Kinematics and Dynamics of Rotating and Moving Systems

Q1. Determine the magnitudes of the velocity and acceleration of point P. The lengths $h_0$ and $l_0$ are fixed. Also, identify the expressions for the centripetal acceleration, tangential acceleration, and Coriolis acceleration.

Background

Topic: Kinematics of Rotating Systems

This question tests your understanding of the velocity and acceleration of a point in a rotating and translating mechanism, specifically an industrial robot arm. You are also asked to identify the different types of acceleration present in such systems.

Key Terms and Formulas

• Velocity ($\vec{v}$): The rate of change of position of point P.

• Acceleration ($\vec{a}$): The rate of change of velocity of point P.

• Centripetal acceleration: $a_c = \omega^2 r$

• Tangential acceleration: $a_t = \alpha r$ (where $\alpha$ is angular acceleration)

• Coriolis acceleration: $a_{cor} = 2 \vec{v}_{rel} \times \vec{\omega}$

Step-by-Step Guidance

1. Identify the position vector of point P in terms of the given variables ($h$, $l$, $\omega$).

2. Express the velocity of point P by differentiating the position vector with respect to time, considering both the rotation and translation.

3. Express the acceleration of point P by differentiating the velocity vector with respect to time. Separate the acceleration into components: centripetal, tangential, and Coriolis.

4. Write the mathematical expressions for each acceleration component using the provided variables.

Try solving on your own before revealing the answer!

Final Answer:

The velocity and acceleration magnitudes are given by:

$\vec{v}_P = \dot{h} \hat{z} + \dot{l} \hat{x} + \omega l \hat{y}$

$\vec{a}_P = \ddot{h} \hat{z} + \ddot{l} \hat{x} + \omega^2 l \hat{x} + 2 \omega \dot{l} \hat{y}$

Centripetal acceleration: $a_c = \omega^2 l$ Tangential acceleration: $a_t = \dot{\omega} l$ Coriolis acceleration: $a_{cor} = 2 \omega \dot{l}$

These expressions account for the fixed lengths $h_0$ and $l_0$ and the time derivatives of $h$ and $l$.

Q2. The point A of the system has polar coordinates varying with time: $r(t) = \alpha_1 - \beta_1 \sin(\gamma_1 \pi t)$ [m], $\theta(t) = \alpha_2 - \beta_2 \cos(\gamma_2 \pi t)$ [rad]. Calculate the velocity and acceleration of point A. For $\alpha_1 = 1$, $\alpha_2 = 1$, $\beta_1 = 1$, $\beta_2 = 1$, plot the trajectory of A for $t \in [0, 100]$ in the following cases: (a) $\gamma_1 = \gamma_2 = 1$, (b) $\gamma_1 = 2$, $\gamma_2 = 1$, (c) $\gamma_1 = 3$, $\gamma_2 = 1$, (d) $\gamma_1 = 4$, $\gamma_2 = 1$, (e) $\gamma_1 = \sqrt{2}$, $\gamma_2 = 1$. Comment on the periodicity of the trajectory in each case.

Background

Topic: Kinematics in Polar Coordinates

This question tests your ability to compute velocity and acceleration in polar coordinates, and to analyze the periodicity of a trajectory based on the time dependence of $r(t)$ and $\theta(t)$.

Key Terms and Formulas

• Velocity in polar coordinates: $\vec{v} = \dot{r} \hat{r} + r \dot{\theta} \hat{\theta}$

• Acceleration in polar coordinates: $\vec{a} = (\ddot{r} - r \dot{\theta}^2) \hat{r} + (r \ddot{\theta} + 2 \dot{r} \dot{\theta}) \hat{\theta}$

• Periodicity: The trajectory is periodic if both $r(t)$ and $\theta(t)$ repeat after a certain time interval.

Step-by-Step Guidance

1. Compute $\dot{r}(t)$ and $\ddot{r}(t)$ by differentiating $r(t)$ with respect to time.

2. Compute $\dot{\theta}(t)$ and $\ddot{\theta}(t)$ by differentiating $\theta(t)$ with respect to time.

3. Substitute these derivatives into the velocity and acceleration formulas for polar coordinates.

4. Set the given values for $\alpha_1$, $\alpha_2$, $\beta_1$, $\beta_2$ and analyze the periodicity for each case of $\gamma_1$ and $\gamma_2$.

Try solving on your own before revealing the answer!

Final Answer:

Velocity: $\vec{v}_A = \dot{r}(t) \hat{r} + r(t) \dot{\theta}(t) \hat{\theta}$ Acceleration: $\vec{a}_A = (\ddot{r}(t) - r(t) \dot{\theta}^2(t)) \hat{r} + (r(t) \ddot{\theta}(t) + 2 \dot{r}(t) \dot{\theta}(t)) \hat{\theta}$

For each case, the periodicity depends on the ratio of $\gamma_1$ to $\gamma_2$. If both are integers and equal, the trajectory is periodic. If $\gamma_1$ is irrational (e.g., $\sqrt{2}$), the trajectory is not periodic.

Q3. The position of the moving suspension point of a simple pendulum is given by $\vec{OO_P} = f(t) \hat{x} + g(t) \hat{y}$. Express the kinematic quantities (position, velocity, and acceleration) of the pendulum’s mass point P: 1) in the frame $(O; \hat{x}, \hat{y})$, 2) in the frame $(O; \hat{e}_r, \hat{e}_\theta)$.

Background

Topic: Kinematics of a Moving Pendulum

This question tests your ability to express the position, velocity, and acceleration of a pendulum whose suspension point is itself moving, in both Cartesian and polar coordinate frames.

Key Terms and Formulas

• Position in Cartesian coordinates: $\vec{r}_P = \vec{OO_P} + \vec{OP}$

• Velocity: $\vec{v}_P = \frac{d}{dt} \vec{r}_P$

• Acceleration: $\vec{a}_P = \frac{d}{dt} \vec{v}_P$

• Position in polar coordinates: $\vec{r}_P = r \hat{e}_r$

• Velocity in polar coordinates: $\vec{v}_P = \dot{r} \hat{e}_r + r \dot{\theta} \hat{e}_\theta$

• Acceleration in polar coordinates: $\vec{a}_P = (\ddot{r} - r \dot{\theta}^2) \hat{e}_r + (r \ddot{\theta} + 2 \dot{r} \dot{\theta}) \hat{e}_\theta$

Step-by-Step Guidance

1. Write the position vector of point P in the $(O; \hat{x}, \hat{y})$ frame by adding the position of the suspension point and the pendulum length.

2. Differentiate the position vector with respect to time to obtain the velocity in the $(O; \hat{x}, \hat{y})$ frame.

3. Differentiate the velocity vector with respect to time to obtain the acceleration in the $(O; \hat{x}, \hat{y})$ frame.

4. Express the position, velocity, and acceleration in the $(O; \hat{e}_r, \hat{e}_\theta)$ frame using the polar coordinate formulas.

Try solving on your own before revealing the answer!

Final Answer:

In the $(O; \hat{x}, \hat{y})$ frame: Position: $\vec{r}_P = f(t) \hat{x} + g(t) \hat{y} + l \sin \theta \hat{x} - l \cos \theta \hat{y}$ Velocity: $\vec{v}_P = \dot{f}(t) \hat{x} + \dot{g}(t) \hat{y} + l \dot{\theta} \cos \theta \hat{x} + l \dot{\theta} \sin \theta \hat{y}$ Acceleration: $\vec{a}_P = \ddot{f}(t) \hat{x} + \ddot{g}(t) \hat{y} + l \ddot{\theta} \cos \theta \hat{x} + l \ddot{\theta} \sin \theta \hat{y}$

In the $(O; \hat{e}_r, \hat{e}_\theta)$ frame: Position: $\vec{r}_P = r \hat{e}_r$ Velocity: $\vec{v}_P = \dot{r} \hat{e}_r + r \dot{\theta} \hat{e}_\theta$ Acceleration: $\vec{a}_P = (\ddot{r} - r \dot{\theta}^2) \hat{e}_r + (r \ddot{\theta} + 2 \dot{r} \dot{\theta}) \hat{e}_\theta$