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Design and experiments of fully compliant bistable micromechanisms
Mechanism and Machine Theory
Mechanism and Machine Theory 40 (2005) 17–31
www.elsevier.com/locate/mechmt
Design and experiments of fully compliant bistable micromechanisms Jinni Tsay, Hsin-An Chang, Cheng-Kuo Sung
*
Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu 300, Taiwan, ROC Received 27 September 2003; received in revised form 16 April 2004; accepted 28 May 2004 Available online 19 August 2004
Abstract This paper proposed a design of a fully compliant bistable micromechanism for the application of switching devices. The topology of a fully compliant four-bar linkage was adopted herein. The central mass of the micromechanism was employed as a carriage to transmit switching components, such as mirror, electrical contact, etc. The equations that predicted the existence of bistable behavior, the extreme positions of the motion range, and the maximum stresses of elastic members were derived. The proposed micromechanisms were fabricated by MUMPs of Cronos Integrated Microsystems and an experimental rig was established for the purpose of verifying the theoretical predictions. The compliant bistable micromechanisms were switched either by a probe or actuators to manipulate the central mass. The experimental results demonstrated that the motions observed approximately met the predicted values. 2004 Elsevier Ltd. All rights reserved. Keywords: Bistable mechanism; Compliant mechanism; Switching devices
1. Introduction To reduce power consumption is an important factor for the design of microsystems. The bistable micromechanism, which has two stationary positions at the two extremes of the *
Corresponding author. Tel.: +886 35742918; fax: +886 35715314. E-mail address: [email protected] (C.-K. Sung).
0094-114X/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2004.05.006
18
J. Tsay et al. / Mechanism and Machine Theory 40 (2005) 17–31
motion range, possesses excellent performance in decreasing power consumption. Power is supplied only at the instant while switching the micromechanism from one stable state to another one [1] because the bistable micromechanism can store and release energy in their moving segments. Therefore, this micromechanism is suitable for the devices operated in low switching rate. According to different design concepts, the bistable micromechanisms may be categorized into three classes: 1. Mechanisms composed of compliant and rigid links are stationary at both positions in the extreme of motion range [1]. 2. Membranes containing residual stresses or loaded stresses induce buckling to obtain out-ofplane bistable motions [2]. 3. A special braking tool clamps the device at the extremes of motion to achieve bistable function [3]. By considering the manufacturability, joint clearance and friction, and mechanical or geometrical advantage, the compliant micromechanism obviously is a good candidate for obtaining the bistable function. According to the configuration, compliant micromechanisms may be classified as partially and fully compliant micromechanisms [4]. The partially compliant micromechanism means that parts of links or joints of the rigid linkage are replaced by elastic members. On the other hand, the fully compliant micromechanism means all the links and joints are elastic [5]. This paper presents the design, fabrication and experiment of fully compliant bistable micromechanisms. The topology of the compliant bistable micromechanism, whose structure is suspended by four anchors fixed on the substrate as shown in Fig. 1, is adopted herein. The compliant parts that result in motions of the micromechanism are two side beams and four elastic hinges. The central mass moves in the vertical direction only. In Section 2, the equations governing the existence of the bistable behavior and their positions were derived and that is followed by a parametric study. In Section 3, the fabrication of the proposed bistable micromechanisms based on MUMPs provided by Cronos Integrated Microsystems is introduced, and the experimental setup as well as results is described.
Fig. 1. A fully compliant bistable micromechanism.
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2. Theory and design A fully compliant four-bar linkage is selected as the structure of the bistable micromechanism for demonstrating the proposed design methodology. Among three basic topologies of four-bar linkages shown in Fig. 2, the double-slider mechanism as shown in Fig. 2(c) is considered to be more suitable for the investigation of compliant and bistable characteristics. The fully compliant bistable micromechanism can be modeled as Fig. 3(a). The center of side beam can only move horizontally, so it can be modeled as a slider moving in x-direction with a translational spring
Fig. 2. Basic topologies of four-bar linkages.
Fig. 3. A symmetric double-slider mechanism. (a) Physical model of the bistable mechanism; (b) a symmetric doubleslider mechanism; (c) motion of the symmetric double-slider mechanism.
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J. Tsay et al. / Mechanism and Machine Theory 40 (2005) 17–31
whose spring coefficient is equal to the stiffness of side beam. The central mass can only move in ydirection if the actuating force exerts along the central line of the mechanism. Therefore, the central mass can be modeled as a slider moving in y-direction. The relatively flexible part can be modeled as a hinge joint with a torsional spring. Fig. 3(b) shows a symmetric double-slider mechanism for the following kineto-static analysis. The theoretical analysis of the compliant bistable micromechanism is based on the following assumptions: 1. The transverse deflection of the elastic hinge is considered only since its longitudinal deformation is too small. 2. The deflection at the center of the side beam caused by the moment of the bent elastic hinge is ignored. 3. The motion of the compliant bistable micromechanism is same as that of the symmetric doubleslider mechanism, which indicates that the central mass moves along the y-direction only and the horizontal and torsional motions of the central mass are ignored. 4. Similar to the assumption 3, the center of the side beam moves along the x-axis only.
2.1. Stable positions Fig. 3(c) illustrates the motion of the compliant bistable micromechanism. The deformation of the center of the side beams yields the displacement of the slider moving in x-direction. The elastic hinges yield angular motion. Then, the central mass translates in y-direction. The double-slider mechanism with solid lines stands for the original position while the one with dotted lines represents the subsequent position. In these two states, the lengths projected in the x- and y-directions are expressed as: x ¼ L cosðh0 hÞ L cos h0 y ¼ L sin h0 L sinðh0 hÞ
ð1Þ
where x and y are the displacements of the sliders in the x- and y-directions with respect to the angular displacement of the coupler link, h, while h0 stands for initial angle of the coupler link that possesses a length L. In Eq. (1), x and y are both functions of h, which indicates that the degree of freedom of the double-slider mechanism is one. As long as h is known, the motion of the double-slider mechanism is determined, that is, the state of the compliant bistable micromechanism is known. In order to calculate the total strain energy of the compliant mechanism, the strain energy of each elastic member has to be derived. By referring to Fig. 4, the force exerting at the central point of side beam, F, to create a deflection x can be written as F ¼
192EI b x l3b
ð2Þ
The notations of the side beam are denoted by subscript ‘‘b’’, and the force exerting at the end of the elastic hinge, P, to yield an angle of rotation h can be expressed as
J. Tsay et al. / Mechanism and Machine Theory 40 (2005) 17–31
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Fig. 4. Strain energy of the elastic members. (a) Side beam; (b) elastic hinge.
P¼
2EI h h l2h
ð3Þ
The notations of elastic hinges are denoted by subscript ‘‘h’’; then, the strain energies of the elastic members are Ub ¼
1 192EI b 2 96EI b 2 x ¼ 3 x 2 l3b lb
ð4Þ
Uh ¼
P 2 EI 3h 2EI h 2 ¼ h 6EI h 3lh
ð5Þ
where E is the elastic modulus of the material and I is the moment of inertia: Ii ¼
tw3i 12
i ¼ b; h
ð6Þ
t and w are the thickness and width of the elastic members, respectively. Since the two side beams and four elastic hinges form the compliant parts of the double-slider mechanism, the total strain energy, which is the summation of the strain energy of all elastic members, in the micromechanism can be written as: U ¼ 2U b þ 4U h ¼
192EI b 2 8EI h 2 x þ h 3lh l3b
By differentiating U with respect to h yields dU 384EI b ox 16EI h x h ¼ þ 3 dh oh 3lh lb where
ox oh
ð7Þ
ð8Þ
can be obtained from (1),
ox ¼ L sinðh0 hÞ oh
ð9Þ
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The stable positions exist as long as the following conditions being satisfied. 8 dU > > > < dh ¼ 0 > d2 U > > : 2 >0 dh
ð10Þ
2.2. Flexural stresses To prevent from yielding of elastic members, the maximum stress occurring in the elastic members cannot exceed the fracture strength of the material. The elastic members of the compliant bistable mechanism, side beams and elastic hinges, have different loading conditions. By observing Fig. 5, because the tangent of the beam at center is vertical, the deflection due to the transverse force, F2 , and the bending moment, M, on the curvature of the beam are identical. The moment at the center of side beam can be computed as following: 1 ð11Þ M ¼ Flb 8 The transverse force causes both normal and shear stresses. The shear stress is so small that it can be neglected. On the other hand, the normal stress caused by both transverse force and bending moment is written as: rb ¼ rF rM ¼
F 2
yM x Ib
ð12Þ
In the longitudinal direction, the maximum stress occurs at the fixed end of the beam, where y ¼ l2b . In the transverse direction, the maximum stress occurs at the edge of the beam, where x ¼ W2b , as shown in Fig. 5(a). Subsequently, the maximum normal stress can be obtained by substituting M ¼ Fl8b into (12), rb;max ¼
Flb wb 16I b
ð13Þ
Fig. 5(b) shows the loading condition of the elastic hinge. The elastic hinge is subjected to a transverse force P that results in an angular displacement, h. Express the normal stress induced by the transverse force in the elastic hinge as:
Fig. 5. Loading conditions on elastic members. (a) Side beam; (b) elastic hinge.
J. Tsay et al. / Mechanism and Machine Theory 40 (2005) 17–31
rh ¼
P ðlh xÞ y Ih
23
ð14Þ
In the longitudinal direction, the maximum stress occurs at the fixed end of the hinge, where x = 0. In the transverse direction, the maximum stress occurs at the edge of the hinge, where y ¼ W2h , as shown in Fig. 5(b). Substitute into (14) and the maximum stress in the elastic hinge is derived: rh;max ¼
Plh wh 2I h
ð15Þ
The maximum stresses occurring in the elastic members must be smaller than the fracture strength of the material; that is, rb, max < rfracture and rh, max < rfracture. On the basis of the formulas derived above, the total strain energy is calculated by setting the parameters into different sets of values. Under the constraint that the maximum stress does not exceed the fracture strength, the curve of the total strain energy has two local minimums where the micromechanism performs bistable behavior within the extent of motion. The parameters of the compliant bistable micromechanism proposed in this paper are listed in Table 1. The curves of total strain energy are different from one set of parameters to another. If little variation of the parameters causes significant changes of the total strain energy, these parameters are considered as sensitive ones. On the contrary, the parameters are regarded as non-sensitive ones. During the design process, the widths of the side beam and the elastic hinge as well as the angle of the coupler link are recognized as dominant parameters for the total strain energy and the stress state. Hence, more attention should be paid to these parameters when dealing with the design of compliant bistable micromechanisms. The proposed micromechanisms are made of polysilicon which elastic modulus and fracture strength are also listed in Table 1. Substitute the parameters in Table 1 into Eqs. (7) and (8), and draw the figures of total strain energy U and the derivative of total strain energy with respect , as shown in Fig. 6. While the curve of dU passes through zero and the curve of U concaves to h, dU dh dh upward, these points are regarded as stable positions because the local minimums of the total strain energy appear. Fig. 6 displays the angle between two stable positions to be about 2.7. The curves with respect to the displacement of central mass, y, is shown in Fig. 7, which shows the displacement between two stable positions is 17.8 lm. Table 1 Parameters of the compliant bistable mechanism Parameter
Denotation
Quantity
t lb lh L wb wh h0 Ib Ih E rfracture
Thickness of polysilicon Length of side beam Length of elastic hinge Length of coupler link Width of side beam Width of elastic hinge Initial angle of coupler link Moment of inertia of side beam Moment of inertia of elastic hinge Elastic modulus of polysilicon Fracture strength of polysilicon
3.5 lm 53 lm 30 lm 385 lm 4 lm 2 lm 1.5 18.67 lm4 2.33 lm4 165 GPa 1.2 GPa
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Fig. 6. Total strain energy and its differential curves w.r.t. the angle of rotation.
Fig. 7. Total strain energy and its differential curves w.r.t. the displacement.
Again, by substituting the parameters listed in Table 1 into Eq. (1), the motions of the compliant bistable micromechanism in x- and y-directions, x and y, can be computed and shown in Fig. 8. The micromechanism starts at the first stable position, where y is 0. While y arrives at 10 lm, x
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Fig. 8. Displacements of elastic members.
reaches the maximum of 0.14 lm approximately. While y arrives at 17.8 lm, the bistable micromechanism reaches the second stable position. To prevent from yielding of the elastic members, the maximum stresses generated in the elastic members cannot exceed the fracture strength of the material. If the elastic members are made of a material that possesses larger fracture strength or less elastic modulus, it will generate a larger deformation, so that the compliant bistable micromechanism creates a larger deflection. However, due to the constraints of the fabrication process, this research selects polysilicon as the material of the compliant bistable mechanism. According to Table 1, the fracture strength of polysilicon doped with phosphorus is 1.2 ± 0.15 GPa [6]. The maximum stresses in the side beams and the elastic hinges are illustrated in Fig. 9. By observing the curves, none of the stresses exceeds the fracture strength of polysilicon, so this set of parameters is feasible for the safety of the compliant bistable micromechanism.
3. Fabrication and experiments 3.1. Fabrication In order to verify the design, the fabrication of the structures described above was carried out by MUMPs (Multi-User MEMS Processes) provided by Cronos Integrated Microsystems. MUMPs are able to deposit seven layers of four different kinds of materials on the substrate, as shown in Fig. 10(a). The fabrication processes include one layer of Si3N4 for electrical isolation, one layer of polysilicon (Poly 0) for the ground electrode, two layers of polysilicon (Poly 1 and
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J. Tsay et al. / Mechanism and Machine Theory 40 (2005) 17–31
Fig. 9. Stresses of elastic members.
Fig. 10. Layers of MUMPs [7]. (a) Before released; (b) after released; (c) the steps of etching.
Poly 2) for mechanical structures, two layers of PSG (Phosphorus Silicate Glass) for sacrificial material in order to suspend the mechanical structures, and one layer of gold for electrical con-
J. Tsay et al. / Mechanism and Machine Theory 40 (2005) 17–31
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tacts or optical mirror. The bistable micromechanisms are built in Poly 1 in combination with Poly 2 in order to enlarge the thickness. By employing the MUMPs process, the only works that need to be done by users are drawing masks and post-processing including etching the sacrificial layers to release the structure layers as shown in Fig. 10(b). This research employs a process of evaporation drying to release the micro devices. The steps of the post-process, as shown in Fig. 10(c), are described in sequence as follows: 1. 2. 3. 4.
Immerse the chip in HF (49%) for 5 min. Rinse the chip with de-ionized (DI) water. Rinse the chip with isopropyl alcohol (IPA) in order to replace DI water. Bake the chip by 200 C hot plate to evaporate the remaining IPA on the chip.
Since IPA has smaller surface tension than DI water, replacing DI water with IPA makes the microstructure less possible being got stuck to the substrate after baking. Therefore, we use IPA to rinse the chip before baking on the hot plate. Fig. 10 illustrates the side view of devices after the sacrificial layers are etched away. Having completed the above works, the experiments commence. 3.2. Experiments The experimental setup composes of several equipment and instruments as stated in the following: 3.2.1. Experimental rig 3.2.1.1. Vibration isolation platform. Vibrations from surroundings can cause enormous disturbances for the observation by using the microscope and consequently raise the difficulties of experiments. Therefore, an isolation platform is necessary to separate the external vibrations. 3.2.1.2. Micrography system. The micrography system includes a microscope, a CCD camera, and a computer. The microscope manufactured by Avimo Precision Instruments (Optem) can magnify the images of micro devices, which are sensed by the CCD camera. The images are captured and recorded by the computer so that one can observe the contours of the micro devices on the monitor and also record the motions of the micro devices. 3.2.1.3. Workbench. The workbench contains a stage on which the chip is laid and moved in x, y and h directions, an air pump that sticks the chip by continuous extraction of air, and several probes that operate the chip in three dimensions. 3.2.1.4. Scanning electronic microscope. The JSM-5610LV scanning electron microscope contributed the SEM photos. The JSM-5610LV SEM has the capacity of high magnification and tilted carriage to scan the side of specimen. The images can be transmitted to the computer and be saved as electronic photo files. In addition, the function of ‘‘Scale’’ supplies the on-line measurement of scale. It is suitable for the observation and measurement of MEMS devices.
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3.2.1.5. Power supply system. A set of electrical instruments for driving the actuators, such as scratch drive actuator (SDA), thermal actuator, etc., include a function generator and a power amplifier. The former generates an alternating current (AC) with adjustable frequency and amplitude, and the signal of AC can be selected as sine wave, triangular wave, or square wave. The later magnifies the amplitude of AC up to 20·, 40·, 100· or 200·; the output of the power supply is an AC of adjustable high voltages with various waveforms to meet the requirement of driving actuators. The highest voltage and current outputs are 250 V and 3 A, respectively. 3.2.2. Experimental processes 3.2.2.1. Switched by a probe. From the computed results, the compliant bistable micromechanism can be stable in two positions. The purposes of this experiment were to observe if the micromechanism was bistable and to measure the deflection of the bistable mechanism. Fig. 11(a) and (b) show the images of the compliant bistable micromechanism before and after being switched. After stirred by the probe, the compliant bistable micromechanism switched from the first stable position to the second one and held still. It indicates that the compliant bistable micromechanism functioned as expected. The displacement of central mass was measured by the attached function of SEM. From the SEM photograph shown in Fig. 12, the displacement between two stable positions is 17.6 lm. This experimental result approximately met the simulation introduced before where 17.8 lm of the displacement was expected.
Fig. 11. SEM images of the compliant bistable mechanism switched by a probe. (a) Before switched; (b) after switched.
Fig. 12. SEM image of displacement between two stable positions (enlarged view).
J. Tsay et al. / Mechanism and Machine Theory 40 (2005) 17–31
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Fig. 13. SEM images of the bistable micromechanism switched by SDA. (a) Before switched; (b) after switched.
3.2.2.2. Switched by SDA. As a sub-system, the compliant bistable micromechanism has to be manipulated by itself. An embedded actuator, SDA, is designed to integrate with the mechanism. The SDA pulls the central mass of the micromechanism. The SEM photograph, Fig. 13, illustrates
Fig. 14. SEM image of displacement between two stable positions (enlarged view).
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J. Tsay et al. / Mechanism and Machine Theory 40 (2005) 17–31
the central mass held at first and second stable states. The calculated displacement of central mass was 30 lm; the measured one met the theoretical calculation, as shown in Fig. 14.
4. Conclusions This research proposed a design method for fully compliant bistable micromechanisms. The topology and dimensions of the compliant micromechanism was first designed based on the requirement of the motion and force transmission. By considering the bistable function, the total strain energy of the compliant micromechanism was derived by summing up the strain energy stored in each elastic member. In addition, the behaviors of compliant bistable micromechanisms were examined. The bistable devices have two stationary positions where no external energy is required to hold the micromechanism in place. Power is needed only when switching the micromechanism into another stable position. As the micromechanism is properly designed, the bistable devices can even further reduce the power consumption. Moreover, the compliant micromechanisms composed of elastic members can simplify the fabrication process and prevent from friction, wear and clearance that may exist in hinge joints. Hence, the compliant micromechanism is suitable for the applications in MEMS. The purpose of the compliant bistable micromechanism is to combine the advantages of both bistable devices and compliant mechanisms. From the viewpoint of energy, there are two local minimums on the curve of the total strain energy of the compliant bistable micromechanism. This paper derived the equations governing the bistable behavior, strain energy and stress states. Besides, the parameters that dominate the variations of the strain energy are learned. It could be very helpful for the future design of the compliant bistable micromechanism. The micro devices designed in this research were fabricated by MUMPs process. By observing the experimental results, the compliant bistable micromechanisms performed bistable functions as expected. As a result, the equations and simulations were verified by the experiments.
Acknowledgement The authors would like to thank the financial support of National Science Council via the contract of NSC91-2212-E-194-036.
References [1] M.S. Baker, S.M. Lyon, L.L. Howell, A Linear displacement bistable micromechanism, Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2000, pp. 1–7. [2] B. Ha¨lg, On a nonvolatile memory cell based on micro-electro-mechanics, in: Proceedings IEEE Workshop on MEMS, 1990, pp. 172–176. [3] X. Sun, K.R. Farmer, M.N. Carr, A bistable microrelay based on two-segment multimorph cantilever actuators, in: Proceedings IEEE Workshop on MEMS, 1998, pp. 154–159. [4] A.G. Erdman, Modern Kinematics––Developments in the Last Forty Years, John Wiley and Sons, 1993.
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[5] L.L. Howell, A. Midha, A method for the design of compliant mechanisms with small-length flexural pivots, Transactions of the ASME 116 (1994) 280–290. [6] W.N. Sharpe, Jr. Bin Yuan, R.L. Edwards, R. Vaidyanathan, Measurements of YoungÕs modulus, PoissonÕs ratio, and tensile strength of polysilicon, in: 10th IEEE International Workshop on Microelectromechanical Systems, Nagaya, Japan, 26–30 January 1997, pp. 424–429. [7] MUMPs Design Handbook, Web Site: www.memsrus.com/cronos/CIMSmain2ie.html. Cheng-Kuo Sung has been a professor of Department of Power Mechanical Engineering, National Tsing Hua University after he received his Ph.D. from Michigan State University, USA in 1986. He was the former Chairman of Taiwan Chapter of the International Federal of Theory of Mechanisms and Machine (IFToMM). His major research interests include micro-mechanism design, machine dynamics, and precision machine design. Jinni Tsay is a Ph.D. candidate of Department of Power Mechanical Engineering, National Tsing Hua University, Taiwan, ROC.
Mechanism and Machine Theory 40 (2005) 17–31
www.elsevier.com/locate/mechmt
Design and experiments of fully compliant bistable micromechanisms Jinni Tsay, Hsin-An Chang, Cheng-Kuo Sung
*
Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu 300, Taiwan, ROC Received 27 September 2003; received in revised form 16 April 2004; accepted 28 May 2004 Available online 19 August 2004
Abstract This paper proposed a design of a fully compliant bistable micromechanism for the application of switching devices. The topology of a fully compliant four-bar linkage was adopted herein. The central mass of the micromechanism was employed as a carriage to transmit switching components, such as mirror, electrical contact, etc. The equations that predicted the existence of bistable behavior, the extreme positions of the motion range, and the maximum stresses of elastic members were derived. The proposed micromechanisms were fabricated by MUMPs of Cronos Integrated Microsystems and an experimental rig was established for the purpose of verifying the theoretical predictions. The compliant bistable micromechanisms were switched either by a probe or actuators to manipulate the central mass. The experimental results demonstrated that the motions observed approximately met the predicted values. 2004 Elsevier Ltd. All rights reserved. Keywords: Bistable mechanism; Compliant mechanism; Switching devices
1. Introduction To reduce power consumption is an important factor for the design of microsystems. The bistable micromechanism, which has two stationary positions at the two extremes of the *
Corresponding author. Tel.: +886 35742918; fax: +886 35715314. E-mail address: [email protected] (C.-K. Sung).
0094-114X/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2004.05.006
18
J. Tsay et al. / Mechanism and Machine Theory 40 (2005) 17–31
motion range, possesses excellent performance in decreasing power consumption. Power is supplied only at the instant while switching the micromechanism from one stable state to another one [1] because the bistable micromechanism can store and release energy in their moving segments. Therefore, this micromechanism is suitable for the devices operated in low switching rate. According to different design concepts, the bistable micromechanisms may be categorized into three classes: 1. Mechanisms composed of compliant and rigid links are stationary at both positions in the extreme of motion range [1]. 2. Membranes containing residual stresses or loaded stresses induce buckling to obtain out-ofplane bistable motions [2]. 3. A special braking tool clamps the device at the extremes of motion to achieve bistable function [3]. By considering the manufacturability, joint clearance and friction, and mechanical or geometrical advantage, the compliant micromechanism obviously is a good candidate for obtaining the bistable function. According to the configuration, compliant micromechanisms may be classified as partially and fully compliant micromechanisms [4]. The partially compliant micromechanism means that parts of links or joints of the rigid linkage are replaced by elastic members. On the other hand, the fully compliant micromechanism means all the links and joints are elastic [5]. This paper presents the design, fabrication and experiment of fully compliant bistable micromechanisms. The topology of the compliant bistable micromechanism, whose structure is suspended by four anchors fixed on the substrate as shown in Fig. 1, is adopted herein. The compliant parts that result in motions of the micromechanism are two side beams and four elastic hinges. The central mass moves in the vertical direction only. In Section 2, the equations governing the existence of the bistable behavior and their positions were derived and that is followed by a parametric study. In Section 3, the fabrication of the proposed bistable micromechanisms based on MUMPs provided by Cronos Integrated Microsystems is introduced, and the experimental setup as well as results is described.
Fig. 1. A fully compliant bistable micromechanism.
J. Tsay et al. / Mechanism and Machine Theory 40 (2005) 17–31
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2. Theory and design A fully compliant four-bar linkage is selected as the structure of the bistable micromechanism for demonstrating the proposed design methodology. Among three basic topologies of four-bar linkages shown in Fig. 2, the double-slider mechanism as shown in Fig. 2(c) is considered to be more suitable for the investigation of compliant and bistable characteristics. The fully compliant bistable micromechanism can be modeled as Fig. 3(a). The center of side beam can only move horizontally, so it can be modeled as a slider moving in x-direction with a translational spring
Fig. 2. Basic topologies of four-bar linkages.
Fig. 3. A symmetric double-slider mechanism. (a) Physical model of the bistable mechanism; (b) a symmetric doubleslider mechanism; (c) motion of the symmetric double-slider mechanism.
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whose spring coefficient is equal to the stiffness of side beam. The central mass can only move in ydirection if the actuating force exerts along the central line of the mechanism. Therefore, the central mass can be modeled as a slider moving in y-direction. The relatively flexible part can be modeled as a hinge joint with a torsional spring. Fig. 3(b) shows a symmetric double-slider mechanism for the following kineto-static analysis. The theoretical analysis of the compliant bistable micromechanism is based on the following assumptions: 1. The transverse deflection of the elastic hinge is considered only since its longitudinal deformation is too small. 2. The deflection at the center of the side beam caused by the moment of the bent elastic hinge is ignored. 3. The motion of the compliant bistable micromechanism is same as that of the symmetric doubleslider mechanism, which indicates that the central mass moves along the y-direction only and the horizontal and torsional motions of the central mass are ignored. 4. Similar to the assumption 3, the center of the side beam moves along the x-axis only.
2.1. Stable positions Fig. 3(c) illustrates the motion of the compliant bistable micromechanism. The deformation of the center of the side beams yields the displacement of the slider moving in x-direction. The elastic hinges yield angular motion. Then, the central mass translates in y-direction. The double-slider mechanism with solid lines stands for the original position while the one with dotted lines represents the subsequent position. In these two states, the lengths projected in the x- and y-directions are expressed as: x ¼ L cosðh0 hÞ L cos h0 y ¼ L sin h0 L sinðh0 hÞ
ð1Þ
where x and y are the displacements of the sliders in the x- and y-directions with respect to the angular displacement of the coupler link, h, while h0 stands for initial angle of the coupler link that possesses a length L. In Eq. (1), x and y are both functions of h, which indicates that the degree of freedom of the double-slider mechanism is one. As long as h is known, the motion of the double-slider mechanism is determined, that is, the state of the compliant bistable micromechanism is known. In order to calculate the total strain energy of the compliant mechanism, the strain energy of each elastic member has to be derived. By referring to Fig. 4, the force exerting at the central point of side beam, F, to create a deflection x can be written as F ¼
192EI b x l3b
ð2Þ
The notations of the side beam are denoted by subscript ‘‘b’’, and the force exerting at the end of the elastic hinge, P, to yield an angle of rotation h can be expressed as
J. Tsay et al. / Mechanism and Machine Theory 40 (2005) 17–31
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Fig. 4. Strain energy of the elastic members. (a) Side beam; (b) elastic hinge.
P¼
2EI h h l2h
ð3Þ
The notations of elastic hinges are denoted by subscript ‘‘h’’; then, the strain energies of the elastic members are Ub ¼
1 192EI b 2 96EI b 2 x ¼ 3 x 2 l3b lb
ð4Þ
Uh ¼
P 2 EI 3h 2EI h 2 ¼ h 6EI h 3lh
ð5Þ
where E is the elastic modulus of the material and I is the moment of inertia: Ii ¼
tw3i 12
i ¼ b; h
ð6Þ
t and w are the thickness and width of the elastic members, respectively. Since the two side beams and four elastic hinges form the compliant parts of the double-slider mechanism, the total strain energy, which is the summation of the strain energy of all elastic members, in the micromechanism can be written as: U ¼ 2U b þ 4U h ¼
192EI b 2 8EI h 2 x þ h 3lh l3b
By differentiating U with respect to h yields dU 384EI b ox 16EI h x h ¼ þ 3 dh oh 3lh lb where
ox oh
ð7Þ
ð8Þ
can be obtained from (1),
ox ¼ L sinðh0 hÞ oh
ð9Þ
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The stable positions exist as long as the following conditions being satisfied. 8 dU > > > < dh ¼ 0 > d2 U > > : 2 >0 dh
ð10Þ
2.2. Flexural stresses To prevent from yielding of elastic members, the maximum stress occurring in the elastic members cannot exceed the fracture strength of the material. The elastic members of the compliant bistable mechanism, side beams and elastic hinges, have different loading conditions. By observing Fig. 5, because the tangent of the beam at center is vertical, the deflection due to the transverse force, F2 , and the bending moment, M, on the curvature of the beam are identical. The moment at the center of side beam can be computed as following: 1 ð11Þ M ¼ Flb 8 The transverse force causes both normal and shear stresses. The shear stress is so small that it can be neglected. On the other hand, the normal stress caused by both transverse force and bending moment is written as: rb ¼ rF rM ¼
F 2
yM x Ib
ð12Þ
In the longitudinal direction, the maximum stress occurs at the fixed end of the beam, where y ¼ l2b . In the transverse direction, the maximum stress occurs at the edge of the beam, where x ¼ W2b , as shown in Fig. 5(a). Subsequently, the maximum normal stress can be obtained by substituting M ¼ Fl8b into (12), rb;max ¼
Flb wb 16I b
ð13Þ
Fig. 5(b) shows the loading condition of the elastic hinge. The elastic hinge is subjected to a transverse force P that results in an angular displacement, h. Express the normal stress induced by the transverse force in the elastic hinge as:
Fig. 5. Loading conditions on elastic members. (a) Side beam; (b) elastic hinge.
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rh ¼
P ðlh xÞ y Ih
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ð14Þ
In the longitudinal direction, the maximum stress occurs at the fixed end of the hinge, where x = 0. In the transverse direction, the maximum stress occurs at the edge of the hinge, where y ¼ W2h , as shown in Fig. 5(b). Substitute into (14) and the maximum stress in the elastic hinge is derived: rh;max ¼
Plh wh 2I h
ð15Þ
The maximum stresses occurring in the elastic members must be smaller than the fracture strength of the material; that is, rb, max < rfracture and rh, max < rfracture. On the basis of the formulas derived above, the total strain energy is calculated by setting the parameters into different sets of values. Under the constraint that the maximum stress does not exceed the fracture strength, the curve of the total strain energy has two local minimums where the micromechanism performs bistable behavior within the extent of motion. The parameters of the compliant bistable micromechanism proposed in this paper are listed in Table 1. The curves of total strain energy are different from one set of parameters to another. If little variation of the parameters causes significant changes of the total strain energy, these parameters are considered as sensitive ones. On the contrary, the parameters are regarded as non-sensitive ones. During the design process, the widths of the side beam and the elastic hinge as well as the angle of the coupler link are recognized as dominant parameters for the total strain energy and the stress state. Hence, more attention should be paid to these parameters when dealing with the design of compliant bistable micromechanisms. The proposed micromechanisms are made of polysilicon which elastic modulus and fracture strength are also listed in Table 1. Substitute the parameters in Table 1 into Eqs. (7) and (8), and draw the figures of total strain energy U and the derivative of total strain energy with respect , as shown in Fig. 6. While the curve of dU passes through zero and the curve of U concaves to h, dU dh dh upward, these points are regarded as stable positions because the local minimums of the total strain energy appear. Fig. 6 displays the angle between two stable positions to be about 2.7. The curves with respect to the displacement of central mass, y, is shown in Fig. 7, which shows the displacement between two stable positions is 17.8 lm. Table 1 Parameters of the compliant bistable mechanism Parameter
Denotation
Quantity
t lb lh L wb wh h0 Ib Ih E rfracture
Thickness of polysilicon Length of side beam Length of elastic hinge Length of coupler link Width of side beam Width of elastic hinge Initial angle of coupler link Moment of inertia of side beam Moment of inertia of elastic hinge Elastic modulus of polysilicon Fracture strength of polysilicon
3.5 lm 53 lm 30 lm 385 lm 4 lm 2 lm 1.5 18.67 lm4 2.33 lm4 165 GPa 1.2 GPa
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Fig. 6. Total strain energy and its differential curves w.r.t. the angle of rotation.
Fig. 7. Total strain energy and its differential curves w.r.t. the displacement.
Again, by substituting the parameters listed in Table 1 into Eq. (1), the motions of the compliant bistable micromechanism in x- and y-directions, x and y, can be computed and shown in Fig. 8. The micromechanism starts at the first stable position, where y is 0. While y arrives at 10 lm, x
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Fig. 8. Displacements of elastic members.
reaches the maximum of 0.14 lm approximately. While y arrives at 17.8 lm, the bistable micromechanism reaches the second stable position. To prevent from yielding of the elastic members, the maximum stresses generated in the elastic members cannot exceed the fracture strength of the material. If the elastic members are made of a material that possesses larger fracture strength or less elastic modulus, it will generate a larger deformation, so that the compliant bistable micromechanism creates a larger deflection. However, due to the constraints of the fabrication process, this research selects polysilicon as the material of the compliant bistable mechanism. According to Table 1, the fracture strength of polysilicon doped with phosphorus is 1.2 ± 0.15 GPa [6]. The maximum stresses in the side beams and the elastic hinges are illustrated in Fig. 9. By observing the curves, none of the stresses exceeds the fracture strength of polysilicon, so this set of parameters is feasible for the safety of the compliant bistable micromechanism.
3. Fabrication and experiments 3.1. Fabrication In order to verify the design, the fabrication of the structures described above was carried out by MUMPs (Multi-User MEMS Processes) provided by Cronos Integrated Microsystems. MUMPs are able to deposit seven layers of four different kinds of materials on the substrate, as shown in Fig. 10(a). The fabrication processes include one layer of Si3N4 for electrical isolation, one layer of polysilicon (Poly 0) for the ground electrode, two layers of polysilicon (Poly 1 and
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Fig. 9. Stresses of elastic members.
Fig. 10. Layers of MUMPs [7]. (a) Before released; (b) after released; (c) the steps of etching.
Poly 2) for mechanical structures, two layers of PSG (Phosphorus Silicate Glass) for sacrificial material in order to suspend the mechanical structures, and one layer of gold for electrical con-
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tacts or optical mirror. The bistable micromechanisms are built in Poly 1 in combination with Poly 2 in order to enlarge the thickness. By employing the MUMPs process, the only works that need to be done by users are drawing masks and post-processing including etching the sacrificial layers to release the structure layers as shown in Fig. 10(b). This research employs a process of evaporation drying to release the micro devices. The steps of the post-process, as shown in Fig. 10(c), are described in sequence as follows: 1. 2. 3. 4.
Immerse the chip in HF (49%) for 5 min. Rinse the chip with de-ionized (DI) water. Rinse the chip with isopropyl alcohol (IPA) in order to replace DI water. Bake the chip by 200 C hot plate to evaporate the remaining IPA on the chip.
Since IPA has smaller surface tension than DI water, replacing DI water with IPA makes the microstructure less possible being got stuck to the substrate after baking. Therefore, we use IPA to rinse the chip before baking on the hot plate. Fig. 10 illustrates the side view of devices after the sacrificial layers are etched away. Having completed the above works, the experiments commence. 3.2. Experiments The experimental setup composes of several equipment and instruments as stated in the following: 3.2.1. Experimental rig 3.2.1.1. Vibration isolation platform. Vibrations from surroundings can cause enormous disturbances for the observation by using the microscope and consequently raise the difficulties of experiments. Therefore, an isolation platform is necessary to separate the external vibrations. 3.2.1.2. Micrography system. The micrography system includes a microscope, a CCD camera, and a computer. The microscope manufactured by Avimo Precision Instruments (Optem) can magnify the images of micro devices, which are sensed by the CCD camera. The images are captured and recorded by the computer so that one can observe the contours of the micro devices on the monitor and also record the motions of the micro devices. 3.2.1.3. Workbench. The workbench contains a stage on which the chip is laid and moved in x, y and h directions, an air pump that sticks the chip by continuous extraction of air, and several probes that operate the chip in three dimensions. 3.2.1.4. Scanning electronic microscope. The JSM-5610LV scanning electron microscope contributed the SEM photos. The JSM-5610LV SEM has the capacity of high magnification and tilted carriage to scan the side of specimen. The images can be transmitted to the computer and be saved as electronic photo files. In addition, the function of ‘‘Scale’’ supplies the on-line measurement of scale. It is suitable for the observation and measurement of MEMS devices.
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3.2.1.5. Power supply system. A set of electrical instruments for driving the actuators, such as scratch drive actuator (SDA), thermal actuator, etc., include a function generator and a power amplifier. The former generates an alternating current (AC) with adjustable frequency and amplitude, and the signal of AC can be selected as sine wave, triangular wave, or square wave. The later magnifies the amplitude of AC up to 20·, 40·, 100· or 200·; the output of the power supply is an AC of adjustable high voltages with various waveforms to meet the requirement of driving actuators. The highest voltage and current outputs are 250 V and 3 A, respectively. 3.2.2. Experimental processes 3.2.2.1. Switched by a probe. From the computed results, the compliant bistable micromechanism can be stable in two positions. The purposes of this experiment were to observe if the micromechanism was bistable and to measure the deflection of the bistable mechanism. Fig. 11(a) and (b) show the images of the compliant bistable micromechanism before and after being switched. After stirred by the probe, the compliant bistable micromechanism switched from the first stable position to the second one and held still. It indicates that the compliant bistable micromechanism functioned as expected. The displacement of central mass was measured by the attached function of SEM. From the SEM photograph shown in Fig. 12, the displacement between two stable positions is 17.6 lm. This experimental result approximately met the simulation introduced before where 17.8 lm of the displacement was expected.
Fig. 11. SEM images of the compliant bistable mechanism switched by a probe. (a) Before switched; (b) after switched.
Fig. 12. SEM image of displacement between two stable positions (enlarged view).
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Fig. 13. SEM images of the bistable micromechanism switched by SDA. (a) Before switched; (b) after switched.
3.2.2.2. Switched by SDA. As a sub-system, the compliant bistable micromechanism has to be manipulated by itself. An embedded actuator, SDA, is designed to integrate with the mechanism. The SDA pulls the central mass of the micromechanism. The SEM photograph, Fig. 13, illustrates
Fig. 14. SEM image of displacement between two stable positions (enlarged view).
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the central mass held at first and second stable states. The calculated displacement of central mass was 30 lm; the measured one met the theoretical calculation, as shown in Fig. 14.
4. Conclusions This research proposed a design method for fully compliant bistable micromechanisms. The topology and dimensions of the compliant micromechanism was first designed based on the requirement of the motion and force transmission. By considering the bistable function, the total strain energy of the compliant micromechanism was derived by summing up the strain energy stored in each elastic member. In addition, the behaviors of compliant bistable micromechanisms were examined. The bistable devices have two stationary positions where no external energy is required to hold the micromechanism in place. Power is needed only when switching the micromechanism into another stable position. As the micromechanism is properly designed, the bistable devices can even further reduce the power consumption. Moreover, the compliant micromechanisms composed of elastic members can simplify the fabrication process and prevent from friction, wear and clearance that may exist in hinge joints. Hence, the compliant micromechanism is suitable for the applications in MEMS. The purpose of the compliant bistable micromechanism is to combine the advantages of both bistable devices and compliant mechanisms. From the viewpoint of energy, there are two local minimums on the curve of the total strain energy of the compliant bistable micromechanism. This paper derived the equations governing the bistable behavior, strain energy and stress states. Besides, the parameters that dominate the variations of the strain energy are learned. It could be very helpful for the future design of the compliant bistable micromechanism. The micro devices designed in this research were fabricated by MUMPs process. By observing the experimental results, the compliant bistable micromechanisms performed bistable functions as expected. As a result, the equations and simulations were verified by the experiments.
Acknowledgement The authors would like to thank the financial support of National Science Council via the contract of NSC91-2212-E-194-036.
References [1] M.S. Baker, S.M. Lyon, L.L. Howell, A Linear displacement bistable micromechanism, Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2000, pp. 1–7. [2] B. Ha¨lg, On a nonvolatile memory cell based on micro-electro-mechanics, in: Proceedings IEEE Workshop on MEMS, 1990, pp. 172–176. [3] X. Sun, K.R. Farmer, M.N. Carr, A bistable microrelay based on two-segment multimorph cantilever actuators, in: Proceedings IEEE Workshop on MEMS, 1998, pp. 154–159. [4] A.G. Erdman, Modern Kinematics––Developments in the Last Forty Years, John Wiley and Sons, 1993.
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[5] L.L. Howell, A. Midha, A method for the design of compliant mechanisms with small-length flexural pivots, Transactions of the ASME 116 (1994) 280–290. [6] W.N. Sharpe, Jr. Bin Yuan, R.L. Edwards, R. Vaidyanathan, Measurements of YoungÕs modulus, PoissonÕs ratio, and tensile strength of polysilicon, in: 10th IEEE International Workshop on Microelectromechanical Systems, Nagaya, Japan, 26–30 January 1997, pp. 424–429. [7] MUMPs Design Handbook, Web Site: www.memsrus.com/cronos/CIMSmain2ie.html. Cheng-Kuo Sung has been a professor of Department of Power Mechanical Engineering, National Tsing Hua University after he received his Ph.D. from Michigan State University, USA in 1986. He was the former Chairman of Taiwan Chapter of the International Federal of Theory of Mechanisms and Machine (IFToMM). His major research interests include micro-mechanism design, machine dynamics, and precision machine design. Jinni Tsay is a Ph.D. candidate of Department of Power Mechanical Engineering, National Tsing Hua University, Taiwan, ROC.