**3.3 Instable deformation behaviour of compliant mechanisms: Snap-through**

In contrast to the stable deformation behaviour of compliant mechanisms, the instable case has more than one equilibrium position for a particular load. The instable deformation behaviour shows snap-through or bifurcation.

In case of snap-though a sudden transition from one equilibrium position to another happens. Thereby a given load corresponds to several equilibrium positions. In Figure 7, a rotational structure is shown having a half-toric curve around the spherical curve in the origin state. One characteristic feature of such a mechanism is the potential bistable deformation behaviour, which can be enforced by the specified geometric parameters (shape, wall thickness, etc.).

Fig. 7. Snap-through of a curved mechanism having a monostable deformation behaviour

asymmetrical actuators), and material properties (actuators with variation of the material properties). A combination of materials of different elasticity and/or anisotropic materials can fulfil this characteristic, too. The numerical calculations approved, that the mechanism behaviour is influenceable geometrically and materially. Hence it can be adjusted to specific

Compared to other mechanisms, more complex motion trajectories can be easily provided

One application for a mechanism having the property of a direction reversion is using it as gripping fingers. Through model based optimisation a novel dependency of the load on the displacement could be achieved (Figure 6 a, broken line). Therein the characteristic displacement will not increase at a defined value of load irrespective of further increase of load. This property puts aside the sensory effort to monitor the gripping force. This force is already defined by the structure's mechanical properties. Such a structure is shown in

Fig. 6. a: I – Dependency of displacement u on internal pressure p of the mechanism introduced in Figure 4, II – p(u) for gripping-fingers with defined gripping-force; b:

**3.3 Instable deformation behaviour of compliant mechanisms: Snap-through** 

In contrast to the stable deformation behaviour of compliant mechanisms, the instable case has more than one equilibrium position for a particular load. The instable deformation

In case of snap-though a sudden transition from one equilibrium position to another happens. Thereby a given load corresponds to several equilibrium positions. In Figure 7, a rotational structure is shown having a half-toric curve around the spherical curve in the origin state. One characteristic feature of such a mechanism is the potential bistable deformation behaviour, which can be enforced by the specified geometric parameters

Fig. 7. Snap-through of a curved mechanism having a monostable deformation behaviour

gripping-fingers with defined gripping-force made of silicone rubber

behaviour shows snap-through or bifurcation.

(shape, wall thickness, etc.).

with unidirectional pressure change with the help of these mechanisms.

tasks.

Figure 6 b.

Fig. 8. Snap-through of a curved mechanism having a bistable deformation behaviour

Two different mechanisms with a big and small wall thickness are presented in Figure 7 and 8, respectively. By increasing the load, the angular point (centre point) of the median curvature of both mechanisms moves outwards up to the critical load (Figure 9 c). Herein the value of the critical load is different to the named structures. An arbitrary small rise of load causes a huge displacement, as soon as the critical load is reached. In this process the median curvature penetrates completely (state 2 in Figure 9). Removing the load causes the first mechanism to reverse to the original position (monostable deformation). This is demonstrated by state 3 in Figure 9 a. The second mechanism switches to another equilibrium position (bistable deformation), named in Figure 9 b with state 3. A sketch of both characteristics and the calculated positions by means of FEM is shown in Figure 9 c.

Fig. 9. Sketch of Snap-through behaviour of a curved mechanism: mechanism with monostable deformation (l.), mechanism with bistable deformation (r.)

Fig. 10. Applications as mechanical valves demonstrated for double-curved rotational mechanisms: a, b – pipe with one output is disabled for p=pcr, c – pipe with two outputs A and B, output B is closed if critical load is reached

Some applications of these mechanisms used as mechanical valves are shown in Figure 10. To generate bistable deformation (Figure 10 a) an opening is inserted in the centre point of

On the Mechanical Compliance of Technical Systems 351

2D-free floating location of the site of application under load. Two possible trajectories of this point and two realisations of the equilibrium are illustrated. Solutions have been determined numerically, a current application is the design of compliant grasping devices.

**-4 -3 -2 -1 0.5 -1**

x/R

Fig. 12. Equilibrium situations of a half-cycle shaped beam under external load; A force

The introduced classification which considers the deformation of compliant mechanisms is supposed to forward their development and to facilitate their implementation in rigid body systems or the functional expanded substitution of individual parts of the rigid body. The meaningful application of compliant mechanisms especially of such structures with instable static behaviour offers a great development potential. The role of the sensor system can be partly or completely adopted by "intelligent" mechanics. With the application of compliant mechanisms and structural elements, which show an instable static behaviour and therefore segue from one state to another depending on external conditions, elementary characteristics of the system can change (Risto et al., 2008; Linß et al., 2008; Risto et al., 2010; Griebel et al., 2010). Hence such systems will autonomously and directly adapt to the

In relation with functional dominating compliant characteristics many application-oriented tasks, for example gripping-fingers with particular characteristics, medical structural

Albanesi, A. E., Fachinotti, V. D., Pucheta, M. A. (2010). A review on design methods for

Beder, S., Suzumori, K. (1996) Elastic materials producing compliant robots. *Robotics and Autonomous Systems*, Vol.18, No.1, (July 1996), pp. 135-140, ISSN 0921-8890

compliant mechanisms. *Mecánica Computacional*, Vol.29, E. Dvorkin, M. Goldschmit,

2R

External Force

Bifurcation

Unloaded Configuration

Configuration Equilibrium I

Configuration Equilibrium II

**0**

constant in amount and directions traces the free end of the beam

M. Storti (Eds.), Buenos Aires, (2010), pp. 59-72.

**1 1.5 2**

y/R

**4. Conclusion** 

working conditions.

**5. References** 

elements and systems are conceivable.

**3**

the mechanism. Because of the critical dynamical pressure the mechanism is deformed and the flow is interrupted. In this case the current position guarantees the closure of the pipe. Low-pressure on the compliant part of the valve makes the flow possible again. The next Figure 10 b shows that the effect of the critical pressure yields to a deformation of the monostable mechanism, so the pipe is completely closed. If the pressure falls under the critical level, the original position is recaptured and the flow rate is reconstituted. The last example in Figure 10 c illustrates a valve installed in a pipe with two outputs. Output B is disabled, as soon as the critical pressure is reached. Decreasing the pressure enables this output.
