Bi-Stable Structures
Three different concepts for bi-stable linear mechanisms have been developed. Prototypes of each concept were manufactured and tested on an Instron materials
testing machine. The experimental results were then compared to analytical results derived for each model.
4 Bar, 2 Spring Mechanism (4B2S).
Fig 1. 4B2S
The 4B2S mechanism is a two dimensional, 4 bar, 2 spring mechanism. All the bars are of equal length and connected to each other by revolute joints with parallel axes. All of the four joints are connected to the joint opposite with a length of steel tape spring. At the joints between the tape springs and the bars the only constrained axis is the one along the tape spring.
The geometry of the linkage makes it impossible for both tape springs to be straight (unbuckled) at any time. When the tape springs are buckled, they exert a relatively small force on the four bar linkage. This force is considerably lower than the force required to buckle the tape springs. As a result, the mechanism is stable and stiff when one tape spring or the other is straight. The mechanism has relatively low stiffness when both tape springs are buckled.
Using the buckling characteristics and moment-rotation characteristics of the tape springs, a force deflection plot could be derived. This can be compared to the plot obtained from the Instron. The dotted line in the analytical plot.
Fig 2. force-length plot
Having determined that a good correlation was achieved between the analytical and experimental results a design plot was created. The output criteria will typically be $F_{max}, F_{min}, L_{max}$ and $L_{min}$. The design variables in the mechanism are the eccentricity of the loading on the two tape springs, the area and second moment of area of the two tape springs, the number to tape springs and the length of the tape springs and the bars. Given the number of variables (11) and the number of known parameters (4), it is not possible to set up a sufficient number of equations to fix all the variables by solving simultaneous equations.
In order to simplify matters, the following assumptions were made. The values of $A$ and $I$ are fixed, in this case given the values of 25.4 mm Sears carpenter's tape. $B$ is set to an arbitrary value of $\frac{L_(max)}{4}\sqrt{5}$. Finally, $n_a$ and $n_b$ are set to the minimum value possible in order to achieve the required forces.
As an example, the design plot for the geometry of the prototype 4B2S mechanism has been included below.
Fig 3. design chart
4B4S mechanism
The four bar four spring (4B4S) mechanism in Figure 4 is constructed from 4 identical pieces of tape spring and 4 pieces of angled steel. It is stable when either pair of opposing tape springs are straight. In between these two states, the mechanism has very low stiffness.
Figure 4 4b4s
The critical upward and downward forces in the extended and compacted states can be calculated in the following manner.
Figure 5 critical force diagram
Knowing what the critical moment of the tape spring is along with the constant moment applied by the buckled tape spring, it is possible to calculate the critical force. The calculated forces are a factor of three lower than those measured experimentally. This discrepancy is due to the poor understanding of the critical bending moment of the tape springs when they are bent so that the free edges are in compression.
6 S mechanism
The 6 Spring (6S) mechanism shown in Figure~\ref{6S_Collage} consists of 6 pieces of tape spring and 4 connector elements. The connector elements are circular discs of 9 mm thick plywood. The discs have thin radial slots cut into them that allow tape springs to be glued in place. In the compact state, stability is provided by the diamond configuration of the four edge springs as in Figure~\ref{6S_Collage}(a). Stability in the extended state is provided by the two central tape springs as in Figure~\ref{6S_Collage}(b). The inside tape springs act in tension/compression whereas the outside tape springs are subjected to shear forces and bending moments.
Figure 6 6S
The central two tape springs behave as the tape springs in the 4B2S mechanism. Using the same analysis, the buckling load of each spring was calculated to be 305 N. The total calculated buckling load of the 6S mechanism in the extended configuration was 610 N. This is some 45% higher than the experimental value of 420 N.
The analysis of the outer tape spring is very similar to the one applied to the 4B4S mechanism, once again, the analysis yields a value that is a factor of three out from the experimental value of 34 N.