9.2: Orientation and position
The first step in the kinematic analysis of a rigid body is to have a unique description of its position and orientation. In planar kinematics, we fully determine the position of a rigid body by fixing the position vectors of 2 points in the rigid body that can be freely chosen, like points \(A\) and \(B\) of the rectangle in Figure 9.1.
If we know position vectors \(\overrightarrow{\boldsymbol{r}}_{A}\) and \(\overrightarrow{\boldsymbol{r}}_{B}\) the position and orientation of the rigid body is fully determined. This requires four coordinates: \(x_{A}, y_{A}, x_{B}\) and \(y_{B}\). However, because we know the distance between the points, we can use constraint equation 9.1 to reduce this to 3 coordinates, namely \(x_{A}, y_{A}\) and \(\phi_{B / A}\), where \(\phi_{B / A}\) is the angle the relative position vector \(\overrightarrow{\boldsymbol{r}}_{B / A}\) makes with the \(x\)-axis. With \(x_{A}, y_{A}\) we can determine \(\overrightarrow{\boldsymbol{r}}_{A}\), and with \(\phi_{B / A}\) and knowledge of the distance \(\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right|\) we can determine \(\overrightarrow{\boldsymbol{r}}_{B / A}\) :
\[\begin{align} \overrightarrow{\boldsymbol{r}}_{A, 2 D} & =x_{A} \hat{\boldsymbol{\imath}}+y_{A} \hat{\boldsymbol{\jmath}} \tag{9.2} \label{9.2}\\[4pt] \overrightarrow{\boldsymbol{r}}_{B / A, 2 D} & =\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right| \cos \phi_{B / A} \hat{\boldsymbol{\imath}}+\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right| \sin \phi_{B / A} \hat{\boldsymbol{\jmath}} \tag{9.3} \label{9.3}\end{align}\]
Now we determine \(\overrightarrow{\boldsymbol{r}}_{B}\) from the 3 coordinates by adding these two vectors as shown in Figure 9.1.
\[\begin{align} \overrightarrow{\boldsymbol{r}}_{B} & =\overrightarrow{\boldsymbol{r}}_{A}+\overrightarrow{\boldsymbol{r}}_{B / A} \tag{9.4} \label{9.4}\\[4pt] \overrightarrow{\boldsymbol{r}}_{B, 2 D} & =\left(x_{A}+\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right| \cos \phi_{B / A}\right) \hat{\boldsymbol{\imath}}+\left(y_{A}+\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right| \sin \phi_{B / A}\right) \hat{\boldsymbol{\jmath}} \tag{9.5} \label{9.5}\end{align}\]