Force Elements and Reaction Forces

Abstract

In the present chapter some of the most relevant applied forces and joint reaction forces are introduced. There are many types of forces that can be present in multibody systems, such as gravitational forces, spring-damper-actuator forces, normal contact forces, tangential or frictional forces, external applied forces and moments, forces due to elasticity of bodies, and thermal, electrical and magnetic forces. However, only the first six types of forces are relevant in the multibody systems of common application.

Figure 11.1 illustrates a body i acted upon by a gravitational field in the negative z direction. The choice of the negative z direction as the direction of gravity is totally arbitrary. However, in the present work, the gravitational field will be considered to be acting in this direction unless indicated otherwise. If w _i is the weight of the body i, resulting from the product of mass of the body by the gravitational constant, then the contribution of this force to the generalized vector of forces of body i is given by (Nikravesh 1988; Shabana 1989)

Fig. 11.1

Gravitational field acting on a body i

$$ {\mathbf{g}}_{i}^{(g)} = \left\{ {\begin{array}{*{20}c} 0 & 0 & { - w_{i} } & 0 & 0 & 0 \\ \end{array} } \right\}^{T} $$

(11.1)

Consider a single body force f _i acting on body i at point P _i , as shown in Fig. 11.2a. This force has three Cartesian components. In addition, a moment with respect to the body center of mass must be computed as (Jalón and Bayo 1994)

Fig. 11.2

A body i acted upon by a a single force b a pure moment

$$ {\mathbf{n}}_{i} = {\tilde{\mathbf{s}}}_{i}^{P} {\mathbf{f}}_{i} $$

(11.2)

Thus, the contribution to generalized vector of forces of a single force is

$$ {\mathbf{g}}_{i}^{(f)} \equiv \left\{ {\begin{array}{*{20}c} {{\mathbf{f}}_{i} } \\ {{\mathbf{n}}_{i} } \\ \end{array} } \right\} $$

(11.3)

When a pure moment with magnitude n _i acts on a body i, as shown in Fig. 11.2b, its contribution to the vector of forces of body i is given by

$$ {\mathbf{g}}_{i}^{(n)} \equiv \left\{ {\begin{array}{*{20}c} {\mathbf{0}} \\ {{\mathbf{n}}_{i} } \\ \end{array} } \right\} $$

(11.4)

Figure 11.3 shows a spring-damper-actuator element connecting bodies i and j through two points of connectivity P _i and P _j . The vector l that connects the points P _i and P _j can be evaluated by

Fig. 11.3

Spring-damper-actuator element connecting bodies i and j

$$ {\mathbf{l}} = {\mathbf{r}}_{j}^{P} - {\mathbf{r}}_{i}^{P} = {\mathbf{r}}_{j} + {\mathbf{A}}_{j} {\mathbf{s}}_{j}^{{{\prime }P}} - {\mathbf{r}}_{i} - {\mathbf{A}}_{i} {\mathbf{s}}_{i}^{{{\prime }P}} $$

(11.5)

The magnitude of this vector is

$$ l = \sqrt {{\mathbf{l}}^{T} {\mathbf{l}}} $$

(11.6)

The unit vector along the spring-damper-actuator element is defined as

$$ {\mathbf{u}} = \frac{1}{l}{\mathbf{l}} $$

(11.7)

The time rate of change of the damper length can be obtained by differentiating Eq. (11.6), yielding

$$ {\dot{l}} = \frac{1}{l}{\mathbf{l}}^{T} {\dot{\mathbf{l}}} $$

(11.8)

where ${\dot{\mathbf{l}}}$, in turn, is found from Eq. (11.5)

$$ {\dot{\mathbf{l}}} = {\dot{\mathbf{r}}}_{j}^{P} - {\dot{\mathbf{r}}}_{i}^{P} $$

(11.9)

Then, the resulting spring-damper-actuator force is evaluated as

$$ f^{sda} = k(l - l^{0} ) + c{\dot{l}}+ f^{a} $$

(11.10)

where the first term on the right-hand side is the spring force, the second term represents the damper force and the third term denotes the actuator force. The spring stiffness is represented by k, l is the deformed length, l ⁰ is the undeformed or natural length of the spring, c is the damping coefficient of the damper and ${\dot{l}}$ is the time rate of change of the damper length.

The forces that act on the bodies i and j can be evaluated as

$$ {\mathbf{f}}_{i}^{sda} = f^{sda} {\mathbf{u}}\quad {\text{and}}\quad {\mathbf{f}}_{j}^{sda} = - f^{sda} {\mathbf{u}} $$

(11.11)

Finally, the contribution to generalized vector of forces is given by

$$ {\mathbf{g}}_{i}^{(sda)} \equiv \left\{ {\begin{array}{*{20}c} {{\mathbf{f}}_{i}^{sda} } \\ {{\tilde{\mathbf{s}}}_{i}^{P} {\mathbf{f}}_{i}^{sda} } \\ \end{array} } \right\}\quad {\text{and}}\quad {\mathbf{g}}_{j}^{(sda)} \equiv \left\{ {\begin{array}{*{20}c} {{\mathbf{f}}_{j}^{sda} } \\ {{\tilde{\mathbf{s}}}_{j}^{P} {\mathbf{f}}_{j}^{sda} } \\ \end{array} } \right\} $$

(11.12)

As it was presented previously, the joint reaction forces and moments are expressed in terms of the Jacobian matrix of the constraint equations and a vector of Lagrange multipliers and expressed by Eq. (10.10). Thus, for instance, for a spherical joint between bodies i and j, the vector of reaction forces is expressed as

$$ {\mathbf{D}}^{T} {\varvec{\uplambda}} = \left[ {\begin{array}{*{20}c} { - {\mathbf{I}}} \\ {{\tilde{\mathbf{s}}}_{i}^{P} } \\ {\mathbf{I}} \\ { - {\tilde{\mathbf{s}}}_{j}^{P} } \\ \end{array} } \right]{\varvec{\uplambda}} $$

(11.13)

Equation (9.5) has been considered in Eq. (11.13). For a spherical joint, λ is a 3-vector representing exactly the reaction force acting at point P _i . The same force but in the opposite direction acts at point P _j , as it is shown in Fig. 11.4. It must be noted that a spherical joint does not produce a reaction moment. However, when a reaction force is moved to the corresponding mass center, the moment associated with that force must be included in the rotational equations of motion. These reaction moments are automatically taken care of by the Jacobian matrix (Nikravesh 1988).

Fig. 11.4

Reaction forces associated with a spherical joint

For the case of a spherical-spherical joint, the reaction force can be expressed in the form

$$ {\mathbf{D}}^{T} {\varvec{\uplambda}} = \left[ {\begin{array}{*{20}c} { - 2{\mathbf{d}}^{T} } \\ {2{\mathbf{d}}^{T} {\tilde{\mathbf{s}}}_{i}^{P} } \\ {2{\mathbf{d}}^{T} } \\ { - 2{\mathbf{d}}^{T} {\tilde{\mathbf{s}}}_{j}^{P} } \\ \end{array} } \right]{\varvec{\uplambda}} = \left[ {\begin{array}{*{20}c} { - {\mathbf{I}}} \\ {{\tilde{\mathbf{s}}}_{i}^{P} } \\ {\mathbf{I}} \\ { - {\tilde{\mathbf{s}}}_{j}^{P} } \\ \end{array} } \right]2{\mathbf{d}}\lambda $$

(11.14)

in which Eq. (10.23) has been employed. This composite joint contains a single Lagrange multiplier which its value is proportional to the magnitude of the reaction force. This reaction force acts exactly along the axis of the link that defines the joint, as it is illustrated in Fig. 11.5. The reaction moments are the result of the reaction forces having arms with respect to their corresponding center of mass (Schiehlen 1990).

Fig. 11.5

Reaction forces associated with a spherical-spherical joint