Bistable compliant mechanism

A compliant, bistable mechanism has a plurality of segments coupled end-to-end in a series to form a continuous chain of segments. The plurality of segments includes at least two rigid segments and at least one relatively flexible and resilient segment. Adjacent rigid segments are coupled by flexible joints or pin joints. The flexible and resilient segment is coupled to adjacent segments either fixedly or by pin joints. There are at least four pin joints, flexible joints, and/or flexible and resilient segments. The joints allow relative movement of the segments while the flexible and resilient segment resists movement and biases the segments. The segments move between first and second stable equilibrium positions. The segments have a pseudo-rigid-body model resembling a four-bar linkage. The segments and flexible joints may be integrally formed. First and second electrical contacts may be coupled to the segments to form an electrical connection as the segments move to one of the positions.

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.

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'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'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'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'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'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.ltoreq.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. 3a). The flexible joints 52 are substantially flexible and may be
 formed by a "living hinge". The pin joints 54 (FIG. 3a) 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. 3a). 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. 3a), 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. 3a and 3b, 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. 3c, 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
EQU V.sub.i =1/2K.sub.i.psi..sub.i.sup.2 (1)
 where V is the potential energy, K is the torsional spring constant, and
 .psi. is the angular deflection of each torsional spring. For each spring
 98' shown in FIG. 3c,
EQU .psi..sub.1 =.theta..sub.2 -.theta..sub.20
EQU .psi..sub.2 =(.theta..sub.2 -.theta..sub.20)-(.theta..sub.3
 -.theta..sub.30)
EQU .psi..sub.3 =(.theta..sub.4 -.theta..sub.40)-(.theta..sub.3
 -.theta..sub.30)
EQU .psi..sub.4 =.theta..sub.4 -.theta..sub.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
EQU V=1/2(K.sub.1.psi..sub.1.sup.2 +K.sub.2.psi..sub.2.sup.2
 +K.sub.3.psi..sub.3.sup.2 +K.sub.4.psi..sub.4.sup.2) (3)
 The values of each .psi. 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.sub.2, as shown in FIG. 3c, 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.sub.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:
 ##EQU1##
 by taking the derivative of this equation, it may be seen that
 ##EQU2##
 assuming that the moment at the initial position is zero. Therefore,
 M.sub.2 is equal to the first derivative of the energy with respect to the
 angle of the input link. This means that
 ##EQU3##
 The derivatives in Equation (6) above may be evaluated using Equation (2)
 and the additional formulas
 ##EQU4##
 As mentioned previously, the value of M.sub.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
 ##EQU5##
 When the value of M.sub.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.sub.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's "stiffness," where a high
 stiffness corresponds to a rapidly increasing restoring force.
 The mechanism shown in FIG. 3c may be further classified according to
 Grashof'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's criterion is mathematically stated as
 s+l.ltoreq.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'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
EQU s+l&gt;p+q non-Grashof
EQU 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. 3a) 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. 2a) or 54 (FIG. 3a)
 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 2a, and a second position 72, as shown in FIG.
 2b. 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 2a, 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'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, 2a and 2b, 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 2a, and a second location 98, as shown in FIGS. 2b. 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. 4a and 4b, 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. 4a, and a second position
 172, as shown in FIG. 4b. 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. 4a, 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. 4b, 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. 4c) 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 .mu.m. 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 .mu.m 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. 4c 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. 5a and 5b, 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. 5b. 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.sub.1, r.sub.2, r.sub.4, .theta..sub.20, .theta..sub.40,
 I.sub.2, and I.sub.4, where each parameter is defined as:
 r.sub.1 --the distance between the centers of the pin joints.
 r.sub.2 --the length of the largest side link of the pseudo-rigid-body
 model. The length l.sub.2 of the associated compliant fixed-pinned segment
 may be found from the equation
 ##EQU6##
 where .gamma. is approximately 0.85, as approximated for any material
 properties, but may be tabulated for a wide range of loading conditions.
 r.sub.4 --the length of the shortest side link of the pseudo-rigid-body
 model. The length l.sub.4 of the associated compliant fixed-pinned segment
 may be found using the same method used to find l.sub.2.
 .theta..sub.20 --the initial value of .theta..sub.2 (defined in FIG. 5b) at
 the undeflected position.
 .theta..sub.40 --the initial value of .theta..sub.4 (defined in FIG. 5b) at
 the undeflected position. An alternate approach to define the mechanism
 would be to specify the value of r.sub.3 rather than one of the two
 initial angles. However, while r.sub.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.sub.2 --the area moment of inertia of the flexible segment associated
 with link 2. For a rectangular cross-section,
 ##EQU7##
 where h is the height of the beam (out of the plane of motion) and t is the
 segment's thickness (within the plane of motion).
 I.sub.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's Young's modulus, the values of the
 torsional spring constants may be calculated from the equations
 ##EQU8##
 where .gamma. and K.sub..theta. 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.sub.1, r.sub.4, .theta..sub.20, .theta..sub.40, and I.sub.4 --same as for
 class I.
 r.sub.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.sub.2 may be defined.
 I.sub.2 --the area moment of inertia of the small-length flexural pivot,
 given by Equation (13).
 I.sub.s --length of the small-length flexural pivot.
 Spring constant K.sub.B is the same as for Class I, but K.sub.A must be
 found from the equation
 ##EQU9##
 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. 5b may be found as a function of .theta..sub.2 using
 rigid-body kinematics textbooks. The potential energy equation may be
 found by summing the energy stored in the two torsional springs:
EQU V=1/2(K.sub.A.psi..sub.A.sup.2 +K.sub.B.psi..sub.B.sup.2) (17)
 where V is the potential energy, K.sub.A and K.sub.B are the torsional
 spring constants, and .psi..sub.A and .psi..sub.B are the relative
 deflections of the torsional springs. These are given by
EQU .psi..sub.A =(.theta..sub.2 -.theta..sub.20)-(.theta..sub.3
 -.theta..sub.30)
EQU .psi..sub.B =(.theta..sub.4 -.theta..sub.40)-(.theta..sub.3
 -.theta..sub.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 .theta..sub.2 is
 ##EQU10##
 where h.sub.32 and h.sub.42 are the kinematic coefficients
 ##EQU11##
 The second derivative of potential energy is
 ##EQU12##
 Any value of .theta..sub.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, .psi..sub.A,max and .psi..sub.B,max. For either compliant
 segment, the maximum nominal stress may be approximated with the classical
 stress equation
 ##EQU13##
 where M.sub.max may be approximated, using the pseudo-rigid-body model as
 the product of K and .psi..sub.max. Assuming a rectangular cross section,
 ##EQU14##
 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's modulus, S.sub.UT /E, was specified to be
 1.05.times.10.sup.-2. This value was determined from prior experience in
 the design of compliant micro-mechanisms.
 Referring to FIGS. 6a and 6b, an alternative embodiment of a bistable
 mechanism, indicated generally at 210, is shown which is characterized as
 a class II Young'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. 6a, and a second
 position 272, as shown in FIG. 6b. 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. 6a, 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. 6b, 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. 4c)
 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.