Planar Rigid Body Kinematics: Velocity

Planar rigid body kinematics: velocity

Planar rigid body kinematics focuses on studying the motion of rigid bodies confined to a two-dimensional plane, considering only the geometry of the motion without involving the forces causing it. This includes translational and rotational movements, examining how position, orientation, and velocity are described in planar motion.

There are three types of planar motion. Translation occurs when all points of a rigid body follow parallel paths. When these paths are straight, the motion is rectilinear translation, and when curved, it is curvilinear translation. In both cases, the body maintains a fixed orientation. Rotation about a fixed axis describes motion where all points move in circular paths around a stationary axis, each point having the same angular velocity but varying linear velocities depending on their distance from the axis. General planar motion combines translation and rotation, where the body changes both position and orientation simultaneously.

To describe motion mathematically, I use position vectors. For instance, the position of a point Q can be expressed as:

\mathbf{r}_{OQ} = \mathbf{r}_{OP} + \mathbf{r}_{PQ}

where \mathbf{r}_{OQ} is the position vector from the origin to Q, \mathbf{r}_{OP} is the vector from the origin to P, and \mathbf{r}_{PQ} represents the relative position of Q to P.

By differentiating these vectors with respect to time, I derive velocities:

\mathbf{v}_Q = \mathbf{v}_P + \omega \, (\mathbf{k} \times \mathbf{r}_{PQ})

Here, \mathbf{v}_Q is the velocity of point Q, \mathbf{v}_P is the velocity of P, and \omega is the angular velocity. This equation effectively captures how translation and rotation combine in planar motion.

For more insights into this topic, you can find the details here.