Analyzing velocity and acceleration in planar two-dimensional motion
Understanding the motion of points relative to multiple reference frames is essential in engineering and physics. In this blog post, I explore a practical problem involving a point constrained to move along a sliding bar within a two-dimensional planar system. The analysis focuses on the velocity and acceleration relationships between two reference frames and the specific point of interest.
The scenario involves two reference frames:
- fixed frame: This is stationary and serves as the global reference;
- moving frame: Attached to a sliding vertical bar, with its origin at the center of rotation.
The point of interest, denoted as \mathbf P, slides along the vertical bar. The angular velocity and acceleration of the bar and the relative velocity and acceleration of point \mathbf P are calculated using the following vector relationships:
\mathbf{v}_P = \mathbf{v}_{O_2} + \mathbf{v}_{\text{rel}} + \omega \mathbf{k} \times \mathbf{r}
where \mathbf{v}_P is the velocity of \mathbf P, \omega is the angular velocity of the moving frame, and \mathbf{v}_{\text{rel}} represents the velocity of \mathbf P relative to the moving frame.
\mathbf{a}_P = \mathbf{a}_{O_2} + \mathbf{a}_{\text{rel}} + \alpha_2 \mathbf{k} \times \mathbf{r} - \omega_2^2 \mathbf{r} + 2\omega_2 \mathbf{k} \times \mathbf{v}_{\text{rel}}
Here, \alpha_2 is the angular acceleration of the moving frame, and the Coriolis and centripetal acceleration terms account for the relative motion within the rotating reference frame.
From the analysis:
- The angular velocity of the bar, \omega_2, was found to be -0.2 \, \text{rad/s}.
- The relative velocity of point P in the moving frame was determined to be 0.1 \, \text{m/s}.
- For accelerations, the angular acceleration \alpha_2 was calculated as 2.73 \, \text{rad/s}^2, and the relative acceleration of point \mathbf P was -0.125 \, \text{m/s}^2.
These calculations illustrate the interplay between rotational motion and relative motion within constrained mechanical systems.
This example provides a framework for analyzing velocity and acceleration in planar systems, emphasizing practical applications in mechanical engineering and physics.
For more insights into this topic, you can find the details here.