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Section 10.4 Frictionless Joints
The contact action screw which results from the connection of two rigid bodies \(\cB_1\) and \(\cB_2\) by a frictionless joint can be simplified due to the fact that the local contact force is directed along the local unit normal vector \(\bn_{12}\) to the surfaces in contact
\begin{equation}
\bof_{1 \to 2} ^c (Q) = N_{1 \to 2} (Q) \bn_{12} (Q)\tag{10.4.1}
\end{equation}
at any point
\(Q\) of the contacting surfaces. We list below the expressions of the action screw
\(\{ \cA^c_{1 \to 2} \}\) for the six lower kinematic pairs described in
Chapter 7 .
Subsection 10.4.1 Frictionless Pivot
Recall that a
pivot is a one-degree-of-freedom joint which permits a rotation about a common axis
\(\Delta \equiv (O_1 , \bz_1 ) = (O_2 , \bz_2)\) of
\(\cB_1\) and
\(\cB_2\text{.}\) The corresponding kinematic screw is given by
\begin{equation}
\{ \cV _{2 / 1} \} =
\left\{
\begin{array}{c}
\bom_{2/1} = \om_z \bz_1 \\
\bze
\end{array}
\right\}_{A\in \Delta}\tag{10.4.2}
\end{equation}
Then, assuming frictionless contact, the corresponding action contact screw \(\{ \cA^c_{1 \to 2} \}\) satisfies
\begin{equation}
\bM^c_{A, 1 \to 2} \cdot \bz_1 = 0\tag{10.4.3}
\end{equation}
for any point
\(A\) of axis
\(\Delta\) as was shown in
Example 10.3.18 . All other components of the action screw are in general non-zero:
\begin{equation*}
\{ \cA_{1 \to 2}^c \} =
\left\{
\begin{array}{c}
R_x \bx_1 +R_y \by_1 +R_z \bz_1 \\
M_{Ax} \bx_1 +M_{Ay} \by_1
\end{array}
\right\}_{A\in \Delta}
\end{equation*}
Figure 10.4.1.
Subsection 10.4.2 Frictionless Slider
In a
slider , a prismatic surface of
\(\cB_2\) coincides with a prismatic surface of
\(\cB_1\) thus restricting the motion to a rectilinear translation along a direction
\(\bx_1 = \bx_2\) common to
\(\cB_1\) and
\(\cB_2\text{.}\)
Figure 10.4.3.
This is a one-degree-of-freedom joint, characterized by the following kinematic screw
\begin{equation}
\{ \cV _{2 / 1 } \} =
\left\{
\begin{array}{c}
\bze \\
v_{2x} \bx_1
\end{array}
\right\}\tag{10.4.4}
\end{equation}
Then, assuming frictionless contact, the corresponding contact action screw \(\{\cA^c_{ 1 \to 2} \}\) satisfies
\begin{equation}
\bR^c_{1 \to 2} \cdot \bx_1 = 0\tag{10.4.5}
\end{equation}
All other components of the action screw are non-zero:
\begin{equation*}
\{ \cA_{1 \to 2}^c \} =
\left\{
\begin{array}{c}
R_y \by_1 +R_z \bz_1 \\
M_{Ax} \bx_1 +M_{Ay} \by_1 +M_{Az}\bz_1
\end{array}
\right\}_A
\end{equation*}
Again one can verify again that \(\{ \cV _{2 / 1} \} \cdot \{ \cA_{1 \to 2}^c \} =0\text{.}\)
Subsection 10.4.3 Frictionless Slider-Pivot
A
slider-pivot is a two-degree-of-freedom joint which permits both a rotation about and a translation along an axis
\(\Delta (O_1 , \bz_1 )\text{.}\)
\begin{equation}
\{ \cV _{2 / 1} \} =
\left\{
\begin{array}{c}
\om_z \bz_1 \\
v_{2z}\bz_1
\end{array}
\right\}_{A \in \Delta}\tag{10.4.6}
\end{equation}
Then, assuming frictionless contact, the corresponding contact action screw \(\{ \cA_{1 \to 2}^c \}\) satisfies
\begin{equation}
\bR^c_{1 \to 2} \cdot \bz_1 = 0, \qquad
\bM^c_{A, 1 \to 2} \cdot \bz_1 = 0\tag{10.4.7}
\end{equation}
for all points \(A\) of axis \(\Delta\text{.}\) All other components of the action screw are non-zero
\begin{equation*}
\{ \cA_{1 \to 2}^c \} =
\left\{
\begin{array}{c}
R_x \bx_1 +R_y \by_1 \\
M_{Ax} \bx_1 +M_{Ay}\by_1
\end{array}
\right\}_{A \in \Delta}
\end{equation*}
Subsection 10.4.4 Frictionless Helical Joint
A
helical joint is a slider-pivot joint in which the rotational about and translational motion along axis
\(\Delta (O_1, \bz_1)\) are constrained to each other by the relation
\(v_{2z} = (p / 2\pi) \om_z\) (where
\(p\) is a constant). Thus, this is a one-degree-of-freedom joint, and the corresponding kinematic screw can be written as
\begin{equation}
\{ \cV _{2 / 1} \} =
\left\{
\begin{array}{c}
\om_z \bz_1 \\
{p\over 2\pi} \om_z \bz_1
\end{array}
\right\}_{A \in \Delta}\tag{10.4.8}
\end{equation}
Then, assuming frictionless contact, the corresponding contact action screw \(\{ \cA_{1 \to 2}^c \}\) satisfies
\begin{equation}
\frac{p}{2\pi} \bR^c_{1 \to 2} \cdot \bz_1 +
\bM^c_{A, 1 \to 2} \cdot \bz_1 = 0\tag{10.4.9}
\end{equation}
for any point \(A\) of axis \(\Delta\text{.}\) This can be found by imposing \(\{ \cV _{2 / 1} \} \cdot \{ \cA^c_{1 \to 2} \} =0\text{.}\)
Figure 10.4.5.
Subsection 10.4.5 Frictionless Spherical Joint
In a
spherical joint , a single point
\(O_2\) of
\(\cB_2\) remains fixed relative to
\(\cB_1\text{.}\) This is a three-degree-of-freedom joint with corresponding kinematic screw
\begin{equation}
\{ \cV _{2 / 1 } \} =
\left\{
\begin{array}{c}
\bom_{2/1} \\
\bze
\end{array}
\right\}_{O_2}\tag{10.4.10}
\end{equation}
Figure 10.4.6.
Then, assuming frictionless contact, the corresponding contact action screw \(\{ \cA^c_{1 \to 2} \}\) satisfies
\begin{equation}
\bM^c_{O_2, 1 \to 2} = \bze\tag{10.4.11}
\end{equation}
All other components of the action screw are non-zero:
\begin{equation*}
\{ \cA_{1 \to 2}^c \} =
\left\{
\begin{array}{c}
R_x \bx_2 + R_y \by_2 +R_z \bz_2 \\
\bze
\end{array}
\right\}_{O_2}
\end{equation*}
Subsection 10.4.6 Planar Frictionless Joint
In a
planar joint between
\(\cB_1\) and
\(\cB_2\text{,}\) a plane
\((O_2 , \bx_2, \by_2)\) of
\(\cB_2\) coincides with a plane
\((O_1 , \bx_1, \by_1)\) of
\(\cB_1\text{.}\) A planar joint is a three-degree-of-freedom joint with corresponding kinematic screw
\begin{equation}
\{ \cV _{2 /1 } \} =
\left\{
\begin{array}{ccc}
\om_z \bz_1 \\
v_{O_2 x} \bx_1 + v_{O_2 y} \by_1
\end{array}
\right\}_{O_2}\tag{10.4.12}
\end{equation}
The corresponding contact action screw \(\{ \cA^c_{1 \to 2} \}\text{,}\) assuming frictionless contact, satisfies
\begin{equation}
\bR^c_{1 \to 2} \cdot \bx_1 = 0, \qquad
\bR^c_{1 \to 2} \cdot \by_1 = 0, \qquad
\bM^c_{A, 1 \to 2} \cdot \bz_1 = \bze\tag{10.4.13}
\end{equation}
about any point \(A\text{.}\) All other components of the action screw are non-zero:
\begin{equation*}
\{ \cA_{1 \to 2}^c \} =
\left\{
\begin{array}{c}
R_z \bz_1 \\
M_{Ax} \bx_2 +M_{Ay}\by_2
\end{array}
\right\}_{A}
\end{equation*}
Figure 10.4.7.
