Chapter 1 Introduction
1.3 Goals
From the discussion above, the goals of this work may be ascribed into two major topics.
The first one is to propose a better analytical model on SDA to improve the prediction capability such as priming voltage, step size, total travel distance, output force and dynamic response simulation. The second one is to propose a novel lower voltage SDA (LVSDA) design in comparison with higher driving voltage of conventional rectangular SDA to extend the
compatibility of SDA with integrated circuit process.
1.4 Approach in this study
To achieve research goals of this work, the current approach is stated as following. In chapter 1, the motivation is first declared and followed by a comprehensive literature review of SDA and related topics. Then the goals are introduced and followed with detail description on research methods and procedures. In chapter 2, a unified analytical method for improved SDA modeling is proposed and derived in sequence. Governing differential equation of fourth-order is presented based on Euler-Bernoulli beam and distributed electrostatic load. From this equation, much valuable information may be extracted in systematic manner; such as deflection curve, step size, bending moment, maximum stress and output force. The output force data just derived is applied into a single degree of freedom model of SDA. By simulating its vibrational motion, total travel distance and dynamic output force can be determined numerically for the first time under given parameters of mass, friction, spring constant and input voltage.
In chapter 3, a novel flexible joint design is proposed to incorporate into the conventional SDA design to improve performance in low-voltage region due to the smaller flexural rigidity.
This novel design of lower voltage scratch drive actuator (LVSDA) is confirmed by qualitative analysis and finite element analyses on stress distribution and deformation of test structures. In chapter 4, micro-electroplated nickel SDA and LVSDA arrays connected with suspended spring are fabricated by a two-mask process developed in NCTU. Then the released devices are tested to verify capability of the proposed SDA model and novel LVSDA design with analytical and simulated results. The test results show good agreement between the prediction of improved SDA model with experiment data, including total travel distance and output force.
Because the step size is in nanometer range, as its measurement is not as easier as the measurement of total travel distance on the basis of thousands of input pulses, the step size
measurement is in stead of total travel distance. The test results also show that the LVSDA can operate well even at as low as 40 volts, a 50% reduction than 80 volts of SDA, which can not be done by conventional rectangular SDA with the same size. In chapter 5, the major contributions of this work are summarized and the future works are also discussed in brief.
Figure 1-1. Operation principles of SDA [1]. (a) Terminologies, (b) SDA placed on insulator and substrate, driven by voltage pulse in capacitor circuit, (c) Positive signal, main plate priming, bushing scratches ahead, (d) Low signal, main plate spring back and sliding forward, (e) Negative signal, main plate priming again, bushing scratches ahead
Figure 1. Typical applications of SDA in microelectromechanical system (MEMS).
Figure 1-2. Typical applications of SDA in microelectromechanical system (MEMS).
(a) Concept of micro-assembly [1]. (b) Optical xyz-stage assembly [5].
(c) Variable Optical Attenuator assembly [6]. (d) Platform assembly [7]
(e) Microrobot actuated [8]. (f) Cam-motor [9]. (g) Micro-fan [10].
(h) Wireless microrobot [13]. (i) Micro-gripper[16]. (j) Micro-translation table [17].
Figure 1-4. Step size estimated from experimental results in Langlet’s work [19].
Figure 1-5. Step size predicted from analytical model in Kazuaki’s work [20].
Figure 1-6. Snap through voltage (a) and priming voltage predicted from analytical model in Linderman’s work [8].
Figure 1-3. Step size estimated from experimental results in Akiyama’s work [2].
Table 1-1. Summary of previous researches on SDA modeling
Property Reference
Priming voltage
Non-contact length
Step size Traveling distance
Output force Dynamical simulation Akiyama,1993 [2] No after testing* after testing after testing No No
Langlet, 1997 [19] No after testing after testing after testing No No
Akiyama,1997 [1] No after testing after testing after testing after testing No
Kazuaki,1998 [20] No Yes Yes No No No
Linderman, 2001 [8]
Yes No after testing after testing after testing No
This work Yes Yes Yes Yes Yes Yes
Note: “after testing” means the property is estimated after testing has been done.
Figure 1-7. Output force estimation methods. (a) Buckling beam method in Akiyama’s work [1], (b) Tether spring method proposed in Linderman’s work [8] and Li’s work [23]
respectively.
Chapter 2 Modeling of Scratch Drive Actuator
The typical structures of SDA are shown in Figure 2-1(a). One usual constraint type at the contact part of support beams with rail is a slider joint without rotation as in Figure 2-1(b) [23].
A systematic approach will be proposed to analyze the static and dynamic behavior of SDA.
2.1 Operation principle of SDA
The operation of SDA from rest to scratch forward is classified into five states shown in Figure 2-2. As the input voltage V increases from zero, the still SDA in Figure 2-2(a) starts rotating about the bushing paw and the free edge of main beam just touches the dielectrics at the pull-in voltage Vpi as in Figure 2-2(b). For input higher than pull-in voltage Vpi, the main beam will be bent and gradually becomes a surface contact with the dielectrics as Figure 2-2(c), so-called priming voltage Vpr. After that, the contact length of main beam to the dielectrics increases as input voltage increases, it is so-called post-priming at voltage Vpp, as shown in Figure 2-2(d). Finally, when the input voltage is fully discharged, the SDA will spring back and keep the one forward step, as shown in Figure 2-2(e).
2.2 Static analysis of SDA
Some assumptions are made first before the derivation. The deformation and electrostatic force are assumed to be same along the width direction of SDA, as the main plate may be treated by beam model [24]. The Euler-Bernoulli beam theory is applied on main plate and support beams; the bushing, dielectrics and rails are assumed to be rigid. The angle between the bushing and main plate remains right angle all over the deformation process, and the electrostatic force acted on bushing is ignored. At pull-in, priming or post-priming state, the contact between main beam and dielectrics remains. In other words, free end of main beam is assumed to be fixed, and only the bushing can deflected and scratch ahead at priming and
post-priming states. In static analysis, it is natural to draw the condition as no slippage occurs during charging and discharging. The vertical displacement of bushing top is negligible during main beam deflection in priming and post-priming states. The fringing effect of electrostatic field is ignored.
For a general parallel plate capacitor, the distributed load q, i.e. the electrostatic force per unit length applied to the plate, is given as
where ε0 is the permittivity of air; V is the input voltage; W is the plate width; d is the gap between two plates. Let the SDA be modeled as plate capacitor, d is defined as the distance from the main beam to the electrode on substrate, including air and the dielectric layers. The distance d varies with the deflection y and electrostatic load q. Due to the nonlinear coupling between the electrostatic load q and main beam deflection y, there is not yet an exact solution for the deflection curve y. A square law of deflection has been proposed to define the deflection curve of a cantilever beam under electrostatic force [8]. By combining Euler-Bernoulli beam theory and the square law model, the governing equation of the main beam with the distributed electrostatic load q(x) at post-priming state can be expressed as a fourth-order differential equation:
where a is equal to the thickness t of the dielectric layer divided by the relative permittivity k; b is equal to h / ln2
; ln is the non-contact length of main beam at the given input voltage; h is the bushing height; E is the Young’s modulus of SDA material; I is the second area moment of main beam; x is the horizontal coordinate of the element in consideration. The downward
1)
deflection is defined as positive. The coordinate system is shown in Figure 2-3 where the origin is defined at the juncture point between contact and non-contact regimes with distance h above.
Integrating equation (2-2) successively four times leads to following equations:
The four integration constants, c1, c2, c3, c4 and non-contact length ln, can be solved with the following five boundary conditions:
Eq. (2-7) means the y position of origin remains to be the bushing height, h. Also, the contact region is flat and fixed to the dielectrics at post-priming state, so the slope and moment at origin O are zero, as implied by Eq. (2-8) and Eq. (2-9), respectively. Eq. (2-10) states that the vertical displacement of bushing is negligible. Eq. (2-11) expresses the balance between the moment EIy”(ln) of main beam and the torque kt y’(ln) from the support beam at x = ln. The parameter kt
aspect ratio constant based on the ratio of the width to thickness of the support beam [25]; Wp is the width of support beam; Lp is the length of support beam; G is shear modulus of SDA material.
Applying the boundary condition of Eq. (2-7) to Eq. (2-9) into characteristic equations Eq.
(2-4) to Eq. (2-6), the integration constants c2, c3 and c4 can be solved as:
Applying Eq. (2-10) into characteristic Eq. (2-3), the integration constant c1 can be solved as:
Rewriting the expression of Eq. (2-11) in terms of Eq. (2-4) and Eq. (2-5) can lead to the
characteristic equation of non-contact length ln for given input V;
Since no explicit solution exists, the non-contact length ln is solved by numerical method. At the priming state, the not-contact length is considered to be the same as the main beam length L, and then the priming voltage Vpr can be determined from reorganizing Eq. (2-16) by replacing ln
with main beam length L as
By definition, the bending moment M at any position x in non-contact part of main beam is
given by definition as following where flexural rigidity EI is constant
There are two methods to determine the position xmax where the maximum moment Mmax
happens along main beam for a specific input voltage. The first one is to decide the coordinate that the shear force, −EI( dy/dx), becomes zero. Then substituting this coordinate into Eq. (2-20) will find the maximum moment. The one is straight forward in applying the optimization toolbox in Matlab to find out the extreme value along main beam. The maximum bending stress
σmax along main beam for corresponding input can be expressed as:
The horizontal output force Fo of charged SDA array along the main beam varies with the input voltage. The friction force along contact regime is assumed to be large enough to keep the contact area stationary. The horizontal output force Fo is basically the horizontal component of 21)
the electrostatic force along main beam. It can be solved by integrating the horizontal component of electrostatic force along the non-contact part as:
where N is the SDA number in the SDA array, θ is the angle between the horizontal and the tangent line along main beam, and can be calculated from the slope of deflection curve.
In order to recognize SDA step size in fully discharge mode, Figure 2-4 shows the SDA deflection and displacement of bushing from still, charge and discharge. At pull-in voltage, the SDA rotates about bushing paw. Due to the slope change on bushing, the bushing top moves back horizontally with magnitude △x1=h*sin(tan−1(h/L)), as shown in Figure 2-4(a). When the input voltage is equal to or larger than the priming voltage, the main beam starts bending and causes the bushing top to move backward more due to so-called curvature shortening effect, which has not been considered in previous literatures on SDA. The lateral displacement due to curvature shortening effect can be formulated as △x2= dx
x
This effect will make the bushing rotate more in counterclockwise, and then the bushing paw moves forward for certain horizontal distance given by △x3=h*｜dy/dx｜as shown in Figure 2-4(c). When SDA is fully discharged, the main beam will spring back and rotate about the bushing paw as Figure 2-4(d). By combining the net effect of three above effects, the step size in this type of SDA motion control becomes
The weakness of driving method in Figure 2-4 is that the main beam may rotate too much.
As a result, SDA is possible to turn over or not easy to pull back by electrostatic force 23)
immediately. An effective alternate driving method is to keep the main beam constantly contact with dielectrics under input levels VP and VPP between priming and some post-priming input.
Figure 2-5 shows the components of step size in this driving mode, i.e., constantly contact mode.
As the slope effect ∆x1 is avoided, the net step size ∆x may be simplified as
the step size at input voltage VPP, ∆xpp minus the step size at input voltage VP , ∆xp.
24) -(2 x
x
x=∆ PP−∆ P
∆
2.3 Dynamic analysis of SDA
To explore the dynamic behavior of SDA including friction effect, a mass-spring-damper model of single degree of freedom (SDOF) is proposed as an approximation. The equation of motion and its normalized form are given as Eq. (2-25) and Eq. (2-26)
The coordinate x and its first and second derivatives of with respect to time are the displacement, velocity and acceleration of SDA, respectively. The lump-mass m defines the total mass of SDA array including the ring around SDA. The coefficient k defines the spring constant of the suspended tether spring. The damper may contain one or two types of friction; the first is viscous damping of coefficient c of contact surface between main beam and insulated layer.
The second one is the Coulomb friction of coefficient µ accounting for the stick-slip phenomenon in contact surfaces. The force FN is the normal force between the contact area of main beam and insulated layer. In MEMS, the body force such as gravity force is often much less than surface traction force as the electrostatic force. So the normal force is simplified to contain only the electrostatic force in the contact area. The force FO is the amplitude of driving force which is the resultant horizontal component of the electrostatic force applied along the main beam of SDA, the force along bushing is ignored here. The driving voltage is in sinusoidal wave sin(ωt) of angular velocity ω. However, it should be noted that the electrostatic force is an attractive force between contact surfaces, no matter the charge on main beam is positive or negative. Therefore, the driving force is expressed as the absolute value of driving voltage waveform, i.e., abs(sin(ωt)) multiplying FO, which is the motive force that makes SDA scratch forward. The friction terms in Eq. (2-25) may be and rearranged as following
26)
In Eq. (2-27), sign(~) is the sign function result of the operand in parenthesis; the gain (c/m) is defined as a regulation factor that modifies viscous friction coefficient. The offset (µFN/m) is the dry friction force that should be overcome in motion. To simplify the derivation, the static and dynamic dry friction coefficients are assumed to be same as µ.
When the Coulomb friction is activated, the analysis of displacement (total travel distance) response is too complex to derive an exact and compact analytical solution. To solve this type of nonlinear vibration problems, some methods have been developed such as perturbation method, averaging method, multiple scales method and direct separation of motions [26-30].
Instead of these methods, direct numerical simulation method as Simulink in Matlab has been adopted to investigate the dynamic response in a straight forward way to simulate the dynamical behavior of SDA.
To evaluate the fatigue behavior of SDA, the information of stress bounds are needed. Eq.
(2-21) may calculate the maximum and minimum stresses as following
The equivalent stress σR, the fully reverse stress of same fatigue life [31], is expressed as
The minimum moment Mmin at specific input, though not easily be derived, lies between zero for fully discharge mode and the maximum moment at Vp in constantly contact mode.
27)
Figure 2-1. Structures and elements of SDA. (a) Support beams are free at contact part.
(b) contact part is constrained to slide without rotation [24].
substrate
dielectrics rail
main plate
bushing support beam W
L
Wp
Lp
t
h
(a)
(b)
Figure 2-2. Five states in SDA operation at different input voltages: (a) initial, zero input; (b) pull-in, voltage Vpi; (c) priming, voltage Vpr; (d) post-priming, voltage Vpp; (e) one-step forward after discharge.
Figure 2-3. Coordinate system and electrostatic force density in post-priming configuration. The electrostatic force density in contact region is constant. In non-contact region, the electrostatic force density follows Petersen model [8].
Figure 2-4. Step size of SDA operated in fully-discharge mode. (a) Initial state. (b) The main plate rotates around bushing paw and causes lateral backward movement
∆x1. (c) The curvature shortening effect causes the bushing top lateral movement
∆x2 backward more. Also, the bushing paw displaces ∆x3 forward. (d) SDA is discharged fully with one net step size ∆x =∆x3−∆x1−∆x2.
Figure 2-5. Step size of SDA operated in constantly-contact mode. (a) Initial state at voltage VPP. (b) Partly discharge to voltage VP , main plate springs back. (c) SDA is again charged to VPP, the step size ∆x =∆x(VPP)−∆x(VP).
Chapter 3
Performance Enhancement of Scratch Drive Actuator with Modified Flexible Joint
3.1 Concept design of low voltage Scratch Drive Actuator (LVSDA)
The concept design of the proposed LVSDA is shown in Figure 3-1, where two narrow beams as flexible joint are placed between the main plate and scratch plate. The operation procedures of LVSDA and conventional SDA are illustrated in Figure 3-2. They are both at rest initially with zero input in Figure 3-2(a). When the input increases to snap-through voltage, the free edge of main plate of LVSDA or SDA will touch the dielectrics (Figure 3-2(b)). At the same input voltage, LVSDA will snap through more due to the smaller flexural rigidity to generate a larger bushing lateral displacement, as shown in Figure 3-2(b). At an even higher input voltage, the main plate will contact with the dielectrics more area and further push the bushing forward, as Figure 3-2(c). After discharging, LVSDA or SDA will bounce back to complete one cycle with one forward step size ∆x, as Figure 3-2(d).
In order to fabricate and test the proposed LVSDA, the geometric parameters in LVSDA are defined in Figure 3-3(a). The design of testing device is illustrated in Figure 3-3(b), where a tether spring consisted of four box springs links four LVSDAs to the contact electrode. The comparison between the proposed LVSDA and conventional SDA is based on the same device size. It means all LVSDAs and SDAs here have the same total plate length L and plate width W, which are fixed as 80 µm and 65 µm, respectively.
L will remain to be 80 µm, as well as the plate length of SDA. The thickness t of plate, spring, and support beams, as well as the bushing height, of LVSDA and SDA is all set at 2 µm.
Each support beam has length Lp= 25 µm and width Wp= 3 µm. Different flexible joint
dimensions and locations are designed to investigate their effects on performance. A shorter or wider flexible joint will provide a more rigid joint. Two different flexible joint widths Wj are designed as 6 µm and 12 µm. The designed flexible joint length Lj includes 15 µm and 20 µm.
The designed scratch plate length Ls includes 10 µm, 15 µm, and 20 µm. A shorter Ls means the flexible joint is closer to the bushing. Due to different combinations on scratch plate length and flexible joint length, the corresponding main plate length Lm of LVSDA includes 50 µm, 45 µm and 40 µm to keep the total plate length as 80 µm. The designed LVSDA dimensions are denoted as LV-Ls-Lj-Lm-Wj. For example, the test device expressed as LV-10-20-50-12 has following dimensions: Ls=10 µm, Lj=20 µm, Lm=50 µm, and Wj=12 µm. Eight types of LVSDA used in this work are listed in Table 3-1.
The spring constant of one box spring can be expressed as [24]:
k1=EtWr3
/(Lr3
), (3-1)
where E is the Young’s modulus of spring material, t and Wr are the thickness and width of spring beam, respectively, and Lr is the half length of box spring. For the tether spring
where E is the Young’s modulus of spring material, t and Wr are the thickness and width of spring beam, respectively, and Lr is the half length of box spring. For the tether spring