**3.1 Stable deformation-behaviour of compliant mechanisms: monotonic deformation**

Figure 3 shows an example of the monotonic deformation behaviour of a pneumatically driven compliant mechanism. By increasing the load (here: internal pressure) the characteristic deformation parameters such as the angle between the longitudinal-axis of the rigid structural parts also increases. This mechanism is used as a finger of a gripper.

On the Mechanical Compliance of Technical Systems 347

The characteristic parameter for such structure is the angle between the tangent of beam-end

**3.2 Stable deformation-behaviour of compliant mechanisms: Behaviour with direction** 

The deformation behaviour with direction reversion of a pneumatically driven compliant mechanism is shown in Figure 5. The geometry is optimised, to achieve a reversion by loading the internal pressure. The horizontal displacement of the working element was chosen as one characteristic parameter, whereat the vertical displacement and the distance of the working element to the clamping are also possible characteristic parameters which can be taken into account. In the present case the increasing load initially yields to an extension of the horizontal displacement of the working element and finally causes a smaller

Fig. 5. Static deformation behaviour with direction reversion of a compliant mechanism: at the top – FEM-calculations; at the bottom – compliant mechanism made of silicone rubber

Figure 6 a graphically presents the change of the characteristic parameters mentioned above. Therein the displacement of the working element u3 equals the reversal point of the

The considered deformation effect, referenced here as reversal effect, consists in reversion of movement, which the working element of the mechanism performs during unidirectional change of the internal pressure (Zentner et al., 2009). Thus movements in two opposite directions depend on the magnitude of the uniformly increasing pressure, and two opposite working directions follow one another. One of the important consequences of this effect is that the movement range of the mechanisms in the first working direction is not limited by the maximum pressure but by the material properties and geometric forming, whereby the sensors utilization can be minimized in grasping applications for example. The main characteristic criteria of the mechanisms considered are: (1) fulfilling movements in two opposite directions, where both movements are caused by unidirectional pressure-activating of the actuator; (2) the movement range in the first working direction being, independently from the size of the pressure load, bounded; (3) a given position of the working element of the mechanism is to be reached at two different pressure loads and therefore with different compliances. The reversal effect can be generally realized in two ways: (1) through cascading of several conventional structures; or (2) with a non-conventional actuator. In both cases such an effect can be achieved by both geometrical properties (geometrically

with characteristic parameters ui, i=1,..,3 (u2>u3>u1)

and the x1-axe. This angle is strictly monotonic increasing, when the pressure rises.

**reversion** 

displacement.

structure.

Fig. 3. Monotonic deformation behaviour of a pneumatically driven compliant mechanism made of silicone rubber

Another example concerns a compliant fluid driven structure, which is applied as a medical probe. The cross-section diameter of probe changes from at the fixed end (3 mm) to the free end (1 mm) linearly. In the probe model there is a hollow with constant diameter of 0.2 mm. An unstretchable thin fibre is embedded in the wall with constant distance h from the symmetry axes of probe-beam. Under inner pressure p in the hollow, with the cross-section of A, the probe structure will bend towards the embedded fibre. Linear material law is supposed. The equation for displacement of probe is calculated analytically in order to examine the possibilities to obtain the required bending.

$$\begin{aligned} Q\_1' - \kappa Q\_2 &= 0 \\ Q\_2' + \kappa \{Q\_1 - pA\} &= 0 \\ EI\_3 \kappa' + Q\_2 &= 0 \\ \theta' &= \kappa \\ u\_1' &= \cos \theta - 1 \\ u\_2' &= \sin \theta \end{aligned} \tag{5}$$

Corresponding boundary conditions are:

Fig. 4. Displacement behaviour of a compliant fluid driven structure by increasing internal pressure (1-5)

Fig. 3. Monotonic deformation behaviour of a pneumatically driven compliant mechanism

Another example concerns a compliant fluid driven structure, which is applied as a medical probe. The cross-section diameter of probe changes from at the fixed end (3 mm) to the free end (1 mm) linearly. In the probe model there is a hollow with constant diameter of 0.2 mm. An unstretchable thin fibre is embedded in the wall with constant distance h from the symmetry axes of probe-beam. Under inner pressure p in the hollow, with the cross-section of A, the probe structure will bend towards the embedded fibre. Linear material law is supposed. The equation for displacement of probe is calculated analytically in order to

 

cos 1

sin

0

3

0.005 0.01 0.015 0.02 0.025 0.03

54 2 3

0

( ) ( )

*pALh <sup>L</sup> EI*

() ()

*Q L pA L*

  0

(5)

(6)

1

x1

0

*Q Q pA*

*Q Q*

*EI Q*

1 2

1 2

*Q L*

1 2

*u u*

0.002 0.004 0.006 0.008 0.01 x2 ( ) ( ) ( )

 

Fig. 4. Displacement behaviour of a compliant fluid driven structure by increasing internal

( )

*u u*

 

made of silicone rubber

examine the possibilities to obtain the required bending.

Corresponding boundary conditions are:

h

fibre

A

pressure (1-5)

The characteristic parameter for such structure is the angle between the tangent of beam-end and the x1-axe. This angle is strictly monotonic increasing, when the pressure rises.

#### **3.2 Stable deformation-behaviour of compliant mechanisms: Behaviour with direction reversion**

The deformation behaviour with direction reversion of a pneumatically driven compliant mechanism is shown in Figure 5. The geometry is optimised, to achieve a reversion by loading the internal pressure. The horizontal displacement of the working element was chosen as one characteristic parameter, whereat the vertical displacement and the distance of the working element to the clamping are also possible characteristic parameters which can be taken into account. In the present case the increasing load initially yields to an extension of the horizontal displacement of the working element and finally causes a smaller displacement.

Fig. 5. Static deformation behaviour with direction reversion of a compliant mechanism: at the top – FEM-calculations; at the bottom – compliant mechanism made of silicone rubber with characteristic parameters ui, i=1,..,3 (u2>u3>u1)

Figure 6 a graphically presents the change of the characteristic parameters mentioned above. Therein the displacement of the working element u3 equals the reversal point of the structure.

The considered deformation effect, referenced here as reversal effect, consists in reversion of movement, which the working element of the mechanism performs during unidirectional change of the internal pressure (Zentner et al., 2009). Thus movements in two opposite directions depend on the magnitude of the uniformly increasing pressure, and two opposite working directions follow one another. One of the important consequences of this effect is that the movement range of the mechanisms in the first working direction is not limited by the maximum pressure but by the material properties and geometric forming, whereby the sensors utilization can be minimized in grasping applications for example. The main characteristic criteria of the mechanisms considered are: (1) fulfilling movements in two opposite directions, where both movements are caused by unidirectional pressure-activating of the actuator; (2) the movement range in the first working direction being, independently from the size of the pressure load, bounded; (3) a given position of the working element of the mechanism is to be reached at two different pressure loads and therefore with different compliances. The reversal effect can be generally realized in two ways: (1) through cascading of several conventional structures; or (2) with a non-conventional actuator. In both cases such an effect can be achieved by both geometrical properties (geometrically

On the Mechanical Compliance of Technical Systems 349

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

Fig. 9. Sketch of Snap-through behaviour of a curved mechanism: mechanism with

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

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

monostable deformation (l.), mechanism with bistable deformation (r.)

and B, output B is closed if critical load is reached

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.

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 tasks.

Compared to other mechanisms, more complex motion trajectories can be easily provided with unidirectional pressure change with the help of these mechanisms.

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 Figure 6 b.

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