- #1

zenterix

- 688

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- Homework Statement
- A torsional oscillator comprises a cylinder with moment of inertia ##I## hanging from a light rod with torsional spring constant ##k##. The cylinder also experiences a drag torque equal to ##-\mu\dot{\theta}## when moving with angular velocity ##\dot{\theta}##. The top of the rod is driven with angular displacement ##\phi(t)=\phi_0\cos{\omega t}##.

Find the steady-state solution for ##\theta(t)##.

Plot the amplitude ##A(\omega)## and phase ##\delta\omega## of your solution for ##\theta(t)## in (1) as a function of ##\omega##.

- Relevant Equations
- For your plot assume that the natural frequency of oscillation of the system is ##\omega_0=1## and plot three curves on the same plot with ##\frac{\mu}{l}=0.25, 1##, and 2. Label your curves to distinguish the three cases.

The "Vibrations and Waves" problem-solving course on MIT OCW has a section on driven harmonic oscillators which can be seen here.

I would like to do the first of the two problems. Unfortunately, there are two issues

1) The latex is not rendering on that website (relatively minor issue, I think I can get parse it).

2) The problem is about a type of oscillator that I am unfamiliar with.

I'd like to show here how I understood the problem to be set up and then how I set up the equations.

My question is if this set up and equations are correct.

I've stated the problem above as I understood it from the website.

Here is how I set it up mathematically.

$$\vec{\tau}=(-k(\theta-\phi)-\mu\dot{\theta})\hat{k}=I\ddot{\theta}\hat{k}\tag{1}$$

$$I\ddot{\theta}+\mu\dot{\theta}+k\theta=k\phi\tag{2}$$

$$\ddot{\theta}+\frac{\mu}{I}\dot{\theta}+\frac{k}{I}\theta=\frac{k}{I}\phi_0\cos{\omega t}\tag{3}$$

In words, the torque experienced by the cylinder has two components. One is due to the angle difference between the cylinder and the rod relative to the equilibrium position, in which this angle difference is zero. This is the ##-k(\theta-\phi)## term.

The other component is drag which is always in the opposite direction to the angular velocity.

If this is correct, this boils down to the same problem as a driven simple pendulum or a driven RLC circuit.

I would like to do the first of the two problems. Unfortunately, there are two issues

1) The latex is not rendering on that website (relatively minor issue, I think I can get parse it).

2) The problem is about a type of oscillator that I am unfamiliar with.

I'd like to show here how I understood the problem to be set up and then how I set up the equations.

My question is if this set up and equations are correct.

I've stated the problem above as I understood it from the website.

Here is how I set it up mathematically.

$$\vec{\tau}=(-k(\theta-\phi)-\mu\dot{\theta})\hat{k}=I\ddot{\theta}\hat{k}\tag{1}$$

$$I\ddot{\theta}+\mu\dot{\theta}+k\theta=k\phi\tag{2}$$

$$\ddot{\theta}+\frac{\mu}{I}\dot{\theta}+\frac{k}{I}\theta=\frac{k}{I}\phi_0\cos{\omega t}\tag{3}$$

In words, the torque experienced by the cylinder has two components. One is due to the angle difference between the cylinder and the rod relative to the equilibrium position, in which this angle difference is zero. This is the ##-k(\theta-\phi)## term.

The other component is drag which is always in the opposite direction to the angular velocity.

If this is correct, this boils down to the same problem as a driven simple pendulum or a driven RLC circuit.