Periodic Response Analysis of A Jeffcott-Rotor System By The Integral Equation Method

In this paper, two modified nonlinear saturation-based controllers and negative velocity feedback controllers are integrated to suppress the horizontal and vertical vibrations of a horizontally supported Jeffcott-rotor system at primary resonance excitation and the presence of 1:1 and 1:2 internal resonances. The second order approximations and the amplitude equations are obtained by applying the integral equation method to analyze the nonlinear behavior of this model. The stability of the steady-state solutions is ascertained based on the Floquet theory. The necessity of adding a negative velocity feedback to the main system is stated. The effects of different control parameters on the frequencyresponse curves and the force-response curves are investigated. Time histories of the whole system are included to show the response with and without control. It is shown that the saturation-based controller can reduce the system response to almost zero and the negative velocity feedback can suppress the transient vibrations and prevent the main system having the large amplitude vibration. The analyses show that analytical solutions are in excellent agreement with the numerical simulations. Finally, a comparison with previously published works is included.

Quadratic velocity coupling term, a four-degree-of-freedom Jeffcott-rotor system

Introduction
Mechanical vibrations are undesired phenomena in dynamical structures because they always cause damages, disturbances, and dangerous accidents. Thus, many research papers focused on how to suppress the nonlinear vibrations in those machines, especially the periodic vibration. To study the periodic motions of a nonlinear vibration system, many methods have been proposed, such as the method of multiple scales, the averaging method, the Lindstedt-Poincare method [1,2], the asymptotic perturbation method [3][4][5][6] and so on. Maccari [3][4][5][6] applied the asymptotic perturbation method to study the parametric resonance and the primary resonance of the van der Pol oscillator with a time delay state feedback. By means of the averaging method and the multiple scales method, El-Bassiouny analyzed the qualitative behavior of the response of a ship rolling in longitudinal waves in [7]. In [8], the HB-AFT method was applied to obtain the periodic solutions of a four-degree-of-freedom Misaligned rotor model. In [9], M.Eissa employed the multiple scales perturbation technique to discuss the vibration suppression of a nonlinear magnetic levitation system via a nonlinear saturation-based controller. In [10][11][12], authors used the method of multiple scales to obtain the evolution equations of the amplitude and the phase of the whole controlled system. Previous published works showed that the multiple scales perturbation method had remarkable accuracy in solving the resonance solution of multi degree of freedom vibration systems. At present, the multiple scales perturbation method is almost the only way to solve the approximate solution of the multi-degree-of-freedom system, especially the four-degree-of-freedom system. However, the calculation process of this method is a little complicated, and the first step using the multiple scales perturbation method is to re-scale the parameters of the system, which has no formula to controller(proportional-derivative controller), P controller and X 3 controller, etc. In [13], the authors utilized four different types controllers(saturation controller, PPF controller, P controller and X 3 controller) to suppress the nonlinear vibrations of a nonlinear beam-like structure and found that PPF and NSC controllers were the most effective. Compared to the PPF controller, using the saturation controller, there would be no double peaks on the frequency-response curve of the controlled system. That is to say, under the saturation control, the controlled system will not have large amplitude vibration in the whole frequency band. In [14], a saturation-based active controller had been proposed to reduce the vibration of a four-degree-or-freedom rotor-AMB at the primary res- Tusset et al. [19] presented two control strategies(NSC,MR) for a parametrically excited pendulum. It showed that saturation control method could suppress the chaotic behavior. Kandil et al. [20] considered the vibration suppression of a compressor blade via a nonlinear saturation controller. It showed that the jump phenomena, saddle-node and Hopf bifurcations could be eliminated by the saturation-based controller. In [22], two modified saturation-based controllers were proposed to reduce the whirling motions of a vertically supported nonlinear Jeffcott-rotor system. The influences of time-delays on the controller performance and system stability were discussed. In order to make up for the narrow frequency band of tranditional saturation controller, Xu et al. [24] improved the saturation controller by replacing quadratic position coupling term with the quadratic velocity coupling term.
In this paper, two modified saturation-based controllers and negative velocity feedbacks are applied to suppress the nonlinear vibrations of a horizontally supported Jeffcott-rotor system subjected to harmonic excitation force. The integral equation method [23][24][25][26], which was introduced by G.Schimidt, is utilized to obtain the second order approximations and the amplitude equations. The stability of the system is investigated by applying a combination of the Floquet theory and Hill's determinant. The stable and unstable solutions are determined depending on the real parts of all eigenvalues of the Hill's determinant. The frequency response curves, the force response curves and time histories are presented to illustrate the performance of the control law. In addition, the effects of the controller parameters on the main systems and controllers are explored and optimal working conditions of the system are extracted from the force-response curves and the frequency-response curves. Numerical simulations are presented to validate the analytical predictions.

Mathematical modeling
After integrating a negative velocity feedback and a nonlinear saturationbased controller to each oscillation mode of the uncontrolled main system, the modified dimensionless equations of motion of a horizontally supported Jeffcottrotor system( [11,27,28]) subjected to harmonic excitation force can be written 5 as follows: Where γ 1 x 2 and γ 3 y 2 are the control forces, γ 2uẋ and γ 4vẏ represent the feedback signals, −2λ 1 ω 1u and −2λ 2 ω 3v represent negative velocity feedback signals. The schematic diagram of the whole controlled system is shown in Fig.1.

Amplitude equations
In this section, we examine the case of the simultaneous resonance (Ω ∼ = ω 1 ∼ = To quantitatively describe the closeness of the considered resonance case, we introduce four detuning parameters σ 1 , σ 2 , σ 3 , σ 4 defined by Ω = ω 1 + σ 1 , ω 2 = ω 1 + σ 3 2 , In this section, the integral equation method (see Appendix) [23] is applied to seek an approximate solutions to Eq.(1). Then, a dimensionless time is introduced by Then Eq.(1) are transformed into the following form: For Eq.(4), the corresponding generalized Green's functions are From Eq.(A.5) and Eq.(A.6) in Appendix, we can obtain a first-order approximate solution to Eqs.(4) as follows: x 1 (t) = r 2 cos t + s 2 sin t (7) v 1 (t) = r 3 cos(2t) + s 3 sin(2t) (8) Substituting Eqs.(6)-(9) into Eqs.(A.5), the second order approximate solution can be obtained as Substituting Eqs. 4f For convenience to investigate the dynamics of the whole controlled system in the following sections, we denote the amplitude of the horizontal oscillation mode as A 1 = r 2 1 + s 2 1 , the amplitude of the vertical oscillation mode as A 3 = r 2 3 + s 2 3 , the amplitude of the controller of the horizontal oscillation mode as A 2 = r 2 2 + s 2 2 , and the amplitude of the controller of the vertical oscillation mode as A 4 = r 2 4 + s 2 4 .

Stability of periodic solutions
To investigate the stability of the periodic solutions of Eq.(1), we suppose are small perturbations about the approximate solu- and z 4 (t) with keeping the linear terms only, yields the following linearized system: According to Floquet theory, the solutions of Eq.(22)-Eq. (25) can be expressed in the form Inserting Eq.(26) into Eqs. (22)- (25), we geẗ The first approximations of Eqs. (27)- (30) can be expressed in the following form    2)). That is to say, it is necessary to keep the natural frequency of the controller of the horizontal oscillation mode (ω 2 ) equal to half of the rotor-spinning speed(Ω). From Fig.8 (b), we observe that, when σ 3 is negative, the frequency-response curve of the controller of the horizontal oscillation mode (σ 1 -A 2 ) moves to the left. And for the positive value of σ 3 , the frequency-response curve moves to the right. From Fig.8 (c), (d), it can be seen that, the change of σ 3 has little effect on the frequency-response   Effects of varying the control signal gains γ 1 , γ 3 on the system frequencyresponse curves are illustrated in Fig.11. It can be seen from Fig.11(a) and (c) that varying the control gain γ 1 , γ 3 has little effect on the amplitudes of the main systems of two oscillation modes and the effective frequency bandwidth of the saturation-based controllers. However, Fig.11 (b) and (d) show that the amplitudes of the two saturation-based controllers decrease as γ 1 , γ 3 increase.

Effects of the control parameters
Therefore, the control signal gains γ 1 , γ 3 can be used as important parameters to prevent the controllers overload risk.  Fig.12(a) and (c), it is noted that decreasing µ 2 , µ 4 can reduce the main system amplitudes of the two oscillation modes better. Meanwhile, Fig.12(b) and (d) show that decreasing µ 2 , µ 4 can increase the amplitudes of the saturation-based controllers due to enhance energy transfer between the main systems and the controllers. That is to say, the smaller the µ 2 , the bigger the overload risk of the saturation controller.  Fig.15 shows that the system exhibits a large oscillation amplitudes before control, but after control, the maximum oscillation amplitude becomes extremely small. From Fig.16, we can see that both amplitudes of the main system A 3 and the controller A 4 at the vertical oscillation mode are independent of the detuning parameter σ 3 = 0. Similarly, Fig.17 shows that the detuning parameter σ 4 has no effect on both amplitudes of the main system and the controller at the horizontal oscillation mode. These figures also confirms a good agreement between the numerical solutions and the approximate analytical solution. 4. Under the combined control, the minimum steady-state amplitudes occur when the natural frequencies of the controllers (ω 2 , ω 4 ) are equal to half the rotor-spinning speed(Ω). So we can select the natural frequencies of the controllers based on the value of the rotor-spinning speed, so that the amplitude of the system can always be reduced to a minimum.

5.
Under the combined control, as the excitations force increases and the controller linear damping coefficient(µ 2 , µ 4 ) decreases, the steady-state amplitudes of the main system saturate to a very small value.
6. Increasing the feedback signal gain(γ 2 , γ 4 ) can broaden the effective frequency bandwidth of the controller but increase the overload risk of the controller. The control signal gain(γ 1 , γ 3 ) has no effect on the amplitude of the main system, but it can suppress the vibration of the controller. So we can use control signal gain as an important parameter to prevent the controller overload.

Comparison with previously published work
In recent years, various types of active controllers [11,21,22,29,30] were designed to suppress the nonlinear vibrations of Jeffcott-rotor system. In [11], two PPF controllers are integrated to mitigate the horizontal and vertical vibrations of a horizontally supported Jeffcott-rotor system. It concluded that PPF controllers could reduce the amplitudes of the considered system to almost zero. However, in the region outside the resonance region, there would be double peaks with large amplitude on the frequency response curves. In [21], a nonlinear PD-controller was proposed to suppress the nonlinear vibrations of a horizontally supported Jeffcott-rotor system. They concluded that the PD controller could effectively eliminate the nonlinear phenomena of the Jeffcott-rotor system even in extremely large eccentricity, and a positive position feedback and a negative velocity feedback controller were considered to be optimal method. In [22], two modified saturation-based controllers are coupled where G i [t, σ] = 1 π ( 1 2λi + ∞ j=1 cos j(t−σ) ϑλi−j 2 ) is the corresponding Generalized Green's function. And if λ i = n 2 i (n i being an integer for the resonance case),the parameters r i , s i can be determined by the periodicity equations x i (t) cos n i t dt, The solutions of (A.2) can be found by successive approximations of x ik (t), k = 1, 2, 3, · · · , which are given by x ik (t) = Frequency-response curves of two main systems in three states: before control (blue,λ1 = λ2 = 0), controlled only by the negative velocity feedback control (red, λ1 = λ2 = 0.1) and the combined control (green, λ1 = λ2 = 0.1).