This paper shows a theoretical vibration analysis regarding the controller’s parameters and the gyroscopic effect, based on a simplified rotordynamic model. Combined with 600 Wh energy storage flywheel rotor system mathematical model, the Campbell diagram of the rotor system was obtained by the calculation of the whirl frequency under different parameters of the controller in MATLAB to analyze the effect of the controller parameter on the whirl frequency and to limit the operating speed and acceleration or deceleration of the rotor. The result of the analysis can be used to set the support position of the rotor system, limit the ratio of transverse moment of inertia and the polar moment of inertia, and direct the flywheel prototype future design. The presented simplified rotordynamic model can also be applied to rotating machines.
Later in the 1970s, flywheel energy storage was proposed as a primary objective for electric vehicles and stationary power backup [
Many problems appear as the development of flywheel energy storage, and one of them is the bearing. Besides, the active magnetic bearing (AMB) implies that bearing forces are actively controlled by means of electromagnets, a welldesigned closed control loop, and other components such as position sensors and power amplifiers [
Many different kinds of excitations exist in rotor system—for example, mechanical unbalance and misalignment of the coupling—which may cause vibrations [
The aim of this paper is to show a theoretical vibration analysis of the rotor system of 600 Wh flywheel energy storage system, based on a simplified rotordynamic model. The result of the analysis can be used to set the support position of the rotor system, limit the ratio of transverse moment of inertia and the polar moment of inertia, and direct the flywheel prototype future design.
The basic layout of a flywheel energy storage system is depicted in Figure
Basic layout of a flywheel energy storage system.
Two radial active electromagnetic bearings (RAMBs) including upper and lower RAMBs (RAMB1 and RAMB2 in Figure
One 70 kg flywheel is mounted on the rotor whose position can be adjusted to modify the rotational characteristics. Furthermore, one motor/generator is equipped between the lower RAMB and flywheel and is responsible for the torque generation or electricity generation based on the rotational speed requirement. However, in the rotor system, the rollingelement auxiliary bearings (auxiliary bearing 1 and auxiliary bearing 2 in Figure
Considering the diameter of the shaft and the loading capacity, the RAMB with 8pole legs was designed in the 600 Wh prototype system. The structural configuration of the RAMB is shown in Figure
Structural configurations and the basic magnetic bearing control loop of the RAMB.
Structural configuration of the RAMB with 8pole legs
The basic magnetic bearing control loop and its elements
Taking the two pairs of poles in the
Additionally, the total nonlinear attractive electromagnetic forces for the
The development of PID control has been around for 90 years and is still popular for industries and academies nowadays [
The block diagram of the total control system of the AMB is shown in Figure
Block diagram of the control system.
The transfer function of the PID controller can be written as
Combined with the gains of the power amplifier and the sensor, the characteristic equation of the system can be obtained as follows:
The parameters
The parameters of the bearing structure are set as
The range of the differential time parameter
In this study, the rotor is assumed to be a rigid and symmetric body. It is assumed that all magnets have identical structure. For simplicity, we neglect the magnetic flux leakage, the fringe magnetic flux, the eddycurrent loss, the saturation and hysteresis of the magnetic core material, and the coupling effects between the electromagnets. The relationship between the center of mass
Geometry relationships of rotor and AMB systems.
Equations (
Equation (
Equations (
Equation (
It is noted that when the rotor is regulated perfectly
When the rotor is suspended steadily, the relative coordinate system of the flywheel rotor coincides with the absolute coordinate system of the AMB’s stator. The potential energy of the stator in the origin
The total kinetic energy of the system
The expression for the translational kinetic energy
Flywheel rotor with the eccentric mass.
By means of Lagrange equations, which state
Transfer function block diagram of the PID controller.
In Figure
The differential equation of the system can be given as
When
From the above analysis,
Suitable values for the parameters involved in the model of (
Model parameters.
Parameter  Description  Value  Unit 


Mass of rotor 

kg 

Length of rotor 

m 

Distance between 

m 

Distance between 

m 

Distance between 

m 

Distance between 

m 

Transverse mass moments of inertia of rotor about the 



Polar mass moment of inertia of rotor about the 



Distance between CG and external disturbances 

m 

Nominal air gaps in the 

mm 

Electromagnet parameter 

H/m 

Bias currents 

A 

Nominal air gap in the 

mm 

Design maximum speed 

rpm 

Rated speed  5000~16000  rpm 
A planer motion of a rotor is called a whirling motion or a whirl. And a circular whirl in the same direction as the shaft rotation is called a forward whirl, and that in the opposite direction is termed a backward whirl. Plot of these natural angular frequencies versus the rotational speed is called a natural angular frequency diagrams (or shortly natural frequency diagram) or a
According to Table
Examination of the results of the simulation using the model of the rotor allows us to argument that rotational modes of vibration have natural frequencies that are considerably larger than those of translational modes. The forward whirling modes that represent critical speeds, that is, the ones that intersect with the line describing the frequencies equal to the running speeds in Figure
Campbell diagram of the system.
In Figure
In Figure
Campbell diagram of the system with the change of the proportional parameter.
In Figure
Campbell diagram of the system with the change of the differential parameter.
The backward whirling modes change distinctly, 3200 RPM in the
The effect of differential parameters on the whirling mode is distinct in running speed
The following analysis shows how the dynamics of the system would change if the distance between the two RAMBs changed in the axial direction. The controller parameter
Campbell diagram of the system with the change of the RAMB positions.
The purpose of the flywheel storage machinery is to store energy as much as possible. The polar mass moment of inertia of rotor determines the energy storage of the flywheel storage machinery. So the effect of the whirling modes of the system is gained as follows. The polar mass moment and transverse mass moments of inertia of rotor depend on the geometry of the rotor. Figure
In Figure
Campbell diagram of the system with the inertia ratio of the polar moment and the transverse mass moment.
With a constant magnetic force and a constant polar mass moment of inertia of rotor, the transverse mass moments of inertia of rotor can be changed just by the change of the distance between the local center to the overall center of the rotor to avoid the work speed in the strong whirling mode, which can greatly improve the stability of the system.
Variation studies were conducted to assess the influence of controller and the rotor geometry, with running speed on rotor dynamic stability. Each point on the lines (or surface) represents the threshold running speed above which the rotor becomes unstable for a certain parameter configuration. This means that there is a distinctive operating speed below which the system is always stable for a given parametric configuration.
Considering the influence of controller, the dynamic model of the rigidity flywheel rotor supporting by AMBs was established to analyze the dynamics characteristic. The influence analysis of the controller’s parameters to the dynamics characteristic was obtained.
The rotational mode will increase as the differential parameter increases. And the translational mode also increases, which showed that the differential parameter also affects the stiffness of the rotor system, but the influence is not very distinctly compared with the influence of the proportional parameter.
The result of the analysis can be used to set the support position of the rotor system, limit the ratio of transverse moment of inertia and the polar moment of inertia, and direct the flywheel prototype future design. The presented simplified rotordynamic model can also be applied to rotating machines.
This work is supported by the National 863 Project of China (no. 2013AA050802).