Abstract
In this paper, we present new modifications for some perturbation procedures used in mathematics, physics, astronomy, and engineering. These modifications will help us to solve the previous problems in different sciences under new conditions. As problems, we have, for example, the rotary rigid body problem, the gyroscopic problem, the pendulum motion problem, and other ones. These problems will be solved in a new manner different from the previous treatments. We solve some of the previous problems in the presence of new conditions, new analysis, and new domains. We let complementary conditions of such studied previously. We solve these problems by applying the large parameter technique used by assuming a large parameter which inversely proportional to a small quantity. For example, in rigid body dynamic problems, we take such quantity to be one of the components of the angular velocity vector in the initial instant of the rotary body about a fixed point. The domain of our solutions will be depending on the choice of a large parameter. The problem of slow (weak) oscillations is considered. So, we obtain slow motions of the bodies instead of fast motions and find the solutions of the problem in present new conditions on both of center of gravity, moments of inertia, and the angular velocity vector or one of these parameters of the body. This study is important for aerospace engineering, gyroscopic motions, satellite motion which has the correspondence of inertia moments, antennas, and navigations.
1. Introduction
In [1], the variation of parameters method is applied to solve the general nonhomogenous linear differential equation of the second order. The author solved the homogenous equation and constructed a particular solution depending on its solutions. He got the first and second differentiation of the particular solutions, substituted them into the considered equation, and performed some investigated steps, and he found the required general solution.
We note that this is an exact solution for a linear differential equation and not an approximated one and so is not the small parameter technique or the large parameter one useful for this problem. Such techniques are useful only for approximated problems consisting of nonlinear differential equations containing a small parameter [2] or a large one [3, 4]. In the following sections, we modify the well-known previous method in terms of the large parameter for solving some systems for some perturbed problems [5, 6] containing quasi-linear systems. This modification gives us the chance to study the considered problems in new conditions and new domains of solutions.
2. The Method of Averaging
We consider the following examples for illustrating this method, and we will find the periodic solutions of each case:
2.1. Van Der Pol’s Technique
This technique is considered in previous works for finding the approximated periodic solutions as power series in terms of a small parameter defined for each problem. In our work, we aim to find new periodic solutions in terms of a large parameter instead of the small one of the equationwhere is imposed to be large and is the movement frequency which is assumed to be small to be different from the natural frequency . By taking the solution to (1), we have assumed the following form:where and are taken to be slow functions that vary with time as well as and . By differentiating (2) twice, we obtain
Using equations (1)–(3), we obtain
Periodic solutions of (1) match solutions:where .
By adding squares (6) and (7) and using (5), we obtain the solution of the frequency equation:
2.2. The Krylov–Bogoliubov Technique
This technique considered through the weak nonlinear equation of the second order in terms of the small parameter was defined in the problem [7–11]. Here, we consider applying the large parameter . Let us consider the following equation:
At , the solution of (9) can be written as follows:where and are constants. To determine the approximated solution of (9) for the large parameter , Krylov and Bogoliubov hypothesized that the solution is still given by (10) but with the time change and , and it is subjected to the condition:
By differentiation (10) w. r. t.t, we obtain
So,
By differentiation (11) w. r. t.t, we obtain
Substituting this expression into (9) and using (10), we obtain
Solving (13) and (15) for and , we obtain that
By integrating (16) into the period [t, t + T], where and can be taken as constants on the right-hand side, we obtainwhere
We also note that and are coefficients of the Fourier series expansion of the function .
As an example, let us consider the Duffing equation.
In this example, we modify the Duffing equation in terms of the large parameter to bewhere
So,
Thus, from (17), it is obtained
Therefore, the first approximation is
For the second example, consider the Van der Pol oscillator:where
In this case,
Hence and from (17), it follows that is obtained in terms of the large parameter, so
By integrating the first equation into (27), it becomes clear that
2.3. The Generalized Method of Averaging
This method was applied for obtaining the periodic solutions of the differential equations applying the small parameter technique which was defined for each case of the motion [12–14]. In our work, we use the large parameter method for obtaining new solutions dependent on the large parameter . Let us consider (10) and (11) as the conversion from and to and in terms of the large parameter, sowhere is called the rapid rotation phase. Let us take the conversions from to such that
Using (30), we find that system (29) takes the following form:where and do not depend on , for each .
Generally, and contain short periodic terms (referred to as superscript s) and long periodic terms (referred to by superscript).
We choose and such that it is equal to the long periodic terms, sothenwhich can be solved, respectively, in and .
As an example, consider the Van der Pol oscillator (24) in which
In this case, equation (29) in terms of the large parameter becomes:
By substituting (30) and (31) into (36) and equating the coefficients of similar powers of we obtain the following.
Order :
Order :
Equally, and by the long periodic terms on the right-hand side in (37) is obtained as
Hence, system (37) becomes
So, we obtain the following solutions
By using (39) and (41), the relation (38) becomes:
Equally, and by the long periodic terms on the right-hand side in (42) is obtained as
Therefore, the solution up to the second approximation becomeswhere
3. Struble’s Technique
Struble developed a technique for treating weak nonlinear oscillatory systems such as those governed by a second-order differential equation in terms of a small parameter [15]. We treat this equation in terms of the large parameter as follows:
Let us consider the solution of (46) in terms of the large parameter , which takes the formwhere and are functions that are slowly changing with time.
As an example, we take Duffing’s equation (19) and use (47) to give
Considering the terms of order and equality of coefficients and on both sides, we get the so-called variation equations:
While the perturbed equations remain, so
For the first order of , equation (49) reduces to
Hence,where and are constants. If we consider and are constants, the solution of (50) for the first order becomes
Hence, the solution for the first-order becomeswhere and are given from equation (52).
Satisfying the solution for second order, we use (53) for calculating the term:
We note that the expression contains a term of order in the following form:
Now, we consider the order leads us to the variation equations:and the perturbed equation is
The solution of equation (57) can be obtained from the recurrence principle of (52) to obtainwhere and are constants. Substituting by from (52) into (58) and solving the resulted equation, we obtain
Thus, we obtain the solution up to the second approximation in the formwhere
The periodic time is given in the formwhere the correction of the period is obtained in the form
4. The Krylov–Bogoliubov–Mitropolski Technique
This technique is devoted to finding approximated periodic solutions using the small parameter as in [7–11]. In our work, we modify the approximated expansions using the large parameter to becomewhere each is a periodic function in , and its period , , and are imposed to change with time as follows:where the functions are arbitrary so that equations (65) to (67) satisfy the differential equation (9). The following derivatives series are obtained:
We will visualize this method in the following example.
4.1. Duffing Equation
We consider a nonlinear oscillator (19), using equations (65) to (69), and equate the coefficients of similar powers to , we obtain
Let be periodic, and the singular terms in the right-hand side of (70) must be deleted. Sincethen equation (70) gives
The solution of equation (70) becomes
Substituting (74) into (71) gives
Canceling the singular terms yields
The solution of (75) becomes
Therefore, the second approximated solution is given in the following form:wherewhere is constant.
5. Multiple Scale Method
This technique is considered for solving the differential equations applying the small parameter [1, 16]. In our work, we modify this method to be suitable for the large parameter technique.
From equation (46), we put , and we obtain
To have accurate and regular expansion over all periods for the solution of this equation, we should be finding an approximated representation of the function as shown in the formwhere
Therefore, the total differential of (80) gives
Substituting expansions (81) and (83) into equation (80), then equalizing the coefficients of equal powers for on both sides, and by solving the resulting equations, the solutions are obtained. Such solutions contain arbitrary functions in the variables and will be determined with a condition and which are bounded to each .
Now, we will apply the modified method to the following example.
5.1. Duffing Equation
Consider the Duffing equation in terms of the large parameter which takes the form
Takingsubstituting (85) in (84), and equating the like powers of to zero, we obtainwhere .
The solution of equation (86) iswhere is the complex conjugate of . Then, equation (87) becomes as follows:
Since is finite for each , the singular terms must be neglected. So,thus
Using into (88) yieldswhere
By eliminating the singular terms, therefore will be finite for all t; then,
Therefore,
A solution regardless of the homogeneous solution is given:
Finally, we get the solution in terms of the large parameter up to the second approximation of in the formwhere and . The obtained solution has the same period (63).
6. Conclusion
In this paper, we modified some of the perturbation methods that are considered ones of the innovations of mathematicians and physicists for finding approximated solutions to important problems. We cannot find a complete solution to these problems. Nayfeh, Poincare, and Krylov are among the prominent scientists in this field as they dealt with many problems in mathematics and physics which are applied in many branches, such as classical mechanics, flexible body mechanics, and many other branches. All these methods depend on a small parameter that was defined for each problem.
The study of such methods shows that the large parameter method is the easiest, most accurate, and one of the methods most used to solve many important problems efficiently. The large parameter method has an advantage over the other methods because it solves the problem in a new domain when it fails all other methods for solving the problem in such a domain. One of the most important application is when we study the slow spin motion of a rigid body in a Newtonian field of force under an external moment [4], the rotational motion of a heavy solid in a uniform gravity field [3], or the gyroscopic motions with a sufficiently small angular velocity component about the major or the minor axis of the ellipsoid of inertia. There are many applications of this technique in aerospace science, satellites, navigations, antennas, and solar collectors. This technique is also useful in all perturbed problems in physics and mechanics, for example, the perturbed pendulum motions and the perturbed mechanical systems.
Data Availability
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.