ACTUATOR 2012, Messe Bremen 1/4
Pre-print version! To be presented at and appear in the proceedings of
ACTUATOR 12 – International Conference and Exhibition on New Actuators and Drive Systems
Bremen, Germany, 18-20 June 2012, www.actuator.de
Analysis of Different Operation Modes
for Inertia Motors
M. Hunstig, T. Hemsel, W. Sextro
University of Paderborn, Mechatronics and Dynamics, Paderborn, Germany
Abstract:
Piezoelectric inertia motors, also known as “stick-slip-drives”, use the inertia of a body to drive it by means of a
friction contact in small steps. While these steps are classically assumed to involve stiction and sliding, the mo-
tors can also operate in a “slip-slip” mode without any phase of stiction. In this contribution, a one degree of
freedom model of an inertia motor is used to analyse the motor characteristics. Appropriate performance indica-
tors for the slider motion are defined: Start-up time, steady state velocity, and smoothness of motion. Inertia
motors reach their theoretical maximum velocity with ideal signals containing harmonics of very high frequen-
cies. With frequency-limited real signals, high motor velocity can only be achieved in “slip-slip” mode. The
maximum motor velocity is reached by superposition of two sinusoidal signals, other signal shapes can improve
other performance criteria. The results help motor designers to choose the appropriate mode of operation and the
best drive parameters for their individual applications.
Keywords: Piezoelectric Motor, Inertia Motor, Stick-Slip, Slip-Slip, Optimization
Introduction
Originally developed for fine positioning applica-
tions in the laboratory, piezoelectric inertia motors
found application in several other fields in the last
years, mainly in miniaturised consumer goods like
digital cameras for mobile phones [1–4]. This was
facilitated by the fact that inertia motors have a sim-
ple construction and are controlled by a single driv-
ing signal, which allows for low production costs
and simplifies miniaturization.
Inertia motors make use of the inertia of a body to
drive it by means of a friction contact in a series of
small steps. Motors of this type are also known as
“stick-slip-drives” because these steps are classically
regarded to be composed of a phase of static friction
between the driving and driven part and a phase
where the two parts slide on each other.
Even though the first piezoelectric inertia motors
were developed in the mid-1980s [5–7], the fact that
these motors can also successfully operate without
phases of static friction has gained wider recognition
only in the last years. This mode of operation with
the parts continuously sliding and only the direction
of relative motion changing is also known as the
“slip-slip” mode. Some authors [4, 8, 9] have de-
scribed inertia motors operating in both stick-slip
and slip-slip mode. But the principal advantages and
disadvantages of the two operation modes and how
to use them advantageously are still unclear. This
contribution aims to give answers to these questions.
Investigated Setup
Figure 1 shows the model which is the basis of the
following analysis. The displacement xR (t) of a rod
is given. A slider of mass mS hangs below the rod.
The contact force Fc between rod and slider results
from the gravitational force Fg and an external force
FM, both assumed to act on the centre of gravity C of
the slider. The friction force Ff between rod and
slider is modelled using a Coulomb friction model
with the coefficients of static and dynamic friction
s and d. The equation of motion of the slider is
S S f - S sin ,
(1)
and in the regime of sliding friction the friction force
is described by
f S c sgn( - S).
(2)
In the investigations described below, horizontal
operation (
= 0) is assumed.
Fig. 1: Rigid Body Model of an Inertia Motor
Ideal Excitation Signals for Maximum Velocity
In many applications of inertia motors, especially in
consumer applications, high velocity is the primary
design goal.
For movement in positive direction of xS the maxi-
mum acceleration of rod and slider that can be
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reached without leaving the regime of stiction is
defined by
S,ma - sin S c
S.
(3)
The movement cycle for maximum velocity in stick-
slip mode is as follows: The slider accelerates with
aR = aS,max until the rod reaches its maximum posi-
tion xR,max. Ideally, it then instantly returns to
xR(t) = 0 and the slider starts to slide and is continu-
ously decelerated. The next acceleration phase of the
rod is started directly after it has returned to xR(t) = 0,
so the slider remains in continuous motion with
alternating phases of stiction and sliding as depicted
in figure 2(a). For such signals the fundamental
signal frequency is determined as
√
,ma .
(4)
If discrete steps of the slider are desired, a phase of
rest can be introduced after each acceleration phase,
allowing the slider to decelerate down to S( ) as
shown in figure 2(b). A more detailed derivation of
these continuous and discrete modes of operation
under the assumption of limited rod acceleration is
presented in [10].
Fig. 2: Rod position and slider velocity for (a) continuous op-
eration and (b) discrete steps
If the rod acceleration aR is increased above aS,max,
there is no stiction between rod and slider and the
motor operates in slip-slip mode. Continuous motion
and discrete steps can be realized in this mode as
well.
Performance Indicators
In many cases, an inertia motor requires several
periods of the drive cycle to reach its maximum
velocity. Figure 3 shows such a typical velocity
increase. is the mean slider velocity in one period
and the steady state velocity ∞ is defined as after
an infinite number of periods.
The start-up time t98 is the time after which reaches
0.98 . It can also be defined for other percentages
of . As is defined over complete periods only, t98
is always a multiple of the drive cycle period.
Fig. 3: Typical velocity increase at start-up
During one period, the slider velocity ranges be-
tween the minimum velocity vmin and the maximum
velocity vmax. The smoothness of the slider motion
can be defined as
- ma - min
.
(5)
Steady state velocity, start-up time and smoothness
of the slider motion are three indicators for the mo-
tional performance of an inertia motor.
Performance with Ideal Excitation Signals
A comparison of the modes with continuous slider
movement shows that rises with aR, see figure 4.
There is a jump in the curve at aR = aS,max due to the
sudden change of the slider acceleration resulting
from the assumption d s. If d s, there is no
jump, but still a kink at the same position. The max-
imum velocity is limited in stick-slip mode because
aR must not exceed aS,max.
Fig. 4: Change of steady state velocity and rod displacement
characteristic with rod acceleration aR
In stick-slip mode, is always reached in the se-
cond period (cp. figure 2(a)) while in slip-slip mode
the slider needs many periods to reach its steady
state velocity. Due to this significant difference
between the two modes, a jump at aR = aS,max is
observed in figure 5 which shows t98.
Fig. 5: Change of start-up time t98 with rod acceleration aR
Figure 6 shows the increase of the smoothness s of
the slider motion with increasing aR. It smoothly
converges towards the unreachable value of 1 for
high rod acceleration.
0
10
20
xR [µm]
0
10
20
0 1 2 3 4
0
20
40
60
80
t [ms]
vS [mm/s]
0 1 2 3 40
20
40
60
80
t [ms]
(a) (b)
0 2 4 6 8 10
0
10
20
30
aR / aS,max
t98 [ms]
stick-slip
slip-slip (smoothened)
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Fig. 6: Change of smoothness s with rod acceleration aR
In summary, higher velocities can be reached and
the slider movement is smoother in slip-slip mode at
the cost of a longer start-up phase and the require-
ment of higher frequencies compared to stick-slip
mode.
Frequency-Limited Real Excitation Signals
Fourier analysis can be used to determine the har-
monics of periodic signals and to reconstruct these
signals by summation of their harmonics. The ideal
excitation signals presented above can only be re-
produced perfectly with an infinite number of har-
monics, but no real actuator can do this.
Frequency limited signals are derived from the ideal
signals using Fourier series of finite length n. Fig-
ure 7 shows resulting characteristics of displacement,
velocity and acceleration for different values of n.
Fig. 7: Rod displacement, velocity and acceleration for the ide-
al signal and for frequency-limited signals of orders 2 and 15
While the displacement signal is approximated well
already with low n, large oscillations are present in
the frequency-limited velocity and acceleration due
to the short phases where velocity or acceleration are
infinite in the ideal signals. Except for a combination
of very low aR and low n, the rod acceleration is
effectively always above aS,max due to these oscilla-
tions. This means that there are no phases of static
friction with significant length.
Performance with Real Excitation Signals
Figure 8 shows the steady state velocity reached by
the simulated motor excited with frequency-limited
signals with different n and aR. While there is no
effective slider movement for n = 1, the steady state
velocity rises with both n and aR. For high values of
n, a further increase of n yields only a small increase
of velocity.
Fig. 8: Steady state velocity with different n and aR
The frequency of the highest harmonic present in the
excitation signals is
ma
√
,ma .
(6)
Real actuators have a maximum frequency up to
which they can be practically operated, so n and aR
cannot be increased arbitrarily. For a given maxi-
mum frequency ma , choosing a higher n means to
have a signal with more harmonics, but a lower
fundamental frequency. Figure 9 illustrates this
relation.
Fig. 9: Three displacement signals with equal fmax
Figures 10 (a) to (c) show steady-state velocity,
start-up time and smoothness of the slider motion,
respectively, reached with equal ma for different
combinations of n and aR.
Fig. 10: Steady state velocity, start-up time and smoothness for
different maximum frequencies
Higher maximum frequencies lead to higher steady
state velocity. For any given maximum frequency
the highest steady state velocity is reached with
n = 2.
0 2 4 6 8 10
0
0.5
1
aR / aS,max
s [-]
stick-slip
slip-slip
0
10
20
xR [µm]
-1
-0.5
0
vR [m/s]
0 0.2 0.4 0.6 0.8 1 1.2
-1
-0.5
0
0.5
1
t [ms]
aR [105 m/s²]
ideal
n = 2
n = 15
0
10
20
t [s]
xR [µm]
n = 2
aR = 16 aS,max
n = 4
aR = 4 aS,max
n = 8
aR = 1 aS,max
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The start-up time is generally larger for larger max-
imum frequencies. With any constant maximum
frequency, increasing n decreases the start-up time.
The smoothness of the slider motion is generally
larger for larger maximum frequencies. Increasing n
slightly decreases the smoothness for constant max-
imum frequency.
Influence of the Actuator Stroke on the Velocity
The investigations described above are true inde-
pendent of the actuator stroke xR,max. But this stroke
has a large influence on the absolute value of the
steady state velocity: It can be shown that if the
fundamental excitation frequency follows equa-
tion (6), √ ,ma . This remains true for the
ideal signals with limited rod acceleration [11]. To
achieve high velocities it is therefore advisable to
use actuators with large stroke. This theoretical
result is supported by the fact that almost all inertia
motors achieving high velocities (here defined as
20 mm/s or larger) use large actuator strokes
achieved either by resonance effects [9, 12] or by
mechanical amplification using compliance mecha-
nisms [13] or bending actuators [14–16]. As far as
known to the authors, only one motor [8] reaches
such velocities with an un-amplified multilayer
actuator.
Conclusions
This contribution showed that, if high velocity is
desired, it is not recommendable to aim for classic
“stick-slip” operation of inertia motors. In “slip-slip”
mode these motors can be operated with signals
containing only harmonics of relatively low fre-
quencies and still reach velocities that are higher
than the theoretical ma imum for “stick-slip” opera-
tion.
With a given maximum signal frequency, the maxi-
mum motor velocity is always reached by superposi-
tion of two sinusoidal signals. This is beneficial as
such a signal can use resonance amplification with
passable complexity, demonstrated for example in
[9, 12]. If other performance criteria are relevant in
the application, the use of a different driving signal
can make sense.
While the frequency-limited signals derived from
the ideal excitation signals produce high velocities,
it has not yet been determined whether these signals
indeed produce the highest velocity. Whether other
signal shapes containing the same harmonic fre-
quencies can be more advantageous, especially re-
garding other performance criteria, is subject to
further investigation.
As the knowledge about the advantages of “slip-
slip” operation of inertia motors is growing and
spreading, the authors expect that the number of
dedicated “slip-slip” inertia motors will grow signif-
icantly in the near future, widening the field of ap-
plication of inertia motors.
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