Tuning of Parametric Excitation for Rotor Balancing

Residual unbalance is the primary source of vibration in rotating machines. In many cases, the time and, consequently, the cost involved in the balancing procedure is high due to the need to accelerate the rotor up to every critical speed to be balanced in every iteration of the traditional influence coefficients method. Recently, a novel method which does not need to accelerate the rotor up to every critical speed to be balanced was proposed. In this method, active magnetic bearings are used to apply a parametric excitation at the combination resonance frequencies inducing an effect similar to transferring kinetic energy from a lower mode to higher modes. Therefore, maximizing the energy transferred is mandatory to obtain a high signal-to-noise ratio and capture the correct responses of the higher modes to successfully balance the rotor. This paper study the sensibility of the kinetic energy transfer between modes to the parameters of the parametric excitation; namely amplitude, frequency, and phase. The analyses are performed via computational simulations of a flexible rotor sustained by two AMBs and with three critical speeds in its operational range.


Introduction
Rotating machines have a wide range of application such as airplanes, factories, laboratories, and power plants. These applications contain shafts rotating at high speeds that must have high trust levels. In order to ensure a safe operation inside the pre-established limits of vibration, rotor balancing is a mandatory procedure before handover of the rotor to the customer. Among many rotor balancing methods available in the literature (e.g. the Modal Balancing and the Unified Balancing Approach) the Influence Coefficient Method is a well-established procedure in the industry (an extensive review of the methods can be found in [1,2], and a detailed description in [3,4]). This method consists of adding test masses in the balancing planes, one at a time, and accelerating the rotor as close as possible to the critical speeds to be balanced. The main drawbacks of the Influence Coefficient Method are the time consumed due to the number of run-up/run-down cycles required, one per test mass, and the need to reach every critical speed to be balanced in every cycle. Besides, high levels of vibration may occur when crossing the critical speeds if the residual unbalance is critical or if the test masses are added in an unfavorable position.
To overcome these limitations, a model-based approach was used by [5] to determine the influence coefficients for a rotor train of a power plant. The disadvantage of this approach is that many errors are involved due to the inaccuracies of the model, mainly due to non-modeled dynamics. In order to  [6] have shown the advantages of using a non-linear model for journal bearings, instead of the traditional equivalent linear coefficients of stiffness and damping.
Recent works [7][8][9][10][11] presented new methods based on the use of parametric excitation to minimize the disadvantages of the traditional influence coefficient method. The central idea of such methods is to transfer kinetic energy to higher modes; thus, making possible to obtain their responses without the need to accelerate the rotor up to the correspondent critical speeds. The parametric excitation triggers the kinetic energy transfer, and it can be mathematically described similarly to equation (1); where ݇ ୮ is the nominal value of the parameter and ߳ is its amplitude variation. (1) Depending on the tuning of the parametric excitation frequency ߥ , , the effect over the system may have a destabilizing or a stabilizing characteristic. In equation (2), the parametric excitation frequency is called a principal parametric resonance for ݇ = ݈, and a parametric combination resonance for ݇ ≠ ݈; where ߱ and ߱ are the k-th and l-th damped natural frequencies of the system, and n denotes the parametric resonance order.
Due to this destabilizing effect, most of the literature [13][14][15][16][17][18] was dedicated to predicting the limits of the instability of parametrically excited systems. Tondl [19] was the first to study the beneficial effect of parametric excitation by using it to suppress self-excited vibrations. The author mathematically formulated the conditions for full vibration suppression in [20]. Later, other authors [21,22,12] presented such conditions using an alternative approach. In the works of [12] and [23], active magnetic bearings (AMB) were used to implement the parametric excitation and increment the system damping. Recently, [24] extended this concept to rotors in fluid-film bearings.
The works of [7][8][9][10] applied a parametric excitation to transfer energy from the first rotational speed harmonic, due to unbalance, to the mode to be balanced, like the effect obtained in the parametric combination resonance. Simultaneously, an unstable response of the correspondent mode is induced to amplify its vibration also using a parametric excitation at the principal parametric resonance of the correspondent mode, and a non-linear force is applied to limit the vibration amplitude.
A simpler approach using parametric excitation was presented in [11]. In the proposed balancing method, the test masses are also used and, instead of accelerating the rotor close to each critical speed of interest, the rotor is accelerated only up to the first critical speed. Once the rotor is close to the first critical speed, there is no need to amplify the mode response or to use a nonlinear term to control any instability or to limit the vibration amplitude. The responses of higher modes are obtained by transferring energy from the first mode to higher modes using parametric excitation at combination resonance frequencies. The parametric excitation is induced in the active magnetic bearings (AMBs), which support the rotor, by a time-periodic PID controller in analogy to [12].
However, the parametric excitation applied in the system should be carefully tuned to maximize the energy transferred to the system. The main parameters to be adjusted are the parametric excitation amplitude ߳, the parametric excitation frequency ߥ , , and, if two AMBs are used simultaneously to induce the parametric excitation, the phase ϕ between the excitation applied by the AMBs.
This work presents a short review of the method developed in [11] and analyzes the influence of the parameters mentioned in the kinetic energy transfer. The results presented were obtained from computational simulations of a flexible rotor sustained by two active magnetic bearings with three critical speeds in its operational range.

Balancing using Parametric Excitation
The balancing procedure proposed by [11] is based on the traditional Influence Coefficient Method and their correlation can be easily understood through figure 1, which presents the Campbell diagram of the rotor used in this work. A complete description of the system under analysis is presented in Section 3. As mentioned before, the traditional balancing consists of several runs, each one with a test mass placed in a different balancing plane. In each run, the rotor response is measured as close as possible of the critical speeds of the modes to be balanced. That means that, in the traditional balancing, the rotor operates over the 1x line (black dashed line in the Campbell Diagram in figure 1) and the measurements are taken at the red dots (critical speeds); i.e. the rotor is accelerated up to each critical speed. The measurements acquired are used to calculate the influence coefficients, which are used to obtain the correction masses. In summary, the rotor must be accelerated up to the last critical speed to be balanced and stopped once per balancing plane, and once with the residual unbalance.
In the proposed balancing method, the rotor is accelerated at the first critical speed only (red dashed line). Once this rotational speed is reached and the response of the first critical speed is measured, parametric excitation at the combination resonance frequencies is used to induce a kinetic energy transfer to the higher modes (blue triangles); to one mode at time. Thus, the higher modes responses can be obtained without the need to accelerate the rotor further than the first critical speed. After the measurements are obtained, the processes of calculating influence coefficients and correction masses are the same used in the traditional balancing method. A limitation of the proposed method, in its present form, is that the relations between the responses of the system obtained via parametric excitation (blue triangles) and by accelerating the rotor up to the correspondent critical speed (red dots) must be known. This means that the rotor must be accelerated up to each critical-speed, just like for the traditional procedure. However, if the same rotor generation (e.g. in a serial production) must be balanced (e.g. in a serial production), the relations obtained for the first rotor could be used for the subsequent rotors; avoiding run-ups to each critical speed and eventually saving time during the balancing procedure. A detailed description of the method and its limitations are found in [11].

Rotor Model
The rotor under analysis is composed by a 400 mm steel shaft of 3.2 mm diameter. The shaft model, presented in figure 2, is divided into 8 elements (red numbers), resulting in 9 nodes (black numbers); the model considers rotatory inertia, gyroscopic moments, and transverse shear effects [25]. The damping matrix of the rotor is proportional to its stiffness matrix by a factor of 1.5·10 -5 s. Five aluminum discs are distributed along the shaft: two at each end (nodes 1 and 9), one at the shaft midspan (node 5), and two discs placed at 75 mm from each bearing (nodes 3 and 7). Due to the complex geometry of the discs, each disc is modeled using three disc elements according to table 1.
The properties considered for aluminum and steel are found in table 2, and the unbalance distribution considered is presented in table 3. Nodes 2, 4, 6, and 8 are in the midspan between the discs.   Two active magnetic bearings are placed at both shaft ends (nodes 1 and 9). The bearings have a nominal gap ݃ of 0.5 mm, a constructive constant ‫ܭ‬ ୫ of 4·10 -6 N.m²/A² and work with a bias current ݅ of 0.04 A. A linearized magnetic force ‫ܨ‬ ୫ was considered, as presented in equation (3) and [12]; where ‫ܭ‬ ୡ is the actuator-gain and ‫ܭ‬ ୶ is the open-loop stiffness. Each axis of the AMBs is governed by a PID controller, according to equation (4) The rotor equation of motion is described by equation (5); where M0 is the mass matrix, K0 is the stiffness matrix, G0 is the gyroscopic matrix, C0 is the damping matrix (proportional to K0), ߗ is the rotational speed, x(t) is the vector of physical coordinates and f is the external force vector containing the unbalance force fu(t) and the AMBs forces fm(t). The system equation can be re-written using the state-space formulation according to equation (6); where 0 is the null matrix and I is the identity matrix.
The system state-space equation (equation (5)) can be connected to the linearized force of the AMBs (equation (3)) resulting in equation (7). Kc and Kx are diagonal matrixes containing, respectively, the actuator-gain and the open-loop stiffness coefficients in the positions correspondent to the d.o.f. where the magnetic forces are applied.
The PID controller can also be written using a state-space formulation (matrixes Ac, Bc, Cc, Dc) according to equation (8); where xc(t) represent the state variables of the controller and the minus signal represent the negative feedback. Finally, the closed-loop system can be obtained by connecting equations (7) and (8), resulting in equation (9).
From equation (9), the closed-loop system in state space can be obtained as presented in equations (10) and (11). The matrixes ‫ۯ‬ሺΩሻ and ۰ are directly obtained from equation (9), matrix ۱ contains ones in the positions of the desired outputs (in this case, the bearings displacements) and matrix ۲ is a null matrix. The closed-loop system matrix ‫ۯ‬ሺΩሻ can be used to obtain the modal response of the system by solving the eigenproblem. From the Campbell diagram of the rotor model, presented in figure 1, the first critical speed of the rotor is 65.93 rad/s, and the damped frequencies of the second and third modes at the first critical speed (blue triangles) are 241.45 rad/s and 429.23 rad/s, respectively. Therefore, the combination resonance frequencies to transfer energy from the first mode to the higher modes, according to equation (1), are ν 12,1 = 175.52 rad/s and ν 13,1 = 363.30 rad/s. Since the unbalance is a forward excitation and the system under analysis is isotropic, it is reasonable to consider only forward modes in the balancing procedure.

Results and Discussion
The objective of the analysis performed is to evaluate the sensitivity of the kinetic energy transfer between modes when the system is subjected to parametric excitation at the combination resonance frequencies. In the balancing procedure, it is desired that the maximum amount of energy is transferred to the mode under analysis, resulting in a higher signal-to-noise ratio.
In the simulations performed, the system was rotating at the first critical speed (65.93 rad/s) and kinetic energy was transferred to the higher modes (blue triangles at the Campbell Diagram in figure 1), at 241.45 rad/s and 429.23 rad/s, by inducing a parametric excitation simultaneously in both directions of both bearings through the PID parameter ‫ܭ‬ ୮ .
The results were obtained from time simulations of the model developed. Once the simulations are ready and the displacements in the bearings positions were obtained, the first two seconds of the results are discarded to avoid transient responses. Next, a time equivalent to 1600 integer periods of the frequency of interest is used to calculate the Discrete Fourier Transform (DFT) of the response. Hanning window was applied to the signals to avoid leakage. The results presented correspond to the horizontal displacement at the first active magnetic bearing (node 1). The displacement in the vertical direction and the displacements in the second bearing presented similar results and were omitted.
Initially, it was analyzed the influence of the phase difference between the parametric excitation applied in both bearings (ϕ = ϕ 1 -ϕ 2), for the range of 0° to 360°, on the energy transfer between modes.
The first simulations considered a parametric excitation amplitude of ߳ = 0.5. Figure 3a presents the response of the system when energy is transferred from the first mode to the second mode; and figure 3b presents the response of the system when energy is transferred from the first mode to the third mode. The entire frequency spectrums of the responses are presented in figure 3, as a function of the phase between the parametric excitations of the bearings ϕ. The red lines highlight the amplitudes of the modes receiving energy. For both cases, there is an optimal phase, in which the energy transferred to the higher modes is maximum. When energy is transferred to the second mode (241.45 rad/s), the optimal phase is 315°; and when it is transferred to the third mode (429.23 rad/s), the optimal phase is 157.5°.
In order to verify the influence of the parametric excitation amplitude ߳, the same set of simulations was repeated considering an amplitude range from 0.1 to 0.9. It is important to recall that the parametric excitation was applied in both bearings with the same amplitude in every case simulated (߳ = ߳ ଵ = ߳ ଶ ). The results obtained are presented in figures 4a and 4b for the second mode, and in figures 4c and 4d for the third mode. The same red lines highlighted in figure 3 are also presented in figure 4 (case ߳ = 0.5).
From the results obtained for both modes (figures 3a and 3c), it can be concluded that, for a given phase difference between the bearings, the response amplitude of the mode receiving energy increases linearly with the increase of the amplitude of the parametric excitation. However, the increase in the amplitude changes as the phase difference between the bearings changes. In the specific case when the AMBs are used to induce the parametric excitation, it is important to analyze if the resulting current due to the amplitude used (߳) does not lead the bearing to a magnetic saturation; which could drive the system unstable. In accordance with the results from figure 3, the second mode presents higher Parametric excitation generates a modulation of the vibration amplitudes with the employed parametric excitation frequency, which explain the peaks in 109.59 rad/s (߱ ଵ െ ߥ ଵଶ,ଵ , reflected in the y-axis) and 285,11 rad/s (߱ ଵ െ 2ߥ ଵଶ,ଵ , reflected in the y-axis) in figure 3a; and the peaks 297,37 rad/s (߱ ଵ െ ߥ ଵଷ,ଵ , reflected in the y-axis) in figure 3b. The presence of these side bands was analytically demonstrated in [12].
In the last set of simulations, the influence of the parametric excitation frequency ߥ on the energy transfer was verified. In this analysis, ߥ is treated as a percentage of the nominal value of the combination resonance frequencies of each mode. It was considered an interval of 80% (0.8) to 120% (1.2) around each parametric resonance frequency (ߥ ଵଶ,ଵ = 175.52 rad/s for the second mode, and ߥ ଵଷ,ଵ = 363.30 rad/s for the third mode). It is important to recall that the parametric excitation was applied in both directions of both bearings with the same amplitude and frequency.
The simulations were performed considering the optimal phase angles found in the previous analysis. First, the results obtained with the optimal phase for the second mode (315°) are presented in figures 5a and 5b (energy transferred to the second mode), and figures 5c and 5d (energy transferred to the third mode). Finally, the results obtained with the optimal phase for the third mode (157.5°) are From the results obtained for both modes in their optimal conditions (figures 5a and 6c), it can be concluded that, for a given parametric excitation frequency, the response amplitude of the mode receiving energy increases with the increase of the amplitude of the parametric excitation; in accordance with the results of the previous analysis. However, the increase rate in amplitude changes as the parametric excitation frequency changes, and the maximum gain in terms of energy transfer is obtained if a small detuning is applied.
The result of the second mode (figure 5b) shows that, when the parametric excitation is applied precisely at the combination resonance frequency (ߥ = ߥ ଵଶ,ଵ ), the energy transfer from the first mode to the second mode is a little lower than if a small detuning is applied to the excitation frequency. If the parametric excitation frequency is set to 91 % of the nominal value (ߥ = 0.91ߥ ଵଶ,ଵ ), the amplitude of the second mode increases only by a factor of 1.17 (17%), considering a parametric excitation amplitude of ߳ = 0.5 (red line in figures 5a and 5b). Considering the third mode in its optimal configuration (figure 6d), the energy transfer is 1.1 times higher (10%) if the parametric excitation frequency is set to 94% of the nominal value (ߥ = 0.94ߥ ଵଷ,ଵ ); a parametric excitation amplitude of ߳ = 0.5 (red line in figures 6c and 6d) was considered However, if the energy transfer is considered for a different mode of the one for which the phase was adjusted (figures 5c and 6a), it can be noted that the energy transfer is very low in the surroundings of the combination resonance frequency; which confirm the previous conclusion that the phase should be adjusted for each mode to receive energy. Figure 5 -Horizontal displacement at the AMB at node 1 in the presence of parametric excitation with a phase 315° between bearings (optimal for second mode) as a function of ϵ and ν: (a) and (b) second mode; (c) and (d) third mode. The red lines represent an equivalent case to the red lines in figure 3.

Conclusions
The beneficial effect of parametric excitation at different combination resonances is related to the effect similar to a kinetic energy transfer between vibration modes. In order to maximize the energy transferred, the parametric excitation must be properly tuned. Specifically, in the case of rotor balancing, a higher energy transfer ensures a higher signal-to-noise ratio, which directly influences the quality of the measurements used in the balancing procedure. This work analyzed the sensibility of the energy transfer between modes in the presence of parametric excitation in a horizontal rotor sustained by two AMBs. The results have shown that the parametric excitation frequency, amplitude and phase should be carefully tuned to maximize the energy transfer between modes. As demonstrated by the simulations, in the case studied, depending on the mode receiving energy, a proper frequency tuning resulted in a response 4 times higher. The results also have shown that a higher amplitude for the parametric excitation results in a higher response. However, it is important to recall that the parametric excitation amplitude should be limited to avoid magnetic saturation in the bearings which could drive the system unstable. Finally, if the parametric excitation is applied simultaneously by the two bearings, there is an optimal phase relation between the excitations that maximize the energy transfer; moreover, this optimal phase is different for each mode to receive energy.
(a) (b) (c) (d) Figure 6 -Horizontal displacement at the AMB at node 1 in the presence of parametric excitation with a phase 157.5° between bearings (optimal for third mode) as a function of ϵ and ν: (a) and (b) second mode; (c) and (d) third mode. The red lines represent an equivalent case to the red lines in figure 3.