,935	149 × 10⁻¹	−175,603
220	−212,369	1,118,617	302 × 10⁻¹	−297,654
230	−188,955	−838 × 10⁻¹	443 × 10⁻¹	−306,731
240	−157,337	−247,699	564 × 10⁻¹	−199,389
250	−118,886	−317,558	660 × 10⁻¹	−164 × 10⁻¹
260	−753 × 10⁻¹	−266,888	728 × 10⁻¹	172,863
270	−284 × 10⁻¹	−114,922	764 × 10⁻¹	2,964,859
280	197 × 10⁻¹	807 × 10⁻¹	767 × 10⁻¹	3,075,783
290	670 × 10⁻¹	245,632	737 × 10⁻¹	2,019,302
300	1,113,256	3,173,723	674 × 10⁻¹	196 × 10⁻¹

Table 2.

The numerical solutions $p_{1}$ and $γ_{1}$ with them derivatives against the time $τ$ when $ℓ_{z}$ = 1000.

$τ$	$p_{1}$	$γ_{1}$	${\dot{p}}_{1}$	${\dot{γ}}_{1}$
0	1,154,700	3,179,793	666 × 10⁻¹	212 × 10⁻¹¹
10	1,544,111	2,576,351	571 × 10⁻¹	−186,372
20	1,866,478	994 × 10⁻¹	452 × 10⁻¹	−302,008
30	2,107,802	−963 × 10⁻¹	314 × 10⁻¹	−303,017
40	2,257,607	−255,701	161 × 10⁻¹	−189,016
50	2,309,386	−317,962	263 × 10⁻³	−327 × 10⁻²
60	2,260,893	−259,540	−156 × 10⁻¹	1,837,095
70	2,114,233	−102,611	−309 × 10⁻¹	3,009,678
80	1,875,773	932 × 10⁻¹	−448 × 10⁻¹	3,039,945
90	1,555,867	2,537,412	−568 × 10⁻¹	1,916,408
100	1,168,406	3,179,118	−663 × 10⁻¹	654 × 10⁻²
110	729 × 10⁻¹	2,614,197	−729 × 10⁻¹	−181,026
120	259 × 10⁻¹	1,057,063	−764 × 10⁻¹	−299,894
130	−220 × 10⁻¹	−900 × 10⁻¹	−765 × 10⁻¹	−304,938
140	−691 × 10⁻¹	−251,753	−733 × 10⁻¹	−194,244
150	−113,403	−317,827	−670 × 10⁻¹	−981 × 10⁻²
160	−152,633	−263,270	−577 × 10⁻¹	1,783,241
170	−185,236	−10,878	−459 × 10⁻¹	2,987,902
180	−209,796	869 × 10⁻¹	−321 × 10⁻¹	3,058,512
190	−225,247	2,497,396	−169 × 10⁻¹	1,968,272
200	−230,917	3,177,094	−105 × 10⁻²	130 × 10⁻¹
210	−226,561	2,650,934	148 × 10⁻¹	−175,602
220	−212,368	1,118,616	301 × 10⁻¹	−297,653
230	−188,954	−837 × 10⁻¹	442 × 10⁻¹	−306,730
240	−157,336	−247,698	563 × 10⁻¹	−199,388
250	−118,885	−317,557	659 × 10⁻¹	−163 × 10⁻¹
260	−752 × 10⁻¹	−266,887	727 × 10⁻¹	172,862
270	−283 × 10⁻¹	−114,921	763 × 10⁻¹	2,964,858
280	196 × 10⁻¹	806 × 10⁻¹	766 × 10⁻¹	3,075,782
290	669 × 10⁻¹	245,631	736 × 10⁻¹	2,019,301
300	1,113,255	3,173,722	673 × 10⁻¹	195 × 10⁻¹

Table 3.

The analytical solutions $p_{1}$ and $γ_{1}$ with them derivatives against the time $τ$ when $ℓ_{z}$ = 2000.

$τ$	$p_{1}$	$γ_{1}$	${\dot{p}}_{1}$	${\dot{γ}}_{1}$
0	−64.92042	63,711.47	−13.44621	−51,016.33
10	−37,468.73	51,512.79	−18.44141	1.29 × 10⁸
20	−60,540.69	19,668.85	−8.829416	−3.07 × 10⁸
30	−60,540.69	−19,668.85	8.829418	−1.03 × 10⁸
40	−37,468.72	−51,512.8	18.44141	−3.61 × 10⁸
50	−64.91486	−63,711.48	13.44622	3.05 × 10⁸
60	37,428.61	−51,574.54	3.315035	5.41 × 10⁸
70	60,645.73	−19,707.02	−5.19 × 10⁻¹	1.21 × 10⁹
80	60,645.73	19,707.02	5.19 × 10⁻¹	1.15 × 10⁹
90	37,428.61	51,574.54	−3.315036	6.21 × 10⁸
100	−64.93157	63,711.48	−13.44623	−7.46 × 10⁸
110	−37,468.73	51,512.8	−18.44141	−1.48 × 10⁹
120	−60,540.7	19,668.84	−8.829412	1.17 × 10⁹
130	−60,540.68	−19,668.88	8.829429	−2.18 × 10⁹
140	−37,468.74	−51,512.79	18.44141	−2.40 × 10⁹
150	−64.91891	−63,711.48	13.44622	9.67 × 10⁸
160	37,428.62	−51,574.53	3.315033	2.56 × 10⁹
170	60,645.73	−19,707.03	−5.19 × 10⁻¹	−5.81 × 10⁸
180	60,645.74	19,707	5.19 × 10⁻¹	−1.06 × 10⁹
190	37,428.6	51,574.54	−3.315038	−4.29 × 10⁸
200	−64.94272	63,711.48	−13.44623	9.34 × 10⁸
210	−37,468.71	51,512.81	−18.44141	1.71 × 10⁹
220	−60,540.69	19,668.86	−8.829419	−3.64 × 10⁹
230	−60,540.69	−19,668.86	8.829421	−2.51 × 10⁹
240	−37,468.71	−51,512.81	18.44141	3.65 × 10⁹
250	−64.93816	−63,711.48	13.44623	−4.16 × 10⁹
260	37,428.65	−51,574.5	3.315025	−2.79 × 10⁹
270	60,645.72	−19,707.05	−5.19 × 10⁻¹	6.46 × 10⁸
280	60,645.75	19,706.98	5.19 × 10⁻¹	−4.31 × 10⁷
290	37,428.61	51,574.53	−3.315034	2.08 × 10⁹
300	−64.92348	63,711.48	−13.44622	−3.62 × 10⁸

Table 4.

The numerical solutions $p_{1}$ and $γ_{1}$ with them derivatives against the time $τ$ when $ℓ_{z}$ = 2000.

$τ$	$p_{1}$	$γ_{1}$	${\dot{p}}_{1}$	${\dot{γ}}_{1}$
0	−64.92042	63,711.47	−13.44621	−51,016.33
10	−37,468.72	51,512.78	−18.44140	1.28 × 10⁸
20	−60,540.68	19,668.84	−8.829415	−3.06 × 10⁸
30	−60,540.68	−19,668.84	8.829417	−1.02 × 10⁸
40	−37,468.71	−51,512.7	18.44140	−3.60 × 10⁸
50	−64.91485	−63,711.47	13.44621	3.04 × 10⁸
60	37,428.60	−51,574.53	3.315034	5.40 × 10⁸
70	60,645.72	−19,707.01	−5.18 × 10⁻¹	1.20 × 10⁹
80	60,645.72	19,707.01	5.18 × 10⁻¹	1.14 × 10⁹
90	37,428.60	51,574.53	−3.315035	6.20 × 10⁸
100	−64.93157	63,711.47	−13.44622	−7.45 × 10⁸
110	−37,468.73	51,512.7	−18.44141	−1.47 × 10⁹
120	−60,540.7	19,668.84	−8.829412	1.16 × 10⁹
130	−60,540.68	−19,668.88	8.829429	−2.17 × 10⁹
140	−37,468.74	−51,512.79	18.44141	−2.39 × 10⁹
150	−64.91891	−63,711.48	13.44622	9.66 × 10⁸
160	37,428.62	−51,574.53	3.315033	2.55 × 10⁹
170	60,645.73	−19,707.03	−5.19 × 10⁻¹	−5.80 × 10⁸
180	60,645.74	19,707	5.19 × 10⁻¹	−1.05 × 10⁹
190	37,428.6	51,574.54	−3.315038	−4.28 × 10⁸
200	−64.94272	63,711.48	−13.44623	9.33 × 10⁸
210	−37,468.71	51,512.81	−18.44141	1.70 × 10⁹
220	−60,540.69	19,668.86	−8.829419	−3.63 × 10⁹
230	−60,540.69	−19,668.86	8.829421	−2.50 × 10⁹
240	−37,468.70	−51,512.80	18.44140	3.64 × 10⁹
250	−64.93815	−63,711.47	13.44622	−4.15 × 10⁹
260	37,428.64	−51,574.4	3.315024	−2.78 × 10⁹
270	60,645.71	−19,707.03	−5.18 × 10⁻¹	6.45 × 10⁸
280	60,645.74	19,706.97	5.18 × 10⁻¹	−4.30 × 10⁷
290	37,428.60	51,574.52	−3.315033	2.07 × 10⁹
300	−64.92347	63,711.47	−13.44621	−3.61 × 10⁸

Figure 2.

The stability diagram for ${\dot{p}}_{1}$ and $p_{1}$ at $ℓ_{z}$ = 1000.

Figure 3.

The stability diagram for ${\dot{p}}_{1}$ and $p_{1}$ at $ℓ_{z}$ = 2000.

Figure 4.

The stability diagram for ${\dot{γ}}_{1}$ and $γ_{1}$ at $ℓ_{z}$ = 1000.

Figure 5.

The stability diagram for ${\dot{γ}}_{1}$ and $γ_{1}$ at $ℓ_{z}$ = 2000.

Conclusion

The rotary motion of a disc in presence of a gyro torque around one of its principal axes with a sufficiently small angular speed component about this axis is considered. The large parameter is achieved due to the smallness of the speed. The geometric integral, the energy integral, and the angular momentum one are obtained with the governed equations of motion depending on such parameter. The derived equations are reduced to two degrees of freedom independent system with aiding the first integrals and the large parameter. Then the three first integrals are also reduced to one independent integral. The large parameter technique is applied for obtaining the analytic approximated periodic solutions (32) for the reduced system. These solutions are reformulated by a computer program see Tables 1 and 3. The investigated solutions are dependent on the large parameter and the low angular speed which update the new domain of the solutions. This problem is considered a general one of such studies in the uniform gravity force field. The angle of nutation is considered a limiting case for this problem. A lot of singularities that appeared previously do not appear in this work due to the large value of the frequency depending on the torque. Using the initial values of the analytic solutions through a computer program depending on the disk parameters and applying the fourth-order Runge–Kutta method, we achieve numeric solutions see Tables 2 and 4. The stability of the attained solutions is investigated in phase plane diagrams through Figures 1–4. An example of the validity of the considered techniques is given in Figure 3. The initial energy required for initializing the motion is very small due to the low initial angular velocity. This problem has many applications in gyros, radars, dynamics systems, satellites, astronomy, aerospace engineering, and many physic and geometric applications. For example, from the recent applications which are useful in this field of study in gyro’s dynamics, the gyroscope system, the flexible rotor’s motion, and their chaos synchronization represent new directions for these works.^55,56

Footnotes

Acknowledgements

The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: (22UQU4240002DSR06)

Author Contributions

All authors have the same contribution to this manuscript.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Deanship of Scientific Research at Umm Al-Qura University.

ORCID iD

AI Ismail

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