Dynamic Model of Contact Interface between Stator and Rotor

Abstract

Based on the equivalent principle, a linear spring contact model was established for the friction layer between stator and rotor. Different contact conditions were described by a distance index δ. Detailed analysis of the nonlinear contact behavior especially the static and dynamic slipping was carried on using a space-time equation. A contact deflection angle was proposed to quantitatively express the influence of friction force on the output performance. A more precision simulation model was established based on the theoretical analysis, and influences of different preload pressures and elastic modulus E _m of friction layer on output performance were analyzed. The results showed the simulation results had very good consistency with experimental results, and the model could well reflect the output characteristics of contact interface.

1. Introduction

Ultrasonic motor possesses some excellent performances compared to the traditional magnetic motor such as high power density, large torque at low speed, quick response, running silently, and standstill force without excitation in the high technique area [1–3]. Due to the broad application potentials, it has become a hot research topic in the electromechanical domain.

Currently, most ultrasonic motors convert the stator's vibration into rotor's rotary or linear motion by the way of frictional interface which has a crucial effect on the motor's performance [4–6]. According to different driving principles, contact force produced by traveling wave is steady and continuous which can generate steady output performance [7]. Hirata and Ueha [8] established a frictional interface model in 1993, and Wallaschek [9] summarized research of the model periodically in 1998. At present, linear model with simple results is the main model of the contact interface between stator and rotor [10]. Studies on nonlinear factors mainly depend on experiments and finite element method [11–13], which cannot systematically expound the influencing factors and the varying law of contact behavior.

Thus, a systematic research for the frictional interface was carried on in this article. The microcosmic driving principle and the stick-slipping phenomenon as well as various frictional losses were analyzed. Thereafter, a simulation model including the nonlinear behavior was established which provided the foundation of further optimum design of the friction interface.

2. Simplified Contact Model of Frictional Interface

Driving mechanism of stator is illustrated in Figure 1. When stator works, a sin wave is produced with microamplitude in a ultrasonic range, and trajectories of particles on the side surface are three-dimensional elliptical motions which can drive rotor moving through a friction layer. Displacement and velocity components of particles along the radial direction will cause radial slipping between stator and rotor which cause system energy dissipation and lower electrical efficiency.

Figure 1:

Driving mechanism of stator.

A relatively soft layer of friction material lies between stator and rotor as shown in Figure 2 where 2a stands for the contact length, W _z is the wave amplitude, and δ represents the distance between unchanged free surface and midaxes x of stator's surface wave. When the motor works, surface of the stator shows a figure of sine wave. Supposing only the friction layer produces deflection which has the same figure as stator surface, the normal contact pressures are different at different contact points between stator and rotor owning to the different normal deflections of the friction layer. Therefore, the friction layer can be simplified as some distributed linear springs. Define the equivalent spring stiffness per unit spin length as

$k_m=\frac{E_m{b}_m}{h_m},$

(1)

where h _m is the depth of friction layer. E _m is the elastic modules of friction material and b _m is its radial width.

Figure 2:

Contact model of friction interface.

Relationship between δ and 2a can be calculated as

$\text{if}\hspace{0.17em}\delta >-W_z,\hspace{1em}a=\frac{{\cos }^{-\mathrm{1}}\delta }{k},$

(2)

$\text{if}\hspace{0.17em}\delta <-W_z,\hspace{1em}a=\frac{\lambda }{\mathrm{2}}.$

(3)

Based on the postulation of linear style, normal pressure on the contact surface can be established as

$F_n=\mathrm{2}n{{\displaystyle \int }}_{\mathrm{0}}^{a}p\left(x\right)dx=\mathrm{2}n{k}_m{W}_z\left(\frac{\mathrm{1}}{k}\sin ka-a\cos ka\right),$

(4)

where n is the traveling wave number.

When the motor works, the rotor moves with a fixed velocity of v _r as shown in Figure 3 while particles on stator's surface run with different tangential velocities in different contact positions. Thus, velocity differences exist between stator and rotor except for two equal-velocity points which subsequently causes slide friction loss on the contact surface. If the tangential velocity is larger than v _r , the stator produces a driving force for the rotor; contrarily, direction of the tangential force produced by friction material is opposite to the rotary direction of rotor, which substantially takes a blocking action as the shadow area shown in Figure 3. Position of the equal-velocity point b can be obtained from the following expression:

$v_x\left(b\right)=k{d}_c\mathrm{2}\pi f_n{W}_z\cos kb=v_r,$

(5)

where d _c is the distance between the midlayer and stator's surface. Power losses caused by pure sliding friction can be calculated as

$P_{\text{slip}}=\mathrm{2}n{\mu }_d{{\displaystyle \int }}_{\mathrm{0}}^{a}p\left(x\right)\left|v_x\left(x\right)-v_r\right|dx,$

(6)

where μ _d is the sliding friction coefficient.

Figure 3:

Driving area distribution of friction in terrace.

Only if the tangential velocity of the stator's surface particle is larger than v _r , the stator produces a driving force for the rotor, and the effective output torque is

$M_n=\mathrm{2}n{R}_m{\mu }_d{{\displaystyle \int }}_{\mathrm{0}}^{a}p\left(x\right)\mathrm{sign}\left(v_x-v_r\right)dx,$

(7)

where R _m is the average driving radius.

3. Nonlinear Dynamic Behavior of the Frictional Interface

Substantially, stick-slipping phenomenon which can cause remarkable influence to motor's startup and load characteristic exists except for the slidingfriction expounded above and should be taken into consideration during establishing the model of the frictional interface. As shown in Figure 4, point B on frictional material surface has the same v _r with rotor because it sticks to the rotor, while point A contacts with stator surface and obtains the tangential velocity. Due to the different velocities of points A and B, the frictional layer produces shear deflection which can be expressed by a deflection angle α and can be calculated as

$\alpha \left(t\right)={{\displaystyle \int }}_{T_{\mathrm{0}}}^t\frac{\left(v_x\left(\tau \right)-v_r\right)}{h_m}d\tau .$

(8)

Figure 4:

Relationship between friction force and deflection angle.

When α is less than a maximum critical deflection angle α _c , no slide exists between point A and stator, and only elastic deflection occurs in the frictional layer which produces static-frictional force. Suppose α(t) < 0 when the displacement of point A lags point B′ and α(t) > 0 as shown in Figure 4; the frictional force produced by stator has the same sign with α. Otherwise, if α(t) > α _c , the sticking state will be destroyed, and sliding friction force occurs.

Figure 5 shows the contact state between stator and rotor at certain time. One tooth can realize all the contact states after one cycle's spreading. As to the instantaneous contact interface shown in Figure 2, the contour line equation of tooth's surface can be established as

$W\left(x\right)=W_z\cos kx.$

(9)

Figure 5:

Distributions of contact state between stator and rotor.

Suppose the normal displacement and tangential velocity of one tooth are

$W\left(t\right)=W_z\sin \omega t,$

(10)

$v_x\left(x,t\right)=d_{c}k\omega W_z\sin \omega t.$

(11)

Obviously, there exists a corresponding space-time relationship comparing the above two expressions. Suppose the initial contact point is at x = a and the displacement is w(a) = W _z coska. Substituting the displacement into (11), time T₁ can be obtained when the tooth has the same displacement. By analogy, time T₂, T₃, T₄ can be obtained corresponding to the first equal-velocity point, the second equal-velocity point, and the disengaging point. By doing this, the spatial contact state equation can be converted to the contact state in time domain, which is more close to the actual condition and much easier to analyze the contact state dynamically and to estimate the sticking and slipping phenomenon.

The contact state can be divided into four steps as follows:

1. t = T₁:when the stator just begins to contact with rotor, point A sticks to stator's surface and obtains its velocity v _x (T₁).

2. $T_1<t<T{}_2$ : before reaching to the equal-velocity, velocity relationship of v _x (t) < v _r makes point A's displacement lags B's, which causes shear deflection in the frictional layer. During this time, the stator produces a block function to rotor because α < 0. When α(t) < α _c , point A on the contact surface keeps sticking to the stator, and only the frictional layer produces elastic deflection, which will transfer static friction force. During this process, the static-friction force is varied which has a direct proportion to α. When α(t) > α _c , slipping occurs between point A and stator surface, and the frictional layer keeps the maximum contact-deflection angle α _c . In this process, slide friction force will be transferred.

3. T₂ < t < T₃: between the two equal-velocity points, stator's velocity is larger than rotor's at all times, that is, v _x (t) > v _r , the frictional layer's deflection angle begins to add from a negative value. When – α _c < α(t) < α _c , the contact surface is in a stick state. During this process, the static friction force varies from negative value to positive value and from blocking condition to driving condition as the changing of α. When α(t) > α _c , it is in a state of sliding driving.

4. After transcending the second equal-velocity point, v _x (t) < v _r .α begins to diminish from a positive value. When – α _c < α(t) < α _c , it is in a state of sticking. During this process, the static friction force varies from a positive value to a negative value and from driving condition to blocking condition. When α(t) > α _c , it is in slide blocking state.

From the above analysis, it is easy to judge the sticking-slipping contact state of the contact area at different positions using space-time corresponding connection. Based on the sign of α(t), the driving and blocking area also can be estimated.

4. Simulation Model of Dynamic Behavior of the Frictional Interface

Based on the analysis of the sticking-slipping phenomenon, a more practical simulating model to calculate the output characteristic of the friction interface can be established and the calculating process are as follows.

1. Setting up certain necessary parameters including the number of traveling wave n, the average contact radius R _m , and parameters of friction layer E _m , b _m , h _m , and d _c , parameters of λ, k, and k _m can be calculated.

2. Specify the preload F _n and stator's amplitude W _z . In fact, the stator's amplitude is influenced by many factors; here F _n and W _z are treated as separated.

3. The contact space δ can be obtained from (2), and the contact range can be judged from the relationship of δ and W _z .

4. Calculate α(t) and judge the stick-slipping area.

   According to the Hook's law, the critical deflection angle can be calculated as

   ${\alpha }_c=\frac{{\tau }_{\max }}{G_m}=\frac{F_{t\max }/\left(b_m\mathrm{\Delta }l\right)}{G_m},$

   (12)

   where F _{t max} is the maximal friction force and G _m is the shear modules. Based on coulomb friction's law ΔF _tmax = μ _d ΔF _n , where ΔF _n is the normal pressure, another expression can be gained as

   ${\alpha }_c=\frac{{\mu }_d\mathrm{\Delta }{F}_n/\left(b_m\mathrm{\Delta }l\right)}{G_m}=\frac{{\mu }_d{k}_m\mathrm{\Delta }w}{G_m{b}_m}.$

   (13)

   So α _c can be rewritten as

   ${\alpha }_c=\frac{{\mu }_d{E}_m\mathrm{\Delta }w}{G_m{b}_m}.$

   (14)

   The static friction force produced in blocking or driving area has a direct proportion to the shear angle, and the friction coefficient can be modified as follows:

   $\text{when}\hspace{0.17em}\left|\alpha \left(t\right)\right|\ge {\alpha }_c,\hspace{1em}\mu \left(t\right)={\mu }_d.$

   (15)
   $\text{otherwise},\hspace{0.17em}\left|\alpha \left(t\right)\right|<{\alpha }_c,\hspace{1em}\mu \left(t\right)={\mu }_s\frac{\left|\alpha \left(t\right)\right|}{{\alpha }_c},$

   (16)

   where μ _s is the static friction coefficient, μ(t) is the friction coefficient of the contact interface which also can be expressed as μ(x), and α(t) can be showed as α(x) as well.

5. Calculate the output torque of the friction interface:

   $M_n=\mathrm{2}n{R}_m{{\displaystyle \int }}_{\mathrm{0}}^a\mu \left(x\right)p\left(x\right)\mathrm{sign}\left(\alpha \left(x\right)\right)dx.$

   (17)

6. Calculate the slide friction loss:

   $P_{\text{slip}}=\mathrm{2}n{\mu }_d{{\displaystyle \int }}_{\text{slip}}^{}p\left(x\right)\left|v_x\left(x\right)-v_r\right|dx.$

   (18)

Taking USM60 type ultrasonic motor as example, the nonlinear dynamic behavior of contact interface between stator and rotor is studied according to the above stimulation model.

Figure 6 shows the contact conditions between stator and rotor under different δ. When δ > 0, exposure range between stator and rotor is confined to the area near wave crest and 2a < λ/2. There is no block area when the contact area is in pure slipping state on the condition that the tangential velocity decrease to 0. When W _z < δ < 0, the contact area is not only in the wave crest range, but also in some valley areas with large stick area and sliding loss decreases. When δ < W _z , the contact area is in the whole wavelength range with stick state and a balance occur between driving area and block area which cause the output speed decrease to 0.

Figure 6:

Contact conditions between stator and rotor under different δ.

Changes of output performance with elastic modulus E _m of friction layer under increasing preload are shown in Figure 7. With the increasing of friction interface, output power and efficiency of contact layer increased with the increasing of E _m , which is gradually offset to the high load segment. The sliding friction losses show a contrary tendency.

Figure 7:

Changes of output performance with elastic modulus of friction layer.

Actually, the preload should be aimed at different application background. Large power output requires increasing pressure which is at the price of efficiency. Enough attention should be paid to the friction layer design in order to guarantee the stability operation.

Figure 8 shows the load characteristics under different preload pressure when amplitude remained unchanged 1 μm, and the load characteristics under different amplitude when pressure remained unchanged 170 N were illustrated in Figure 9. The experimental results and the simulation results have very good consistency with each other in changing tendency, and the model can well reflect the output characteristics of contact interface between stator and rotor.

Figure 8:

Load characteristics under different preloads when W _z = 1 μm.

Figure 9:

Load characteristics under different amplitudes when F _n = 170 N.

5. Conclusion

The nonlinear dynamic behavior of contact interface between stator and rotor was studied based on a simplified linear spring model. Results showed that different contact states which produced static and dynamic sliding friction force were easily to be described by a space-time corresponding relation and the distance index instead of contact length. Output performance and various friction losses could be illustrated by a deflection angle. Results of the simulation model are in good agreement with experiment data which is accurate to analyze the output characteristics of the nonlinear frictional interface.