On Numerical Simulation Approach for Multiple Resonance Modes in Servo Systems

1School of Mechanical Engineering, Suzhou University of Science and Technology, Suzhou 215009, China 2School of Electrical and Automation Engineering, East China Jiaotong University, Nanchang 330013, China 3School of Electronic and Information Engineering, Suzhou University of Science and Technology, Suzhou 215009, China 4School of Mechanical and Electronic Engineering, Jingdezhen Ceramic Institute, Jingdezhen 333001, China


Introduction
Servo system is a feedback control system that is used to follow or reproduce a process accurately.One of the basic requirements in control systems is that they have the ability to regulate the controlled variables to reference commands without a steady-state error against unknown and unmeasurable disturbance inputs.Its main task is to achieve the power amplification, transformation, and control under the command requirements, so that the output torque, speed, and position control is very flexible and convenient.In many cases, the servo system means that output of the system is a feedback control system of mechanical displacement or displacement velocity and acceleration and its role is to enable the mechanical displacement output (or angular displacement) to accurately track the input (or angle).Its structures have no difference with other forms of feedback control systems in principle.The servo system is originally used for national defense, such as gun control, automatic driving ships, aircraft, and missiles.Later it gradually extended to many sectors of the national economy such as automatic machine, wireless tracking control, and semiconductor manufacturing equipment [1][2][3][4][5][6].
Mechanical resonance is one of the most common problems designers face when trying to maximize either command response or dynamic stiffness [7].In the case of rotation, the drive and the drive shaft are generally considered rigid.In fact, the mechanical transmission shaft has different degrees of elastic torsion deformation.With the broadening of the frequency band of the system, the effect of the mechanical resonance caused by the elastic deformation on the dynamic characteristics of the system is obviously strengthened.From the transfer function of the system, it often contains a pair of conjugate zeroes and poles, which easily lead to mechanical resonance phenomenon.And this phenomenon is more obvious in the junction of feedback sensor and load connection in servo system [8].
Resonance is harmful in most cases.It causes mechanical deformation and greatly reduces the mechanical structure and even produces environmental pollution, and so on.In order to make the system have good dynamic performance, it is necessary to pay attention to control and eliminate resonance in servo control system.When resonance happened, the work of the excitation input system is balanced with the dissipated work phase, and the shape of the resonance peak is closely related to the damping.From the mechanical point of view, resonance precaution has the following measures: (1) improving the structure or the mechanical excitation to make the natural frequency avoid the excitation frequency; (2) adopting damping devices; (3) passing the resonance fast when there is a mechanical starting or stopping.From the control point, the low-pass or notch filter is often used to reduce the influence of resonance in the system [9][10][11][12].Other theories and approaches such as resonance ratio control [13] and acceleration feedback [14] have been put forward to decrease the influence of resonance mode [15,16].The simulation of closed-loop is requisite when analyzing the influence of the resonance modes in servo system.And the mathematical model of the resonance mode must be considered in the simulation; otherwise the closed-loop simulation results will differ from the actual servo system greatly.Therefore the study of numerical simulation of resonant modes has great significance.For the simulation of the resonance mode, only the simulation of single resonance mode is reported [17].And the simulation of multiple resonance modes is still open.This motivates us to do this study.
Based on the analysis of the mathematical model of resonance mode, the resonance mode is perceived as a product of a second-order differentiation element and an oscillating element.And the second-order differentiation element is proposed to simulate the resonant part and the oscillating element is proposed to simulate the antiresonant part.The simulation approach for multiple resonance modes in servo systems is proposed in this paper.

Mathematical Model of Resonance Mode
Mechanical resonance is one of the most pervasive problems in motion control.Resonance is caused by springiness or compliance between two inertias.Most often, resonance is caused by compliance in transmission between motor and load.It is shown in Figure 1 [10,15], where   is electromagnetic torque,   is motor position,   is motor velocity,   is motor acceleration,   is load position,   is load velocity,   is motor acceleration,   is the moment of inertia for motor,   is the moment of inertia for load,   is velocity compliance coefficient, and   is position compliance.Once the motor shaft begins to turn, the combined transmission compliance winds up, which will produce torque to the load in proportion to the difference between motor and load positions.
When there is only one resonance mode in the system, the model of the resonance is expressed as follows [9,10,18]: The Bode diagram of this model is shown in Figure 2.

The Simulation of Single Resonance Mode
It is very difficult to get the value of   and   in (1) and users care little about these two values in applications.But the values of resonant frequency  R , resonant amplitude  R , antiresonant frequency  AR , and antiresonant amplitude  AR should be cared about.We can design the resonance mode shown in (1) if these four parameters are given.In the simulation of resonance mode, the following model can be considered: We now consider setting the values of  1 ,  2 ,  1 , and  2 to make the frequency response of () similar to its expected frequency response.
This model is a product of a second-order differentiation element and an oscillating element.At low frequency, coefficient  2 2 / 2 1 makes the magnitude of the model's frequency character turn to zero, which will not affect the low frequency character of the position loop object.
Before determining the undetermined coefficients in (2), we would consider the oscillating element Given  2 = 0.02 and  2 = 200 rad/s, the frequency character of ZD() is shown in Figure 3.This Bode diagram shows a resonance clearly, and our purpose is to use the magnitude of this resonance  2 to generate the resonant magnitude  R in resonance model.It is easy to prove that the resonant frequency of this oscillating element occurs at  2 , and it follows that [19] If the resonant magnitude is known, we can use it to get damping by rewriting formula (4) into the following: But the resonant magnitude unit in ( 4) and ( 6) is not dB, and if we want to use the known resonant magnitude  R (unit is dB) to calculate proper damping, the resonant magnitude whose unit is dB should be changed into corresponding resonant magnitude whose unit is not dB.The transformation formula is shown as follows: Substitute ( 7) into (6), to obtain When resonant magnitude of resonance mode is small, it has little influence in servo systems.The resonant magnitudes which we need to consider generally have large value.Since the damping  2 is small in (8), we can argue through (5) that When simulating resonance modes, we can simply consider the resonant frequency as  2 .If the resonant frequency  R and the resonant magnitude  R are known, we can know  2 and  2 ; thus the model of oscillating element shown in (3) can be defined.
Then we consider the second-order differentiation element in (2).
Given  1 = 0.03 and  1 = 140 rad/s, the frequency character of WF() is shown in Figure 4.The Bode diagram clearly shows an antiresonance, and our purpose is to use the magnitude of this resonance  1 to generate the antiresonant magnitude  AR .It can be easily proved that the antiresonant frequency of the element occurs at  1 , and it follows that But the unit of antiresonant magnitude in (11) is not dB, and if antiresonant magnitude  AR (unit dB) is known and we use it to calculate proper damping, the antiresonant magnitude whose unit is dB should be changed into corresponding antiresonant magnitude whose unit is not dB.The transformation formula is Substitute ( 13) into (11), to obtain Since the sign of the antiresonant magnitude  AR is minus, the damping rate  1 is small.We can argue through (12) that When simulating resonance modes, we can simply consider the antiresonant frequency as  1 .If antiresonant frequency  AR and antiresonant magnitude  AR are known, we can calculate  1 and  1 ; that is to say, the model of second-order differentiation element shown in ( 10) is defined.Calculate (3) to obtain Calculate (10) to obtain Add formulas ( 16) and ( 17), and set  =  1 and  =  2 ; then  AR ,  R can be calculated as Simplify formula (18) to obtain This is a system of binary quadric equations about  2 1 ,  2 2 and one can calculate  1 ,  2 through solving the equality system.When the values of  1 ,  2 ,  1 , and  2 are gotten, the model of formula (2) can be gotten.That is to say, the mathematical model of a resonance mode is gotten.

The Simulation of Multiple
Resonance Modes If the system has two resonance modes, the main idea of the simulation is as follows: (1) Setting up the mathematical model of these two resonance modes separately (2) Multiplying these two mathematical models together (3) Fine-tuning the values of damping ratio.
The flow chart for the simulation of two resonance modes is shown in Figure 5.What we need to explain is that two resonance modes have coupling effect at the resonant frequency and antiresonant frequency when the Bode diagrams of two resonance modes are superposed, so the given  AR1 and  R1 or designed  11 and  12 and the given  AR2 and  R2 or designed  21 and  22 will not meet the requirements.In this case, we could fine-tune the values of  11 ,  12 ,  21 , and  22 by a way of circulation based on the sign of the difference of  AR1 and  AR2 or  R1 and  R2 .The principle of fine tuning is that the smaller the damping ratio is, the larger the resonance magnitude is and the larger the damping ratio is, the smaller the resonance magnitude is.
Similarly, this approach can be extended to the simulation of three or even more resonances in servo systems.

Numerical Simulation Examples
Example 1. Generate one resonance mode by the approach in this paper.The resonance mode has the antiresonant frequency of 200 Hz and the antiresonant magnitude of 20 dB, while the resonant frequency is 300 Hz and the resonant magnitude is 20 dB.Using the approach in this paper, the numerical model of this resonance mode can be gotten by  () = 3.553 × 10  The Bode diagrams of the modes are shown in Figure 7.We can see clearly from the Bode diagrams that there is an antiresonance which has a magnitude of −19.9 dB at the frequency of 220 Hz, a resonance which has a magnitude of 19.4 dB at the frequency of 309 Hz, an antiresonance of −10 dB at the frequency of 540 Hz, and a resonance magnitude of 15 dB at 695 Hz that meet the requests.

Conclusions
The mathematical model of resonance modes is introduced in this paper.The second-order differentiation element is proposed to simulate the resonant part and the oscillating element is proposed to simulate the antiresonant part; the numerical simulation of a single resonant mode is realized in this way.In the case of the simulation of multiple resonance modes, one needs to multiply the simulated single resonance modes and fine-tuning damping to meet accuracy requirements.When fine-tuning the damping ratio, one should use a method of circulation based on the sign of the difference of two resonant magnitudes.The principle of fine tuning is that the smaller the damping ratio is, the larger the resonant magnitude is and the larger the damping ratio is, the smaller the resonant magnitude is.
This approach of simulation can be applied to the research on eliminating or compensating the effects of mechanical resonance in servo system and also the optimization of servo system, and so on.An interesting topic for further study is the method of fine-tuning damping since the multiple resonances will have the coupling effect.

Figure 1 :
Figure 1: Diagram of compliant coupling between motor and load.

Figure 4 :
Figure 4: Bode diagram of typical differential model.

Figure 6 :
Figure 6: Bode diagram of one resonance mode.
6 2 + 2.499 × 10 8  + 5.611 × 10 12 1.579 × 10 6  2 + 3.755 × 10 8  + 5.611 × 10 12 .(20) Generate two resonance modes by the approach in this paper.One resonance mode has the antiresonant frequency of 220 Hz and the antiresonant magnitude of 20 dB, while the resonant frequency is 310 Hz and the resonant magnitude is 20 dB.Another resonance mode has the antiresonant frequency of 540 Hz and the antiresonant M ！２1 , w n11 ; M ２1 , w n21 ; M ！２2 , w n12 ; M ２2 , w n22 .Set w ！２1 as w n11 , w ！２2 as w n12 , w ２1 as w n21 , w ２2 as w n22 .M ！２1 , w ！２1 ; M ２1 , w ２1 to generate superposition of two Multiply two links of mathematical models decided by M ！２2 , w ！２2 , M ２2 , w ２2 to generate superposition of two resonances And fine-tune damping  21 ,  22 by a way of circulation.Fine-tune the damping ratio  21 ,  22 ,  11 ,  12 by a way of circulation based on the sign of the difference of M ！２1 and M ！２2 , or M ２1 and M ２2 .