Patent Publication Number: US-6215081-B1

Title: Bistable compliant mechanism

Description:
This application claims the benefit of U.S. Provisional Application No. 60/098,633, filed Aug. 31, 1998. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to a mechanism which is compliant and stable in two positions, and which is particularly well suited for use with electrical switches. More particularly, the present invention relates to a mechanism having a plurality of segments coupled end-to-end in series with at least two rigid segments and at least one flexible and resilient segment. 
     2. Prior Art 
     Switches are used to activate or adjust an electrical or mechanical system. A toggle switch is one that permits adjustment only to a certain limited number of settings; a bistable switch is further limited in that only two settings are available. As such, bistable switches are very useful for electric circuits, in which it is desirable to open a circuit to cut off the power to an electric device, thereby turning it off. Bistable switches are similarly useful in mechanical systems where the switch is to maintain the system in one of two states. 
     Many different bistable toggle switches have been invented. The majority are either of the push-button type, such as jumper switches for fuse boxes, the rotary type, as found in many appliances such as stoves and ovens, or the rocker type, which are most commonly mounted on walls to control household electric devices. Both types of switches are in wide use in electrical applications. Switches include some surface or member situated for the transmission of external forces into the switch. In the case of an ordinary household light switch, for example, this can take the form of a post designed to be pushed up or down by a hand or finger. Additionally, mechanical joints such as hinges often require a bistable rocking, rotating, or translating action; this can be accomplished by a bistable switch mechanism. Although the switches are typically inexpensive and small in size, the large number of these switches in common use provides the incentive for reduction of the costs involved in manufacturing them. 
     Many switches function using some type of linkage to transform the input force to the desired output motion. A linkage is a mechanical system made up of four or more members, or links, which are connected to each other by means of joints that allow the links to pivot or slide with respect to each other. Traditionally, the links were rigid and the joints between them utilized pinned joints, sockets, or mechanical sliders to effect the relative motion. The length of the links and the nature of the joints could be adjusted to obtain the desired output motion in one link from a given input motion or force on another link. 
     Such a linkage system can be made bistable by the insertion of a device that exerts a linear or torsional force on a sliding or pivoting joint, respectively. These devices are often simple springs; the stable linkage positions are those in which the spring deflection is at a relative minimum. Therefore, the stable points for the linkage system are those in which motion of the linkage in either direction will increase the total potential energy stored in the mechanisms. 
     There are many disadvantages associated with traditional mechanical linkage systems. One disadvantage with traditional mechanisms is that the links must be separately made and assembled with the joints; as a result, the cost of manufacturing linkages on a large scale is considerable. In addition, there are the usual difficulties associated with surfaces that slide against each other. These difficulties include wear, friction losses, and the need for lubrication. 
     Therefore, it would be advantageous to develop a bistable mechanism capable of movement between two stable positions. It would also be advantageous to develop such a bistable mechanism capable of simple and inexpensive manufacture. It would also be advantageous to develop such a bistable mechanism with a reduced number of parts. It would also be advantageous to develop such a bistable mechanism with few or no wear surfaces. It would also be advantageous to develop such a bistable mechanism capable of use with electrical switches. 
     OBJECTS AND SUMMARY OF THE INVENTION 
     It is an object of the present invention to provide a bistable mechanism. 
     It is another object of the present invention to provide a bistable mechanism movable between two stable positions. 
     It is a further object of the present invention to provide a bistable mechanism with few parts. 
     It is a further object of the present invention to provide a bistable mechanism with few wear surfaces. 
     It is a further object of the present invention to provide a bistable mechanism for use with electrical switches. 
     These and other objects and advantages of the present invention are realized in a compliant, bistable mechanism having a plurality of segments coupled end-to-end in series to form a continuous chain of segments. The plurality of segments includes at least two relatively rigid segments, and at least one relatively flexible and resilient segment. 
     Adjacent rigid segments are coupled by either flexible joints or pin joints. The relatively flexible and resilient segment is coupled to adjacent segments either fixedly or by pin joints. The sum of the pin joints, the flexible joints and/or the relatively flexible and resilient segments is at least four. 
     The relatively flexible and resilient segment operates to resist relative movement of the segments, but allows the segments to be selectively moved. The plurality of segments are biased by the at least one relatively flexible and resilient segment. The plurality of segments are cooperatively movable relative to one another between (i) a first, stable, static, equilibrium position, and (ii) a second, stable, static, equilibrium position. 
     In accordance with one aspect of the present invention, the first position is a low-energy position in which the at least one relatively flexible and resilient member is substantially undeflected, and stores substantially no energy, or low energy relative to surrounding positions. The second position is a force loaded position in which the at least one relatively flexible and resilient segment is deflected, and stores energy such that the mechanism exerts a force in the second position. Alternatively, the at least one relatively flexible and resilient segment may be deflected in one or both of the first and second positions. In addition, both first and second positions may be low-energy positions in which the relatively flexible and resilient segment is undeflected. 
     In accordance with another aspect of the present invention, the at least two relatively rigid segments are coupled by, and formed integrally with, a substantially flexible joint. In addition, all of the plurality of segments may be integrally formed from a single piece of material. The single piece of material has cross sectional dimensions of (i) relatively wide portions, (ii) relatively thin portions, and (iii) at least one portion with an intermediate width. The relatively rigid segments are formed of the relatively wide portions, and thus are generally rigid. The substantially flexible segments are formed of the relatively thin portions, and thus are generally compliant. The relatively flexible and resilient segment is formed of the portion of intermediate width, and thus is both flexible and resilient. 
     In accordance with the preferred embodiment of the present invention, the plurality of segments includes four relatively rigid segments coupled end-to-end in series by three substantially flexible joints, or pivot joints, and one relatively flexible and resilient segment. The relatively flexible and resilient segment is fixedly coupled to adjacent rigid segments. 
     In accordance with the preferred embodiment of the present invention, two electrical contacts are coupled to the plurality of segments including first and second electrical contacts. The first electrical contact is movable with one of the segments between (i) a first location, and (ii) a second location. In the first location, the first electrical contact contacts the second electrical contact, and defines an on position. In the second location, the first electrical contact is in a non-contacting relationship with the second electrical contact, and defines an off position. 
     In accordance with one aspect of the present invention, the plurality of segments has a pseudo-rigid-body model resembling a four-bar linkage. In addition, the mechanism may be a Young mechanism, a Grashof mechanism, or a non-Grashof mechanism. In addition, the mechanism may be a MEMS (micro-electo-mechanical system), and each segment has a length less than 500 microns. In addition, each segment may have a thickness less than 3 microns. 
     These and other objects, features, advantages and alternative aspects of the present invention will become apparent to those skilled in the art from a consideration of the following detailed description taken in combination with the accompanying drawings. 
    
    
     DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a perspective view of a preferred embodiment of a bistable switch mechanism of the present invention. 
     FIG. 2 a  is a side view of the preferred embodiment of the bistable switch mechanism of the present invention shown in a first off position. 
     FIG. 2 b  is a side view of the preferred embodiment of the bistable switch mechanism of the present invention shown in a second on position. 
     FIGS. 3 a  and  3   b  are schematic views of the preferred embodiment of the bistable switch mechanism of the present invention showing its corresponding pseudo-rigid-body model. 
     FIG. 3 c  is a pseudo-rigid-body model of a general four-link mechanism with torsional springs at each joint. 
     FIG. 4 a  is a side view of an alternative embodiment of a bistable mechanism of the present invention shown in a first position. 
     FIG. 4 b  is a side view of an alternative embodiment of a bistable mechanism of the present invention shown in a second position. 
     FIG. 4 c  is a cross-sectional side view of a pin joint of an alternative embodiment of a MEMS (micro-electric-mechanical system) of the present invention. 
     FIG. 5 a  is a schematic view of the alternative embodiment of the bistable mechanism of the present invention showing its corresponding pseudo-rigid-body model. 
     FIG. 5 b  is a pseudo-rigid-body model of the alternative embodiment of the bistable mechanism of the present invention. 
     FIG. 6 a  is a side view of an alternative embodiment of a bistable mechanism of the present invention shown in a first position. 
     FIG. 6 b  is a side view of an alternative embodiment of a bistable mechanism of the present invention shown in a second position. 
     FIG. 7 is a schematic view of the alternative embodiment of the bistable mechanism of the present invention showing its corresponding pseudo-rigid-body model. 
     FIG. 8 is a perspective view of an alternative embodiment of a bistable mechanism of the present invention. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Reference will now be made to the drawings in which the various elements of the present invention will be given numerical designations and in which the invention will be discussed so as to enable one skilled in the art to make and use the invention. 
     As illustrated in FIG. 1, a bistable switch mechanism, indicated generally at  10 , in accordance with a preferred embodiment of the present invention is shown. The switch mechanism  10  has a plurality of segments, indicated generally at  14 , coupled end-to-end in series to form a continuous chain of segments. 
     Several terms are used to describe and characterize mechanisms and their components, which are defined as follows. Rigid-body mechanisms are constructed of rigid links joined with kinematic pairs, such as pin joints and sliders. These components are easily identified and characterized. Since compliant mechanisms gain at least some of their motion from the deflection of flexible members, components such as links and joints are not as easily distinguished. Identification of such components is necessary to allow the accurate communication of design and analysis information. 
     A “link” is defined as the continuum connecting the mating surfaces of one or more kinematic pairs. Revolute (pin or turning) joints and prismatic (sliding) joints are examples of kinematic pairs. Links can be identified by disassembling the mechanism at the joints and counting the resulting links. 
     A mechanism with no traditional joints has zero links. Such mechanisms are termed “fully compliant” mechanisms, since all of their motion is obtained from the deflection of compliant members. Compliant mechanisms that contain one or more traditional kinematic pairs along with compliant members are called “partially compliant” mechanisms. 
     For a rigid link, the distances between joints are fixed, and the shape of the link is kinematically unimportant regardless of the applied forces. The motion of a compliant link, however, is dependent on link geometry and the location and magnitude of applied forces. Because of this difference, a compliant link is described by its structural type and its functional type. 
     The structural type is determined when no external forces are applied and is similar to the identification of rigid links. A rigid link that has two pin joints is termed a “binary link.” A rigid link with three or four pin joints is a “ternary” or a “quaternary link,” respectively. A compliant link with two pin joints has the same structure as a binary link, and is called a “structurally binary link,” and so on for the other types of links. 
     A link&#39;s functional type takes into account the structural type and the number of pseudo joints. Pseudo joints occur where a load is applied to a compliant segment. If a force is applied on a compliant link somewhere other than at the joints, its behavior may change dramatically. A structurally binary link with force or moment loads only at the joints is termed “functionally binary.” A compliant link with three pin joints is “structurally ternary,” and if loads are only applied at the joints, it is also “functionally ternary.” The same applies for quaternary links. If a link has two pin joint connections and also has a force on a compliant segment, it is “structurally binary” and “functionally ternary” due to the added pseudo joint caused by the force. 
     While the definition of a link used above is consistent with that for rigid-body kinematics, it is not very descriptive of a compliant link. The application of a force or moment to a compliant link affects the deformation of the link, and therefore, its contribution to the mechanism&#39;s motion. Link characteristics that influence its deformation include cross-sectional properties, material properties, and magnitude and placement of applied loads and displacements. Thus, a compliant link is further characterized into “segments.” 
     A link may be composed of one or more “segments.” The distinction between segments is a matter of judgement, and may depend on the structure, function, or loading of the mechanism. Discontinuities in material or geometric properties often represent the end points of segments. Since the distance between the end points of a rigid segment remains constant, it is considered a single segment, regardless of its size or shape. 
     The characteristics of individual segments and links may also be described. A segment may be either rigid or compliant. This is referred to as a segment&#39;s “kind.” A compliant segment may be further classified by its category of either simple or compound. A simple segment is one that is initially straight, has constant material properties, and a constant cross-section. All other segments are compound. 
     A link may be either rigid or compliant (its kind) and may consist of one or more segments. A rigid link needs no more characterization. A compliant link may be either simple or compound (its category). A simple compliant link consists of one simple compliant segment; all others are compound links. A compound link may be either homogeneous or nonhomogeneous. This is its “family.” A homogeneous link is one that consists of all rigid segments or all compliant segments. Therefore, rigid links and simple compliant links are special cases of homogeneous links. Nonhomogeneous links contain both rigid and flexible segments. 
     Traditional mechanism analysis employs the assumption that the deflections of a mechanism&#39;s parts are negligible compared to the overall motion of the mechanism. If the parts are rigid, the mechanism motion is not a function of the shape of the links or the applied forces. This allows motion analysis (kinematics), and the analysis of motion and the forces that produce it (kinetics), to be analyzed independently, thus simplifying the analysis. 
     The minimum number of variables required to describe the configuration of a mechanism completely is called its “degrees of freedom.” An unconstrained planar rigid link has three degrees of freedom because three displacement variables are required to describe its position and orientation. Therefore, the total possible degrees of freedom in a plane of n unconstrained links is 3n. By definition, a mechanism has one fixed link, which has zero degrees of freedom. The maximum possible degrees of freedom in a plane of an n-link mechanism is then 3(n−1). 
     When links are connected together with joints it is called a “kinematic chain.” The chain is considered a mechanism if one of the links is considered to be the fixed link, which means that it is chosen as the reference link. The fixed link is usually the frame or base link connected to ground. The basic kinematic chain has the same relative motion between links, regardless of which link is fixed. A kinematic inversion is obtained when a different link is fixed. This does not change the relative motion between links, but can drastically change the absolute motion of the mechanism. 
     Grashof&#39;s law states that for at least one link of a four-bar mechanism to have full rotation, the following inequality must hold: s+l≦p+q , where s is the length of the shortest link, 1 is the length of the longest link, and p and q are the lengths of the remaining links. The shortest link of a Grashofian mechanism is allowed full rotation relative to its adjacent links. Different types of mechanisms are based on which link is the shortest link. For example, if a side link is the shortest link in a Grashofian mechanism, then it is called a “crank rocker” mechanism; the shorter side link (the crank) is able to revolve, and the other side link (the rocker) rocks between two limit positions. 
     The plurality of segments  14  includes at least two relatively rigid segments and at least one relatively flexible and resilient segment. As shown, the switch mechanism  10  preferably has four rigid segments  18 ,  22 ,  26  and  30 , and one relatively flexible and resilient segment  34 . The segments  14  are coupled at coupling points. 
     In the preferred embodiment of the switch mechanism  10 , the plurality of segments  14  includes a first relatively rigid base segment  26 , a second relatively rigid coupling segment  18 , and first and second arm segments  22  and  30 . The base segment  26  may be fixed and has first and second ends  40  and  42 . Similarly, the coupling segment  18  has first and second ends  46  and  48 . The first arm segment  22  is coupled between the first ends  40  and  46  of the base segment  26  and the coupling segment  18 . Similarly, the second arm segment  30  is coupled between the second ends  42  and  48  of the base segment  26  and the coupling segment  18 . 
     An engagement member  50  may extend from the coupling segment  18  for a user to engage the mechanism  10 . In the application of an electrical switch, many of the segments  14 , such as the segments  22 ,  26  and  30 , are disposed in a wall or panel behind a face plate (not shown) while the engagement member  50  protrudes from the face plate, as is common in typical household switches. 
     The rigid segments  18 ,  22 ,  26  and  30  are coupled to adjacent rigid segments by either flexible joints, indicated generally at  52 , or pin joints  54  (FIG. 3 a ). The flexible joints  52  are substantially flexible and may be formed by a “living hinge”. The pin joints  54  (FIG. 3 a ) are typical pin joints and are well known in the art. 
     Extremely short and thin small-length flexural pivots are often called “living hinges.” The pseudo-rigid-body model, as discussed more fully below, of a pin joint at the center of the flexible segment is highly accurate for living hinges. In systems with both living hinges and other compliant segments, the rigidity of the living hinges is often so low, compared with the other flexible segments in a system, that their torsional springs are ignored. However, if a system contains only living hinges, then their rigidity should be considered in the analysis. 
     A pin joint allows rotation about one axis, but does not allow rotation in any other axis or translation in any direction. A door hinge is a common example of a pin joint. Small-length flexural pivots have behavior similar to pin joints, but they use the deflection of flexible members to obtain motion rather than pure rotation of parts about a pin. The “hinge” of a cover of a hardcover book is an example of a small-length flexural pivot. The rigidity of the flexible portion is much smaller than the more rigid part due to a change in both material and geometry. 
     There are many types of small-length flexural pivots, and a living hinge is a special case small-length flexural pivot. They are very small in length, offer little resistance to deflection, and approximate very closely the behavior of a pin joint. They offer so little resistance to bending, that they are often modeled with the pseudo-rigid-body model as a pin joint without a torsional spring. 
     Polypropylene is the most commonly used material for living hinges. Other materials may be used but will usually result in a shorter life. In some applications, life is not a major concern since the hinge may only be expected to flex once. For example, many containers are constructed of a single piece of material and then folded at living hinges to make the container. In such cases, the designer has many acceptable options in material and geometry choices. In most compliant mechanism designs, however, living hinges are expected to endure many cycles without failure. The discussion that follows assumes that a long life is required. The recommendations are summarized from the experience of several plastics suppliers and other sources. Living hinges made using these methods have been tested to undergo millions of cycles without failure. 
     Hinges may be made by injection molding, extrusion, hot-stamping, and blow molding. When injection molded, the molten plastic should be caused to flow perpendicular to the hinge. This causes a good fill and also helps align the material in a favorable direction. Extruded hinges will have a much shorter life because the material flow is parallel to the hinge axis. 
     The hinge should be flexed immediately after molding while the heat from the mold is still present. It should be flexed once slowly then rapidly several times. Flexing will stretch the hinge area considerably (a 0.010 in thickness may thin down to less than 0.005 in.). The elongation orients the material and dramatically increases the tensile strength. A thin, white line will appear on the hinge after flexing. This is normal and does not mean that the hinge has been weakened. 
     Some molding considerations are as follows: Cylinder temperature—450-550 degrees F.; injection speed—fast; mold temperatures—120 to 150 degrees F.; gate opening—if possible make up to 50% larger than for non-hinged parts. If using a single gate, locate it to ensure smooth flow to hinge area, make the flow perpendicular to the hinge axis, place the gate slightly to the rear of the center lines of the largest cavity, and center it if the flow to the hinge is greater than 8 in. For multiple gates: ensure that gates on the same side of the hinge are no farther apart than twice the distance from gate to hinge; if the flow on the opposite side of the hinge is greater than 8 in., the part should be gated in both sides; locate so a weld line does not form at the hinge. The hinge should be an insert machined from hardened steel to resist the stresses of the flowing resin. 
     In the preferred embodiment of the switch mechanism  10 , the base segment  26  is coupled to the first arm segment  22  by a first flexible joint  58 ; the coupling segment  18  is coupled to the first arm segment by a second flexible joint  60 ; and the coupling segment  18  is coupled to the second arm segment  30  by a third flexible joint  62 . 
     In the preferred embodiment of the switch mechanism  10 , the plurality of segments  14  includes one relatively flexible and resilient segment  34 . The relatively flexible and resilient segment  34  is compliant, or is able to bend or deflect. 
     The relatively flexible and resilient segment  34  is coupled to adjacent segments either fixedly, or by a pin joint  54  (FIG. 3 a ). In the preferred embodiment of the switch mechanism  10 , the relatively flexible and resilient segment  34  is fixedly coupled to and between the base segment  26  and the second arm segment  30 . 
     The sum of the pin joints  54  (FIG. 3 a ), the flexible joints  52 , and the relatively flexible and resilient segments  34  is at least four. In the preferred embodiment of the switch mechanism  10 , there are three flexible joints  58 ,  60  and  62 , and one relatively flexible and resilient segment  34 , which sum to four. 
     Referring to FIGS. 3 a  and  3   b,  a pseudo-rigid-body model, indicated generally at  10 ′, of the mechanism is shown. The pseudo-rigid-body model  10 ′ resembles, or corresponds to, a four-bar linkage. 
     The purpose of the pseudo-rigid-body model is to provide a simple method of analyzing systems that undergo large, nonlinear deflections. The pseudo-rigid-body model concept is used to model the deflection of flexible members using rigid-body components that have equivalent force-deflection characteristics. Rigid-link mechanism theory may then be used to analyze the compliant mechanism. In this way, the pseudo-rigid-body model is a bridge that connects rigid-body mechanism theory and compliant mechanism theory. The method is particularly useful in the design of compliant mechanisms. Different types of segments require different models. 
     For each flexible segment, a pseudo-rigid-body model predicts the deflection path and force-deflection relationships of a flexible segment. The motion is modeled by rigid links  14 ′ attached at pin joints  54 . Springs  98  are added to the model  10 ′ to accurately predict the force-deflection relationships of the compliant segments  34  (FIG.  1 ). The key for each pseudo-rigid-body model is to decide where to place the pin joints and what value to assign the spring constants. 
     As indicated above, the pseudo-rigid-body model  10 ′ resembles a four-bar mechanism. Referring to FIG. 3 c,  a moment acts on link two, the input link. A torsional spring  98 ′ at each of the four pin joints  54  allows energy to be stored as the mechanism  10 ′ moves. The torsional springs  98 ′ represent the stiffness of a compliant segment ( 34  in FIG.  1 ), as specified in the pseudo-rigid-body model. The energy stored in each spring may be found from 
     
       
           V   i =½ K   i ψ i   2   (1)  
       
     
     where V is the potential energy, K is the torsional spring constant, and ψ is the angular deflection of each torsional spring. For each spring  98 ′ shown in FIG. 3 c,    
     
       
         ψ 1 =θ 2 −θ 20    
       
     
     
       
         ψ 2 =(θ 2 −θ 20 )−(θ 3 −θ 30 )  
       
     
     
       
         ψ 3 =(θ 4 −θ 40 )−(θ 3 −θ 30 )  
       
     
     
       
         ψ 4 =θ 4 −θ 40   (2)  
       
     
     where the “0” subscripts symbolizes the initial (undeflected) value of the angle. The total potential energy of the system may then be given as 
     
       
           V   =½(   K   1 ψ 1   2   +K   2 ψ 2   2   +K   3 ψ 3   2   +K   4 ψ 4   2 )  (3)  
       
     
     The values of each ψ may be found using kinematic analysis for all positions of the mechanism, allowing a graph of potential energy to be constructed. Any positions corresponding to local minima are stable positions; any local maxima represent unstable equilibrium positions. 
     The stability of the mechanism  10 ′ can also be determined analytically. The principle of virtual work can be used to find the values of arbitrary moments or forces required to keep a mechanism in a particular position. For analyzing the bistable characteristics of the mechanism, however, only the value of M 2 , as shown in FIG. 3 c,  is necessary. This moment represents the moment that must be applied to the input link to keep the mechanism in a given position. At the equilibrium positions, its value will be zero. The M 2  curve may be found by realizing that it is the first derivative of the energy curve with respect to the angle of the input link. This may be proved by considering the equation for work put into the system:              V   =       ∫     θ   20       θ   2              M   2          d      θ                 (   4   )                         
     by taking the derivative of this equation, it may be seen that                     V            θ   2         =     M   2             (   5   )                         
     assuming that the moment at the initial position is zero. Therefore, M 2  is equal to the first derivative of the energy with respect to the angle of the input link. This means that                M   2     =         K   1          ψ   1       +       K   2          ψ   2                 ψ   2              θ   2           +       K   3          ψ   3                 ψ   3              θ   2           +       K   4          ψ   4                 ψ   4              θ   2                     (   6   )                         
     The derivatives in Equation (6) above may be evaluated using Equation (2) and the additional formulas                       θ   3              θ   2         =       h   32     =         r     2                       sin        (       θ   4     -     θ   2       )             r     3                       sin        (       θ   3     -     θ   4       )                     (   7   )             and                                  θ   4              θ   2         =       h   42     =         r     2                       sin        (       θ   3     -     θ   2       )             r   4                     sin        (       θ   4     -     θ   3       )                     (   8   )                         
     As mentioned previously, the value of M 2  will be zero at all equilibrium positions. The stability of the equilibrium position may be determined by considering the sign of the second derivative of the energy curve at that point. The second derivative is                       2        V            θ   2   2         =       K   1     +       K   2          (     1   -     2        h   32       +     h   32   2     -       ψ   2          h   32   ′         )       +       K   3          [       h   42   2     -     2        h   42          h   32       +     h   32   2     +       ψ   3          (       h   42   ′     -     h   32   ′       )         ]       +       K   4          (       h   42   2     +       ψ   4          h   42   ′         )                 (   9   )               where                                        h   32   ′     =              h   32              θ   2         =         r   2       r   3            [           cos        (       θ   4     -     θ   2       )         sin        (       θ   3     -     θ   4       )              (       h   42     -   1     )       -           sin        (       θ   4     -     θ   2       )            cos        (       θ   3     -     θ   4       )             sin   2          (       θ   3     -     θ   4       )              (       h   32     -     h   42       )         ]                 (   10   )               and                                        h   42   ′     =              h   42              θ   2         =         r   2       r   4            [           cos        (       θ   3     -     θ   2       )         sin        (       θ   4     -     θ   3       )              (       h   32     -   1     )       -           sin        (       θ   3     -     θ   2       )            cos        (       θ   3     -     θ   4       )             sin   2          (       θ   4     -     θ   3       )              (       h   32     -     h   42       )         ]                 (   11   )                         
     When the value of M 2  is zero, the equilibrium position will be stable if the second derivative of potential energy is positive. If the second derivative of potential energy is negative, the equilibrium position is unstable, and if it is zero, the equilibrium position is neutrally stable. 
     As the mechanism  10 ′ moves from one stable position to another, the absolute value of M 2  will increase to some maximum before decreasing down to zero at the unstable position. This maximum moment represents the largest moment that must be applied to the input link to make the mechanism snap into its second position. This important value may be called the “critical moment,” or, if a force is applied instead, the “critical force.” 
     In addition, a high value of the second derivative at a stable position means that the energy curve is changing very rapidly at that point. This means that the restoring force returning the mechanism to that position is relatively high. Thus, the value of the second derivative at a stable position may be called the stable position&#39;s “stiffness,” where a high stiffness corresponds to a rapidly increasing restoring force. 
     The mechanism shown in FIG. 3 c  may be further classified according to Grashof&#39;s criterion as a Grashof or non-Grashof mechanism. In a Grashof mechanism, the shortest link can rotate through a full revolution with respect to either link connected to it. In a non-Grashof mechanism, no link can rotate through a full revolution with respect to any other links. Recall that Grashof&#39;s criterion is mathematically stated as s+l≦p+q, where s is the length of the shortest link, 1 is the length of the longest link, and p and q are the lengths of the intermediate links. If the mechanism&#39;s link lengths satisfy this inequality, it is a Grashof mechanism. Crank rockers, double cranks, and double rockers are examples of Grashof mechanisms. If the inequality is not satisfied, the mechanism is non-Grashof. These mechanisms are triple rockers. If the sum of the lengths of the longest and shortest links is equal to the sum of the lengths of the other two links, the mechanism is a special case of a Grashof mechanism known as a change-point mechanism. Mathematically 
     
       
         s+l&gt;p+q non-Grashof  
       
     
     
       
         s+l=p+q change point  
       
     
     The requirements for bistable behavior will be different for Grashof and non-Grashof mechanisms. A Grashof four-bar link mechanism will be bistable if the torsional spring in the pseudo-rigid-body model is placed at either position opposite the shortest link. A change-point of non-Grashof mechanism will be bistable if a spring is placed at any one of the four joint positions. When more than one torsional spring is present in the pseudo-rigid-body model then an analysis of the potential energy is required to determine its stability. 
     Referring again to FIG. 1, the plurality of segments  14  advantageously may be integrally formed. In addition, the plurality of segments  14  (including rigid segments  22 ,  26  and  30 , and the relatively flexible and resilient segment  34 ) and the flexible joints  52  (including the first, second and third flexible joints  58 ,  60  and  62 ) may be integrally formed. Thus, the plurality of segments  14  and flexible joints  52  may be formed from a single piece of material  80  having cross sectional dimensions including relatively wide portions  82 , relatively thin portions  84 , and portions with an intermediate width  86 . The relatively rigid segments  18 ,  22 ,  26  and  30  are formed by the relatively wide portions  82 . The substantially flexible joints  58 ,  60  and  62  are formed by the relatively thin portions  84 . The relatively flexible and resilient segment  34  is formed by the portion of intermediate width  86 , and is thus both flexible and resilient. 
     The flexible joints  52  and pin joints  54  (FIG. 3 a ) allow the plurality of segments  14  to move relative to one another. Adjacent segments  14  pivot with respect to one another about the joint  52  (FIG. 2 a ) or  54  (FIG. 3 a ) coupling them. As indicated above, the relatively flexible and resilient segment  34  operates to resist relative movement of the segments  14 , but allows the segments  14  to be selectively moved. The plurality of segments  14  cooperatively move with respect to one another between a first position  70 , as shown in FIGS. 1 and 2 a,  and a second position  72 , as shown in FIG. 2 b.  In addition, the relatively flexible and resilient segment  34  biases the plurality of segments  14  between the two positions  70  and  72 . 
     Referring to FIGS. 1 and 2 a,  the first position  70  preferably is a stable, static, equilibrium position, or the plurality of segments are in a position in which they are stable, static, and in equilibrium. The first position  70  may be a low-energy position in which the relatively flexible and resilient segment  34  is substantially undeflected and stores substantially no energy. Alternatively, the first position  70  may be a force loaded position in which the relatively flexible and resilient segment  34  is deflected and stores energy. 
     Referring to FIG. 2b, the second position  72  may be a stable, static, equilibrium position, or the plurality of segments are in a position in which they are stable, static, and in equilibrium. The second position  72  also may be a low-energy position in which the relatively flexible and resilient segment  34  is substantially undeflected and stores substantially no energy. Alternatively, the second position  72  may be a force loaded position in which the relatively flexible and resilient segment  34  is deflected and stores energy. Thus, the mechanism  10  or segments  14  exert a force in the second position  72 . The first arm segment  22  pivots towards the base segment  26  in the second position  72 , or as the segments  14  move between first and second position  70  and  72 . In addition, the second arm segment  30  pivots away from the base segment  26 . 
     When a system has no acceleration, it may be said to be in a state of equilibrium. The state of equilibrium is stable if a small external disturbance causes oscillations about the equilibrium state. However, if a small external disturbance causes the system to diverge from its equilibrium state, then the equilibrium position is unstable. If, on the other hand, the system reacts to the disturbances and stays in the disturbed position, then the equilibrium position is neutral. 
     The stability of a system may be explained using the “ball on the hill” analogy which utilizes a position of a ball with respect to a hill flanked on both sides by valleys. A ball positioned in the valley is in a stable equilibrium position. If it is shifted from this position by a small amount, it will tend to return to the bottom of the valley or oscillate around it. However, a ball positioned on the top of the hill is in an unstable equilibrium position. Although the ball will stay in position if placed precisely on top of the hill, it will move to a different position if any disturbance occurs. Likewise, a ball positioned on the other side of the hill in the other valley is in a stable equilibrium position. 
     Because this system has two stable equilibrium positions, it is bistable. Because two local minima enclose a local maximum, two stable equilibrium positions will have an unstable position between them. Therefore, a bistable mechanism will have two stable equilibrium positions and at least one unstable equilibrium position. 
     Note that a ball positioned on the side of the hill is not in an equilibrium position. However, placing a stop on the side of the hill creates a new equilibrium position by the application of an external load. The stop could also be represented by a force of the proper magnitude and direction. This new equilibrium position is also stable. 
     Several methods have been developed to determine the stability of a system. The energy method, based on the Lagrange-Dirichlet theorem, states that a stable equilibrium position occurs at a position where the potential energy has a local minimum. Therefore, to establish the stability of a mechanism, the potential energy of the mechanism may be plotted over the mechanism&#39;s motion and any local minima represent stable positions. The potential energy curve is similar to the hill topography in the ball on the hill analogy. 
     Compliant bistable mechanisms gain their bistable behavior from the energy stored in the flexible segments which deflect to allow mechanism motion. This approach integrates desired mechanism motion and energy storage to create bistable mechanisms with dramatically reduced part count compared to traditional mechanisms incorporating rigid links, joints, and springs. 
     A bistable mechanism has two stable equilibrium positions within its range of motion. It achieves this behavior by storing energy during part of its motion, and then releasing it as the mechanism moves toward a second stable state. Compliant mechanisms, which gain motion through the deflection of their members, offer an economical way to accomplish bistable behavior. Because flexible segments store energy as they deflect, a compliant mechanism can use the same segments to gain both motion and two stable states, allowing a significant reduction in part count. 
     Bistable mechanisms offer two distinct, repeatable stable positions, allowing devices which utilize bistable mechanisms to require no power input to keep them in each position. Specific energy storage characteristics are necessary in these mechanisms to obtain the bistable behavior. 
     Referring to FIGS. 1,  2   a  and  2   b,  the switch mechanism  10  further includes two electrical contacts, a first electrical contact  90  and a second electrical contact  92  coupled to the segments  14 . Preferably, the first electrical contact  90  is disposed on the first arm segment  22  while the second electrical contact  92  is disposed on the base segment  26 . The first electrical contact  90  moves with the first arm segment  22  as the segments  14  move between the first and second positions  70  and  72 . Thus, the first electrical contact  90  moves between a first location  96 , as shown in FIGS. 1 and 2 a,  and a second location  98 , as shown in FIGS. 2 b.  In the first location  96 , the first electrical contact  90  is in a non-contacting relationship with the second electrical contact  92  and defines an “off” position. In the second location  98 , the first electrical contact  90  contacts the second electrical contact  92  and defines an “on” position. It is of course understood that the contacts  90  and  92  may be disposed on any appropriate segments  14 . 
     Referring to FIGS. 4 a  and  4   b,  an alternative embodiment of a bistable mechanism, indicated generally at  110 , is shown. Similar to the above described mechanism  10 , the alternative mechanism  110  has a plurality of segments, indicated generally at  114 , coupled end-to-end in series to form a continuous chain of segments. 
     The plurality of segments  114  includes a first relatively rigid base segment  126 , a second relatively rigid coupling segment  118 , and first and second arm segments  122  and  130 . The base segment  126  has first and second ends  140  and  142 . Similarly, the coupling segment  118  has first and second ends  146  and  148 . The first arm segment  122  is coupled between the first ends  140  and  146  of the base segment  126  and the coupling segment  118 . Similarly, the second arm segment  130  is coupled between the second ends  142  and  148  of the base segment  126  and the coupling segment  118 . 
     The first and second arm segments  122  and  130  are relatively flexible and resilient. The rigid coupling segment  118  is fixedly coupled to the adjacent flexible and resilient arm segments  122  and  130 . The rigid base segment  126  is coupled to the adjacent flexible and resilient arm segments by either flexible joints (not shown), or pin joints  154  and  155 . The pin joints  154  and  155  may be typical pin joints, as are well known in the art. The base segment  126  is coupled to the first arm segment  122  by a first pin joint  154 ; the coupling segment  118  is fixedly coupled to the first arm segment; the coupling segment  118  is fixedly coupled to the second arm segment  130 ; and the base segment  126  is coupled to the second arm segment  130  by a second pin joint  155 . 
     The sum of the pin joints  154  and  155 , the flexible joints (none shown), and the relatively flexible and resilient segments  122  and  130  is at least four. In the alternative mechanism  110 , there are two pin joints  154  and  155 , and two relatively flexible and resilient segments  122  and  130 , which sum to four. 
     As with the preferred embodiment of the mechanism  10 , the plurality of segments  114  in the alternative embodiment of the mechanism  110  may be integrally formed. The rigid coupling segment  118  and the first and second arm segments  122  and  130  are integrally formed. It is of course understood that the rigid base segment  126  may be integrally formed with the arm segments  122  and  130 , and that the pin joints  154  and  155  may be replaced with flexible joints. 
     The pin joints  154  and  155  allow the plurality of segments  114  to move relative to one another. At least one of the relatively flexible and resilient segments  122  and  130  operate to resist relative movement of the segments  114 , but allows the segments  114  to be selectively moved. The plurality of segments  114  cooperatively move with respect to one another between a first position  170 , as shown in FIG. 4 a,  and a second position  172 , as shown in FIG. 4 b.  In addition, at least one of the relatively flexible and resilient segments  122  and  130  biases the plurality of segments  114  between the two positions  170  and  172 . 
     Referring to FIG. 4 a,  the first position  170  preferably is a stable, static, equilibrium position, or the plurality of segments are in a position in which they are stable, static, and in equilibrium. The first position  170  is a low-energy position in which the relatively flexible and resilient segments  122  and  130  are substantially undeflected and store substantially no energy. Alternatively, the first position  170  may be a force loaded position in which the relatively flexible and resilient segments  122  and  130  are deflected and store energy. 
     Referring to FIG. 4 b,  the second position  172  is a force loaded position in which the first relatively flexible and resilient arm segment  122  is deflected and stores energy. Thus, the mechanism  110  or segments  114  may exert a force in the second position  172 . Alternatively, the second position  172  may be a stable, static, equilibrium position, or the plurality of segments are in a position in which they are stable, static, and in equilibrium. The second position  172  also may be a low-energy position in which the relatively flexible and resilient segments  122  and  130  are substantially undeflected and store substantially no energy. 
     The mechanism  110  also may be a micro-mechanism, or formed as a MEMS (micro-electro-mechanism system), as shown. Each segment  114  may have a length L which is less than 500 microns and a thickness +(FIG. 4 c ) less than 3 microns. MEMS mechanisms may be fabricated using a Multi-User MEMS Process (MUMPS) at MCNC. This process uses two released layers of polysilicon. The first layer has a thickness of 2.0 μm. In addition, the “stacked polysilicon” method as described by Comtois, John H. and Bright, Victor M. “Applications for Surface-Micromachined Polysicon Thermal Actuators and Arrays, “ Sensors and Actuators,  January 1997, pp. 19-25, Vol. 58, No. 1.” may be used to make small-length flexural pivots as thick as both layers, or 3.5 μm thick. FIG. 4c shows a cross-section of a pin joint, indicated generally at  190 , fixed to a substrate. The pin joint may be formed as shown in FIG. 4 c  with a disk  192  formed from the first layer  194  of polysilicon, and a post  196  formed from the second layer  198 . 
     Referring to FIGS. 5 a  and  5   b,  a schematic and a pseudo-rigid-body model, indicated generally at  110 ′, of the mechanism is shown. The pseudo-rigid-body model  110 ′ resembles, or corresponds to, a four-bar linkage. 
     To design compliant bistable planar MEMS, a specific class of mechanisms was defined, known as Young mechanisms. A Young mechanism is one that: has two revolute joints  154 ′ and  155 ′, and therefore, two links, where a link is defined as the continuum between two rigid-body joints; has two compliant segments  122 ′ and  130 ′, both part of the same link; and has a pseudo-rigid-body model which resembles a four-bar mechanism. 
     The first and second conditions, taken together, imply that the two pin joints  154 ′ and  155 ′ are connected with one completely rigid link  126 ′, while the other link consists of two compliant segments  122 ′ and  130 ′ and one or more rigid segments  118 ′. A general pseudo-rigid-body model of a Young mechanism  110 ′ is shown in FIG. 5 b.  In this model, the two revolute joints  154 ′ and  155 ′ are connected to ground (or rigid base segment  126 ′), while Pin A and Pin B represent complaint segments modeled by the pseudo-rigid-body model. 
     Young mechanisms make sense for MEMS for several reasons. For example, pin joints connected to the substrate (ground) can easily be fabricated with two layers of polysilicon, but true pin joints connecting two moving links require more layers. Also, the two pin joints help the mechanism to achieve larger motion, in general, by reducing the stress in the compliant segments. In addition, the two compliant segments give the mechanism the energy storage elements it needs for bistable behavior. 
     Three main classes of Young mechanisms may be defined, depending on the type of compliant segments used. These are: 
     Class I: Both compliant segments are fixed-pinned segments. 
     Class II: One compliant segment is a fixed-pinned segment, and the other is a small-length flexural pivot. 
     Class III: Both compliant segments are small-length flexural pivots. 
     A unique Young mechanism of Class I may be described using the seven parameters r 1 , r 2 , r 4 , θ 20 , θ 40 , I 2 , and I 4 , where each parameter is defined as: 
     r 1 —the distance between the centers of the pin joints. 
     r 2 —the length of the largest side link of the pseudo-rigid-body model. The length l 2  of the associated compliant fixed-pinned segment may be found from the equation                l   2     =       r   2     γ             (   12   )                         
     where γ is approximately 0.85, as approximated for any material properties, but may be tabulated for a wide range of loading conditions. 
     r 4 —the length of the shortest side link of the pseudo-rigid-body model. The length l 4  of the associated compliant fixed-pinned segment may be found using the same method used to find l 2 . 
     θ 20 —the initial value of θ 2  (defined in FIG. 5 b ) at the undeflected position. 
     θ 40 —the initial value of θ 4  (defined in FIG. 5 b ) at the undeflected position. An alternate approach to define the mechanism would be to specify the value of r 3  rather than one of the two initial angles. However, while r 3  describes the length of the third link in the pseudo-rigid-body model, it has little physical significance in the actual compliant mechanism. In addition, if only one angle is specified, the mechanism could take either the leading or the lagging form based on the link lengths, so that the definition of the mechanism would be less precise. 
     I 2 —the area moment of inertia of the flexible segment associated with link  2 . For a rectangular cross-section,              I   =       ht   3       1      2               (   13   )                         
     where h is the height of the beam (out of the plane of motion) and t is the segment&#39;s thickness (within the plane of motion). 
     I 4 —the area moment of inertia of the flexible segment associated with link  4 . It is given by Equation (13). 
     Given these values and the material&#39;s Young&#39;s modulus, the values of the torsional spring constants may be calculated from the equations                K   A     =         γ      K     θ            EI   2       l   2                 (   14   )                 K   B     =         γ      K     θ            EI   4       l   4                 (   15   )                         
     where γ and K θ  are approximately 0.85 and 2.65, as approximated for any material properties, but may be tabulated for a wide range of loading conditions. 
     Similar parameters are required to define mechanisms of Class II, but an additional variable is needed to define the length of the small-length flexural pivot. The parameters defining a Class II mechanism are: 
     r 1 , r 4 , θ 20 , θ 40 , and I 4 —same as for class I. 
     r 2 —the length of pseudo-link  2 , defined as the distance from the pin joint to the center of the small-length flexural pivot. No associated value of l 2  may be defined. 
     I 2 —the area moment of inertia of the small-length flexural pivot, given by Equation (13). 
     I s —length of the small-length flexural pivot. 
     Spring constant K B  is the same as for Class I, but K A  must be found from the equation                K   A     =       EI   2       l   s               (   16   )                         
     To design bistable Young mechanisms, equations must be used which relate the motion and potential energy of the mechanism. The motion of the model shown in FIG. 5 b  may be found as a function of θ 2  using rigid-body kinematics textbooks. The potential energy equation may be found by summing the energy stored in the two torsional springs: 
     
       
           V ={fraction ( 1 / 2 )}( K   A ψ A   2   +K   B ψ B   2 )  (17)  
       
     
     where V is the potential energy, K A  and K B  are the torsional spring constants, and ψ A  and ψ B  are the relative deflections of the torsional springs. These are given by 
     
       
         ψ A =(θ 2 −θ 20 )−(θ 3 −θ 30 )  
       
     
     
       
         ψ B =(θ 4 −θ 40 )−(θ 3 −θ 30 )  (18)  
       
     
     where the “0” subscript denotes the initial (undeflected) value of each angle. The minima of Equation (17) may be found by locating zeroes of the first derivative of V where the second derivative is positive. The first derivative of V with respect to θ 2  is                     V            θ   2         =         K   A            ψ   A          (     1   -     h   32       )         +       K   B            ψ   B          (       h   42     -     h   32       )                   (   19   )                         
     where h 32  and h 42  are the kinematic coefficients                h   32     =              θ   3              θ   2         =         r     2                       sin        (       θ   4     -     θ   2       )             r     3                       sin        (       θ   3     -     θ   4       )                     (   20   )                 h   42     =              θ   4              θ   2         =         r     2                       sin        (       θ   3     -     θ   2       )             r   4                     sin        (       θ   4     -     θ   3       )                     (   21   )                         
     The second derivative of potential energy is                       2        V            θ   2   2         =         K   A          (     1   -     2        h   32       +     h   32   2     -       ψ   A          h   32   ′         )       +       K   B          [       h   42   2     -     2        h   42          h   32       +     h   32   2     +       ψ   B          (       h   42   ′     -     h   32   ′       )         ]                 (   22   )               where                                        h   32   ′     =              h   32              θ   2         =         r   2       r   3            [           cos        (       θ   4     -     θ   2       )         sin        (       θ   3     -     θ   4       )              (       h   42     -   1     )       -           sin        (       θ   4     -     θ   2       )            cos        (       θ   3     -     θ   4       )             sin   2          (       θ   3     -     θ   4       )              (       h   32     -     h   42       )         ]                 (   23   )                 h   42   ′     =              h   42              θ   2         =         r   2       r   4            [           cos        (       θ   3     -     θ   2       )         sin        (       θ   3     -     θ   4       )              (       h   32     -   1     )       -           sin        (       θ   3     -     θ   2       )            cos        (       θ   3     -     θ   4       )             sin   2          (       θ   3     -     θ   4       )              (       h   32     -     h   42       )         ]                 (   24   )                         
     Any value of θ 2  for which Equation (19) is zero and Equation (22) is positive identifies a relative minimum of potential energy, and, thus, a stable equilibrium position. 
     The maximum nominal stress in the compliant segment during motion is another important quantity to consider. Compliant mechanism theory can be used to find this stress from the maximum angular deflection of each segment, ψ A,max  and ψ B,max . For either compliant segment, the maximum nominal stress may be approximated with the classical stress equation                σ     0      max       =         M   max        c     I             (   25   )                         
     where M max  may be approximated, using the pseudo-rigid-body model as the product of K and ψ max . Assuming a rectangular cross section,                σ     0      max       =       6          K      ψ     max         ht   2               (   26   )                         
     where h is the height of the compliant beam (the dimension out of the plane of motion) and t is its thickness (the dimension within the plane of motion). This nominal stress is the stress calculated without taking stress at fracture of previously-tested devices with similar stress concentrations. 
     To design the mechanisms presented here, the seven (Class I) or eight (Class II) parameters described above were varied to find mechanism configurations with two stable positions, as determined by the potential energy equation, without exceeding the polysilicon strength during motion. To avoid fracture, a maximum strain, equal to the ratio of ultimate strength to Young&#39;s modulus, S UT /E, was specified to be 1.05×10 -2 . This value was determined from prior experience in the design of compliant micro-mechanisms. 
     Referring to FIGS. 6 a  and  6   b,  an alternative embodiment of a bistable mechanism, indicated generally at  210 , is shown which is characterized as a class II Young&#39;s mechanism. Similar to the above described mechanisms  10  and  110 , the alternative mechanism  210  has a plurality of segments, indicated generally at  214 , coupled end-to-end in series to form a continuous chain of segments. 
     The plurality of segments  214  includes a first relatively rigid base segment  226 , and second and third relatively rigid segments  218  and  222 . The plurality of segments  214  includes first and second relatively flexible and resilient segments  228  and  230 . 
     The first rigid segment  222  is pivotally coupled to the base segment  226  by a pin joint  254 . The first flexible and resilient segment  228  is fixedly coupled to and between the first and second rigid segments  222  and  214 . The second flexible and resilient segment  230  is pivotally coupled to the rigid base segment  226  by a pin joint  255 , and fixedly coupled to the second rigid segment  218 . The first flexible and resilient segment  230  is coupled between the rigid base segment  226  and the second rigid segment  218 . 
     The sum of the pin joints  254  and  255 , the flexible joints (none shown), and the relatively flexible and resilient segments  228  and  230  is at least four. In the alternative mechanism  210 , there are two pin joints  254  and  255 , and two relatively flexible and resilient segments  228  and  230 , which sum to four. 
     As with the preferred embodiment of the mechanism  10 , the plurality of segments  214  in the alternative embodiment of the mechanism may be integrally formed. The rigid first and second segments  222  and  218 , and the first and second flexible and resilient segments  228  and  230 , are integrally formed. It is of course understood that the rigid base segment  226  may be integrally formed with the first rigid segment  222  and the second flexible and resilient segment  230 , and that the pin joints  254  and  255  may be replaced with flexible joints. 
     The plurality of segments  214  cooperatively move with respect to one another between a first position  270 , as shown in FIG. 6 a,  and a second position  272 , as shown in FIG. 6 b.  In addition, at least one of the relatively flexible and resilient segments  228  and  230  biases the plurality of segments  214  between the two positions  270  and  272 . 
     Referring to FIG. 6 a,  the first position  270  preferably is a stable, static, equilibrium position, or the plurality of segments are in a position in which they are stable, static, and in equilibrium. The first position  270  is a low-energy position in which the relatively flexible and resilient segments  228  and  230  are substantially undeflected and store substantially no energy. 
     Referring to FIG. 6 b,  the second position  272  is a force loaded position in which the second relatively flexible and resilient arm segment  230  is deflected and stores energy. Thus, the mechanism  210  or segments  214  may exert a force in the second position  272 . 
     As with the alternative embodiment of the mechanism  110  described above, this alternative embodiment of the mechanism  210  may be a micro-mechanism, or formed as a MEMS (micro-electro-mechanism system). Each segment  214  may have a length L which is less than 500 microns and a thickness +(FIG. 4 c ) less than 3 microns. 
     Referring to FIG. 7, a pseudo-rigid-body model, indicated generally at  210 ′, of the mechanism is shown. The pseudo-rigid-body model  210 ′ resembles, or corresponds to, a four-bar linkage. 
     Referring to FIG. 8, the preferred embodiment of the bistable mechanism  10  is shown in an application as a hinge, as opposed to an electrical switch. Thus, one segment, such as the base segment  26  is coupled to a cabinet or box  400 , while another segment, such as segment  18 , is coupled to a door or lid  410 . 
     It is to be understood that the described embodiments of the invention are illustrative only, and that modifications thereof may occur to those skilled in the art. Accordingly, this invention is not to be regarded as limited to the embodiments disclosed, but is to be limited only as defined by the appended claims herein.