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SMA actuated compliant bistable mechanisms
Mechatronics 14 (2004) 421–437
SMA actuated compliant bistable mechanisms Hiroya Ishii, Kwun-Lon Ting
*
Center for Manufacturing Research, Tennessee Technological University, P.O. Box 5077, Cookeville, TN 38505-6345, USA
Abstract This article introduces the concept, analysis, as well as the design of shape memory alloy (SMA) actuated compliant bistable mechanisms. Compliant bistable mechanisms and SMA linear actuators have noteworthy advantages. When they are combined, not only their advantageous characteristics are preserved, but also some disadvantages can be eliminated. Thus, comparing to a stand-alone SMA actuator, SMA actuated compliant mechanisms have the advantages of requiring no input to stay at the stable positions, high repeatability in positioning, lightweight and simple control. Complexity in analysis, design, and control of SMA actuators and other shortcomings of SMA actuators, such as short output stroke and small output force, are either eliminated or improved. 2003 Elsevier Ltd. All rights reserved. Keywords: Bistable mechanism; SMA actuator; Compliant mechanism
1. Introduction Compliant mechanisms have advantages such as reducing friction, wear, weight, and the number of parts [1–3]. Due to the compliant element, a compliant mechanism can be made to stay at a stable position without external input force. At a stable position, the mechanism is capable of returning to the stable position by itself after a disturbance. A compliant bistable mechanism possesses two stable positions in its range of motion. This type of mechanisms [4–6] is usually manually operated and used in many places such as shampoo caps and switches. This paper presents the design and analysis of shape memory alloy (SMA) actuated bistable mechanisms as well as the effects of the SMA actuator on the bistable mechanism.
*
Corresponding author. Tel.: +1-931-372-3362; fax: +1-931-372-6345. E-mail address: [email protected] (K.-L. Ting).
0957-4158/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0957-4158(03)00068-0
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In this paper, a titanium–nickel SMA linear actuator is utilized to drive a compliant bistable mechanism. There are mainly two types of SMA linear actuators. A straight wire type actuator can deliver a very large force but have small output displacement. The coil spring type is used in this article. A coil spring type actuator can produce a larger output displacement in comparison to a straight wire type. SMA linear actuators have advantageous features, such as excellent power/weight ratio, low noise in operation, lightweight, high corrosion resistance, and good electrical and thermal conductivities. One disadvantage of SMA actuators is the nonlinear relationship of the temperature dependent stress–strain curve. This nonlinearity leads to difficulties in analysis, modeling, design, as well as position control [7–10]. By combining an SMA actuator and a compliant bistable mechanism, the advantages of the mechanism and actuator can be enhanced and some of their drawbacks can be eliminated.
2. Compliant bistable mechanisms Fig. 1 shows a few types of compliant bistable mechanisms. A compliant bistable mechanism possesses two stable positions within the range of motion and requires no input power to stay at both stable positions. Such mechanisms can be analyzed with the energy method [5,6]. This article considers a compliant bistable mechanism that converts a linear input to a rotary output. Fig. 1(a) shows the mechanism in consideration at the first stable position. The slider is the input, and the rocker is the output of the mechanism. One end of the flexible link is attached to the slider and the other end is pinned to the rocker. At a stable position, the flexible link is not deformed. As an input force is applied to the slider and rocker rotates from A to B, the flexible link, which is regarded as a cantilever beam attached to the slider, will be deflected because the tip of the cantilever will be lifted. The following assumptions are made for analysis. • The output rocker is rigid. • The flexible link deforms in a linear range (small deformation). • Mass and friction are negligible. The strain energy of the system will be stored in and released from the flexible link. The slider end of the cantilever moves along the sliding path while the other end travels along a circular path. 2.1. Output position Fig. 2 shows a position of a compliant bistable mechanism QAF , in which QA is the rigid output rocker and FA is the flexible link. At a stable position, the nondeformed flexible link is at F0 A0 , where A0 is at (x0 ; y0 ). As the slider travels a displacement d from F0 , the output rocker rotates to QA. The output link position can be determined as follows.
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Fig. 1. Some compliant bistable mechanisms.
Let Aðx; yÞ be the position of the floating pin joint corresponding to the slider input displacement d. A is on the circle (Fig. 2), which can be described by x2 þ y 2 ¼ l20 :
ð1Þ
Assume that the flexible link is disconnected from the rocker and travels from the stable position F0 A0 through a distance d to F1 A1 without deformation. Then A1 is at (s; y0 ), where s ¼ x0 d (Fig. 2). Because small deformation is assumed, A1 A will be perpendicular to F1 A1 . The equation of line A1 A can be expressed as y y0 ¼ ð1= tan uÞðx sÞ;
ð2Þ
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δ
A (x, y)
A2(x2,y0)
A1(s,y )
A0(x0,y0)
ϕ
x
Q
ϕ
ϕ
θ
y
s=(x0- d)
F2 s
F0
F1 d x0
Fig. 2. Deflection analysis of the flexible link.
where u is the orientation angle of the flexible link measured from the slider path and 1= tan u is the slope of A1 A. Solving Eqs. (1) and (2) simultaneously yields
x¼
y¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 tan u y0 þ s þ tan u ðs þ tan u y0 Þ þ l20 ð1 þ tan2 uÞ ð1 þ tan2 uÞ tan2 u y0 þ tan u s þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðs þ tan u y0 Þ þ l20 ð1 þ tan2 uÞ ð1 þ tan2 uÞ
ð3aÞ
;
:
ð3bÞ
Therefore, the output rocker angular position is at h ¼ tan1 ðy=xÞ:
ð4Þ
2.2. Deflection d The amount of the deformation is d ¼ A1 A (Fig. 2), where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 d ¼ ðs xÞ þ ðy0 yÞ :
ð5Þ
The deflection d is caused by a force P exerted from the rocker. The relationship between d and P is [11] d¼
Pl3s ; 3EI
ð6Þ
where ls is the length of the flexible link, E is the Young’s modulus, and I is the moment inertia. It is noted that unless the deflection is quite large, the above linear analysis offers good accuracy.
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2.3. Strain energy The strain energy U of the mechanism at position QAF1 is [11] U¼
1 P 2 l3s : 6 EI
ð7Þ
From Eq. (6), it can be expressed as U¼
3d2 EI : 2l3s
ð8Þ
Substituting Eq. (5) into (8) yields ( ) 2 3EI s2 x2 þ ðy0 yÞ U¼ 3 xsþ ; ls 2 2
ð9Þ
which represents the strain energy of the bistable mechanism at the input displacement s. 2.4. Input force From Eq. (9), the required force to the slider to produce the deformation of the flexible element can be found by taking derivative with respect to s [5,6]. Thus, the input force to the slider is dU =ds, where dU 3EI dx dx dy dy ¼ 3 s x s þ x y0 þ y Fin ¼ : ð10Þ ds ls ds ds ds ds For a given mechanism at a position, Fin is the necessary input force to keep the compliant bistable mechanism in static equilibrium. 2.5. Stability The derivative of Eq. (10) represents the stability [5,6] of the mechanism. ( ) 2 2 d2 U 3EI dx d2 x dx d2 x d2 y dy d2 y ¼ 3 12 2sþ þ x 2 y0 2 þ þy : ds2 ls ds ds ds ds ds ds ds ð11Þ Fig. 1(a) gives the example of a compliant bistable mechanism, in which the offset (e) is zero and the length of the rocker is 25 mm. The flexible link has the following properties: Young’s modulus: E ¼ 75; 000 MPa (Super Elastic SMA); Width: w ¼ 4 mm; thickness: t ¼ 1:06 mm; length: ls ¼ 35:45 m. For the given mechanism, the deflection, strain energy, input force, and stability versus displacement curves are plotted in Fig. 3. At a stable position, such as
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3 2 1 0 -1
-10
-5
0 5 displacement (mm)
10
-10
-5
0 5 displacement (mm)
10
(a)
energy (N/mm)
10
5
0
(b) 2 1 force (N)
3 0
2
1
-1 -2 -10
(c)
-5
0
5
10
5
10
displacement (mm)
stability (N/mm)
2 1.5 1 0.5 0
-0.5
(d)
-10
-5 0 displacement (mm)
Fig. 3. (a) Deflection–displacement curve, (b) strain energy–displacement curve, (c) input force– displacement curve and (d) stability–displacement curve.
positions 1 and 3 in Fig. 3(c), no input force is needed to keep the system in equilibrium and the stability is positive. At an unstable position, as position 2 in Fig. 3(c), no input force is required to keep the system in equilibrium, the stability is negative, and a small input force there would make the mechanism collapse to a stable position. 2.6. Critical force The critical (input) force refers to the force required to snap a compliant bistable mechanism from one stable position to the other. The critical force is the required
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input force to keep the system in equilibrium at a zero stability position (Fig. 3(c) and (d)). The maximum input force must be greater than the critical force to switch from one stable position to the other.
3. SMA actuated compliant bistable mechanisms An SMA actuated compliant bistable mechanism (Fig. 4) can be considered as a system of two spring sets. One spring set is a compliant bistable mechanism, which can be regarded as a non-linear spring with a sinusoidal type spring force pattern between the stable positions (Fig. 5), and the other is an SMA actuator. The SMA linear actuator provides input force in both positive and negative directions to the bistable mechanism. The force–displacement relationship of an SMA differential type linear actuator is shown in Fig. 5, in which the solid line is the force– displacement relationship in the normal (low temperature) condition and the dotted lines represents the force–displacement relationship when one of the SMA springs is at high temperature and the other at normal temperature. When an SMA actuator and a compliant bistable mechanism are combined to form a system (Fig. 4) the resulting force–displacement relationship is shown in Fig. 5, in which S1 and S3 represent the two stable positions and S2 is the unstable position. S10 is the stable position when the left SMA spring is at the high temperature and the other SMA spring is at the normal temperature. Similarly, S30 is the stable
Fig. 4. SMA actuated compliant bistable mechanism.
F
F
F Rig ht
δ
Both
in
Hig h
in Lo
S1
δ
w
Le
A
B Required Force of Bistable Mechanism
S3' b S3
S1'
ft i
S2
a
δ
nH
igh
Stroke of SMA Stroke of System
C Producing Force of SMA Actuator
Combined Force-Displacement Relationship
Fig. 5. Combining two springs.
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position when the right SMA spring is at the high temperature and the other SMA spring is at the normal temperature. In the normal condition, the system is at a stable position S1 or S3 . When the left SMA is actuated (heated), the system will reach S10 position and as the temperature returns to normal, the system will reach the stable position at S1 . Likewise, when the right SMA is actuated, the system will reach S30 position and as the temperature returns to normal, the system will reach the stable position at S3 . Therefore, the distance between S1 and S3 represents the normal stroke of the SMA bistable mechanism. There will be a slight overshoot when the mechanism is switched from one stable position to the other. There are some remarkable features in SMA actuated compliant bistable mechanisms. When an SMA actuator is used alone, the position control is fully dependent on the control capability of the SMA actuator. In some applications, position may have to be controlled continuously and therefore the temperature dependent stress– strain relationship has to be taken into account in the control system. The temperature dependent stress–strain relationship makes the analysis and design difficult. In addition, the temperature of the SMA springs must maintain constant at the desired position. This demanding temperature and position control system will not be needed when an SMA linear actuator is combined with a compliant bistable mechanism. The resulting system has at least the following advantageous features. 1. The system can serve to provide a rotary output. 2. The stable positions are determined by the internal energy of the system. Therefore, the position or temperature precision requirement of the SMA actuator is minimal. In addition, because the position is controlled by internal energy, positioning repeatability will be high. 3. No input power is required to maintain the system at the destination positions. The force from the SMA actuator is used to shift the position from one stable position to the other only. Even during the process of position change, a very simple temperature control for a short period would be sufficient. 4. Another significant advantage is the stroke. An SMA actuator is operated in a limited range set by the SMA actuator. In the combined system, as shown in Fig. 5, although the stroke (ab) of the SMA actuator is small, the stroke (S1 S3 ) of the resulting bistable mechanism is much longer. A small SMA actuator may generate a long stroke. 5. One of the drawbacks of an SMA actuator is that when the SMA actuator is reaching the destination of the actuator, the actuator cannot produce large output force (Fig. 5). Assuming that the system is moving from S3 to S1 , as the system reaches near position a, the SMA actuator generates very small force to move the system. With the combined system, when the actuator is near position a or b, the bistable mechanism does not require any force from the actuator to change the position. After a bistable mechanism passes the unstable position, the mechanism itself can bring the system to the stable position by itself. 6. Other important features may also include lightweight as well as maintenance-free and simplicity of the design.
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3.1. Analysis When an SMA actuator is used to control a compliant bistable mechanism, the stable positions of the bistable mechanism will be affected. The analysis method is described step by step below. In the system, the compliant bistable mechanism described in the previous section is used and the actuator contains Ti–Ni alloy SMA springs, which have the following dimensions and properties. Wire diameter: d ¼ 0:5 (mm); mean diameter: D ¼ 4:9 (mm); Number of coils: Na ¼ 15; free length: Ls ¼ 7:5 (mm); Shear modulus: GH ¼ 23; 000 (MPa); GL ¼ 8000 (MPa); Maximum shear stress; smax ¼ 120 (MPa); Low temperature spring constant: KL ¼ 0:03542 (N/mm); High temperature spring constant: KH ¼ 0:10182 (N/mm). The bistable mechanism: The compliant bistable mechanism is analyzed in the previous section and the resultant input force–displacement curve is shown in Fig. 3(c). The SMA actuator: Let the two identical non-deformed SMA springs be placed at a distance h (h 6 s 6 h, say h ¼ 10 mm) in both sides of a point that serves as the origin of the force–displacement curve (Fig. 6). The spring has spring constant KL and KH at low and high temperature respectively. When they are connected, at a distance s on the right side of the origin, the resulting actuator generates the following spring forces. (a) When both springs are at low temperature, Fout ¼ F1L þ F2L ¼ KL ðh sÞ KL ðh þ sÞ ¼ 2KL s:
ð12aÞ
(b) When only SMA 1 is heated, Fout ¼ F1H þ F2L ¼ KH ðh sÞ KL ðh þ sÞ:
ð12bÞ
(c) When only SMA 2 is heated, Fout ¼ F1L þ F2H ¼ KL ðh sÞ KH ðh þ sÞ:
ð12cÞ
OUTPUT PORT
SMA 2
SMA1
10 mm
10 mm
Fig. 6. SMA differential type linear actuator.
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Force (N)
C 2 1 A
S2
S1
S3
0 S1'
B
-1 S3' -2 -10
-5
0 5 Displacement (mm)
10
D
Fig. 7. Combined force–displacement curves.
Line segments AB, CB and AD in Fig. 7 show the force–deflection relationships in above equations. The SMA actuated compliant bistable mechanism: Since the output of the actuator is the input slider of the bistable mechanism, when the SMA actuator and the bistable mechanism are combined, one can have Fin ¼ Fout ;
ð13Þ
where Fin and Fout are defined in Eqs. (10) and (12) respectively. The equilibrium positions between the compliant bistable mechanism and the SMA actuator can be determined by Eqs. (10), (12) and (13). At normal (low temperature) condition, S1 and S2 are the two stable positions and S3 is the unstable position. The system will stay at S1 or S3 normally and the distance between S1 and S3 is the normal stroke of the system. When SMA 1 (or SMA 2) is activated or heated, the stable position is at S30 (or S10 ). When the temperature of an SMA spring drops to normal, the stable position will shift to S3 or S1 . It is noted that to switch between S1 and S3 , one spring must remain at the high temperature until passing the unstable position S2 . After the system passes the unstable position, the system will release its strain energy and reach the stable position by itself. 3.2. Effects of offset The effect of the slider offset parameter e (Fig. 1) is illustrated with an example. Fig. 8 shows the relationship of the input force with respect to the displacement s when e ¼ 15. From these figures, the followings are observed. 1. When an offset is introduced to the system, the unstable position will shift away from the center. No offset should be used if similar properties during the forward and return strokes are desired. 2. When the offset increases, the maximum strain energy stored in the system will increase. As shown in Fig. 9, the defection of the flexible element increases and so does the strain energy when the offset is introduced to the system. As the result,
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2
Force (N)
1 0 -1 -2 -10
-5
0 Displacement (mm)
5
10
Fig. 8. Force–displacement curve with e ¼ 15.
the maximum critical force that is required to switch the mechanism between stable positions will increase. In a compliant bistable mechanism, the existence of an external torque to the rocker has the effect of shifting the force curve vertically [12]. However, this force curve has an upper and lower bounds (Fig. 8) when the bistable mechanism is driven by an SMA actuator. This implies that the allowable external torque that the mechanism may carry will decrease when offset increases. 3. As shown in Fig. 8, using offset will increase the slope of force curve at one stable position and decrease the slope at the other stable position. This implies that the stability of the system increase at one stable position and decrease at the other stable position. If the stabilities at both stable positions are to be increased, increase the stiffness of the flexible link will be more effective. This will also lead to a larger critical force, which is restricted by the SMA actuator as discussed above. 4. A compliant bistable mechanism is a non-linear spring system. As soon as the mechanism passes the unstable position, the system starts releasing the energy. A large critical force means that the mechanism is capable of generating a large force. The use of offset may offer a way to regulate the ratio of the output force to the input force of a bistable mechanism. For instance, the ratio of input/output
Trace of Crank
δ m2
δ m1
a Non-Deformed Flexible Link with e = 15
Non-Deformed Flexible Link with e = 0
Fig. 9. Effect of offset on deflection.
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may be controlled by giving an offset without replacing any part of the mechanism. 3.3. Design of SMA actuated bistable mechanisms The temperature–stress relationship of the SMA springs can be ignored completely. The design procedure of an SMA bistable mechanism (Fig. 4) is explained and demonstrated with the following example. An SMA actuated compliant bistable mechanism is to be designed. The output angular stroke between the two stable positions is a ¼ 50, and no offset is desired. The flexible link has the same material property of the flexible link used in Section 3.1, as well as the SMA wire used for the springs. An iterative design process is outlined step by step below. (1) Determine the stroke of the SMA actuator and the output link length. The relationship between the output link length and the stroke of the SMA actuator can be expressed as (Fig. 10) st ¼ 2 l0 sinða=2Þ: One may assume the value of st or l0 and determine the other. Let l0 ¼ 25 mm. Then st ¼ 2 l0 sinða=2Þ ¼ 2ð25Þ sin 25 ¼ 21:131 mm: (2) Select an output displacements sta and aa of the bistable mechanism. It is noted that the output stroke of a bistable mechanism is usually 10–20% longer than that of an SMA actuated compliant bistable mechanism. Generally, if the difference is small, the load capacity will be lower. Let 15% be chosen. The output displacement of the bistable mechanism will be sta ¼ 1:15 st ¼ 1:15ð21:131Þ ¼ 24:301 mm: sta 24:301 aa ¼ 2 sin1 ¼ 58:1586 degree: ¼ 2 sin1 2ð25Þ 2l0 st a st s
Desired Stable Position of the Combined System
TRACE OF CRANK
α
αa lo
Stable Position of the Bistable Mechanism
y x
Fig. 10. An output crank.
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(3) Determine the approximate maximum deformation dm (Fig. 9). aa 58:159 ¼ 3:151346 mm: dm ¼ l0 l0 cos ¼ 25 25 cos 2 2 (4) Select the maximum strain emax and the thickness–length ratio t=ls of the flexible link. A suitable thickness–length ration t=ls can be chosen. Let t=ls ¼ 0:03. Super elastic SMA is capable of recovering about 8% of strain. For a longer life span, let emax ¼ 0:4%. Now, use the relationship of the strain and deflection [11], the length and thickness of flexible link can be determined. 3 dm t 3 3:151 emax ¼ ¼ ð0:03Þ ¼ 0:004: 2 ls ls 2 ls Thus, ls ¼
3ð0:03Þð3:151Þ ¼ 35:45 mm: 2ð0:004Þ
t ¼ 0:003 ls ¼ 0:003ð35:45Þ ¼ 1:063 mm: (5) Select the width of the flexible link. The width of the flexible link can be chosen arbitrary. However, the width affects on the capacity of the strain energy. Therefore, a large width will lead to a high maximum strain energy and require a large force from the actuator. On the other hand, a narrow width will lead to low stability of the system. Let w ¼ 4 mm be selected. (6) Determine the force curve of the bistable mechanism. Now, the key dimensions of the bistable mechanism are determined. The force–deflection curve of the mechanism can be determined by using Eq. (10). The result is shown in Fig. 11. (7) Determine the normal force–displacement relation Fout of the SMA actuator. Fout is the output spring force of the SMA actuator when the both SMA springs are at
2
Force (N)
PL
1 KL
B
FA
10.5655 0 -10.5655 A
KL
-1
-2 - 10
-5
0 Displacement (mm)
5
10
Fig. 11. Force–displacement curve of the designed bistable mechanism.
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normal temperature. Because no offset is desired, only one stable position needs to be considered. Let FA (Fig. 11) be the force at the desired stable position of the SMA actuated compliant bistable mechanism. Since the desired total stroke st is 21.131 mm, the displacement at the stable position should be s ¼ 10:5655 mm, which is equal to half of the total stroke. Thus, from the equation of the force curve (10), with s ¼ 10:5655 mm, FA is determined as 0.75644 N (Fig. 11). Therefore, the stiffness Fout is Fout ¼
0:75644 ¼ 0:071595 N=mm: 10:5655
(8) Determine PL . PL is the output force of the actuator when one of the SMA springs is not deformed in normal temperature. As shown in Fig. 11, the left and right SMA springs of the SMA actuator have no deformation at A and B respectively, and both positions A and B are the stable positions of the compliant bistable mechanism. Therefore, sta PL ¼ Fout A ¼ Fout ¼ 0:071595ð12:15057Þ ¼ 0:870 N: 2 (9) Determine the desired SMA spring properties at normal temperature. Referring to Fig. 11, the ideal spring constant of the SMA spring at low temperature is KL ¼
PL 0:870 ¼ 0:03580 N=mm: ¼ sta 24:301
Now, the desired properties of the SMA spring, which has GL ¼ 8000 N/mm2 , at low temperature are found and listed below. PL ¼ 0:870 N;
dL ¼ sa ¼ 24:301 mm;
KL ¼ 0:03580 N=mm:
(10) Determine the desired SMA spring properties at high temperature. The relationship between the properties of the SMA spring at high and low temperature is governed by following equation [12,13]. PH PL ¼ : dH G H dL G L
ð14Þ
Therefore, the spring constant of the SMA spring at high temperature is KH ¼
PH GH PL ð23; 000Þð0:86992Þ ¼ 0:10293 N=mm: ¼ ¼ ð24:301Þð8000Þ dH dL GL
dH ¼ 19:4781 mm is used because the bistable mechanism reaches the critical force at this position (Fig. 11) and the SMA spring at high temperature generate a force PH , where PH ¼ dH KH ¼ ð19:4781Þð0:10293Þ ¼ 2:005 N: The critical force of the bistable mechanism must be less than the force PH of the SMA spring (at high temperature) to have bistable positions with certain load capacity. To make sure the SMA actuator can generate enough force to switch the bistable mechanism, it is recommended that the value of PH is about 20% higher than
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the critical force of the bistable mechanism. If the resulting PH value is the same as the critical force at this position, the output force curve will intersect the required input force curve of the bistable mechanism. Therefore, depending on the resulting value of PH , one may return to step 4, or 5 to adjust the bistable mechanism until a reasonable PH to critical force ratio is obtained. A higher ratio will lead to a mechanism with a higher external load capacity. In this example, the critical force of the mechanism is 1.26218 N and PH ¼ 2:005 N. Therefore, the actuator can produce 160% of the critical force, which indicates that the force generated by the actuator will be enough to overcome the critical force to switch between the stable positions. The obtained properties of the SMA spring, which has the property of GH ¼ 23; 000 N/mm2 at high temperature, are summarized below: PH ¼ 2:005 N;
dH ¼ 19:4781 mm;
KH ¼ 0:10293 N=mm:
(11) Design the spring. Because the SMA spring actuator is used to drive a compliant bistable mechanism, conventional SMA spring design method [12–14] is sufficient for this design purpose. In order to use the design method, the design parameters, the spring index C and the maximum shear strain cmax , are chosen. In this example, the spring index C ¼ 8 is selected. Because dmax , which is the deformation of an SMA spring at a stable position of bistable mechanism, is very close to dL , it is assumed cmax cL . For a long life span, it is chosen that cmax ¼ 1:2%. Let the KLR and KHR be the resultant spring stiffness in low and high temperatures respectively. By following the design process in [12–14], the dimensions and the SMA spring constants are obtained below. d ¼ 0:5 mm;
D ¼ 4:6 mm;
KLR ¼ 0:04281 N=mm;
Na ¼ 15 turns;
KHR ¼ 0:12307 N=mm;
Ls ¼ 7:5 mm:
One may notice that the resultant spring stiffness, KLR and KHR , are about 20% greater than the ideal spring stiffness, KL and KH . In order to minimal this design error, an iterative process is used to adjust the resultant SMA spring constants [12] by changing a dimension of the spring. In this example, the mean diameter D is increased by 0.1 mm in each iteration process. It is found that when D ¼ 4:9 mm, the spring constants become KLR ¼ 0:03542 N/mm and KHR ¼ 0:10182 N/mm, which are within 1.1% of the ideal spring constant. (12) Evaluate the final design. The dimensions of the SMA actuated bistable mechanism are determined and summarized below. The bistable mechanism: ls ¼ 35:45 mm, w ¼ 4 mm, t ¼ 1:06 mm, l0 ¼ 25 mm and stroke ¼ 24.301 mm. The SMA actuator: d ¼ 0:5 mm, D ¼ 4:9 mm, Na ¼ 15 turns, KLR ¼ 0:03542 N/ mm, KHR ¼ 0:10182 N/mm and Ls ¼ 7:5 mm. The scaled figure is shown in Fig. 4. The force–displacement curves of the resulting SMA actuated compliant bistable mechanism are shown in Fig. 7, which has an output angular stroke of 50.10.
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Fig. 12. (a) A two-position control system, and (b) a four-position control system.
4. Conclusion This article presented the first comprehensive work on SMA actuated compliant bistable mechanisms. The concept, analysis, as well as design procedure of SMA actuated compliant bistable mechanisms is discussed. Although the discussion is focused on one type of bistable mechanisms, the same method is applicable to most of the bistable mechanisms. Because of the combination with a bistable mechanism, the conventional analysis and design method can be used, which allows one to ignore the temperature dependent stress–strain relationship of the SMA and greatly simplify the design and analysis. The combination preserves the advantages of SMA actuators and compliant bistable mechanisms while reduces or eliminates most of their shortcomings. An SMA actuated compliant bistable mechanism is a non-linear spring system. The system stores and releases energy as it moves from a stable position to the unstable position and continues in the same direction toward the other stable position. The phenomenon occurs in both directions of motion. An SMA actuated compliant bistable mechanisms can be also used as a driver. When the system is used with a single degree of freedom mechanism, such as four-bar mechanism, the entire system can be used as two-position control system (Fig. 12a). Since each actuator would generate two stable positions, if more are used in a multiple degree of freedom mechanism, more stable positions can be generated (Fig. 12b). References [1] Howell LL, Midha A. The development of force–deflection relationships for compliant mechanisms. ASME Machine Elements Machine Dynamics DE 1994;71:501–8.
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[2] Salamon BA. Mechanical Advantage Aspects in Compliant Mechanisms Design, M.S. thesis, Purdue University, 1989. [3] Sevak N, McLarnan C. Optimal Synthesis of Flexible Link Mechanisms with Large Static Deflectors, ASME Paper no. 74-DET-83, 1974. [4] Jensen BD, Howell LL. Identification of compliant pseudo-rigid-body mechanism configurations resulting in bistable behavior. In: Proceedings of DETC’00/MECH-14147, ASME 2000 Design Engineering Technical Conferences and Computers and Information in Engineering Conferences, Baltimore, Maryland, 10–13 September 2000. p. 1–10. [5] Jensen BD, Howell LL, Salmon LG. Introduction of two-link, in-plane, bistable compliant MEMS. In: Proceedings of DETC’98/MECH-5837, 1998 ASME Design Engineering Technical Conferences, Atlanta, GA, 13–16 September 1998. [6] Opdahl PG, Jensen BD, Howell LL. An investigation into compliant bistable mechanisms. In: Proceedings of DETC’98/MECH-5914, 1998 ASME Design Engineering Technical Conferences, Atlanta, GA, 13–16 September 1998. p.1–10. [7] Song G, Kelly B, Agrawal BN, Lam PC, Srivatsan TS. Application of shape memory alloy wire actuator for precision position control of a composite beam. ASM Int, J Mater Eng Perform 2000;9(3):330–3. [8] Auricchio F, Sacco E. A SMA thermomechanical finite-element beam model. In SMST-97: Proceedings of the Second International Conference on Shape Memory and Superelastic Technologies, Asilomar Conference Center, Pacific Grove, California, USA, 1997. p. 485–90. [9] Madill DR, Wang D. Modeling and L2-stability of a shape memory alloy position control system. IEEE Trans Control Systems Technol 1998;6(4):473–81. [10] Prahlad H, Chopra I. Experimental characterization of Ti–Ni shape memory alloy wires under uniaxial loading conditions. J Intell Mater Systems Struct 2000;11:272–82. [11] Benham PP, Crawford RJ. Mechanics of engineering materials. Longman Scientific & Technical; 1987. [12] Ishii H. SMA Actuated Compliant Bistable Mechanisms, M.S. thesis, Tennessee Technological University, 2002. [13] Funakubo H. Shape memory alloys. New York: Gordon and Breach Science Publishers; 1987. [14] Ohkata I, Suzuki Y. The design of shape memory alloy actuators and their applications. In: Otsuka K, Wayman CM, editors. Shape memory materials. Cambridge, UK: Cambridge University Press; 1998. p. 240–66.
SMA actuated compliant bistable mechanisms Hiroya Ishii, Kwun-Lon Ting
*
Center for Manufacturing Research, Tennessee Technological University, P.O. Box 5077, Cookeville, TN 38505-6345, USA
Abstract This article introduces the concept, analysis, as well as the design of shape memory alloy (SMA) actuated compliant bistable mechanisms. Compliant bistable mechanisms and SMA linear actuators have noteworthy advantages. When they are combined, not only their advantageous characteristics are preserved, but also some disadvantages can be eliminated. Thus, comparing to a stand-alone SMA actuator, SMA actuated compliant mechanisms have the advantages of requiring no input to stay at the stable positions, high repeatability in positioning, lightweight and simple control. Complexity in analysis, design, and control of SMA actuators and other shortcomings of SMA actuators, such as short output stroke and small output force, are either eliminated or improved. 2003 Elsevier Ltd. All rights reserved. Keywords: Bistable mechanism; SMA actuator; Compliant mechanism
1. Introduction Compliant mechanisms have advantages such as reducing friction, wear, weight, and the number of parts [1–3]. Due to the compliant element, a compliant mechanism can be made to stay at a stable position without external input force. At a stable position, the mechanism is capable of returning to the stable position by itself after a disturbance. A compliant bistable mechanism possesses two stable positions in its range of motion. This type of mechanisms [4–6] is usually manually operated and used in many places such as shampoo caps and switches. This paper presents the design and analysis of shape memory alloy (SMA) actuated bistable mechanisms as well as the effects of the SMA actuator on the bistable mechanism.
*
Corresponding author. Tel.: +1-931-372-3362; fax: +1-931-372-6345. E-mail address: [email protected] (K.-L. Ting).
0957-4158/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0957-4158(03)00068-0
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In this paper, a titanium–nickel SMA linear actuator is utilized to drive a compliant bistable mechanism. There are mainly two types of SMA linear actuators. A straight wire type actuator can deliver a very large force but have small output displacement. The coil spring type is used in this article. A coil spring type actuator can produce a larger output displacement in comparison to a straight wire type. SMA linear actuators have advantageous features, such as excellent power/weight ratio, low noise in operation, lightweight, high corrosion resistance, and good electrical and thermal conductivities. One disadvantage of SMA actuators is the nonlinear relationship of the temperature dependent stress–strain curve. This nonlinearity leads to difficulties in analysis, modeling, design, as well as position control [7–10]. By combining an SMA actuator and a compliant bistable mechanism, the advantages of the mechanism and actuator can be enhanced and some of their drawbacks can be eliminated.
2. Compliant bistable mechanisms Fig. 1 shows a few types of compliant bistable mechanisms. A compliant bistable mechanism possesses two stable positions within the range of motion and requires no input power to stay at both stable positions. Such mechanisms can be analyzed with the energy method [5,6]. This article considers a compliant bistable mechanism that converts a linear input to a rotary output. Fig. 1(a) shows the mechanism in consideration at the first stable position. The slider is the input, and the rocker is the output of the mechanism. One end of the flexible link is attached to the slider and the other end is pinned to the rocker. At a stable position, the flexible link is not deformed. As an input force is applied to the slider and rocker rotates from A to B, the flexible link, which is regarded as a cantilever beam attached to the slider, will be deflected because the tip of the cantilever will be lifted. The following assumptions are made for analysis. • The output rocker is rigid. • The flexible link deforms in a linear range (small deformation). • Mass and friction are negligible. The strain energy of the system will be stored in and released from the flexible link. The slider end of the cantilever moves along the sliding path while the other end travels along a circular path. 2.1. Output position Fig. 2 shows a position of a compliant bistable mechanism QAF , in which QA is the rigid output rocker and FA is the flexible link. At a stable position, the nondeformed flexible link is at F0 A0 , where A0 is at (x0 ; y0 ). As the slider travels a displacement d from F0 , the output rocker rotates to QA. The output link position can be determined as follows.
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Fig. 1. Some compliant bistable mechanisms.
Let Aðx; yÞ be the position of the floating pin joint corresponding to the slider input displacement d. A is on the circle (Fig. 2), which can be described by x2 þ y 2 ¼ l20 :
ð1Þ
Assume that the flexible link is disconnected from the rocker and travels from the stable position F0 A0 through a distance d to F1 A1 without deformation. Then A1 is at (s; y0 ), where s ¼ x0 d (Fig. 2). Because small deformation is assumed, A1 A will be perpendicular to F1 A1 . The equation of line A1 A can be expressed as y y0 ¼ ð1= tan uÞðx sÞ;
ð2Þ
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δ
A (x, y)
A2(x2,y0)
A1(s,y )
A0(x0,y0)
ϕ
x
Q
ϕ
ϕ
θ
y
s=(x0- d)
F2 s
F0
F1 d x0
Fig. 2. Deflection analysis of the flexible link.
where u is the orientation angle of the flexible link measured from the slider path and 1= tan u is the slope of A1 A. Solving Eqs. (1) and (2) simultaneously yields
x¼
y¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 tan u y0 þ s þ tan u ðs þ tan u y0 Þ þ l20 ð1 þ tan2 uÞ ð1 þ tan2 uÞ tan2 u y0 þ tan u s þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðs þ tan u y0 Þ þ l20 ð1 þ tan2 uÞ ð1 þ tan2 uÞ
ð3aÞ
;
:
ð3bÞ
Therefore, the output rocker angular position is at h ¼ tan1 ðy=xÞ:
ð4Þ
2.2. Deflection d The amount of the deformation is d ¼ A1 A (Fig. 2), where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 d ¼ ðs xÞ þ ðy0 yÞ :
ð5Þ
The deflection d is caused by a force P exerted from the rocker. The relationship between d and P is [11] d¼
Pl3s ; 3EI
ð6Þ
where ls is the length of the flexible link, E is the Young’s modulus, and I is the moment inertia. It is noted that unless the deflection is quite large, the above linear analysis offers good accuracy.
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2.3. Strain energy The strain energy U of the mechanism at position QAF1 is [11] U¼
1 P 2 l3s : 6 EI
ð7Þ
From Eq. (6), it can be expressed as U¼
3d2 EI : 2l3s
ð8Þ
Substituting Eq. (5) into (8) yields ( ) 2 3EI s2 x2 þ ðy0 yÞ U¼ 3 xsþ ; ls 2 2
ð9Þ
which represents the strain energy of the bistable mechanism at the input displacement s. 2.4. Input force From Eq. (9), the required force to the slider to produce the deformation of the flexible element can be found by taking derivative with respect to s [5,6]. Thus, the input force to the slider is dU =ds, where dU 3EI dx dx dy dy ¼ 3 s x s þ x y0 þ y Fin ¼ : ð10Þ ds ls ds ds ds ds For a given mechanism at a position, Fin is the necessary input force to keep the compliant bistable mechanism in static equilibrium. 2.5. Stability The derivative of Eq. (10) represents the stability [5,6] of the mechanism. ( ) 2 2 d2 U 3EI dx d2 x dx d2 x d2 y dy d2 y ¼ 3 12 2sþ þ x 2 y0 2 þ þy : ds2 ls ds ds ds ds ds ds ds ð11Þ Fig. 1(a) gives the example of a compliant bistable mechanism, in which the offset (e) is zero and the length of the rocker is 25 mm. The flexible link has the following properties: Young’s modulus: E ¼ 75; 000 MPa (Super Elastic SMA); Width: w ¼ 4 mm; thickness: t ¼ 1:06 mm; length: ls ¼ 35:45 m. For the given mechanism, the deflection, strain energy, input force, and stability versus displacement curves are plotted in Fig. 3. At a stable position, such as
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H. Ishii, K.-L. Ting / Mechatronics 14 (2004) 421–437 deflection (mm)
3 2 1 0 -1
-10
-5
0 5 displacement (mm)
10
-10
-5
0 5 displacement (mm)
10
(a)
energy (N/mm)
10
5
0
(b) 2 1 force (N)
3 0
2
1
-1 -2 -10
(c)
-5
0
5
10
5
10
displacement (mm)
stability (N/mm)
2 1.5 1 0.5 0
-0.5
(d)
-10
-5 0 displacement (mm)
Fig. 3. (a) Deflection–displacement curve, (b) strain energy–displacement curve, (c) input force– displacement curve and (d) stability–displacement curve.
positions 1 and 3 in Fig. 3(c), no input force is needed to keep the system in equilibrium and the stability is positive. At an unstable position, as position 2 in Fig. 3(c), no input force is required to keep the system in equilibrium, the stability is negative, and a small input force there would make the mechanism collapse to a stable position. 2.6. Critical force The critical (input) force refers to the force required to snap a compliant bistable mechanism from one stable position to the other. The critical force is the required
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input force to keep the system in equilibrium at a zero stability position (Fig. 3(c) and (d)). The maximum input force must be greater than the critical force to switch from one stable position to the other.
3. SMA actuated compliant bistable mechanisms An SMA actuated compliant bistable mechanism (Fig. 4) can be considered as a system of two spring sets. One spring set is a compliant bistable mechanism, which can be regarded as a non-linear spring with a sinusoidal type spring force pattern between the stable positions (Fig. 5), and the other is an SMA actuator. The SMA linear actuator provides input force in both positive and negative directions to the bistable mechanism. The force–displacement relationship of an SMA differential type linear actuator is shown in Fig. 5, in which the solid line is the force– displacement relationship in the normal (low temperature) condition and the dotted lines represents the force–displacement relationship when one of the SMA springs is at high temperature and the other at normal temperature. When an SMA actuator and a compliant bistable mechanism are combined to form a system (Fig. 4) the resulting force–displacement relationship is shown in Fig. 5, in which S1 and S3 represent the two stable positions and S2 is the unstable position. S10 is the stable position when the left SMA spring is at the high temperature and the other SMA spring is at the normal temperature. Similarly, S30 is the stable
Fig. 4. SMA actuated compliant bistable mechanism.
F
F
F Rig ht
δ
Both
in
Hig h
in Lo
S1
δ
w
Le
A
B Required Force of Bistable Mechanism
S3' b S3
S1'
ft i
S2
a
δ
nH
igh
Stroke of SMA Stroke of System
C Producing Force of SMA Actuator
Combined Force-Displacement Relationship
Fig. 5. Combining two springs.
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position when the right SMA spring is at the high temperature and the other SMA spring is at the normal temperature. In the normal condition, the system is at a stable position S1 or S3 . When the left SMA is actuated (heated), the system will reach S10 position and as the temperature returns to normal, the system will reach the stable position at S1 . Likewise, when the right SMA is actuated, the system will reach S30 position and as the temperature returns to normal, the system will reach the stable position at S3 . Therefore, the distance between S1 and S3 represents the normal stroke of the SMA bistable mechanism. There will be a slight overshoot when the mechanism is switched from one stable position to the other. There are some remarkable features in SMA actuated compliant bistable mechanisms. When an SMA actuator is used alone, the position control is fully dependent on the control capability of the SMA actuator. In some applications, position may have to be controlled continuously and therefore the temperature dependent stress– strain relationship has to be taken into account in the control system. The temperature dependent stress–strain relationship makes the analysis and design difficult. In addition, the temperature of the SMA springs must maintain constant at the desired position. This demanding temperature and position control system will not be needed when an SMA linear actuator is combined with a compliant bistable mechanism. The resulting system has at least the following advantageous features. 1. The system can serve to provide a rotary output. 2. The stable positions are determined by the internal energy of the system. Therefore, the position or temperature precision requirement of the SMA actuator is minimal. In addition, because the position is controlled by internal energy, positioning repeatability will be high. 3. No input power is required to maintain the system at the destination positions. The force from the SMA actuator is used to shift the position from one stable position to the other only. Even during the process of position change, a very simple temperature control for a short period would be sufficient. 4. Another significant advantage is the stroke. An SMA actuator is operated in a limited range set by the SMA actuator. In the combined system, as shown in Fig. 5, although the stroke (ab) of the SMA actuator is small, the stroke (S1 S3 ) of the resulting bistable mechanism is much longer. A small SMA actuator may generate a long stroke. 5. One of the drawbacks of an SMA actuator is that when the SMA actuator is reaching the destination of the actuator, the actuator cannot produce large output force (Fig. 5). Assuming that the system is moving from S3 to S1 , as the system reaches near position a, the SMA actuator generates very small force to move the system. With the combined system, when the actuator is near position a or b, the bistable mechanism does not require any force from the actuator to change the position. After a bistable mechanism passes the unstable position, the mechanism itself can bring the system to the stable position by itself. 6. Other important features may also include lightweight as well as maintenance-free and simplicity of the design.
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3.1. Analysis When an SMA actuator is used to control a compliant bistable mechanism, the stable positions of the bistable mechanism will be affected. The analysis method is described step by step below. In the system, the compliant bistable mechanism described in the previous section is used and the actuator contains Ti–Ni alloy SMA springs, which have the following dimensions and properties. Wire diameter: d ¼ 0:5 (mm); mean diameter: D ¼ 4:9 (mm); Number of coils: Na ¼ 15; free length: Ls ¼ 7:5 (mm); Shear modulus: GH ¼ 23; 000 (MPa); GL ¼ 8000 (MPa); Maximum shear stress; smax ¼ 120 (MPa); Low temperature spring constant: KL ¼ 0:03542 (N/mm); High temperature spring constant: KH ¼ 0:10182 (N/mm). The bistable mechanism: The compliant bistable mechanism is analyzed in the previous section and the resultant input force–displacement curve is shown in Fig. 3(c). The SMA actuator: Let the two identical non-deformed SMA springs be placed at a distance h (h 6 s 6 h, say h ¼ 10 mm) in both sides of a point that serves as the origin of the force–displacement curve (Fig. 6). The spring has spring constant KL and KH at low and high temperature respectively. When they are connected, at a distance s on the right side of the origin, the resulting actuator generates the following spring forces. (a) When both springs are at low temperature, Fout ¼ F1L þ F2L ¼ KL ðh sÞ KL ðh þ sÞ ¼ 2KL s:
ð12aÞ
(b) When only SMA 1 is heated, Fout ¼ F1H þ F2L ¼ KH ðh sÞ KL ðh þ sÞ:
ð12bÞ
(c) When only SMA 2 is heated, Fout ¼ F1L þ F2H ¼ KL ðh sÞ KH ðh þ sÞ:
ð12cÞ
OUTPUT PORT
SMA 2
SMA1
10 mm
10 mm
Fig. 6. SMA differential type linear actuator.
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Force (N)
C 2 1 A
S2
S1
S3
0 S1'
B
-1 S3' -2 -10
-5
0 5 Displacement (mm)
10
D
Fig. 7. Combined force–displacement curves.
Line segments AB, CB and AD in Fig. 7 show the force–deflection relationships in above equations. The SMA actuated compliant bistable mechanism: Since the output of the actuator is the input slider of the bistable mechanism, when the SMA actuator and the bistable mechanism are combined, one can have Fin ¼ Fout ;
ð13Þ
where Fin and Fout are defined in Eqs. (10) and (12) respectively. The equilibrium positions between the compliant bistable mechanism and the SMA actuator can be determined by Eqs. (10), (12) and (13). At normal (low temperature) condition, S1 and S2 are the two stable positions and S3 is the unstable position. The system will stay at S1 or S3 normally and the distance between S1 and S3 is the normal stroke of the system. When SMA 1 (or SMA 2) is activated or heated, the stable position is at S30 (or S10 ). When the temperature of an SMA spring drops to normal, the stable position will shift to S3 or S1 . It is noted that to switch between S1 and S3 , one spring must remain at the high temperature until passing the unstable position S2 . After the system passes the unstable position, the system will release its strain energy and reach the stable position by itself. 3.2. Effects of offset The effect of the slider offset parameter e (Fig. 1) is illustrated with an example. Fig. 8 shows the relationship of the input force with respect to the displacement s when e ¼ 15. From these figures, the followings are observed. 1. When an offset is introduced to the system, the unstable position will shift away from the center. No offset should be used if similar properties during the forward and return strokes are desired. 2. When the offset increases, the maximum strain energy stored in the system will increase. As shown in Fig. 9, the defection of the flexible element increases and so does the strain energy when the offset is introduced to the system. As the result,
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431
2
Force (N)
1 0 -1 -2 -10
-5
0 Displacement (mm)
5
10
Fig. 8. Force–displacement curve with e ¼ 15.
the maximum critical force that is required to switch the mechanism between stable positions will increase. In a compliant bistable mechanism, the existence of an external torque to the rocker has the effect of shifting the force curve vertically [12]. However, this force curve has an upper and lower bounds (Fig. 8) when the bistable mechanism is driven by an SMA actuator. This implies that the allowable external torque that the mechanism may carry will decrease when offset increases. 3. As shown in Fig. 8, using offset will increase the slope of force curve at one stable position and decrease the slope at the other stable position. This implies that the stability of the system increase at one stable position and decrease at the other stable position. If the stabilities at both stable positions are to be increased, increase the stiffness of the flexible link will be more effective. This will also lead to a larger critical force, which is restricted by the SMA actuator as discussed above. 4. A compliant bistable mechanism is a non-linear spring system. As soon as the mechanism passes the unstable position, the system starts releasing the energy. A large critical force means that the mechanism is capable of generating a large force. The use of offset may offer a way to regulate the ratio of the output force to the input force of a bistable mechanism. For instance, the ratio of input/output
Trace of Crank
δ m2
δ m1
a Non-Deformed Flexible Link with e = 15
Non-Deformed Flexible Link with e = 0
Fig. 9. Effect of offset on deflection.
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may be controlled by giving an offset without replacing any part of the mechanism. 3.3. Design of SMA actuated bistable mechanisms The temperature–stress relationship of the SMA springs can be ignored completely. The design procedure of an SMA bistable mechanism (Fig. 4) is explained and demonstrated with the following example. An SMA actuated compliant bistable mechanism is to be designed. The output angular stroke between the two stable positions is a ¼ 50, and no offset is desired. The flexible link has the same material property of the flexible link used in Section 3.1, as well as the SMA wire used for the springs. An iterative design process is outlined step by step below. (1) Determine the stroke of the SMA actuator and the output link length. The relationship between the output link length and the stroke of the SMA actuator can be expressed as (Fig. 10) st ¼ 2 l0 sinða=2Þ: One may assume the value of st or l0 and determine the other. Let l0 ¼ 25 mm. Then st ¼ 2 l0 sinða=2Þ ¼ 2ð25Þ sin 25 ¼ 21:131 mm: (2) Select an output displacements sta and aa of the bistable mechanism. It is noted that the output stroke of a bistable mechanism is usually 10–20% longer than that of an SMA actuated compliant bistable mechanism. Generally, if the difference is small, the load capacity will be lower. Let 15% be chosen. The output displacement of the bistable mechanism will be sta ¼ 1:15 st ¼ 1:15ð21:131Þ ¼ 24:301 mm: sta 24:301 aa ¼ 2 sin1 ¼ 58:1586 degree: ¼ 2 sin1 2ð25Þ 2l0 st a st s
Desired Stable Position of the Combined System
TRACE OF CRANK
α
αa lo
Stable Position of the Bistable Mechanism
y x
Fig. 10. An output crank.
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433
(3) Determine the approximate maximum deformation dm (Fig. 9). aa 58:159 ¼ 3:151346 mm: dm ¼ l0 l0 cos ¼ 25 25 cos 2 2 (4) Select the maximum strain emax and the thickness–length ratio t=ls of the flexible link. A suitable thickness–length ration t=ls can be chosen. Let t=ls ¼ 0:03. Super elastic SMA is capable of recovering about 8% of strain. For a longer life span, let emax ¼ 0:4%. Now, use the relationship of the strain and deflection [11], the length and thickness of flexible link can be determined. 3 dm t 3 3:151 emax ¼ ¼ ð0:03Þ ¼ 0:004: 2 ls ls 2 ls Thus, ls ¼
3ð0:03Þð3:151Þ ¼ 35:45 mm: 2ð0:004Þ
t ¼ 0:003 ls ¼ 0:003ð35:45Þ ¼ 1:063 mm: (5) Select the width of the flexible link. The width of the flexible link can be chosen arbitrary. However, the width affects on the capacity of the strain energy. Therefore, a large width will lead to a high maximum strain energy and require a large force from the actuator. On the other hand, a narrow width will lead to low stability of the system. Let w ¼ 4 mm be selected. (6) Determine the force curve of the bistable mechanism. Now, the key dimensions of the bistable mechanism are determined. The force–deflection curve of the mechanism can be determined by using Eq. (10). The result is shown in Fig. 11. (7) Determine the normal force–displacement relation Fout of the SMA actuator. Fout is the output spring force of the SMA actuator when the both SMA springs are at
2
Force (N)
PL
1 KL
B
FA
10.5655 0 -10.5655 A
KL
-1
-2 - 10
-5
0 Displacement (mm)
5
10
Fig. 11. Force–displacement curve of the designed bistable mechanism.
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normal temperature. Because no offset is desired, only one stable position needs to be considered. Let FA (Fig. 11) be the force at the desired stable position of the SMA actuated compliant bistable mechanism. Since the desired total stroke st is 21.131 mm, the displacement at the stable position should be s ¼ 10:5655 mm, which is equal to half of the total stroke. Thus, from the equation of the force curve (10), with s ¼ 10:5655 mm, FA is determined as 0.75644 N (Fig. 11). Therefore, the stiffness Fout is Fout ¼
0:75644 ¼ 0:071595 N=mm: 10:5655
(8) Determine PL . PL is the output force of the actuator when one of the SMA springs is not deformed in normal temperature. As shown in Fig. 11, the left and right SMA springs of the SMA actuator have no deformation at A and B respectively, and both positions A and B are the stable positions of the compliant bistable mechanism. Therefore, sta PL ¼ Fout A ¼ Fout ¼ 0:071595ð12:15057Þ ¼ 0:870 N: 2 (9) Determine the desired SMA spring properties at normal temperature. Referring to Fig. 11, the ideal spring constant of the SMA spring at low temperature is KL ¼
PL 0:870 ¼ 0:03580 N=mm: ¼ sta 24:301
Now, the desired properties of the SMA spring, which has GL ¼ 8000 N/mm2 , at low temperature are found and listed below. PL ¼ 0:870 N;
dL ¼ sa ¼ 24:301 mm;
KL ¼ 0:03580 N=mm:
(10) Determine the desired SMA spring properties at high temperature. The relationship between the properties of the SMA spring at high and low temperature is governed by following equation [12,13]. PH PL ¼ : dH G H dL G L
ð14Þ
Therefore, the spring constant of the SMA spring at high temperature is KH ¼
PH GH PL ð23; 000Þð0:86992Þ ¼ 0:10293 N=mm: ¼ ¼ ð24:301Þð8000Þ dH dL GL
dH ¼ 19:4781 mm is used because the bistable mechanism reaches the critical force at this position (Fig. 11) and the SMA spring at high temperature generate a force PH , where PH ¼ dH KH ¼ ð19:4781Þð0:10293Þ ¼ 2:005 N: The critical force of the bistable mechanism must be less than the force PH of the SMA spring (at high temperature) to have bistable positions with certain load capacity. To make sure the SMA actuator can generate enough force to switch the bistable mechanism, it is recommended that the value of PH is about 20% higher than
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the critical force of the bistable mechanism. If the resulting PH value is the same as the critical force at this position, the output force curve will intersect the required input force curve of the bistable mechanism. Therefore, depending on the resulting value of PH , one may return to step 4, or 5 to adjust the bistable mechanism until a reasonable PH to critical force ratio is obtained. A higher ratio will lead to a mechanism with a higher external load capacity. In this example, the critical force of the mechanism is 1.26218 N and PH ¼ 2:005 N. Therefore, the actuator can produce 160% of the critical force, which indicates that the force generated by the actuator will be enough to overcome the critical force to switch between the stable positions. The obtained properties of the SMA spring, which has the property of GH ¼ 23; 000 N/mm2 at high temperature, are summarized below: PH ¼ 2:005 N;
dH ¼ 19:4781 mm;
KH ¼ 0:10293 N=mm:
(11) Design the spring. Because the SMA spring actuator is used to drive a compliant bistable mechanism, conventional SMA spring design method [12–14] is sufficient for this design purpose. In order to use the design method, the design parameters, the spring index C and the maximum shear strain cmax , are chosen. In this example, the spring index C ¼ 8 is selected. Because dmax , which is the deformation of an SMA spring at a stable position of bistable mechanism, is very close to dL , it is assumed cmax cL . For a long life span, it is chosen that cmax ¼ 1:2%. Let the KLR and KHR be the resultant spring stiffness in low and high temperatures respectively. By following the design process in [12–14], the dimensions and the SMA spring constants are obtained below. d ¼ 0:5 mm;
D ¼ 4:6 mm;
KLR ¼ 0:04281 N=mm;
Na ¼ 15 turns;
KHR ¼ 0:12307 N=mm;
Ls ¼ 7:5 mm:
One may notice that the resultant spring stiffness, KLR and KHR , are about 20% greater than the ideal spring stiffness, KL and KH . In order to minimal this design error, an iterative process is used to adjust the resultant SMA spring constants [12] by changing a dimension of the spring. In this example, the mean diameter D is increased by 0.1 mm in each iteration process. It is found that when D ¼ 4:9 mm, the spring constants become KLR ¼ 0:03542 N/mm and KHR ¼ 0:10182 N/mm, which are within 1.1% of the ideal spring constant. (12) Evaluate the final design. The dimensions of the SMA actuated bistable mechanism are determined and summarized below. The bistable mechanism: ls ¼ 35:45 mm, w ¼ 4 mm, t ¼ 1:06 mm, l0 ¼ 25 mm and stroke ¼ 24.301 mm. The SMA actuator: d ¼ 0:5 mm, D ¼ 4:9 mm, Na ¼ 15 turns, KLR ¼ 0:03542 N/ mm, KHR ¼ 0:10182 N/mm and Ls ¼ 7:5 mm. The scaled figure is shown in Fig. 4. The force–displacement curves of the resulting SMA actuated compliant bistable mechanism are shown in Fig. 7, which has an output angular stroke of 50.10.
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Fig. 12. (a) A two-position control system, and (b) a four-position control system.
4. Conclusion This article presented the first comprehensive work on SMA actuated compliant bistable mechanisms. The concept, analysis, as well as design procedure of SMA actuated compliant bistable mechanisms is discussed. Although the discussion is focused on one type of bistable mechanisms, the same method is applicable to most of the bistable mechanisms. Because of the combination with a bistable mechanism, the conventional analysis and design method can be used, which allows one to ignore the temperature dependent stress–strain relationship of the SMA and greatly simplify the design and analysis. The combination preserves the advantages of SMA actuators and compliant bistable mechanisms while reduces or eliminates most of their shortcomings. An SMA actuated compliant bistable mechanism is a non-linear spring system. The system stores and releases energy as it moves from a stable position to the unstable position and continues in the same direction toward the other stable position. The phenomenon occurs in both directions of motion. An SMA actuated compliant bistable mechanisms can be also used as a driver. When the system is used with a single degree of freedom mechanism, such as four-bar mechanism, the entire system can be used as two-position control system (Fig. 12a). Since each actuator would generate two stable positions, if more are used in a multiple degree of freedom mechanism, more stable positions can be generated (Fig. 12b). References [1] Howell LL, Midha A. The development of force–deflection relationships for compliant mechanisms. ASME Machine Elements Machine Dynamics DE 1994;71:501–8.
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