CC BY
3Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 5, pp. 873-883. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd221226
NONLINEAR ENGINEERING AND ROBOTICS
MSC 2010: 34H15
Reducing Rotor Vibrations in Active Conical Fluid Film Bearings with Controllable Gap
Yu . N . Kazakov, A . V . Kornaev, D . V . Shutin, E . P. Kornaeva, L . A . Savin
Despite the fact that the hydrodynamic lubrication is a self-controlled process, the rotor dynamics and energy efficiency in fluid film bearing are often the subject to be improved. We have designed control systems with adaptive PI and DQN-agent based controllers to minimize the rotor oscillations amplitude in a conical fluid film bearing. The design of the bearing allows its axial displacement and thus adjustment of its average clearance. The tests were performed using a simulation model in MATLAB software. The simulation model includes modules of a rigid shaft, a conical bearing, and a control system. The bearing module is based on numerical solution of the generalized Reynolds equation and its nonlinear approximation with fully connected neural networks. The results obtained demonstrate that both the adaptive PI controller and the DQN-based controller reduce the rotor vibrations even when imbalance in the system grows. However, the DQN-based approach provides some additional advantages in the controller designing process as well as in the system performance.
Keywords: active fluid film bearing, conical bearing, simulation modeling, DQN-agent, adaptive PI controller
Received September 06, 2022 Accepted December 09, 2022
The study was supported by the Russian Science Foundation grant No. 22-19-00789, https://rscf.ru/en/project/22-19-00789/.
Yuri. N. Kazakov [email protected] Denis V. Shutin [email protected] Elena P. Kornaeva [email protected] Leonid A. Savin [email protected] Orel State University
ul. Komsomolskaya 95, Orel, 302026 Russia
Alexey V. Kornaev [email protected]
Innopolis University
ul. Universitetskaya 1, Innopolis, 420500 Russia
1. Introduction
Active bearings are often developed based on magnetic bearings [1-5]. However, there are also a number of works devoted to active fluid film bearings. The control systems in such bearings are aimed at solving the following problems: minimizing vibration and noise, reducing friction losses, and increasing reliability and service life [6, 7]. Active fluid film bearings come in a variety of designs. Often, bearings with built-in active components are used [8-12]. This design of the bearing allows one to change the geometry of the lubricant film in the bearing. Models of bearings with moving pads are presented in [8-10]. In [13-15], the authors deal with active self-aligning bearings. Z. Kai et al. [13] and A. Wu et al. [14] have improved bearing performance using nonlinear controllers. The authors showed that the proposed nonlinear controllers require less control power compared to a PID controller [14]. Another type of active fluid film bearings is based on the adjustable lubrication principle. In [15] the authors have presented the concept of an active bearing with adjustable supply pressure in the radial direction using PD controllers.
Researchers often use simple control algorithms for active bearings. They pay great attention to the design and structure of bearings. However, despite the rather simple structure of the controller, its configuration is a complex process. It often requires a lot of efforts, experience and even some fortune. Intelligent control systems can become an alternative. Such controllers are often based on the analysis of a large amount of experimental data [16, 17]. Quite a lot of work is devoted to controllers based on reinforcement learning. One of such controllers is the discrete algorithm called Deep Q Network (DQN) [18-21]. Tarun [18] used DQN to control the movement of the manipulator. J.B.Kim [20] developed a transfer learning algorithm for the DQN agent. He used simulation models to train the control system. In [21] the authors used DQN for the control system of hydraulic units. Thus, DQN agents are used in many applications and they can also be implemented in controlled fluid film bearings. It is worth noting that DQN agents require a relatively small set of training data compared to other deep reinforcement learning agents.
This work presents a theoretical study on a controllable rotor system where active conical fluid film bearings reduce rotor vibrations. The control algorithms based on a PI controller and reinforcement learning methods are compared. The results show features of both controllers and differences in their performance. Also, the simulation results are useful for determining a rational configuration of the hardware facilities to be manufactured for the further experimental part of the study.
2. Conception of controlled shaft bearing system
The operation of fluid film bearings is based on the hydrodynamic effect. This effect depends on many factors. For example, it is necessary to achieve a very thin lubricant film to obtain large bearing capacity. Film thickness is connected with several parameters such as shaft eccentricity and average bearing clearance [22]. The main idea of the adjustable bearing presented in this study is that the gap in a conical fluid film bearing can be controlled by axial displacement of the bearing.
The proposed configuration of the active rotor system includes a shaft on a conical bearing. The motor end of the shaft rests on a coupling, and its free end rests on a conical bearing (see Fig. 1) with a sleeve with an adjustable axial position. The coupling on which the shaft rests is compliant and able to take loads. This configuration was used to conduct computational experiments and test the developed rotor motion controllers.
Pressure sensor
Fig. 1. Schematic of a shaft-bearing system with an active conical bearing
The bearing can be lubricated with both water and oil. The lubricant is supplied under pressure p0. The pressure is provided by a pump and can be adjusted using a servo valve.
In a screw-nut transmission, the nut has a conical surface. The movement of the nut allows an axial force to be applied to the bearing surface. This action causes the bearing to move in axial direction. The reactions in the damping element depend on the amount of displacement and the axial speed of the bearing. Such a bearing design allows one to adjust the average clearance and modify some bearing parameters such as load capacity and friction torque.
The presented concept includes displacement sensors to monitor the position of the shaft on all axes, a pressure sensor to measure the supply pressure and a torque sensor.
The following system parameters were used for the simulation. The small bearing diameter is 40 mm, its taper angle is a = 3 degrees, and the bearing width is 26 mm. Water was used as a lubricant, the supply pressure is p0 = 1.2 • 105 Pa. The shaft is 380 mm long and 40 mm in diameter. The shaft mass is 3 kg and the imbalance is 0 ^ mud ^ 1_4 kgm. The shaft rotates at a constant speed of 3000 rpm. The stiffness of the damping element and the damping coefficients are K = 40 000 N/m and B = 50 N • s/m, respectively. The bearing is able to move in the axial direction due to the applied control pressure in the range from —0.5 • 10_3 m to 0.7 • 10_3 m. Accordingly, the gap in the bearing changed from 73 • 10_6 m to 138 • 10_6 m. The mean friction torque under these conditions varies from 0.0061 Nm to 0.0032 Nm.
3. Simulation model
The simulation model of the presented active conical bearing system was developed in the Simulink environment. For this, the following Simscape Multibody modules and the Deep Learning, Reinforcement Learning, and Signal Processing toolboxes were used.
3.1. Rotor dynamics
A simulation model of an active bearing is shown in Fig. 2. Force ANN and Torque ANN blocks were used to calculate the reactions of the lubricating layer of the bearing and the friction torque. These blocks contain artificial neural networks for approximating the physical equations. They receive data on the speed and position of the rotor in the bearing from the bearing joint block. The imbalance force block was used to generate the centrifugal force according to the given unbalance value. The damping element reaction block calculates the damping element according to Fig. 1. The end face force block was designed to calculate the axial force depending on the pressure of the lubricant supply. The clutch block contains the corresponding stiffness and damping coefficients. They were selected in such a way that at p0 the shaft displacement was about 0.
Electric motor
Shaft
Tj . T . , Torque ANN Bearing Joint ^
Fig. 2. Structure of the simulation model
The use of such a modeling method makes it possible to model objects as equivalent to real objects. It also makes it possible to perform a long process of training agents on a simulation model without using real object data.
3.2. Hydrodynamic lubrication
A series of rotor trajectories were calculated to train ANNs approximating the responses of the fluid film. Fully connected ANNs [23] were used for approximation. As an approximation, we can represent the bearing response and friction torque as a function of the position and speed of the shaft in the bearing [22] and imbalance: Fb = Fb(Xi, Vi, mud), M = M(Xi, Vi, mud) [24], where Xi are the coordinate axes, V\ is the velocity in the coordinate axes, and mud is imbalance. It is known that artificial neural networks work well even under conditions of high nonlinearity [23]. We have used the neural net fitting programming tool in MATLAB to train the ANNs [23]. Two datasets of approximately 384 000 samples each were generated based on simulations to train, validate, and test networks at a ratio of 0.8:0.15:0.05, respectively.
As a result, the ANNs obtained provide an approximation error of less than 1 %. Outside the controlling region, the shaft position error is 3.5%, and for the friction torque it is 3%. However, the errors increase as the axial displacement of the shaft increases.
4. Design of controllers
4.1. Adaptive PI controller model
The adaptive PI controller is based on a simple PI controller:
p(z)
uAPI = u(z) = Pe.(z) + Its-^-L, (4.1)
1—z
where P and I are the proportional and integral coefficients, respectively, tS is the sample time, z is a complex number, u is an output controller signal, and e(z) is error control.
The controller has been upgraded for the use in the shaft-bearing system. The control error of the adaptive PI controller is
\pos\ — hg™ if X3 > Xmin A pos > hg™, eAPI = <( 0 if X3 > X^n A pos < hR™, (4.2)
\X3\—X3min if X3 <X3min,
where pos = \/Xf + is eccentricity, h™ax is the desired control area in a bearing, and X™ is the minimal admissible position on the X3 axis.
4.2. Simulation model with an adaptive PI controller
The simulation environment includes a controller block, an environment, a button, and a lamp. The button is intended to turn the controller on and off. The lamp notifies the observer about the output of the oscillations of the rotor beyond the specified limits (see Fig. 3).
Fig. 3. Adaptive PI controller
The control error value is the input of the controller, which is calculated by (4.2). The values h0 and k are used to select the trusted control area. The cumulative sum accumulates the control signal. The control signal of the controller is the force with which the actuator acts on the bearing. The output signal of the controller is limited by the limits Lmax and Lmin.
4.3. DQN agent model
The DQN agent is a reinforcement learning algorithm. At each time step T the controller (agent) receives a feedback from the system (environment) in the form of a state signal ST, then takes an action AT and a reward rT in response. It is supposed that a current state completely characterizes the state of the system.
The agent trains a critic q(S, A) to estimate the return of the future reward [25]:
Qt = rT + YrT +i + Y rT+2 + •••
(4.3)
where Y is discount.
During the training process it is necessary to achieve the minimization of the error between the trained function q(S, A) and the optimal function q*(S, A) that can be estimated with the Bellman equation [25]:
QT (St , AT ) = rT + Y max[qT+I(St+i, AT+I )]•
The critic is normally an ANN that minimizes the loss function while training:
1 m 2
(4.4)
(4.5)
i=1
where 0(fc) are the weights of the network, m is the number of training samples in the minibatch, and yT = qT is the estimation for the future reward.
4.4. Simulation model with DQN controller
The control system is a DQN agent block with the observation, reward and interrupt functions as the input parameters, and with the control signal as the output (see Fig. 4) [25].
Fig. 4. DQN controller
The control system generates a discrete control signal in the range of the preset values. The designed control system uses 5 levels of the control signal: —1, —0.5, 0.5, 1, 0 N. The control signal at each time step is added to the accumulated signal value. The frequency of renewal of the control signal is 10 Hz.
5. Results and discussion
Malfunctions of rotary machines can lead to growing oscillations in time. Such phenomena adversely affect the system and can lead to its failure. The simulation test series were performed in order to obtain the qualitative estimations of the proposed control systems. The described control systems were tested in the task of minimizing the amplitudes of rotor oscillations in time.
5.1. Adaptive PI controller
A simulation environment was created to test the controller. It was assumed that, during the simulation, the amplitude of rotor oscillations would increase. The growth of the amplitude was set by the increasing imbalance. The imbalance varied linearly from 0 to 9.3 • 10_5 kgm. The simulation time was 35 s. The adaptive PI controller had the following settings: P = 0.0001, I = 0.001, Lmax = 0.1, Lmin = —0.1, k = 0.8. The critical area of rotor operation is the eccentricity exceeding 85 /m, after which the light comes on.
It is assumed in the considered scenario that the rotary machine is operating under fault conditions. This leads to growing amplitudes of rotor oscillations. After the rotor exits the selected critical area, a signal light is activated. The operator has the option to turn on the controller. When the controller is turned on, a control action is generated. The simulation results are shown in Fig. 5. It can be seen that the final amplitude of rotor oscillations under control is less than if there was no control system. When the controller is turned on, at about 23 seconds, the rotor starts moving towards the geometric center of the bearing. With this movement, there is a decrease in the thickness of the lubricant film and an increase in its stiffness and damping coefficients. This leads to a decrease in the amplitude of the rotor oscillations. This decrease in amplitude is about 15 percent. However, the decrease in the amplitude is due to a decrease in the gap of the bearing. This can greatly affect the dynamics of the rotor at small gap values.
5.2. DQN agent
The DQN agent training process has the following settings: the maximum number of iterations is 1000, the maximum episode duration is 5 s, the learning rate is 0.001, the experience buffer length is 100 000 time steps, the discount factor is 0.85, and the minibatch size is 250. The DQN agent network architecture is shown in Fig. 4. The number of neurons in hidden layers is [[14, 18, 18]]. At each time step, the DQN agent receives the reward of +1 if the oscillations are smaller than 0.9h0. Otherwise, the reward is equal to +0. A penalty of —50 is applied when the axial displacement of the shaft is more than 1.5 mm, or less than —0.7 mm, or the shaft touches the bearing. The DQN agent was trained at 630 iterations. The test results are presented in Fig. 6.
In the test simulation scenario the imbalance varied linearly from 2 • 10_5 to 9.4 • 10_5. The simulation time was 5 s. The figure shows that the final trajectory of the rotor oscillations with the activated DQN controller is smaller by about 40 % than without control. However, the gap at this point is small for safe rotor movement. This probably requires a partial improvement in
Bearing clearance at the end time of simulation gpo Initial Bearing clearance h0 = 100 fim
The preferable area 0.9h0 120° 1^0
i=ls t = 2 s
Fig. 5. Rotor trajectories when controlled by the adaptive PI controller. (a) with control, (b) without control
the reward function. The resulting trajectory lies very close to the bearing surface. This is due to the error of the model rework (see Fig. 6).
The main advantage of the DQN based approach over PID control is that the DQN is an optimal controller. It is able to adjust the control action dynamically under changing conditions. While the adaptive PID controller minimizes the shaft oscillation around the fixed setpoint, the DQN controller is able to adjust the setpoint maximizing the minimum film thickness in the bearing. It is not necessary to set such behavior in an explicit manner for the DQN agent, unlike in the case of PID control. Using the task for the DQN agent as an example, we do not set a clear boundary which the agent needs to keep. This limit is automatically selected by the system during the controller training process and depends on the operating conditions of the system at a certain moment. This makes the controller flexible and more convenient in design.
6. Conclusions
The proposed simulation model of a rotating machine with an adjustable conical fluid film bearing allows an estimation of the efficiency of control systems based on an adaptive PI controller and a DQN-agent. The following points can be highlighted from the simulation results.
270°
270°
Fig. 6. Rotor trajectories when controlled by the DQN-agent. (a) with control, (b) without control
1. Oscillations in an active conical fluid film bearing can be reduced using both control techniques considered above. However, this is due to a change in the gap and the risk of violation of the hydrodynamic regime of friction.
2. The DQN control provides more flexible and reliable system behavior in the task of minimization of shaft oscillations as compared to PID control because the DQN agent implements an optimal control strategy. Also, unlike PID control, DQN control does not require setting all control parameters in explicit form. The complex constraints and requirements can be set in a simple way as rewards and penalties, and an optimal control strategy taking them in account will be found automatically during the training process.
3. The main disadvantage of the DQN control as compared to the PID control is associated with a long training process. However, this disadvantage may be partially reduced by using digital twins of the machines.
Conflict of interest
The authors declare that they have no conflict of interest.
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