**2. Influence factors of compliance**

First, the factors will be considered that determine the compliance of mechanisms generally, without a specific application.

The compliance of a mechanism is determined with respect to a displacement of a specific selected reference point or area of the mechanism as a result of an external force. That approach is necessary because the deformation of a mechanism is usually associated with varying displacements for differing areas of a mechanism. Accordingly, the compliance of a

On the Mechanical Compliance of Technical Systems 343

3 2

4

The compliance would be demonstrated by means of one point for constant bending stiffness in one dimension domain. Consequently, the constant compliance is the compliance

The mathematical model for the large displacements is based on the theory of curved beams. The following equations are nonlinear equations of equilibrium and constitutive

 

*Q Q Q Q EI Q*

1

*R*

cos 1

0

 

sin

Where - curvature of the loaded beam, Qi – internal forces, EI3 - bending stiffness, ui – displacements on the directions of x1 and x2, - angle between the tangent and the axis x1. All these parameters depend on the beam coordinate s (0sL). This system of nonlinear

0 0 0

1 2

equations was solved with the program MATHEMATICA with boundary conditions:

1 2

*Q L QL F L*

( ) ( ) ( ) ( ) ( ) ( )

1 2

As for the large displacements, we have different compliances depending on the particular position. To each position, a different force dF corresponds, which displaces a beam point on du. This situation is presented in Table 1 as a curve in 2D domain. The different compliance is characterized by ∂u/∂F dependent on u. By means of changing the prestressing of a compliant structure its compliance can be changed. Another example of this is a structure in the ring shape as a prestressed spring. While changing the clamping with the help of the parameter h, the compliance of the spring can be purposefully set onto

In addition to this, if we also take the temperature into consideration concerning a mechanism made of a temperature sensitive material, the compliance will depend on the

*u u*

the point P. Such compliance can be called the compliance of the first degree.

*u u*

 

*u R F EI*

3

4

(2)

(3)

(4)

The compliance applied of the end of beam can be given by:

equations of a curved beam (Zentner, 2003).

of the zero degree.

mechanism depends on the location of the reference point. At the same mechanism for the same load and boundary conditions, the evaluation of compliance with respect to different points leads to different results. For the reference usually the force application point or area is chosen. Depending on the application, the amount of the displacement vector of the reference point or its components are used when specifying the compliance.

The compliance is not a pure structure-related property, defined only by the initial geometric configuration and initial material properties. Geometric boundary conditions (location, type), the loading situation (location, type, magnitude, direction and loading history) and environmental conditions (e.g. thermal, chemical) must be also considered in order to formulate the compliance of a mechanism. On the material side the compliance is influenced by actual and previous environmental conditions and by the loading history (elastic or plastic behaviour). The geometric configuration for a given load is dependent on the material properties and geometric boundary conditions. Therefore the compliance of the mechanism depends on its actual geometric configuration and actual material properties and is valid only for the considered reference point by the given actual load and boundary conditions (Figure 1).

In practical applications, the boundary conditions, load levels and the reference point are given. In this case, the adjustment of the compliance can be achieved by appropriate design and material selection. The effort for the design depends on the variety of possible future applications of the mechanism.

Fig. 1. Influence factors on the mechanical compliance of mechanisms

#### **2.1 Variability of compliance**

Generally, the compliance can be either constant or variable. The constant compliance is impossible in the nature. However, we can use the theoretical models with constant compliance, for example, in the linear theory of small bending of beams. In this case the force is linear proportional to the displacement. Table 1 shows deferent compliance for a compliant quarter-circle shaped beam with radius R=20 mm. For this problem we use the Castigliano's theorem for describing the displacement of the end of beam with the geometric linear theory:

$$
\mu = \sqrt{\mu\_1^2 + \mu\_2^2} = \frac{FR^3}{2EI\_3} \sqrt{\frac{\pi^2}{4} + 1} \tag{1}
$$

mechanism depends on the location of the reference point. At the same mechanism for the same load and boundary conditions, the evaluation of compliance with respect to different points leads to different results. For the reference usually the force application point or area is chosen. Depending on the application, the amount of the displacement vector of the

The compliance is not a pure structure-related property, defined only by the initial geometric configuration and initial material properties. Geometric boundary conditions (location, type), the loading situation (location, type, magnitude, direction and loading history) and environmental conditions (e.g. thermal, chemical) must be also considered in order to formulate the compliance of a mechanism. On the material side the compliance is influenced by actual and previous environmental conditions and by the loading history (elastic or plastic behaviour). The geometric configuration for a given load is dependent on the material properties and geometric boundary conditions. Therefore the compliance of the mechanism depends on its actual geometric configuration and actual material properties and is valid only for the considered reference point by the given actual load and boundary

In practical applications, the boundary conditions, load levels and the reference point are given. In this case, the adjustment of the compliance can be achieved by appropriate design and material selection. The effort for the design depends on the variety of possible future

Generally, the compliance can be either constant or variable. The constant compliance is impossible in the nature. However, we can use the theoretical models with constant compliance, for example, in the linear theory of small bending of beams. In this case the force is linear proportional to the displacement. Table 1 shows deferent compliance for a compliant quarter-circle shaped beam with radius R=20 mm. For this problem we use the Castigliano's theorem for describing the displacement of the end of beam with the geometric

> 2 2 1 2

*uuu*

3 2

1

(1)

3

*EI*

2 4 *FR*

reference point or its components are used when specifying the compliance.

Fig. 1. Influence factors on the mechanical compliance of mechanisms

conditions (Figure 1).

applications of the mechanism.

**2.1 Variability of compliance** 

linear theory:

The compliance applied of the end of beam can be given by:

$$\frac{\mu}{F} = \frac{R^3 \sqrt{\pi^2 + 4}}{4EI\_3} \tag{2}$$

The compliance would be demonstrated by means of one point for constant bending stiffness in one dimension domain. Consequently, the constant compliance is the compliance of the zero degree.

The mathematical model for the large displacements is based on the theory of curved beams. The following equations are nonlinear equations of equilibrium and constitutive equations of a curved beam (Zentner, 2003).

$$\begin{aligned} Q\_1' - \kappa Q\_2 &= 0 \\ Q\_2' + \kappa Q\_1 &= 0 \\ EI\_3 \kappa' + Q\_2 &= 0 \\ \theta' &= \kappa - \frac{1}{R} \\ \mu\_1' &= \cos \theta - 1 \\ \mu\_2' &= \sin \theta \end{aligned} \tag{3}$$

Where - curvature of the loaded beam, Qi – internal forces, EI3 - bending stiffness, ui – displacements on the directions of x1 and x2, - angle between the tangent and the axis x1. All these parameters depend on the beam coordinate s (0sL). This system of nonlinear equations was solved with the program MATHEMATICA with boundary conditions:

$$\begin{aligned} Q\_1(L) &= 0\\ Q\_2(L) &= F\\ \kappa(L) &= 0\\ \theta(0) &= 0\\ \mu\_1(0) &= 0\\ \mu\_2(0) &= 0 \end{aligned} \tag{4}$$

As for the large displacements, we have different compliances depending on the particular position. To each position, a different force dF corresponds, which displaces a beam point on du. This situation is presented in Table 1 as a curve in 2D domain. The different compliance is characterized by ∂u/∂F dependent on u. By means of changing the prestressing of a compliant structure its compliance can be changed. Another example of this is a structure in the ring shape as a prestressed spring. While changing the clamping with the help of the parameter h, the compliance of the spring can be purposefully set onto the point P. Such compliance can be called the compliance of the first degree.

In addition to this, if we also take the temperature into consideration concerning a mechanism made of a temperature sensitive material, the compliance will depend on the

On the Mechanical Compliance of Technical Systems 345

In case of a joint with a local compliance, the rigid body joint is introduced in the most cases into the middle of the compliant part. With a compliant joint possessing the distributed compliance, it is important that the position of the rigid body joint is determined for a substitute model. It can be admitted that for the compliant joints the following conditions are available: if the extension of the joint is 10 or more times smaller than the biggest dimension of the whole mechanism, it is classified as a joint with a distributed compliance. Mechanisms with concentrated compliance behave like classic rigid link mechanisms, where kinematic joints are replaced with flexible hinges, and in consequence methods conceived to design rigid body mechanisms can be modified and applied successfully in this case (Albanesi et al., 2010). Mechanisms with distributed compliance are treated as a continuum flexible mechanism, and Continuum Mechanics design methods are used instead of rigid body kinematics (Albanesi et al., 2010). An overview of calculation methods of compliant

mechanisms is indicated in (Albanesi et al., 2010; Shuib et al., 2007).

Fig. 2. Classification of the static deformation of compliant mechanisms.

**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.

**3. Classification of compliant mechanisms concerning the deformation** 

In case the deformation-behaviour is chosen as a criterion for classifying compliant mechanisms, two subgroups – dynamic and static deformation – can be distinguished (Zentner & Böhm, 2009). Furthermore the static deformation behaviour of compliant mechanisms having a fixed compliance are considered, whereat influences caused by inertia are neglected. The static deformation behaviour is divided into stable and instable behaviour (Figure 2). Stable deformation behaviour is characterised by a surjective mapping of a particular load F on the deformation u. Thereby one can differentiate between a monotonic behaviour and the behaviour with a singular smooth reversion. In case of instable behaviour of compliant mechanisms snap-through (deformation-behaviour with jump-discontinuities) and bifurcation (local bifurcation of the behaviour) are possible.

two parameters, namely the displacement position and the temperature. In such a case we have compliance of the second degree. The surface F=F(u, T) would reflect the compliance depending on the two parameters, as a value for compliance ∂u/∂F for a definite temperature T and a particular displacement u.

The list of such parameters, which influence the compliance, can continue to be developed. If the compliance depends on the N parameter, we deal with the case of the compliance of the N degree.


Table 1. Compliance of three degrees, from zero till two, for a compliant quarter-circle shaped beam (EI3=100 Nmm2, F=1N)
