Approximate Analytical Solutions to Nonlinear Oscillations of Horizontally Supported Jeffcott Rotor

: The present paper focuses on nonlinear oscillations of a horizontally supported Jeffcott rotor. An approximate solution to the system of governing equations having quadratic and cubic nonlinearities is obtained in two cases of practical interest: simultaneous and internal resonance. The Optimal Auxiliary Functions Method is employed in this study, and each governing differential equation is reduced to two linear differential equations using the so-called auxiliary functions involving a moderate number of convergence-control parameters. Explicit analytical solutions are obtained for the ﬁrst time in the literature for the considered practical cases. Numerical validations proved the high accuracy of the proposed analytical solutions, which may be used further in the study of stability and in the design process of some highly performant devices.


Introduction
The nonlinear dynamics of rotors have long attracted attention, being an interesting subject with considerable technical depths and breadths. The theory of oscillations was intensively developed in the field of high-speed machinery and can be used particularly in studies of a disk on a massless shaft; power generation; land, sea, and air transportation; aerospace; textiles; home appliances; or various military systems. For an analysis of simple machinery, one has to take into consideration the accurate forms of excitation, heating and supports, the complicated geometry of the rotor, and so on. There are many types of rotating machines, with different rotor sizes, complexities, speeds, loads, powers, and rigidities [1].
The nonlinear oscillations of rotating machines were studied by many researchers. Muszynska [2] proposed many possible responses of rotor-stator systems. Karlberg and Aidanpää [3] considered the nonlinear vibrations of a rotor system with clearance, analyzing the two-degree-of-freedom unbalanced shaft in relation to a non-rotating massless housing. The rotor start-up lateral vibration signal is investigated by Patel and Darpe [4]. Vibration responses are simulated for the Jeffcott rotor having two lateral degrees of freedom. The Hilbert-Huang transform is applied to investigate the coast-up rub signal, and the wavelet transform is employed for comparison purposes.
The chaotic vibration analysis of a disk-shaft system with rub impact was performed by Khanlo et al. [5], including a consideration of the Coriolis and centrifugal effect. Yabuno et al. [6] explored nonlinear normal modes which considered the natural frequencies in vertical and horizontal directions, investigating the characteristics with primary resonance. Theoretical and experimental investigations are presented by Lahriri et al. [7], considering the impact motion of the rotor against a conventional annular backing guide, and an unconventional annular guide built with four adjustable pins. Various analytical does not imply the presence of a small or large parameter in the governing equations, or the boundary/initial conditions, and can be applied to a variety of engineering domains. The validity of this original method is proved by comparing the results with numerical integration results. We deal with the OAFM in a proper manner and completely differently in comparison with other known techniques. The cornerstone of the validity and flexibility of this approach is in the choice of linear operators and optimal auxiliary functions, which both contribute to obtaining highly accurate results. The convergence-control parameters involved in our procedure are optimally identified in a rigorous mathematical way. Each nonlinear differential equation is reduced to two linear differential equations that do not depend on all terms of the nonlinear equation.
The present study provides accurate explicit analytical solutions which may be used further in the study of stability, and in the design process of some highly performant devices.

The Governing Equations of Motion
In this research, we consider the horizontally supported Jeffcott rotor presented in Figure 1. approximate solution by means of a moderate number of convergence-control parameters. Our technique does not imply the presence of a small or large parameter in the governing equations, or the boundary/initial conditions, and can be applied to a variety of engineering domains. The validity of this original method is proved by comparing the results with numerical integration results. We deal with the OAFM in a proper manner and completely differently in comparison with other known techniques. The cornerstone of the validity and flexibility of this approach is in the choice of linear operators and optimal auxiliary functions, which both contribute to obtaining highly accurate results. The convergence-control parameters involved in our procedure are optimally identified in a rigorous mathematical way. Each nonlinear differential equation is reduced to two linear differential equations that do not depend on all terms of the nonlinear equation.
The present study provides accurate explicit analytical solutions which may be used further in the study of stability, and in the design process of some highly performant devices.

The Governing Equations of Motion
In this research, we consider the horizontally supported Jeffcott rotor presented in Figure 1. The origin O of the inertial coordinate system, Ouvz, is the intersection of the disk and the bearing center line. The whirling motion is assumed to occur on the U-V plane. The mass of the disk is m, its center of gravity G(u,v) deviates slightly from the geometric center with eccentricity ed. If ω is the angular velocity of the rotor spinning, the restoring force F can be a symmetric nonlinear cubic function with respect to the vertical deflection r of the shaft: where k1 and k3 are positive constants. The nonlinear differential equations that describe the horizontal and vertical oscillations of horizontally supported Jeffcott rotor system are expressed as follows [6,15]: where ( ) and ( ) is a nonlinear restoring force due to the bearing clearance, cu and cv are the damping coefficients in the U and V directions, g is the gravity acceleration, and the dot represents the derivative with respect to time.
From Equation (3), the deflection of the shaft due to the gravity in the static equilibrium state satisfies: The origin O of the inertial coordinate system, Ouvz, is the intersection of the disk and the bearing center line. The whirling motion is assumed to occur on the U-V plane. The mass of the disk is m, its center of gravity G(u,v) deviates slightly from the geometric center with eccentricity e d . If ω is the angular velocity of the rotor spinning, the restoring force F can be a symmetric nonlinear cubic function with respect to the vertical deflection r of the shaft: where k 1 and k 3 are positive constants. The nonlinear differential equations that describe the horizontal and vertical oscillations of horizontally supported Jeffcott rotor system are expressed as follows [6,15]: where k 3 u u 2 + v 2 and k 3 v u 2 + v 2 is a nonlinear restoring force due to the bearing clearance, c u and c v are the damping coefficients in the U and V directions, g is the gravity acceleration, and the dot represents the derivative with respect to time. From Equation (3), the deflection of the shaft due to the gravity in the static equilibrium state satisfies: where v st is the static displacement of the geometric center G due to the disk weight.
as a consequence, the motion of geometrical center G in terms of deviations u d and v d from the static equilibrium can be rewritten in the directions U-and V-as: and therefore, the resulting equations are: introducing the dimensionless parameters: one can get the dimensionless nonlinear differential equations of motion: where the prime denotes the derivative with respect to τ, and: From Equations (10) and (11), we remark that the linear natural frequencies of the horizontal and vertical directions are slightly different due to the nonlinearity of the restoring force and the static deflection v st given by Equation (5). Furthermore, the same effects produce an asymmetric nonlinear quadratic component.
In what follows, an approximate analytical solution will be determined to the asymmetric system (10) and (11) using the Optimal Auxiliary Functions Method (OAFM).

Basics of the OAFM
The nonlinear differential Equations (10) and (11) can be written in a general form as [22][23][24][25][26][27]: where L is a linear operator, N is a nonlinear operator, and X(τ) is an unknown function. In our particular case, X(τ) = (u(τ),v(τ)). The corresponding boundary/initial conditions for Equation (13) are: We suppose that the approximate analytical solution X(τ) of Equation (13) can be rewritten in the form: where the initial approximation X 0 (τ) and the first approximation X 1 (τ) can be determined as follows. Inserting Equation (15) into Equation (13) we are led to: The initial approximation X 0 (τ) is obtained by solving the linear differential equation: and the first approximation X 1 (τ) follows to be determined from the nonlinear equation: The nonlinear operator N is expanded in the form: To avoid the difficulties which appear when solving Equation (18), accelerating the convergence of the approximate solutions needs, instead of the last term from Equation (18), the employment of another expression. As such, Equation (18) can be rewritten: where F i (τ), i = 1,2, . . . ,p and p are known auxiliary functions depending on the initial approximation X 0 (τ), on the functions which appear in the composition of N[X 0 (τ)], or the combination of such expressions. We remark that the p and the auxiliary functions Fi(τ) are not unique. Accordingly, X 0 (τ) and N[X 0 (τ)] are sources for the auxiliary functions, and it should be emphasized that we have a large amount of freedom to choose these auxiliary functions. In expression (20), C i , i = 1,2, . . . ,p and p are unknown parameters at this moment. We remark that the nonlinear differential Equation (13) is reduced to only two linear differential Equations, namely (17) and (20). Now, using the results obtained from the theory of differential equations, the variation of parameters method, Cauchy method, Kantorovich method, or the integral factor method [28], we have the freedom to choose the first approximation in the form: where F j are the auxiliary functions defined in Equation (20) and f i are n functions depending on the functions F j , satisfying the boundary/initial conditions: As a consequence, the first approximation X 1 can be determined from Equations (21) and (22). Finally, the unknown parameters C i are optimally identified via rigorous mathematical approaches, such as the collocation method, Galerkin method, Ritz method, the least square method, or by minimizing the residual error. In this way, the approximate solution X(τ) is well determined after the identification of the optimal values of the initially unknown convergence-control parameters C i , i = 1,2, . . . , n.
We will prove that our approach is a very powerful tool for solving nonlinear problems without the presence of small or large parameters in the initial Equation (13) or the boundary/initial conditions (14).

Application of OAFM to Nonlinear Oscillations of Jeffcott Rotor in the Case of Internal Resonance
Taking into account that Ω = ω 1 ≈ ω 2 , the functions f i from Equation (42) and g i from Equation (44) will depend on the p 1 , p 2 , and Ω, as follows: so that the approximate analytical solution for Equations (26) and (27) where C i , i = 7,8, . . . 12 are unknown parameters and f i , g i are given by Equations (48) and (49), respectively.

Numerical Example
In order to prove the accuracy of our approach, we consider that the data for Equations (25)- (27) for every case (simultaneous resonance and internal resonance) are as follows:

The Case of Internal Resonance.
In the case of internal resonance, the parameters are chosen as:

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The optimal values of the convergence-control parameters in this case are:   In Figures 4 and 5, we compared the numerical solutions of Equations (10) and (11),

The Case of Internal Resonance
In the case of internal resonance, the parameters are chosen as: The approximate solution in the case of the internal resonance of Equations (10), (11), and (23) becomes: In Figures 4 and 5, we compared the numerical solutions of Equations (10) and (11), and the approximate solutions (60) and (61), respectively, for the case of internal resonance.

Conclusions
The objective of this research is the study of the nonlinear vibration of a horizontally supported Jeffcott rotor with quadratic and cubic nonlinearity, where the nonlinear restoring force, due to the bearing clearance and the rotor weight, is considered. The linear natural frequencies in the horizontal and vertical directions have small differences due to the nonlinearity of the restoring force and disk weight.
The nonlinear vibrations of the horizontally supported Jeffcott rotor are generated by the rotor eccentricity.
Explicit analytical solutions for the two cases are established using our original Optimal Auxiliary Functions Method (OAFM). Our approach considerably simplifies calculations because any nonlinear differential equation is reduced to two linear ordinary differential equations using the so-called auxiliary functions. This idea does not appear in any other methods known in the scientific literature. Our technique is different from other traditional procedures, especially concerning the optimal auxiliary functions that depend on some initially unknown parameters. We have a large degree of freedom to choose the auxiliary functions and the number of convergence-control parameters.
The obtained approximate analytical solutions are in excellent agreement with the numerical integration results in all cases. Our technique is valid, even if the nonlinear governing equations do not contain small or large parameters. The construction of the first iterations is completely different from other known methods. The optimal values of the

Conclusions
The objective of this research is the study of the nonlinear vibration of a horizontally supported Jeffcott rotor with quadratic and cubic nonlinearity, where the nonlinear restoring force, due to the bearing clearance and the rotor weight, is considered. The linear natural frequencies in the horizontal and vertical directions have small differences due to the nonlinearity of the restoring force and disk weight.
The nonlinear vibrations of the horizontally supported Jeffcott rotor are generated by the rotor eccentricity.
Explicit analytical solutions for the two cases are established using our original Optimal Auxiliary Functions Method (OAFM). Our approach considerably simplifies calculations because any nonlinear differential equation is reduced to two linear ordinary differential equations using the so-called auxiliary functions. This idea does not appear in any other methods known in the scientific literature. Our technique is different from other traditional procedures, especially concerning the optimal auxiliary functions that depend on some initially unknown parameters. We have a large degree of freedom to choose the auxiliary functions and the number of convergence-control parameters.
The obtained approximate analytical solutions are in excellent agreement with the numerical integration results in all cases. Our technique is valid, even if the nonlinear governing equations do not contain small or large parameters. The construction of the first iterations is completely different from other known methods. The optimal values of the From Figures 2-5, a very good agreement can be observed between the approximate solutions and numerical integration results, which confirms the great potential of the OAFM.

Conclusions
The objective of this research is the study of the nonlinear vibration of a horizontally supported Jeffcott rotor with quadratic and cubic nonlinearity, where the nonlinear restoring force, due to the bearing clearance and the rotor weight, is considered. The linear natural frequencies in the horizontal and vertical directions have small differences due to the nonlinearity of the restoring force and disk weight.
The nonlinear vibrations of the horizontally supported Jeffcott rotor are generated by the rotor eccentricity.
Explicit analytical solutions for the two cases are established using our original Optimal Auxiliary Functions Method (OAFM). Our approach considerably simplifies calculations because any nonlinear differential equation is reduced to two linear ordinary differential equations using the so-called auxiliary functions. This idea does not appear in any other methods known in the scientific literature. Our technique is different from other traditional procedures, especially concerning the optimal auxiliary functions that depend on some initially unknown parameters. We have a large degree of freedom to choose the auxiliary functions and the number of convergence-control parameters.
The obtained approximate analytical solutions are in excellent agreement with the numerical integration results in all cases. Our technique is valid, even if the nonlinear governing equations do not contain small or large parameters. The construction of the first iterations is completely different from other known methods. The optimal values of the convergence-control parameters are identified by means of a rigorous mathematical procedure, providing a fast convergence of the approximate analytical solutions using only the first iteration.
It is proved that the OAFM is very effective and efficient in practice. This research provides helpful guidance to solve dynamic problems, and may help to design and manufacture more reliable engineering products.