A compliant bistable mechanism is a device that can maintain two distinct positions without power input . A multitude of different compliant bistable mechanisms has been proposed and studied . These mechanisms have many potential applications such as switches , closures, relays , reconfigurable robots , and landing gear mechanisms . Due to their negative-stiffness behaviors around the unstable equilibrium positions (a bistable mechanism has two stable equilibrium positions and an unstable equilibrium position), they have been used as negative stiffness building blocks for achieving static balancing in compliant mechanisms . Bistable mechanisms of different types have also been used as fundamental building blocks for synthesis of multistable mechanisms that have more than two stable positions . also utilized a bistable mechanism for energy harvesting. The wide use of bistable mechanisms motivates the development of an approach for accurately modeling their kinetostatics (i.e., the load-displacement relationship).

However, the nonlinearity associated with the bistable behavior often complicates the modeling. The pseudo-rigid-body method (PRBM) , which approximates the nonlinear deflection as motion of rigid links, is particularly useful in the early design phases of bistable mechanisms, but inaccurate for the later design phases. Currently nonlinear finite element analysis (NFEA) is still an important tool for obtaining accurate kinetostatic behavior of bistable mechanisms during the late design phases.

In this paper, we propose a rational function called “tri-root bistable function” for the purpose of easily and explicitly representing the complete kinetostatic behaviors of bistable mechanisms. This function provides bistable mechanism designers an explicit and easy way to represent their design requirements for bistable behaviors. The tri-root bistable function has a 3rd degree polynomial numerator and a 2nd degree polynomial denominator, and is capable of capturing the key features of a complete bistable kinetostatic curve (including the external kinetostatic behaviors, ) accurately with fewer parameters. If the key points of a kinetostatic behavior are given as the design requirements of a bistable mechanism, the tri-root bistable function enables the construction of the whole bistable curve. Using the constructed function as the target bistable curve, an optimization process can be employed to search for a desired bistable mechanism design . Moreover, for a given type of bistable mechanism, a regression analysis from available NFEA and/or experimental results can identify the relationships between the tri-root bistable function parameters and the mechanism's design parameters (including both the geometric and the physical parameters). Once the relationships are known for a specific type of bistable mechanisms the design of future mechanisms of this type is greatly simplified.

The rest of this paper is organized as follows: Sect.  reviews different ways for describing the complete bistable kinetostatic behaviors and presents this new rational function. Section  illustrates how accurate the rational function is in representing the complete bistable kinetostatic behaviors using three different kinds of compliant bistable mechanisms. Section  presents the closed-form equations of the kinetostatic behavior of a classic fully-compliant bistable mechanism, which are achieved using a multi-variable nonlinear regression (a mathematical optimization and analysis software called 1stOpt is used). In Sect. , a fully-compliant statically balanced mechanism consisting of three different classic fully-compliant bistable mechanisms is illustrated to validate the accuracy of the six closed-form equations. The concluding remarks are made in the last section.

In this section, we will take the fully-compliant bistable mechanism  shown in Fig.  as an example to demonstrate the representation of bistable behaviors. Figure  plots the kinetostatic behavior curve of a bistable mechanism whose design parameters are listed in Table , which are achieved using nonlinear finite element analysis (NFEA). The whole curve can be divided into three sections: the bistable behavior section, the reverse behavior section  and the post-bistable behavior  section. We denote the whole force-deflection characteristics as F=F(x) where x is the travel distance of the shuttle from its first stable equilibrium position (i.e., the as-fabricated position) along the y axis. As illustrated in Fig. , the function should have three roots, i.e., x1, x2 and x3, where x1 and x3 correspond to the two stable equilibrium positions while x2 corresponds to the unstable equilibrium potion. Moreover, x4 and x5 correspond to the two switching positions named by .

In order to model this highly nonlinear behaviors of this type of mechanism, employed a piecewise third-order function to approximate the bistable behaviors: F1(x)=f1(x)x1≤x<x4f2(x)x4≤x<x5f3(x)x5≤x≤x3 satisfying F1(x1)=F1(x3)=F1(0)=0dF1dxx=x4=dF1dxx=x5=0f1(x4)=f2(x4),f2(x5)=f3(x5)

It was suggested that subfunctions f1(x) and f2(x) were second-order polynomials, and f3(x) was a third-order polynomial. As shown in Fig. , the function F1(x) can be yielded as: F1(x)=f1(x)=-0.8588x2+4.0191x0≤x<2.34f2(x)=0.1019x3-1.5686x2+5.6666x-1.27472.34≤x<7.92f3(x)=3.6603x2-57.9791x+225.4467.92≤x≤8.985

As shown in Fig. , Eq. () satisfies the key points of the bistable behavior accurately, but cannot describe the whole bistable behavior well.

used a ninth-order polynomial to model the nonlinear spring stiffness of the mechanism as F2(x)=∑i=09kixi Third, fifth, seventh, eighth and ninth-order polynomials have been used to model the highly nonlinear force-displacement curve. However, the curves generated by the polynomials with order less than nine do not fit the force-displacement curve well, as shown in Fig. , and the expression can not capture the bistable behaviors directly.

To address these problems, this paper proposes a continuous rational function called “tri-root bistable function” to accurately describe the complete kinetostatic behaviors of the bistable mechanisms with fewer parameters. Considering that the bistable curve contains three zeros, the tri-root bistable function has a cubic polynomial numerator and quadratic polynomial denominator (we empirically chose the denominator based on the testing results among different functions) and is expressed as F3(x)=p1(x-x1)(x-x2)(x-x3)x2+q1x+q2=p1x3+p2x2+p3x+p4x2+q1x+q2 where, (4q2-q12)>0 (i.e., (x2+q1x+q2)>0), p1>0 and x1<x2<x3 (x1, x2 and x3 correspond to the three zeros of F3). According to Vieta's formulas, the relationships between the three zeros and parameters pi(i=1,2,3,4) are given as: x1+x2+x3=-p2p1x1x2+x2x3+x1x3=p3p1x1x2x3=-p4p1

For the five key points shown in Fig. , we have F3(x1)=p1x13+p2x12+p3x1+p4x12+q1x1+q2=p1x13+p2x12+p3x1+p4=0F3(x2)=p1x23+p2x22+p3x2+p4x22+q1x2+q2=p1x23+p2x22+p3x2+p4=0F3(x3)=p1x33+p2x32+p3x3+p4x32+q1x3+q2=p1x33+p2x32+p3x3+p4=0

Taking the first derivative of F3 with respect to x yields the stiffness of the compliant bistable mechanism: F3′(x)=(3p1x2+2p2x+p3)(x2+q1x+q2)-(2x+q1)(p1x3+p2x2+p3x+p4)(x2+q1x+q2)2 thus we have F3′(x4)=(3p1x42+2p2x4+p3)(x42+q1x4+q2)-(2x4+q1)(p1x43+p2x42+p3x4+p4)(x42+q1x4+q2)2=0 F3′(x5)=(3p1x52+2p2x5+p3)(x52+q1x5+q2)-(2x5+q1)(p1x53+p2x52+p3x5+p4)(x52+q1x5+q2)2=0 and F3(x4)=p1x43+p2x42+p3x4+p4x42+q1x4+q2=FmaxF3(x5)=p1x53+p2x52+p3x5+p4x52+q1x5+q2=Fmin At the first and second switching positions (x4 and x5), the stiffnesses of the mechanism are equal to zero.

Furthermore, the total stain energy stored in a deflected bistable mechanism can be obtained by integrating the tri-root bistable function:

V(x)=∫x1xF3(x)dx=∫x1xp1x3+p2x2+p3x+p4x2+q1x+q2dx=2p4-p3q1+p2q12-p1q13-2p2q2+3p1q1q2-q12+4q2arctan⁡(2x+q1-q12+4q2)+12[p1x2+2p2x-2p1q1x+(p3-p2q1+p1q12-p1q2)ln⁡(x2+q1x+q2)] At the first and second stable equilibrium positions (x=x1 and x=x3), the total energies are minimum.

For a compliant bistable mechanism, if the five key points of the kinetostatic behavior curve are given as the design requirements (i.e. the five key points are known), the whole bistable curve can be plotted by the tri-root bistable function, which can be achieved easily by solving Eqs. (), (), () and () simultaneously. Because there are six parameters in Eq. (), only six equations are required among these seven equations. In this paper, we take Eqs. (), () and () to obtain the solutions for the six parameters. In addition, when the fitting curves passing through the origin (i.e., F(x1)=0 when x1=0), the accuracy of the regression analysis may be ruined , which can be avoided by translating the initial position to (-1, 0).

Conversely, for a given compliant bistable mechanism, the bistable kinetostatic behaviors can be captured easily and explicitly by the tri-root bistable function instead of the PRBM and NFEA. This requires us to find the relationships among the five key points and the design parameters firstly, which will be discussed in the following part of this paper.

In this section, three different types of compliant bistable mechanisms are utilized to illustrate the use of the tri-root bistable function in representing kinetostatic behaviors of the bistable mechanisms.

In this section, with the help of a mathematical optimization and analysis software called 1stOpt, we make the regression analysis for the classic fully-compliant bistable mechanism shown in Fig.  (its design relies heavily on NFEA because there is no appropriate model for this kind of bistable mechanism). The regression analysis can obtain expressions that explicitly express the relationships between the design parameters and the key performances of bistable designs in the form of polynomials.

As shown in Fig. , the classic fully-compliant bistable mechanism has six main design parameters (L1, L2, L3, θ1, θ2 and θ3). Given a group of design parameters, to model its bistable kinetostatic behaviors with the tri-root bistable function (Eq. ), we have to find out the relationships among the five key points and design parameters. Here, we propose six closed-form equations to describe the relationships. Here x2 is expressed as: x2=a2L12+b2L32+c2θ13+d2θ33+∑i=02∑j=01∑k=02∑l=01∑m=01∑n=01gijklmnθ1iθ2jθ3kL1lL2mL3n where a2, b2, c2, d2 and gijklmn are the unknown coefficients of all the terms in Eq. (), which can be obtained by 1stOpt, which is introduced in Appendix A.

Similarly, the key points x3, x4, x5, F(x4) and F(x5) can be given as x3=a3L13+b3L33+c3θ12+d3θ32+∑i=01∑j=01∑k=01∑l=02∑m=01∑n=02hijklmnθ1iθ2jθ3kL1lL2mL3nx4=a4L14+b4L34+c4θ12+d4θ32+∑i=01∑j=01∑k=01∑l=03∑m=01∑n=03fijklmnθ1iθ2jθ3kL1lL2mL3nx5=a5L14+b5L34+c5θ12+d5θ32+∑i=01∑j=01∑k=01∑l=03∑m=01∑n=03pijklmnθ1iθ2jθ3kL1lL2mL3nF3(x4)=a40L13+b40L33+c40θ12+d40θ32+∑i=01∑j=01∑k=01∑l=02∑m=01∑n=02rijklmnθ1iθ2jθ3kL1lL2mL3nF3(x5)=a50L13+b50L33+c50θ13+d50θ33+∑i=02∑j=01∑k=02∑l=02∑m=01∑n=02qijklmnθ1iθ2jθ3kL1lL2mL3n respectively.

Four hundred fifteen classic fully-compliant bistable mechanism designs were analyzed by NFEA in order to achieve the five key points for nonlinear regression. The scopes of the geometric parameters of the designs are: 4 mm ≤ L1 ≤ 16 mm, 15 mm ≤ L2 ≤ 25 mm, 4 mm ≤ L3 ≤ 16 mm, 0 ≤θ1≤ 12∘, 7∘ ≤θ2≤ 16∘, and 0 ≤θ3≤ 12∘. The FEA model of each design was meshed with SOLID187 elements in ANSYS. A SOLID187 element is defined by ten nodes, each of which has three degrees of freedom. The model was meshed automatically by ANSYS. To ensure the accuracy of the FEA results, the mesh of each model was refined in the compliant segments so that no less than four elements were generated across the thinnest part, as shown in Fig. 9. Each model was fixed on the both ends of the compliant limb, loaded with a displacement on the shuttle, and the geometric nonlinearity option was turned on (Command “NLGEOM, ON”). When the solution was done, the corresponding force-displacement behaviors were recorded, which were then used to obtain the five key points.

Equations ()–() were obtained by 1stOpt through fitting the FEA results of 415 designs. These equations explicitly express the relationships between the design parameters and the key performances of bistable designs in the form of polynomials, thus can serve as a mathematical model for this kind of bistable mechanism. The regression results are presented in Figs. – in Appendix B, and greatly simplify the design of future mechanisms of the classic fully-compliant bistable mechanism.

Regressing through the origin may ruin the accuracy of the fitting results . To avoid this problem, we chose (-1, 0) as the initial positions of the bistable curves in this paper. Once the kinetostatic behavior curve is determined, we translate the whole curve to the right of one unit (i.e. put the initial position of the curve at (0,0)).

Given the design parameters listed in Table , then the five key points of the bistable kinetostatic behavior curve can be calculated with the six closed-form equations (Eqs. –). Thus the tri-root bistable function can be determined, which is plotted in Fig.  and expressed as: F3(x)=-11.0208x3+129.1472x2-276.8704x-417.0385x2+5.4928x-134.5362 where x is the travel displacement of the shuttle from its initial equilibrium position. The total strain energy stored in the whole bistable mechanism at any position is given as: V(x)=∫-1x-11.0208x3+129.1472x2-276.8704x-417.0385x2+5.4928x-134.5362dx-1≤x≤7.985

where, x=7.985 is the second stable equilibrium position. The curve of strain energy is plotted compared with nonlinear finite element analysis (NFEA) in Fig. . It is obvious that the errors are very little and the tri-root bistable function can mathematically model the classic fully-compliant bistable mechanism accurately. This may greatly simplify the modeling of future mechanisms of this type.

In this section, to validate the regression results, we present a fully-compliant statically-balanced mechanism combining three different classic fully-compliant bistable mechanisms that can ideally provide zero stiffness free motion. This mechanism has been proposed and studied by .

As shown in Fig. , the fully-compliant statically-balanced mechanism is designed with a specific tristable mechanism (a) and a balancing bistable mechanism (b): the tristable mechanism consists of two different compliant bistable mechanisms (i.e., Part A and Part B) connected in series, as shown in Fig. (a). Its design parameters are shown in Fig.  and listed in Table . The force-displacement characteristics of the tristable mechanism are achieved using nonlinear finite element analysis (NFEA) and plotted in Fig. . To statically balance the whole mechanism, proposed a method that: first supposes there exit an ideal balancing bistable mechanism which can counterbalance the given tristable mechanism. Then we can obtain the force-displacement characteristics of the supposed bistable mechanism and the approximate statically-balanced domain (from y1 to y2, shown in Fig. ). Finally, a group of ideal design parameters for the balancing bistable mechanism is found using a particle swarm optimizer (PSO)  integrated with ANSYS. Reference  gives a detailed description for this process.

To avoid the optimization process, we propose a straightforward method combining the tri-root bistable function and regression analysis results introduced above: First we suppose an ideal balancing bistable mechanism and obtain its force-displacement characteristics as shown in Fig. , then we get the five key points x1=0, x2=10.93 mm, x3=16.87 mm, x5=14.5 mm and F(x5)=-4.7. We assume H=6 mm, θ1=θ3=0, w1=w3=0.8 mm and w2=3 mm for the balancing bistable mechanism, thus the unknown design parameters we want are reduced to 4 (L1, L2, L3 and θ2). By solving the four closed-form equations (Eqs. , , and ) simultaneously, the four design parameters can be determined, which are listed in Table .

The force-displacement characteristics of the tristable mechanism, balancing bistable mechanism and the balanced domain of the whole statically-balanced mechanism are plotted in Fig. .

Figure  shows that the whole mechanism has approximately balanced behavior during y1 and y2. Compared with the optimized results obtained by , the statically-balanced behaviors achieved in this paper are accurate and convenient. It indicates that this method can substitute the optimization process in some respects. Meantime, it also validates the high accuracy of the regression analysis made above.

In this paper, we proposed a rational function called “tri-root bistable function” to describe the complete bistable kinetostatic behaviors briefly and accurately with few parameters. The tri-root bistable function has been shown to have a high precision by three different kinds of compliant bistable mechanisms. Then, six closed-form equations capturing the relationships among the five key points and design parameters were proposed and obtained by a multi-variable nonlinear regression analysis for the classic fully-compliant bistable mechanism. This process was achieved by 1stOpt. Finally, a fully-compliant statically-balanced mechanism consisting of three different classic fully-compliant bistable mechanisms was presented and designed with the proposed method to show the capability of the tri-root bistable function in design and analysis of bistable mechanisms.

It should be noted that, the tri-root bistable function and regression analysis can also be employed in other types compliant bistable mechanisms (such as the Young bistable mechanism and FTBM illustrated in this paper) for modeling and design.