REVIEW OF SCIENTIFIC INSTRUMENTS
VOLUME 73, NUMBER 10
OCTOBER 2002
Three-axis lever actuator with flexure hinges for an optical disk system
Chang-Soo Hana)
Intelligence and Precision Machine Department, Korea Institute of Machinery and Materials,
171 Jang-dong, Yuseong, Daejeon, Korea
Soo-Hyun Kim
Department of Mechanical Engineering, KAIST, Yuseong, Daejeon, Korea
共Received 4 April 2002; accepted for publication 15 July 2002兲
A three-axis lever actuator with a flexure hinge has been designed and fabricated. This actuator is
driven by electromagnetic force based on a coil-magnet system and can be used as a high precision
actuator and, especially as a pickup head actuator in optical disks. High precision and low sensitivity
to external vibration are the major advantages of this lever actuator. An analysis model was found
and compared to the finite element method. Dynamic characteristics of the three-axis lever actuator
were measured. The results are in very close agreement to those predicted by the model and finite
element analysis. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1505098兴
I. INTRODUCTION
of the four-wire actuator. In the case of high-density recording devices such as digital versatile disks 共DVDs兲, the
pickup actuator should improve the tilt characteristics. However, the tilt motion of a four-wire actuator is caused by
coupled forces. If high performance of optical disks such as
high density DVDs 共HD-DVDs兲 require a high degree of
stability to vibration and active tilt motion for its pickup
actuator, it is very difficult to satisfy the requirements.
In this article, a three-axis lever actuator that has tilt
motion is proposed. Since a lever mechanism was chosen to
increase the system’s stability to external vibration, there is
After the development of the piezoelectric actuator, precision translation systems have been conducted including
studies of precision X–Y stage,1,2 piezoelectric motors,3 and
other precision positioning devices.4 – 6 In spite of the large
output forces of piezoelectric actuators, the typical 10–15
␮m displacement of a stack-type actuator is not sufficient for
most general engineering applications. Therefore, lever systems are used to amplify the displacement of the piezoelectric actuator.
In most cases, a flexure hinge has also been used as a
smooth guide for the lever system,7–9 because it has many
advantages such as good linearity, no friction, small hysteresis, and a simple structure.
Schematics of the optical disk and the four-wire pickup
head actuator are shown in Fig. 1. Fundamentally, a pickup
head actuator is required to accurately maintain the relative
distance between the track of the disk and the objective lens
in the focusing and tracking directions, since the rotation and
eccentricity of the disk and movement of the feeding motor
cause variation of the relative distance.
Generally, the pickup head actuator should have high
precision positioning accuracy 共about 10 nm兲 and about 0.5
mm movement range. Voice coil motors are widely used in
optical pickup head actuators because of their small size, low
cost, and proper range of movement. A four-wire actuator,
shown in Fig. 1共b兲, widely used due to its low cost and
simple structure, is composed of the following: an electromagnetic component, the magnet-coil yoke; a guide unit,
four spring wires; and a moving element, the lens and lens
holder. In the case of portable optical systems 共for cars, bicycles, jogging, etc.兲, stability under external vibration could
be regarded as one of the most significant performance requirements of the pickup head actuator.10 Thus the weak stability under external vibration may be a major disadvantage
FIG. 1. Optical disk system. 共a兲 Driving mechanism. 共b兲 Pickup head actuator four-wire type.
a兲
Author to whom correspondence should be addressed; electronic mail:
[email protected]
0034-6748/2002/73(10)/3678/9/$19.00
3678
© 2002 American Institute of Physics
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Rev. Sci. Instrum., Vol. 73, No. 10, October 2002
Three-axis lever actuator with flex hinges
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FIG. 4. Kinds of flexure hinge 共t: minimum width, a x : the radius of the
major axis, a y : the radius of the minor axis, R: the radius of curvature兲; 共a兲
rectangular, 共b兲 corner filleted, 共c兲 ellipse, and 共d兲 right circle.
FIG. 2. Three kinds of lever system with 共a兲 three, 共b兲 four, and 共c兲 five
hinges.
no amplification of either displacement or force. A voice coil
motor, not a piezoelectric actuator, was used as the drive for
this system to obtain proper range of movement. First, several topologies for the hinge alignment method were studied.
Next, we constructed theoretical models of the lever actuator
and executed a dynamic analysis using the finite element
method, and the prototype of the actuator was manufactured.
The performance of the actuator was evaluated in views of
its frequency response function and absolute transmissibility.
II. DESIGN OF THE LEVER ACTUATOR
A. Topology of the lever structure with flexure hinges
Considering the number and arrangement of the flexure
hinges we suggest three topologies in Fig. 2 and three lever
structures in the case of six hinges in Fig. 3. The structures
shown in Fig. 2 have lever motion about the middle fixed
parts. One moving part goes to the x axis, and the moving
part on the opposite side moves to the minus x axis. Also,
one moving part goes up to the z axis, and the other moving
part goes down to the minus z axis. However, these structures do not move like a lever for the tilt direction, ␪ y .
The six-hinge structure has greater structural stability
than those with three, four, or five hinges, and also lever
motion for the tilting. When a lever system moves in the z
direction, a twisting moment is applied on hinges of E type
and a bending moment is applied on hinges of I type. When
a lever system moves in the x direction, a bending moment
about the z axis is applied to all hinges of all three types. In
the case of T type hinges, the hinges at both sides twist and
hinges in the middle position bend. When two-axis move-
FIG. 3. Lever structures with six hinges: 共a兲 E type, 共b兲 T type, and 共c兲 I
type.
ment in the x and z directions is needed, a twofold lever
structure 共in the z direction兲 can be used for pure translation.
Consequently, a 12-hinge structure is used for this lever actuator.
B. Selection of hinge
Hinges can be divided into four kinds in view of their
profiles shown in Fig. 4 and research has been done on the
different kinds of hinges and their applications.11–13
We have learned from previous studies that rectangular
hinges have the largest displacement, and right circle hinges
have the highest accuracy for the same length hinge and
same minimum width 共t兲. With regard to stress conditions, it
has been reported that corner filleted hinges show better results than any other kind.
In optical disk systems, accuracy should be the more
important design factor, because stress is not a significant
factor due to little movement. We, therefore, selected the
right circle hinge.
FIG. 5. Structure of the three-axis lever actuator.
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Rev. Sci. Instrum., Vol. 73, No. 10, October 2002
C.-S. Han and S.-H. Kim
the lever arm is a flexible element. The focusing actuator is
similar to the tracking actuator.
Based on the suggested model, a Newtonian method is
used to derive the equation of motion. Paros and Weisbord14
have given an analytical solution for the bending stiffness of
a right circle hinge. Rong et al.12 have presented the twisting
stiffness for a flexure hinge. For computation of the compliance of a single-axis flexure hinge, the following equations
are used:
FIG. 6. Schematic of the 共a兲 single flexure hinge and 共b兲 lever actuator
共tracking兲.
c x⫽
␪x
1
⫽
f 共 ␦ 兲,
T x Gt ␤ 共 ␰ 兲 w 2 1
共1兲
c y⫽
␪y
12
⬵
f 共 ␦ 兲,
M y Ew 3 2
共2兲
c z⫽
␪z
12
⫽ 2 f 1共 ␦ 兲 .
M z Et w
共3兲
Then
f 1共 ␦ 兲 ⫽
C. Structure of the lever actuator
The basic structure of a three-axis lever actuator with
flexure hinges is shown in Fig. 5. Moving parts are located
on both sides about the middle fixed part. If any of the moving parts is driven, the other moving part is always translated
to the opposite direction in the manner of a lever. If any of
the moving parts rotates in a clockwise direction, the other
moving part rotates in a counterclockwise direction. The
hinges are bended or twisted according to the movement
direction. For example, in the case of tracking motion, all
hinges are bended about the z axis. When the actuator is
moved to the focusing or tilting direction, all hinges are
twisted about the x axis. Deformed parts in this actuator are
almost always the hinges, because the lever arm has enough
stiffness to not deform from its original shape. Figure 13
shows the deflection shape of focusing and tracking.
This actuator had a symmetric configuration about the x,
y, and ␪ y directions. One lens holder is used as a pickup
head, while the other holder is designed to have the same
configuration just for balance. The electromagnetic system is
composed of 8 focusing magnets, 4 tracking magnets, 4 focusing coils, 4 tracking coils, and 12 yokes. For guiding
movements in the x, z, and ␪ y directions, four levers are used
and three hinges are attached to each lever. In order to enhance the lever effect, the lever arm except for the flexure
hinges would be designed to have a high degree of stiffness
for the bending and twisting directions. And the lever arm
should be treated with a damping material because of its
resonance.
1
共 2⫹ ␦ 兲 2
⫹
冋
冋
3⫹4 ␦ ⫹2 ␦ 2
2 共 1⫹ ␦ 兲
3 共 1⫹ ␦ 兲
冑␦ 共 2⫹ ␦ 兲
f 2共 ␦ 兲 ⫽ ⫺
冑 ␦ ␦册
␦
␦
冑
册
␦
␦
tan⫺1
2⫹
␲
2 共 1⫹ 兲
⫹
tan⫺1
2 冑␦ 共 2⫹ 兲
,
2⫹
,
␦ ⫽t/2R, ␤ (t/H)⬍ ␤ ( ␰ )⬍ ␤ (t/w) is a geometric related
constant, and H⫽2R⫹w. For computation of maximum
stresses 共␴ max and ␶ max , respectively兲 at the thinnest area of
the flexure hinge for the x and z directions, the following
equations are used:
␴ max⫽
␶ max⫽
6M z
tw 2
Tx
␣ tw 2
,
共4兲
.
共5兲
M z and T x are applied moments for the x and z directions,
and constant ␣ can be found from the material mechanics
table. Maximum stress should be designed to be smaller than
the allowable stress of the hinge.
Under the assumption that the holders are not deformed,
the equations of motion below are obtained for three axis.
The displacement relation into which the lens holder and the
balance holder are shifted is presented at Fig. 7共a兲. x 1 , z 1 ,
and ␾ 1 for three axes denote the displaced positions, respectively.
Displaced positions 共A ⬘ ,B ⬘ ,C and D ⬘ ) are written as
⬘
lever A:共 z a1 ,x a1 , ␾ a1 兲 ⫽ 共 z 1 ⫹ ␾ 1 H,x 1 ⫺ ␾ 1 V, ␾ 1 兲 ,
III. ANALYSIS OF THE LEVER ACTUATOR
A. Theoretical modeling of the lever actuator
We constructed a model for the tracking actuator, shown
in Fig. 6, assuming the flexure hinge is a spring element and
lever B:共 z b1 ,x b1 , ␾ b1 兲 ⫽ 共 z 1 ⫹ ␾ 1 H,x 1 ⫹ ␾ 1 V, ␾ 1 兲 ,
lever C:共 z c1 ,x c1 , ␾ c1 兲 ⫽ 共 z 1 ⫺ ␾ 1 H,x 1 ⫺ ␾ 1 V, ␾ 1 兲 ,
共6兲
lever D:共 z d1 ,x d1 , ␾ d1 兲 ⫽ 共 z 1 ⫹ ␾ 1 H,x 1 ⫹ ␾ 1 V, ␾ 1 兲 ,
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Rev. Sci. Instrum., Vol. 73, No. 10, October 2002
Three-axis lever actuator with flex hinges
3681
FIG. 8. Freebody diagram for the case of base vibration.
M 1 z̈ 1 ⫽
兺 F z ⫽F z1 ⫺ 共 f az1 ⫹ f bz1 ⫹ f cz1 ⫹ f dz1 兲 .
共7c兲
Subscripts 1 and 2 denote the lens holder and balance holder,
respectively. J y1 denotes the moment of inertia of the lens
holder. M yi and F ki denote the moments and forces applied
for i⫽1, 2, and k⫽x, z. m pqr and f pqr denote the internal
moments and forces for p⫽a, b, c, d, and q⫽x, y z, r
⫽1 and 2, respectively.
Using the symmetric conditions, Eq. 共7兲 is rearranged as
M 1 z̈ 1 ⫹2 共 f az1 ⫹ f cz1 兲 ⫽F z1 ,
M 1 ẍ 1 ⫹2 共 f ax1 ⫹ f bx1 兲 ⫽F x1 ,
共8兲
J y1 ␾¨ 1 ⫹4m x1 ⫹2V 共 ⫺ f ax1 ⫹ f bx1 兲
⫹2H 共 f az1 ⫺ f cz1 兲 ⫽M y1 ,
V and H denote the width and height of the holders, respectively. As for the balance holder, the results obtained in a
similar way are
M 2 z̈ 2 ⫹2 共 f az2 ⫹ f cz2 兲 ⫽F z2 ,
M 2 ẍ 2 ⫹2 共 f ax2 ⫹ f bx2 兲 ⫽F x2 ,
共9兲
J y2 ␾¨ 2 ⫹4m x2 ⫹2V 共 ⫺ f ax2 ⫹ f bx2 兲
⫹2H 共 f az2 ⫺ f cz2 兲 ⫽M y2 .
Figure 7共c兲 shows the freebody diagram of the lever for the
focusing.
Arranging the above equations and boundary conditions,
the following equations are obtained:
lens holder:
FIG. 7. Diagram of 共a兲 displaced position at the corner of the lens holder,
共b兲 force of the lens holder, and 共c兲 the lever for focusing.
The displaced position of the balance holder is also obtained
by the same method.
Since the forces for three axis apply to the holder, a
freebody diagram is presented in Fig. 7共b兲. From force and
moment equilibrium conditions, the following equations are
obtained:
J y1 ␾¨ 1 ⫽
M 1 ẍ 1 ⫽
M 1 z̈ 1 ⫹K z11z 1 ⫹K z12z 2 ⫽F z1 ,
共10a兲
M 1 ẍ 1 ⫹K x11x 1 ⫹K x12x 2 ⫽F x1 ,
共10b兲
J y1 ␾¨ 1 ⫹ 共 K y11⫹VK x11⫹HK z11兲 ␾ 1
⫹ 共 K y12⫹VK x12⫹HK z12兲 ␾ 2 ⫽M x1 ;
balance holder:
M 2 z̈ 2 ⫹K z21z 1 ⫹K z22z 2 ⫽F z2 ,
共11a兲
M 2 ẍ 2 ⫹K x21x 1 ⫹K x22x 2 ⫽F x2 ,
共11b兲
兺 M 0 ⫽M y1 ⫺ 共 m ay1 ⫹m by1 ⫹m cy1 ⫹m dy1 兲
J y2 ␾¨ 2 ⫹ 共 K y21⫹VK x21⫹HK z21兲 ␾ 1
⫹V 共 f ax1 ⫺ f bx1 ⫹ f cx1 ⫺ f dx1 兲
⫹ 共 K y22⫹VK x22⫹HK z22兲 ␾ 2 ⫽M x2 ,
⫹H 共 ⫺ f az1 ⫺ f bz1 ⫹ f bz1 ⫹ f dz1 兲 ,
共7a兲
兺 F x ⫽F x1 ⫺ 共 f ax1 ⫹ f bx1 ⫹ f cx1 ⫹ f dx1 兲 ,
共7b兲
共10c兲
共11c兲
where K ti j ⫽4k ti j (t⫽x, y, ␾ , i⫽1, 2, and j⫽1, 2).
If the lever is stiff, the equations for both the lens holder
and the balance holder are decoupled from each other, and
these equations become simple second order systems for the
three axis.
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3682
Rev. Sci. Instrum., Vol. 73, No. 10, October 2002
C.-S. Han and S.-H. Kim
FIG. 9. Sensitivity analysis for the design parameter of 共a兲 focusing and 共b兲
tracking.
When vibration is applied to the actuator’s base, the
freebody diagram is that given in Fig. 8.
The displacement of the excitation’s displacement is denoted by z 3 .
From the two holders, we obtain the following equations:
M 3 z̈ 1 ⫹4 f z1 ⫽0,
共12a兲
M 2 z̈ 2 ⫹4 f z2 ⫽0.
共12b兲
FIG. 11. FRF analysis of the focusing actuator: 共a兲 open side excitation and
共b兲 excitation of both sides.
The equilibrium equations for the lever are as follows:
FIG. 10. Relationship between resonant frequencies for minimum width w
of the hinge.
FIG. 12. Relative transmissibility of the actuator for the focusing direction
共t e : the thickness of the additional member兲.
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Rev. Sci. Instrum., Vol. 73, No. 10, October 2002
Three-axis lever actuator with flex hinges
3683
TABLE I. Material constants of Be–Cu.
Description
Abbreviation
Value
E
␳
␯
␴t
␴y
␴tf
1.1⫻105 共N/mm2 )
8.23 共g/cm3 )
0.3
1205 共N/mm2 )
895 共N/mm2 )
241 共N/mm2 )
Elastic modulus
Density
Poisson ratio
Tensile strength
Yield strength
Fatigue strength 共108 cycles兲
兺 F z ⫽ f z1 ⫺ f z2 ⫺ f z3 ⫽0,
共13a兲
兺 M 0 ⫽m y1 ⫹m y2 ⫹m y3 ⫺ 共 L 1 f z1 ⫹L 2 f z2 兲 ⫽0.
共13b兲
The relation between force and displacement is as follows:
f z1 ⫽k z11共 z 1 ⫺z 3 兲 ⫹k z12共 z 2 ⫹z 3 兲 ,
共14a兲
f z2 ⫽k z21共 z 1 ⫺z 3 兲 ⫹k z22共 z 2 ⫹z 3 兲 .
共14b兲
From the above equations, the following equations are obtained:
M 1 z̈ 1 ⫹K z11共 z 1 ⫺z 3 兲 ⫹K z12共 z 2 ⫹z 3 兲 ⫽0,
共15a兲
M 2 z̈ 2 ⫹K z21 共 z 1 ⫺z 3 兲 ⫹K z22 共 z 2 ⫹z 3 兲 ⫽0.
共15b兲
Applying the Laplace transform to Eq. 共15兲, the following
equations are obtained:
Z 1 共 M 2 s 2 ⫹K z11兲共 K z11⫺K z12兲 ⫺K z12共 K z21⫺K z22兲
⫽
,
Z3
共 M 1 s 2 ⫹K z11兲共 M 2 s 2 ⫹K z22兲 ⫺K z12K z21
共16a兲
Z 2 共 M 1 s 2 ⫹K z22兲共 K z21⫺K z22兲 ⫺K z21共 K z11⫺K z12兲
⫽
.
Z3
共 M 1 s 2 ⫹K z11兲共 M 2 s 2 ⫹K z22兲 ⫺K z12K z21
共16b兲
The above equations represent the absolute transmissibility
of the actuator for the focusing.
The absolute transmissibility for tracking is easily found
in a similar way:
FIG. 13. Modal analysis of the lever actuator for 共a兲 the tracking and 共b兲 the
focusing.
The relative transmissibility is given as 1 minus absolute
transmissibility.
In the ideal case in which the lever is rigid, L 1 ⫽L 2 and
M 1 ⫽M 2 , and Z 1 /Z 3 become 1 and ⫺1, respectively, that is,
if the relative distance 共from the lens to the disk兲 is unchanged for any external vibration input. However, because
the stiffness of the lever does not have infinite magnitude,
the dynamic characteristics differ from ideal ones due to the
lever effect.
B. Analysis results for the design parameters
Using the above dynamic equations, design parameter
analysis is investigated. In the design of a pickup actuator,
X 1 共 M 2 s 2 ⫹K x11兲共 K x11⫺K x12兲 ⫺K x12共 K x21⫺K x22兲
⫽
,
X3
共 M 1 s 2 ⫹K x11兲共 M 2 s 2 ⫹K x22兲 ⫺K x12K x21
共17a兲
X 2 共 M 1 s 2 ⫹K x22兲共 K x21⫺K x22兲 ⫺K x21共 K x11⫺K x12兲
⫽
.
X3
共 M 1 s 2 ⫹K x11兲共 M 2 s 2 ⫹K x22兲 ⫺K x12K x21
共17b兲
TABLE II. Dimensions of the lever actuator.
Description
Radius of hinge
Minimum width of hinge
Thickness of hinge
Moving mass
Length of connecting rod
Abbreviation
Value
R
w
t
M 1, M 2
L 1, L 2
0.6 共mm兲
0.1 共mm兲
0.1 共mm兲
2.5⫻10⫺3 共kg兲
9.75 共mm兲
FIG. 14. Prototype of the lever actuator.
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3684
Rev. Sci. Instrum., Vol. 73, No. 10, October 2002
C.-S. Han and S.-H. Kim
FIG. 15. FRF of the lever actuator for 共a兲 the tracking,
共b兲 the focusing, and 共c兲 the tilting.
the most important performance parameter is the first resonant frequency for each axis. The first resonant frequency of
the lever actuator is determined by its stiffness, because the
moving mass is constant. Figure 9 gives the results of analysis for the first resonant frequency.
The first resonant frequency for the three axis is the most
sensitive for the minimum width of the hinge as shown in
Fig. 10. The tilting stiffness is dependent on the focusing and
tracking stiffness as shown in Eqs. 共10c兲 and 共11c兲.
Considering the force applied and the thickness of the
lever, we analyzed the frequency response function 共FRF兲
and the relative transmissibility of the actuator is analyzed.
Figure 11 shows the FRF and Fig. 12 shows the relative
transmissibility for the focusing direction.
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Rev. Sci. Instrum., Vol. 73, No. 10, October 2002
Three-axis lever actuator with flex hinges
3685
total lever system was modeled in an I-DEAS dynamic modeling analysis task to study its dynamic behavior.
All holders are lumped together and connected to hinges
by rigid line elements. Beryllium–copper 共Be–Cu兲 is used
for hinges and lever arms. Material constants for Be–Cu are
given in Table I. The dimensions of the analysis model are
given in Table II. The FEM results are given in Fig. 13.
The first mode shows focusing movement and the second mode shows tracking movement. The tilt mode of the
lever actuator appeared at 71 Hz, and the resonance modes of
the lever arm appear at 101 Hz in tracking and at 320 Hz in
focusing, respectively.
Three kinds of lever structure 共E type, T type, and I type兲
were simulated by the finite element method. The results
indicated that the first two resonance modes are focusing and
tracking for the three structures. Also, the three types of lever
actuator shown in Fig. 3 differ from each other in terms of
resonant frequency by 5%. Higher mode shapes show similar
patterns for each of the three types. Only I type has more
mode shapes than the other types within 1 kHz. We selected
the E type for fabrication.
IV. EXPERIMENTS
FIG. 16. Absolute transmissibility function of the lever actuator for 共a兲 the
tracking and 共b兲 the focusing.
When force is given to both sides, a simple second order
system is achieved. Otherwise, a second resonant exists as
shown in Fig. 11. If an additional member is attached to the
lever, the second resonant frequency increases, whereas the
first resonant frequency is changed only slightly. The FRF
results of the tracking direction have almost the same patterns as those of the focusing direction because of the similarity between the focusing and the tracking. The second
resonant frequency is dependent upon the width of the lever.
The first resonant frequency of the relative transmissibility can be changed according to the thickness of the additional member. That is, the larger the stiffness of the lever,
the higher the frequency of the relative transmissibility. For
other kinds of actuators, the first resonant frequency is the
frequency of the relative transmissibility, and it cannot be
modified. However, the lever type of actuator can easily
widen the range of reduction by modification of the lever
stiffness.
C. Finite element analysis
The finite element method 共FEM兲 has usually been used
for flexure hinge design and analysis.7,12,13 In this article, a
TABLE III. Comaprisons between simulation and experiment.
First resonance
Theoretical
value
Finite element
value
Experimental
value
Focusing 共Hz兲
Tracking 共Hz兲
Tilting 共Hz兲
23.1
40.7
75.2
23.3
41.6
71
23.9
41.6
73
The prototype of the actuator in Fig. 14 was manufactured from the following components: the moving parts of
the lever actuator are fabricated from aluminum; the kind of
magnet is rare earth 共Nd–B–Fe兲; the focusing coil is made
from aluminum to lessen its weight; the tracking coil is made
from copper; the iron yoke material is carbon steel; the attached members on the lever arms are made from polycarbonate 共PC兲.
The experiment was carried out in two parts: the FRF
and the transmissibility function. Figure 15 shows the frequency function for the tracking, the focusing, and the tilting.
The three-axis lever actuator for tracking and focusing
has the simple dynamics of a second order system under 1
kHz, which is the bandwidth of the optical disk servo. There
is no other resonance within 1 kHz. Parasitic resonance in
high frequency is found at 20 kHz for tracking and at 8 kHz
for focusing. The required bandwidth of tilting is lower than
in the other moving direction, because the rotation frequency
of the disk is about 20 Hz. The first resonant frequency of the
tilting is 73 Hz. The undesired peak due to focusing resonance is negligible.
Figure 16 shows a comparison of the absolute transmissibility function for the two types of actuators. For tracking,
the first resonant frequency is located at 329 Hz in the lever
and at 20 Hz in the four wire. For focusing, the first resonant
frequency is located at 95 Hz in the lever actuator and at 20
Hz in the four-wire type actuator. Figure 16 shows that the
lever actuator has a wider reduction bandwidth for external
vibration.
Table III shows comparisons of theoretical analysis, finite element analysis, and experimental results. The results
show good agreement with each other. This demonstrates
that both theoretical analysis and finite element are available
to design the lever actuator.
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