Analyzing the dynamics of a piston mechanism in planar motion
In this post, I analyze the dynamics of a piston mechanism, an essential example of planar rigid body motion frequently encountered in engineering. The piston mechanism consists of a rotating crankshaft, a connecting rod, and a piston undergoing linear motion. This system demonstrates the interaction of rotational and translational motions, which I studied by calculating velocities and accelerations at specific points.
Starting with the crankshaft, I used the angular velocity \omega_{OA} = -10 \, \text{rad/s} and angular acceleration \alpha_{OA} = 5 \, \text{rad/s}^2, combined with the position vector \mathbf{r}_{OA} = -0.072 \, \mathbf{i} + 0.096 \, \mathbf{j}, to find the velocity of the connecting rod’s end:
\mathbf{v}_A = \mathbf{v}_O + \omega_{OA} \, \mathbf{k} \times \mathbf{r}_{OA}
The acceleration was derived using:
\mathbf{a}_A = \mathbf{a}_O + \alpha_{OA} \, \mathbf{k} \times \mathbf{r}_{OA} - \omega_{OA}^2 \mathbf{r}_{OA}
By incorporating the geometry and relative motion constraints, I computed the velocity and acceleration of the piston. For example, setting the vertical component of the piston’s velocity to zero provided insights into the angular velocity of the connecting rod, which turned out to be \omega_{AB} = -2.27 \, \text{rad/s}.
This analysis highlights how fundamental kinematic equations govern the motion in such systems, offering insights into engineering applications.
For more insights into this topic, you can find the details here.