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In this paper, a new piezoelectric dynamic balance regulator, which can be used in motorised spindle systems, is presented. The dynamic balancing adjustment mechanism is driven by an inplane bending vibration from an annular piezoelectric stator excited by a highfrequency sinusoidal input voltage. This device has different construction, characteristics and operating principles than a conventional balance regulator. In this work, a dynamic model of the regulator is first developed using a detailed analytical method. Thereafter, MATLAB is employed to numerically simulate the relations between the dominant parameters and the characteristics of the regulator based on thedynamic model. Finally, experimental measurements are used to certify the validity of the dynamic model. Consequently, the mathematical model presented and analysed in this paper can be used as a tool for optimising the design of a piezoelectric dynamic balance regulator during steady state operation.
As a novel type of piezoelectric actuator, ultrasonic motors have been studied by researchers and companies all over the World for nearly 40 years. These devices use the converse piezoelectric effect of piezoceramics and convert the ultrasonic vibration of the stator into linear or rotational motion in a rotor using the friction force. They inherently posses slow speed/high torque output, high holding torque and rapid response characteristics, all of which combine for the potential to be used as a precise and accurate positioning actuator. Because standing wavetype ultrasonic motors have the advantage of accurate positioning without feedback, a new idea is proposed in this paper, an ultrasonic piezoelectric dynamic balance regulator based on the basic principles of a standing wavetype ultrasonic motor. This regulator is the key component of an
There are many different types of dynamic balance regulators that have evolved over the years in the field of rotor balancing. Van de Vegte [
In this paper, a new type of piezoelectric dynamic balance regulator is introduced, which can overcome some of the disadvantages of existing dynamic balance regulators. As pointed out before, this regulator based on the basic principles of a standing wavetype ultrasonic motor, and combines features such as high driving torque at low rotational speed, high holding torque without an applied electric power, extremely low noise in operation, accurate positioning without feedback, simple mechanical design and rapid response. Because the regulator utilises the inplane bending vibration mode of the piezoelectric stators to drive the rotor, a contact interface exists on the circumferential side face. This type of structure further minimises the regulator thickness and miniaturises the regulator.Meanwhile, because the regulator is driven with singlephase AC voltage, this type of drive mode simplifies the supply power and enhances the system reliability. Above all, a particularly attractive feature of the regulator is that it possesses high displacement resolution and can achieve control precision on the order of microns, thus it can achieve very high adjustment accuracy.
The topic of this paper is to perform highefficiency dynamic modelling and to optimise the key parameters of this regulator. Analytical, numerical and experimental methods have been employed in these investigations. For simplicity, the nonlinear contact mechanics are studied under the assumption that the vibrational characteristics of the resonating structure are not affected by the contact process. This analysis process simplifies the investigations considerably and quite often results in a good description of the mechanism of motion for the regulator. For an experimental illustration of the theoretical and numerical results, a sample device was fabricated and characterised based on this mechanical analysis. The experimental results reported on here confirm the validity and applicability of the regulator structure and can be used for guidance in further optimisation of the regulator.
In Section 2 of this paper, the overall structural characteristics and the operating principles of the piezoelectric dynamic balance regulator are described in detail. In Section 3, a dynamic analysis on the stator, the rotor, the friction layer and the statorrotor contact model is performed, which permits the energy conversion characteristics of the regulator to be obtained. In Section 4, MATLAB is employed to numerically simulate the characteristics of the regulator based on the above mathematical model, and optimisation of core parameters for the regulator is conducted according to the simulation results. Afterwards, a prototype is fabricated using the optimised parameters, and its performance is tested experimentally. Comparing the experimental and simulated load torque/rotational speed relation for the regulator, the validity of the dynamic model is proven. Finally, conclusions and future research prospects are detailed in Section 5.
There are two piezoelectric actuating devices installed in the regulator housing. Each device consists of a rotor and a piezoelectric stator with six driving teeth. The stator is fixed on the mounting surface of the regulator. The driving teeth on the stator press against the inner circumference of the rotor, which rotates in a groove on the mounting surface (
The stator and the rotor are pretensioned in the radial direction to avoid an increase in the motor thickness during operations. Additionally, a counterweight block is installed on each rotor and rotates with the rotor. When the piezoelectric stator resonates in an inplane bending vibration mode under highfrequency sinusoidal voltage excitation, the stator will act as a friction drive to move the rotor structure to the desired position to alter the balancing vector. The combined action of the two piezoelectric actuating devices creates a resultant balancing vector that achieves dynamic balance adjustment by controlling the positions of the two counterweight blocks. The dynamic balancing principle is shown in
The piezoelectric stator consists of an annular elastic metal body with two annular piezoceramic discs bonded to the top and bottom. Six driving teeth are distributed equally around the outer circumference of the metal body. Each bidirectionally polarised piezoceramic disc has six uniformly polarised regions, as shown in
Under the excitation of a singlephase sinusoidal voltage at a specific frequency, a single wavelength of a standing wave is induced in each pair of neighbouring regions on each piezoceramic disc; consequently, the composite piezoelectric stator is three wavelengths long in the circumferential direction. When three identical highfrequency sinusoidal electrical signals are applied to the two piezoceramic discs, respectively, using the metal body as one electrode and the unbonded surfaces of the two piezoceramic discs as the other electrode, each disc generates its own standing wave pattern according to its polarisation pattern. The two standing waves are in phase and can excite the composite piezoelectric stator to produce the desired inplane bending vibration modes (3,1), (3,2) and (3,3), respectively. Because the (3,1) mode can be excited at relatively low frequency and has greater level radial and circumferential displacement components than the other two modes, it is chosen as the operating vibrational mode of the regulator.
In the inplane bending vibration mode (3,1), each point on the outer circumference of the annular stator has both radial and circumferential motion components. The circumferential displacement is nonzero for all points, except those at extreme radial displacements,
Based on motion analysis in the previous section, it is obvious that the stator intermittently contacts the rotor, so the actuation direction of the regulator is unique. The analysis includes a significant number of nonlinear and uncertain factors, and the whole drive process is very complicated. To simplify the model, the following assumptions are made [
the rotor is a rigid body;
thefriction material is viscoelastic, its surface is smooth and the influence of surface roughness is neglected;
the Coulomb friction law is valid at the contact interface between the stator and rotor; and
the regulator runs in steady state (the starting and stopping processes of the regulator are not considered).
The displacement behaviour for the annular stator can be derived approximately using the analytical method for the inplane vibration of the thin elastic plate [
In
To simplify the analysis, an orthogonal coordinate system is set up, as shown in
Two points are employed to perform the dynamic analysis and are located at the root and top of the stator tooth, respectively (the two points are called the root point and the top point in this paper, respectively). The variables
For small values of
Accordingly,
Additionally, the swing angle
According to the above analysis, the vertical deflection of outer circumferential surface of the stator can be expressed as:
Substituting
For sin
For this case,
According to
In the piezoelectric dynamic balance regulator, the highfrequency intermittent friction contact between the stator and the rotor is essential to generate the driving force. Tribological processes occurring in highfrequency friction contacts determine the torquespeed characteristics, lifetime and longterm behaviour of the regulator. The appropriate choice of materials for the statorrotor interface is important in the design of the regulator. Until now, intermittentcontact type ultrasonic motors have successfully used hard contact materials such as Al_{2}O_{3}. However, in this paper, a special hard composite material coating is proposed for the regulator, which is softer than the base material and has a high wear resistance and stable mechanical properties with respect to temperature and environmental changes. The composite material employs an epoxy resin as base material, MoS_{2} and Al_{2}O_{3} as padding materials. The curing temperature is possessed at 80 °C.
If the contact mechanics are studied assuming that the vibration characteristics of the stator are not affected by the contact processes, deformation will be produced on the friction layer when the rotor is pressed against the stator. The intermittent contact condition is assumed during steady state, and thus, there is not a contact gap between the interface at the driving teeth and the friction layer. Therefore, the displacement in the ydirection at a point on the friction layer is equal to the ydirection displacement of the top point. The ydirection displacement
As mentioned above, the rotor is assumed to be a rigid body. Additionally, the friction layer solidifies directly on the inner circumferential surface of the rotor. Therefore, the angular velocity of the rotor is equal to the angular velocity of the friction layer:
Because the friction layer is softer than the base material of the rotor, the contact deformation is limited to the friction layer region. Thus, the rotor can be seen as a rigid body rotating on a fixed axis, and the equation describing the rotational inertia of the rotor can be given as:
The momentum equation for a rigid rotor rotating on a fixed axis is given as:
The dynamic differential equation describing the rotor used in this research is written as:
The mechanics of the statorrotor contact for the regulator are complicated due to the many parameters that must be taken into account. For the case of the rotor being regarded as a rigid body and the motion of the stator is assumed independent of the contact conditions, a viscoelastic foundation model can be employed to describe the statorrotor contact characteristics. In the proposed regulator, the normal contact resultant force
To further simplify the analysis, the essential assumption that the distributed contact force locally depends only on the local deformation at the contact points is used. The KelvinVoigt constitutive equation is therefore used to describe the stressstrain relationship of the friction material and is given by:
For a contact area with a width of
Substituting
Because the friction forces depend mainly on factors such as the coefficient of static/dynamic friction, and considering the forces and stresses at the interface between the stator and the rotor, the normal contact pressure and the relative motion of interfaces, two different types of “friction” should be distinguished [
The maximum tangential resisting force at the contact interface before sliding begins is called the maximum static friction force
Once relative motion between the stator and the rotor begins, a certain tangential resistance force
At the contact interface, while the linear velocity of the points on top of the driving teeth are greater than the corresponding points on the surface of the friction layer, the rotor is driven by the stator, and it can be thought that the friction force creates positive work, as shown in
Because the piezoelectric stator has six drive teeth and makes use of only three teeth to drive the rotor, according to the Coulomb friction law, the driving torque
A common feature of all piezoelectric actuators is their twostage energy conversion process. In the first stage, the piezoelectric elements convert electric power energy to mechanical vibration power by the piezoelectric ceramics employed in the stator to induce standing waves in the stator at frequencies in the ultrasonic range. In the second stage, the wave energy in the stator is transferred to the rotor by means of the friction contact force between them. The energy conversion efficiency of the second stage is discussed in detail in this paper. Because the intermittent contact between the stator and the rotor is cyclical, it is possible to choose a contact cycle to analyse the energy conversion characteristics [
The frictional losses at the interfaces are:
The energy dissipation from damping in the friction layer is:
Finally, the conversion efficiency of the interface can be given by:
In the proposed research on the piezoelectric dynamic balance regulator, the qualitative relationships between the structural parameters and the energy conversion characteristics must be determined. Therefore, MATLAB is employed to numerically simulate characteristics of the regulator based on the above mathematical model. The model parameters are found in
When some parameters are analysed, they do not use the values from
The coefficient of dynamic friction is one of the most important properties of the friction material, and it is necessary to analyse its influence on the output characteristics of the regulator. By using the related parameters of the regulator depicted in
Young's modulus is also an important property of the friction material, and it is important to research its effects on the regulator characteristics. By using the related parameters of the regulator depicted in
As a result, plots of the output power, loss power and conversion efficiency
By using the related parameters of the regulator depicted in
By using the related parameters of the regulator depicted in
By using the related parameters of the regulator depicted in
It can be seen that the output power and conversion efficiency show similar upward trends with an increase in the load torque. However, the loss power decreases gradually. Therefore, an increase in the load torque will not increase the loss power, which is produced by the friction between the stator and the rotor.
Through the analysis of the dynamic model of the regulator by means of MATLAB, the simulation results allow one to determine the key parameters of the regulator. Using the parameters shown in
It can be seen that the simulation accuracy is approximately 10% lower than the measured experimental rotational speed of the regulator. The simulated result of the stall torque is approximately 3% lower than the measured experimental stall torque. Several reasons can be given for this error between experimental and simulated values. The proposed dynamic model is an approximation of the actual regulator and a parametric model in which design and optimisation processes are possible. To achieve a balance between computational efficiency and accuracy, some of the necessary assumptions may cause modelling error. For example, a possible reason for the modelling error could arise from the fact that the rotor model does not include the rotor flexion because it was assumed to behave as a rigid body. Moreover, the inplane bending vibration waves are assumed to be generated by a distributed force acting on the stator. This approximation also causes some degree of modelling error. However, the results from the simulations and experiments generally exhibit the same trend, and thus, the validity of the dynamic model is proven. To summarise, the mathematical model presented and analysed in this paper can be used as a tool for optimising the design of a piezoelectric dynamic balance regulator in steady state operation.
In this paper, an ultrasonic piezoelectric dynamic balance regulator using the basic principles of a standing wavetype ultrasonic motor is proposed. Its overall structure and the actuation principle are introduced. The FEM and analytical analysis based modelling studies of the regulator are investigated in detail. The proposed model is simple and suitable for the study on friction behavior at the contact interface of regulator, and can simulate and analyze the effects of thickness and Young's modulus of friction layer, dynamic friction coefficient of contact interface, preload and load torque,
The authors would like to acknowledge the support of the National Basic Research Program (973 Program) of China (No. 2009CB724405) and the Program for Changjiang Scholars and Innovative Research Team in University, and the partial support from the National Natural Science Foundation of China (No. 51075321).
Crosssectional view of the
Contact friction drive scheme of the stator and the rotor.
Principle of dynamic balancing.
Structural diagrams of the piezoelectric stator.
Displacement distribution diagram of the stator.
Vibration deformation diagram of the inplane bending vibration mode (3,1) of the structure of the annular stator and the characteristics of the vibration displacement of each point on the circumference.
Local deformation diagram of the inplane bending vibration mode (3,1) of the stator.
Diagram of contact between the stator and the rotor.
Diagram of the effects of the coefficient of dynamic friction on the regulator characteristics.
Diagram of the effects of the Young's modulus of the friction material on the regulator characteristics.
Diagram of the effects of the thickness of the friction layer on the regulator characteristics.
Diagram of the effects of the preload force on the regulator characteristics.
Diagram of the effects of the load torque on the regulator characteristics.
Schematic of the experimental measurement of the resonant frequency response of the piezoelectric stator.
Impedance characteristics of the regulator in the 55 kHz to 80 kHz frequency band.
Schematic of the regulator load torque/rotational speed relational measurement test bench.
Diagram of the experimental and simulated load torque/rotational speed relation.
Structural dimensions of the FEM model.
Outer diameter of stator  80  mm 
Inner diameter of stator  51  mm 
Thickness of stator metal body  2.4  mm 
Thickness of piezoelectric ceramic piece  0.5  mm 
Material parameters of the stator.
Material of stator mental body  Aluminium alloy  Physical properties  Young's modulus  7 × 10^{10} N/m^{2} 
Material density  2,730 kg/m^{3}  
Poisson's ratio  0.3  
Material of piezoceramicdisk  P81  Physical properties  Young's modulus  13.2 × 10^{10} N/m^{2} 
Material density  7,450 kg/m^{3}  
Poisson's ratio  0.32  
Mechanical quality factor 
800  
Electrical properties  Relative dielectric constant

1,000  
Relative dielectric constant

1,400  
Dielectric loss 
0.5%  
Planar electromechanical coupling factor 
0.50  
Piezoelectric strain constants of the piezoceramic 
90 × 10^{−12} C/N  
Piezoelectric strain constants of the piezoceramic 
200 × 10^{−12} C/N  
Piezoelectric strain constants of the piezoceramic 
410 × 10^{−12} C/N 
Structural and material parameters of the regulator.
Outer diameter of stator  mm  80 
Inner diameter of stator  mm  51 
Outer diameter of rotor  mm  97.6 
Inner diameter of rotor  mm  87.6 
Width of stator tooth  mm  8 
Length of stator tooth  mm  4.5 
Density of stator  kg/m^{3}  2,730 
Number of nodal diameters, 
3  
Resonant frequency of stator, 
kHz  65 
Vibration amplitude, 
mm  0.08 
Vibration amplitude, 
mm  0.02 
Young's modulus of metal body  N/m^{2}  7 × 10^{10} 
Young's modulus of friction material  N/m^{2}  1.5 × 10^{9} 
Dynamic friction coefficient of friction material  0.3  
Thickness of the friction layer  mm  0.6 
Viscosity coefficient of friction material  Ns/m^{2}  1.86 × 10^{−6} 
Equivalent damping coefficient of friction material  Ns/m^{2}  1,000 
Preload force  N  20 