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50
УДК 539.62
Безызносный нанодвигатель на основе знакопеременного качения
R. Wetter1, В.Л. Попов123
1 Берлинский технический университет, Берлин, 10623, Германия
2 Национальный исследовательский Томский государственный университет, Томск, 634050, Россия
3 Национальный исследовательский Томский политехнический университет, Томск, 634050, Россия
В настоящей статье предложена концепция высокоточного актуатора, основанного на эффектах инерции и знакопеременного качения. Движущим элементом актуатора является сфера, прижимаемая к подвижному элементу конструкции (бегуну). Суперпозиция нормальных осцилляций сферы со знакопеременным качением приводит к направленному движению бегуна. Асимметрия колебаний в совокупности с инерционными свойствами бегуна позволяет точно контролировать его перемещение. Благодаря вращательной компоненте движения сферы поступательное движение бегуна происходит без проскальзывания, поэтому актуатор работает потенциально безызносно. Для численного расчета актуатора используется метод редукции размерности. Найдены аналитические выражения для стационарных рабочих точек системы, которые находятся в согласии с результатами численного моделирования.
Ключевые слова: актуатор прерывистого скольжения, нанодвигатель, фрикционный инерционный актуатор, метод редукции размерности, знакопеременное качение, упругий контакт качения
A wear-reduced nanodrive based on oscillating rolling
R. Wetter1 and V.L. Popov123
1 Berlin University of Technology, Berlin, 10623, Germany 2 National Research Tomsk State University, 634050, Russia 3 National Research Tomsk Polytechnic University, 634050, Russia
In this article we introduce a promising new concept for a high precision actuator. It is based on inertia effects and oscillating rolling. A sphere acts as the drive and is pressed on a movable substrate that acts as the runner. A combination of oscillating translation and rotation of the sphere induces motion of the runner. A varying normal force leads to varying indentation depth and contact area. This asymmetry together with the inertia of the runner enables accurate control of its displacement. As slip is completely omitted here, in theory the actuator works principally wearless. We use the method of dimensionality reduction to conduct a quasistatic numerical simulation of the system. In addition we derive analytical expressions for the steady working points of the system that are in perfect agreement with the simulation results.
Keywords: stick-slip actuators, nanodrives, friction inertial actuators, method of dimensionality reduction, oscillating rolling, elastic rolling contact
1. Introduction
The ongoing miniaturization in science and technology increases the demand for precise displacement and manipulation devices. High-precision actuators are found for example in nanoscale data storage [1] optical components [2], and xy-stages for micropositioning of probes [3]. The simultaneous demand for accurate resolution in positioning in the range of nanometers and for long operation range of millimeters is a special challenge in the technical realization [4]. A widely used type of actuators combines friction
and inertia effects [5]. In these so-called friction inertial actuators typically a high-resolution piezoelectric motor uses the stick-slip effect [6] to drive the runner [5]. The working principle of these stick-slip actuators is accompanied by a steady wear, which decreases the controllability and can lead to failure of the drive [7, 8].
In the present paper we introduce a promising new approach for a low-wear actuator that combines inertia effects with the principle of oscillating rolling. This allows generating a well-to-control high-precision motion. In this case slip effects can theoretically be completely omitted.
© Wetter R., Popov V.L., 2015
Fig. 1. Hertzian contact of rigid sphere with radius R and movable elastic substrate. The normal force N leads to indentation d and contact length 2a
2. Models and methods
Our concept consists of a rigid sphere with radius R and a movable elastic substrate as depicted in Fig. 1. The sphere acts as the drive and the substrate acts as the runner. In the normal direction the system corresponds to a Hertzian contact of a rigid sphere and an elastic half-space. It is assumed to be uncoupled what means that variations in normal forces will not induce any tangential displacement and vice versa. This requires Dundurs' constant P = 0 as it is the case for frictionless contacts, similar materials, incompressible materials or if one body is rigid and the other one is incompressible [9].
A description for the initial static tangential contact can be found, for example in Popov [10]. The elastic properties of the substrate E* and G* are chosen as effective quantities of a contact of two elastic bodies with particular shear modules Gt and Poisson ratios v t :
\-i / - - \-i
1 -v1 + 1 -v2
2Gi 2G-,
2-vt + 2-v2
4G
4G,
• (1)
The indentation depth d, which is defined as the vertical displacement of the rigid sphere, results for a given normal force N to
3 N
2/3
4 ER )
Both bodies only touch within a circular area that is delimited by the contact radius
a _4Rd. (3)
We assume dry friction of the Coulomb type with constant ^ between the contacting bodies.
The actuation is based on an oscillating rolling of the sphere with varying normal force and is divided into two steps. In the so-called forth step, the center of the sphere moves with velocity v to the right and rotates clockwise with Q as shown in Fig. 2, a. All quantities of the forth step are denoted with a subscript letter (.)1 in the following.
As qR > v applies, the resulting creep ratio for a stationary runner will be positive: _ qR - v
sstat1 _"
N,
b
////////////
////////////
Fig. 2. Concept of the rolling sphere drive: forth step with normal force N (a), back step with lower normal force N2 < N (b)
where R denotes the radius of the sphere. This effect would push the runner to the left. After that follows the so-called back step. All quantities of the back step are denoted with subscript letter (.)2. In this, the sphere moves to the left with velocity -v and rotates counter-clockwise with -q as shown in Fig. 2, b. The according creep ratio is then
-qR + v n ...
Sstat2 _-< 0, (5)
v
and the runner would be pushed to the right. In case of a constant normal force, this would not induce any net-tangential force and the average position of the runner would be stationary.
As shown in Fig. 3, the oscillating rolling is superposed by a varying normal force, i.e. indentation depth d. During the forth-step, the sphere is pushed on the substrate by N1 while during the back step it is pushed by a lower force N2 < N1. Together with the tangential creep during rolling, this asymmetric force path will induce a rigid body motion of the runner. In the following we will identify the working points of the system that are needed for a constant motion of the runner.
2.1. Oscillating steady rolling
In a first step, we assume the mass of the runner to be infinite, so that it moves with constant velocity v* to the left as shown in Fig. 2. In addition we assume infinite friction with ^ ^ It should be highlighted here that we neglect adhesion, i.e. transmission of tangential force requires positive normal force in the sense of a pressure.
Initially, in order to determine the input parameters that are necessary to maintain a constant leftward motion of the
> 0,
(4)
Fig. 3. Rolling wRt and varying indentation depth d over time t/ t with t being the time of oscillation
runner, we assume a high rolling amplitude. Therefore, we can neglect the transition between the two steps and we will only consider the steady left- and rightward rolling of the drive. With the given velocities, the creep-ratio for the two cases in the steady rolling state, i.e. when the runner moves to the left and the sphere leaves the initial area of contact, yield for the forth step
wR-v-v* > 0 (6)
«1 = -
v
and for the back-step
-roR + v - v*
S2 =-
v
< 0.
(7)
In steady rolling state, the creep ratio determines the tangential force T between sphere and runner, i.e. the driving force. For small ratios of T/^N, as for infinite friction, the creep-ratio function can be expanded as a Taylor series, what gives [10] aT
s =--. (8)
2 RN
Now inserting (6) and (7) into (8) gives the tangential forces for the two steady rolling states:
2 RN1 œR - v - v
<0
and
t =■
2RN2 -œR + v - v*
>0.
a2
v
(9)
(10)
In order to maintain constant motion of the runner with velocity v*, the forces of the steady rolling states must be equal. The velocity of the runner in this working point then results to
* v ^ 1 -d2/d1
wR ,
T + T2 = 0:
v
œR
1-
(11)
1 + d 2/ d1
Equation (11) gives the condition for the input parameters that are needed to maintain constant motion of the runner with velocity v* for the case of high rolling amplitudes.
2.2. Model of the method of dimensionality reduction
The method of dimensionality reduction enables an exact mapping of uncoupled, rotationally symmetric tangential contacts with Coulomb friction without loss of essential properties [11, 12]. Using the method of dimensionality reduction, the initial system is replaced by an equivalent model consisting of an elastic foundation of independent springs and a rigid sphere as described in [13]. The stretched radius of the sphere: 1
RlD = 2 R
(12)
as well as the stiffnesses in the normal and tangential direction:
kN = E * Ax, kT = G* Ax (13)
are chosen according to the rule of Popov [13]. Here Ax
Fig. 4. The method of dimensionality reduction model of the drive with moving coordinate translation vAt does not affect the normal deflection of the springs uN (a); rotation RwAt changes the deflection uN (b)
denotes the distance between adjacent springs, i.e. the grid size. The method of dimensionality reduction model enables a fast and precise quasistatic simulation of the oscillatory rolling. In addition, it is possible to derive analytical results on basis of the kinematic relations between the rolling motion and the spring deflections [14, 15]. This enables us to take into account the transition between the two steps of the rolling as shown in Sect. 3.2.
We choose the coordinate £ that moves with the sphere and has its origin in the centerline of the sphere as depicted in Fig. 4. As the translation vt does not affect the normal deflections, we can approximate uN in case that w = 0 with (12):
uN = d (t ) -
-2— = d (t ). 2 R1D r
(14)
An additional rolling, i.e. w ^ 0, changes the deflection of a spring at time t as
(%-wRt )2
Un (£, t) = d (t) —
R
(15)
The tangential deflections are computed stepwise on basis of the velocities of the application points of the springs as depicted in Fig. 5.
In the forth step the tangential displacement at time t for all springs in contact, i.e. uN(t) > 0, yields:
uT 1 t) = uT t -At) + At (wR - v - v*), (16)
where At denotes the time step. In case of the back step the tangential deflections read:
uT 2 (t) = uT (t -At) + At (- wR + v - v*). (17) Equations (15)-( 17) are used in the simulation for a stepwise computation of the deflections. In addition it is possible to
,(£>RAt,
777777777777
777777777777
777777777777
Fig. 5. Change of tangential deflection uT at application points of a spring: translation of the sphere (a), translation of the runner (b) and rotation of the sphere (c) all contribute to the tangential deflection uT
compute the normal force at time t that is given by the integral over all springs in contact:
a+mRt
N(t) _ J e\n & t)dt (18)
- a +mRt
With (15) this yields the well known Hertzian relation between normal force and indentation [10]:
«(,) = 3 Г R*2 d
(19)
indicating that the method of dimensionality reduction model correctly reproduces the normal contact.
3. Results
Given the time of oscillation t we can define the rolling amplitude as
W _ mR1/2 t. (20)
In addition we introduce a normalization as
w _ W/a1, (21)
where aj is the contact radius for N,. On basis of the infinite friction assumption, we restrict ourselves to cases where W > 2aj. (22)
This means that during one cycle of back and forth rolling, every spring is completely released once, i.e. deflection cannot accumulate to infinity.
3.1. Oscillating steady rolling
After a transition process between the two steps, the motion corresponds to a steady rightward respectively leftward rolling of the sphere. Comoving coordinates are chosen for further consideration.
In case of the forth step, we use the coordinate that has its origin on the right side of the contact area at the actual leading edge. For steady rolling, the tangential displacement uTj of a spring at the position corresponds to the sphere-substrate distance Avjt, where
Avj _ wR - v - v*. (23)
With r|i = ffRt it follows that
*T1
Av1t
= Av,t =—— n = 1 wRt
1-
v
v
wR wR
n.
(24)
In the case of the back step, we use the coordinate n2 that has its origin on the left side of the contact area at the new leading edge. The distance between the sphere and substrate then yields
Av2 _-wR + v - v* (25)
and the displacement results to
UT 2 = Av^t =
Av2t wRt
(
П2 =
-1 +
v
v
wR wR
Л2-
(26)
As for the normal force, the tangential force can be determined by integrating over the entire contact area. With (24) for the forth step this gives:
2a, ( „* A
T1 = J G*uT1dn1 = 2G*0[
1 —
wR
v
wR
(27)
and for the back step with (26) this gives:
2a2 (
T2 = J G*Ut2dn2 = 2G*a2
-1+-
wR
v
wR
Л
(28)
Again, constant motion of the runner requires equal forces in steady rolling state, i.e. T, + T2 _ 0. With (27) and (28) this yields the velocity of the runner for the method of dimensionality reduction model as
v* __v_\1 -<4/dj
mR
(29)
mR 11 + d2/d,
This corresponds exactly to the result of the 3D case as stated in Eq. (11), testifying that the method of dimensionality reduction model also correctly reproduces the tangential problem and the corresponding kinematics.
In addition, we conducted a simulation with the method of dimensionality reduction model as described in Sect. 2.2. The forces are computed as the sum of spring displacement times, the stiffnesses as defined in Eq. (13). Figure 6 shows the sum of steady rolling state forces as a function of v/mR and v*/mR for d2/d, _ 0.93 and w = 2.2. The dotted line shows Eq. (29), which gives the zero force relation and is in strict agreement with the simulation. Additionally, Eq. (29) shows that v* increases with decreasing ratio of v/mR. Thus, the higher the difference between translation and rotation of the sphere, the higher the resulting runner velocity.
3.2. Working point of the drive
In principle, the results of Sect. 3.1 give the correct working point for high amplitude w >> 2. In order to give an equivalent to the zero force relation for amplitudes in the range of the contact length 2a,, i.e. w ~ 2, the transition between forth and back step must be taken into account.
Figure 7 shows the spring deflections in four characteristic states: beginning of the forth step 1, rightward steady rolling 2, beginning of the back step 3 and leftward steady
a
*
0.2 v/(mR) 0.6
0.02
0.10
v*/(mR)
Fig
(Tj
for
. 6. Sum of normalized steady rolling state forces + T2)/(2G*aj2) as a function of v/mR and »*/mR d2/d, _ 0.93 and w = 2.2
Fig. 7. Spring deflections in different states: beginning of forth-step (1), rightward steady rolling (2), beginning of back step (3) and leftward steady rolling (4)
rolling 4. The method of dimensionality reduction model enables the derivation of uN and uT as a function of t from simple kinematic relations. We do not give the resulting expressions here, as they are quite complicated, illegible and of no benefit to the reader. However, we can compute the integral of force over one period, i.e. the impulse, as
t t 2 a (t)
It=JT (t )dt = J J G*uT (n, t)dn dt. (30)
0 0 0
The equivalent to the equal force condition is then given by vanishing impulse over one period:
It= 0. (31)
Finally, this yields the velocity of the runner in the working point as
(1 - d 2/ d1 )(1 + J d 2/ d1 - 2 w)
1 (32)
wR ^ wR J 4/3 - 4 d 2/ d1 - 2w(1 + d2/d1)
Equation (32) gives the relation between the input parameters of the drive and the resulting constant velocity of the runner taking into account the transition between the steps. For large rolling amplitudes w >> 2 the limit value of Eq. (32) corresponds exactly to expression (29):
1 - d 2 / d1
lim^ mR
1 --
MR
(33)
1 + d2/ d1
Again we performed a numerical study to compute the impulse over one oscillation period as a function of v/wR and vVwR for d2/d1 = 0.93 and w = 2.2. Figure 8 gives the simulation results and Eq. (32). Both are in perfect agreement. Comparison of Figs. 7 and 8 shows no qualitatively differences.
Figure 9 gives Eq. (32) for different values of d2/d1. It shows that the velocity of the runner in the working point highly increases for decreasing d2/ d1, i.e. for increasing difference of indentation between forth and back step. It
0.00
0.2 v/(mR)
0.06
0.03 v*/(wR)
Fig. 8. Normalized impulse of tangential force over one period I J (2G *a2 t) as a function of v/ wR and v*/wR for d2/ d1 = = 0.93 and w = 2.2
also increases with decreasing velocity ratio v/ wR. Thus, as for the steady rolling case, the higher the velocity difference between rolling and translation, the higher the velocity of the runner in the working point. The influence of the rolling amplitude w is relatively weak. This means that the drive does not require large amplitudes w in order to induce large motion of the runner v*.
4. Conclusion and outlook
We introduced a new concept for a nanodrive based on inertia effects and the principle of oscillating rolling with varying indentation depth. We identified the working point for a steady motion of the runner.
The results show that our concept enables an exact actuation with complete prevention of wear as no slipping occurs at any time. The drive provides well-adjustable input parameters, such as the translational and rotational speed and the indentation depth. We derived the relation of the input parameters and the output, i.e. the velocity of the runner for the quasistatic case. It shows that the velocity increases with increasing difference between the indentation
0.5 v/(MR)
1.0 10
as a
Fig. 9. Velocity of the runner in the working point v* /wR __ _ function of w and v*/wR for d2/d1 = 0.9 (1), 0.7 (2) and 0.5 (3)
of forth and back step and with increasing difference between translation and rotation of the sphere. As the amplitudes of the rolling and the translation are in principle of the same size as the contact length, only small displacements are needed to induce motion of the runner.
This first draft of a novel actuator must be followed by further theoretical examinations taking into account the inertia properties of the runner and effects of finite friction. It is then possible to determine the dynamic properties of the drive, i.e. the ability to accelerate the runner. In addition, an experimental proof of concept must be carried out in order to give practical guidelines for the further development of this type of actuator.
Acknowledgement
This work was supported in part by Tomsk State University Academic D.I. Mendeleev Fund Program, grant No. 8.2.19.2015.
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Поступила в редакцию 25.05.2015 г.
Ceedenun 06 aemopax
Robbin Wetter, Dipl.-Ing., Berlin University of Technology, [email protected]
Valentin L. Popov, Prof. Dr. of Berlin University of Technology, Prof. of Tomsk State University, Prof. of Tomsk Polytechnic University, [email protected]
