Research on key technologies of bearing preload of spindle based on piezoelectric actuators

: This study analysed the influence of bearing preload on the stiffness of spindle bearings. MATLAB/Simulink is used to calculate the vibration of simplified spindle system. The vibration of the simplified spindle system decreases with the increase of bearing preload. A kind of force output characteristics of piezoelectric actuators test device was designed. Experimental analysis of the maximum force output characteristic, hysteresis characteristic, creep characteristic and stability characteristic of PSt 150/4/7 VS9-type piezoelectric actuators under different initial loads, voltage frequencies, cycles working and voltage travel. The evaluating indexes are the maximum output force, average hysteresis, maximum deviation, drift amount and cycle delay. In conclusion that the piezoelectric actuator has better repeatability and stability under the initial load is around 150 N and low-voltage frequency. It is not suitable for multiple cycles working. The increment of output force is proportional to starting voltage at same voltage travel. MATLAB/Simulink is used to calculate the feasibility of variable bearing preload strategy based on a piezoelectric actuator. It provided scientific references for the research of high performance and intelligent spindle.


Introduction
Bearings are the key component of the spindle, which affect spindle performance directly. So there are a lot of researches about bearings that aim at improving the performance of spindle. Ceramic bearings can improve the performance of spindle. Spindle equipped with intelligent detection systems and control systems can improve the performance effectively. Mazak from Japan use intelligent spindle detection function intelligent processing system (IPS) to self-detect the temperature, vibration, displacement and other performance parameters of the spindle can prevent the spindle component failures. Fischer from Switzerland use sensors and smartVision software to monitor the spindle operating status and predict the remaining life of spindle bearings. German GMN, Swiss IBAG, Japanese NSK and other companies use small displacement hydraulic cylinder pumps providing variable bearing preload. American TIMKEN company designed a liquid chamber between the inner and outer rings of the tapered roller bearing to achieve variable bearings preload. The research above can improve the quality of the products. Impact caused by the temperature rise and vibration is minimised on the spindle [1].
Piezoelectric actuators as a new type of actuators with advantages of large output force, high accuracy, fast response and so on. However, the hysteresis, non-linear and creep characteristics directly affect the precision force output of piezoelectric actuators. So there is a lot of research on output characteristics and precise modelling of piezoelectric actuators. According to the ferroelectric theory, the distribution of electric domain can be changed in the external electric field and external stress and other factors which can change the output characteristics and hysteresis phenomenon of piezoelectric actuators [2]. Wiping-out property and equal vertical chord property provide theoretical bases for the prediction model of piezoelectric actuators [3]. Experiments show that an appropriate initial load can increase output displacement and hysteresis phenomenon [4,5]. The number of domain switching and output displacement increase with the initial voltage increase [6].
Mathematical modelling method is difficult to achieve the ideal results. The prediction model based on experimental data of the Preisach model is widely used. The accuracy and application scope of the hysteresis model can be improved by modification of the Preisach model. Preisach model is identified by obtaining the distribution function, but the distribution function is usually very complicated [7,8]. The dynamic Preisach model of tangent function hysteresis operator effectively extends the generalisation ability of the hysteresis model [9]. The process identifier (PI) model based on the working principle of the Preisach model can effectively predict the hysteresis output of piezoceramic actuator [10,11]. The hysteresis model is composed of back propagation (BP) neural network and Preisach model, in which input layer is determined by the local maximum and the local minimum at different stages [12,13].
This paper calculates the influence of bearing preload on the vibration of the spindle. Analyse the force output characteristics of piezoelectric actuators under different initial loads and different voltage frequencies. The initial conditions of the best output are obtained. The hysteresis model of piezoelectric is predicted by BP neural network, in which input layer is based on the principle of the Preisach model. MATLAB/Simulink is used to calculate the feasibility of variable bearing preload based on piezoelectric actuators. It provides a theoretical basis for preload control of bearings based on piezoelectric actuators.

Influence of bearing preload on spindle performance
Bearing preload directly affects the stiffness and vibration of the spindle. It is necessary to study the influence of bearing preload on the performance of spindle. The stiffness of bearing is shown as (1) and (2) under the condition of contact angle and contact deformation is determined [14] K α = 3.37 × 10 6  where K α is the bearing axial stiffness; K γ is the bearing radial stiffness; F P is the bearing preload; Z is the rolling elements number; D W is the diameter of rolling element; and α is the contact angle.
As is shown in (1) and (2), bearing preload can significantly increase the stiffness of bearing. The radial vibration of spindle under different bearing preloads is simulated. Since spindle is mainly subjected to radial loads during operation, 7009C angular contact ball bearings are used in this spindle. The shaft and bearings of spindle weight are 6.836 kg. Moreover, the weights of shaft and bearings are equally distributed to four bearings. This paper ignores the effects of damping. The dynamic equation of simplified spindle is constructed according to (3), in which the speed is 3000 r/min. The weight of the simplified spindle is 1.709 kg. The minimum bearing preload is 30.5 N. So the stiffness of simplified spindle is 7678800 N/m where FC is the grinding force; x is the vibration displacement; and x¨ is the vibration acceleration. The vibration of the simplified spindle system is analysed by MATLAB/Simulink. Exciting force is sine force, in which amplitude is 70 N and cycle period is 2 s. The vibration of simplified spindle under 30.5 N bearing preload and 7,678,800 N/m stiffness conditions is shown in Fig. 1. The vibration of simplified spindle under 60 N bearing preload and 19,137,000 N/m is shown in Fig. 2.
As is shown in Fig. 1, the maximum amplitude is 9.572 μm. As is shown in Fig. 2, the maximum amplitude is 3.832 μm. It is shown that increasing bearing preload can effectively control the vibration of the spindle.

Research on force output characteristics of the piezoelectric actuator
This paper designs a kind of device for testing the output characteristics of the piezoelectric actuator based on 9257B plane dynamometer and HVA-150.A1 piezoelectric actuator driving power. Experimental analysis of the force output characteristics of PSt 150/4/7 VS9-type piezoelectric actuator under different initial loads and voltage frequencies. The test device is shown as Fig. 3.
Testing piezoelectric actuator force output characteristics in 0-150 N initial load, step length is 30 N, drive voltage from 0 to 150 N and voltage frequency is 1 Hz. Testing piezoelectric actuator at voltage frequencies of 1, 10, 20, 30, 40 and 50 Hz under 150 N initial load and drive voltage from 0 to 150 V. Testing piezoelectric actuator force output characteristics in cycle working condition under 150 N initial preload and 1 Hz voltage frequency. Testing piezoelectric actuator force output characteristics under 10 V voltage travel and 50 V voltage travel. Experimental analysis of the force output, hysteresis, creep and stability of piezoelectric actuator by the maximum output force, average hysteresis degree, maximum deviation, drift amount, cycle delay of curves and other parameters as evaluating indexes.

Maximum force output characteristic
As is shown in Fig. 4, the maximum output force of the piezoelectric actuator shows a tendency to increase first and then decrease with the increase of initial load. The value reaches the maximum when the initial load is 150 N.
As is shown in Fig. 5, the maximum output force of the piezoelectric actuator decreases with the increase of the voltage frequency.
As is shown in Fig. 6, the maximum output force of the piezoelectric actuator increases with the increase of starting voltage and voltage travel. Moreover, the voltage travel is below 75 V.
The experimental analysis showed that the maximum output force of the piezoelectric actuator is affected by the initial load, voltage frequency and voltage travel. The initial load has the most obvious effect. According to the ferroelectric theory, the electric domain of piezoelectric actuator will be polarised under the effect of the electric field. In conclusion, the proper initial load can promote electric domain polarisation, promoting the force output of the piezoelectric actuator. However, the force output will be suppressed if the initial load is too large. Friction exists in domain polarisation and internal obstacles can cause energy loss during cycle working. So the voltage frequency and the maximum output force are inversely proportional. The piezoelectric actuator has different polarisations at different starting voltages. So the voltage frequency and the maximum output force are directly proportional under the same voltage travel.

Hysteresis characteristic
As is shown in Fig. 7, the hysteresis phenomenon of the piezoelectric actuator is different in cycle working condition. The hysteresis of each cycle varies around a value. The fluctuation of hysteresis shows a divergent trend under cycle working condition. Therefore, the average value of the hysteresis in the previous seven   cycles is used as the evaluation index in this paper. Using the maximum deviation as the auxiliary evaluation index, which is the percentage of the maximum value of the maximum and minimum hysteresis curves deviations for the maximum output force.
As is shown in Table 1, the average hysteresis of the first seven working cycles is 10.74 and the maximum deviation is 3.89%. The average hysteresis of 14 working cycles is 10.94 and the maximum deviation is 5.68%. It is shown that the average hysteresis of the piezoelectric actuator is maintained within a stable range. The maximum deviation of the piezoelectric actuator increases with cycle working. The repeatability of the piezoelectric actuator decreases with multiple cycles working.
The initial load has no obvious effect on hysteresis characteristics of the piezoelectric actuator, as is shown in Table 2.
The average hysteresis increases with increasing of voltage frequency. However, the average hysteresis is slightly decreased at 50 Hz voltage frequency.
The hysteresis is due to the irreversible effect of some domains in the alternating electric field. So the output force in the voltage drop stage is bigger than the voltage rise stage. Both voltage frequency and cycle working can affect the hysteresis characteristics of the piezoelectric actuator because the domains have inertia and friction during movement. The experiment coincides with the theory in this section.

Creep characteristic
The creep characteristics of piezoelectric actuator affect the precision of the output force. Creep deformation is larger in the starting phase of driving voltage and smaller in the ending phase of the driving voltage. Cycle working, initial load, voltage frequency, starting voltage and voltage travel have an effect on the creep characteristics. The creep characteristic of the piezoelectric actuator can be predicted and corrected by feedback compensation [15].
The adjacent cycle delay is about 0.01 s which is a cumulative increase in cycle working. The cumulative delay of the seventh cycle is 0.062 s. The cumulative delay of the 14th cycle is 0.175 s. So repeated cycle work has an impact on the precise output of the piezoelectric actuator.

Stability characteristic
The piezoelectric actuator has its own natural frequency and often works with stiffness and damping. The piezoelectric system can excite vibrations at different voltage frequencies. As is shown in Fig. 8, hysteresis curves appear wavy in shape at input voltage frequencies 10, 20 and 40 Hz. The fluctuation of wave shape in the starting phase, the ending phase of the driving voltage is smaller than the middle stage of the driving voltage. The wave frequency is 204 times of the voltage frequency after spectrum analysis. The first-order nature frequency of test device beam is around 3743 Hz. So the wave shape of hysteresis curves is not caused by the vibration of the test device. The wave shape of hysteresis curves is caused by the vibration characteristics of the piezoelectric actuator.
Drift amount of piezoelectric actuator is the distance between the minimum output force and the origin of coordinate. The size of the drift amount is evaluated by the deviation amount. The drift amount decreases in cycle working. As is shown in Fig. 9, the drift amount of piezoelectric actuator is smaller under the initial load from 150 to 210 N. The drift amount is larger because the initial load is too large or small. As is shown in Fig. 10, the drift amount of piezoelectric actuator increases with the increase of voltage frequency.
This section combines the ferroelectric characteristics and structure performance of the piezoelectric actuator. The vibration and stability characteristic of piezoelectric actuator were analysed by the shape of the output force curve and drift amount. It is shown that the initial load and voltage frequency can affect the repeatability and stability of the piezoelectric actuator. Appropriate initial load and low-voltage frequency should be selected when operating the piezoelectric actuator.

Research on bearing preload control method of spindle based on a piezoelectric actuator
This section studied the preload control strategy of spindle based on the piezoelectric actuator. ORBIT36CNC surface grinder and Kistler9257B plane dynamometer were used to collect grinding force signal. The silicon nitride specimen cutting depth is 5 μm, grinding speed is 40 m/s and feed speed is 3000 mm/min. Grinding force data are collected as excitation signal source in 2 s. This section ignores the influence of grinding speed on the grinding force. To reflect the influence of different bearing preloads on spindle vibration. The grinding force is shown in Fig. 11. As is shown in Fig. 11, the maximum grinding force is 324.43 N. The grinding force is evenly distributed to the four bearings which are based on the simplified spindle model. So the maximum grinding force for the simulation analysis is 81.11 N. The stiffness of 7009C bearing is 12,056 N/mm when the installation bearing preload is 60 N. The simulation result is shown in Fig. 12.
This section proposes two piezoelectric actuators controlling one bearing. The bearing preload of the spindle is increased to 120 N on the basis of 60 N by piezoelectric actuators. The stiffness of 7009C bearing is 12,056 N/mm and the input voltage of the piezoelectric actuator is 66.3 V. The simulation result is shown in Fig. 13.
As is shown in Fig. 12, the maximum vibration amplitude is 24.26 μm when the bearing preload is 60 N. As is shown in Fig. 13, the maximum vibration amplitude is 9.7 μm when the bearing preload is 120 N and the maximum vibration amplitude is reduced by 60.02%. The influence of bearing preload on stiffness and thermal proposes setting of initial preload and providing preload increment according to spindle operating conditions which   is based on the influence of bearing preload on stiffness and thermal. It can improve the working performance and life of spindle.
As is shown in Fig. 14, 12 groups of preload step increments are provided for vibration which is based on the 66.3 V voltage, and the resolution of preload step increment is 0.001 s. The simulation result is shown in Fig. 15.
The result of the simulation shows that less number of preload step increments can effectively suppress vibration. It provides a theoretical basis for bearing preload control method of spindle based on the piezoelectric actuator.

Conclusion
(1) Spindle vibration and bearing preload have an inverse relation.
(2) The initial load, voltage frequency, cycle working and voltage have an effect on the output characteristics. There is a condition for better repeatability and stability of the piezoelectric actuator.
(3) Bearing preload control based on piezoelectric actuators can effectively improve the performance of spindle.