B The amplitude of a forced undamped ocillation

[anonymous24]

Hello,
We learned in class that for a simple harmonic damped forced oscillation, the amplitude decreased exponentially over time. And for a completely undamped situation, the amplitude grows uncontrollably when the driver frequency matches the natural frequency of the driven. However, I wonder what will happen if the driver and driven frequency is not matched in an undamped situation. Would the amplitude also increase over time as no energy is lost but at a lower rate than when they match? Thank you in advance.

 

[tech99]

Hello,
We learned in class that for a damped forced oscillation, the amplitude decreased exponentially over time. And for a completely undamped situation, the amplitude grows uncontrollably when the driver frequency matches the natural frequency of the driven. However, I wonder what will happen if the driver and driven frequency is not matched in an undamped situation. Would the amplitude also increase over time as no energy is lost but at a lower rate than when they match? Thank you in advance.

In the undamped condition, the Q of the circuit is infinite and the bandwidth is zero, so a generator which differs in frequency cannot couple energy into the resonant circuit.

 

[vanhees71]

Just solve the equations of motion:
$$\ddot{x}+\omega^2 x=A \exp(-\mathrm{i} \Omega t).$$
At the end we take the real part to get a real solution.

The general solution is the general solution of the homogeneous equation,
$$x_h(t)=C_1 \exp(-\mathrm{i} \omega t)+C_2 \exp(\mathrm{i} \omega t),$$
plus any particular solution of the inhomogeneous equation. Physically it's clear that the ansatz
$$x(t)=C \exp(-\mathrm{i} \Omega t)$$
should lead to a solution. Now it's easy for you to find the solution yourself.

Also the particular solution of the singular case of resonance, i.e., $\Omega=\omega$ can be found by taking the limit $\Omega \rightarrow \omega$ at fixed initial conditions (e.g., choosing as initial conditions $x(0)=\dot{x}(0)=0$).

 

[anonymous24]

In the undamped condition, the Q of the circuit is infinite and the bandwidth is zero, so a generator which differs in frequency cannot couple energy into the resonant circuit.

Thank you for your reply but sorry I don't understand what do you mean by circuit. I meant in a simple harmonic system. Perhaps I should edit my post.

 

[anonymous24]

Just solve the equations of motion:
$$\ddot{x}+\omega^2 x=A \exp(-\mathrm{i} \Omega t).$$
At the end we take the real part to get a real solution.

The general solution is the general solution of the homogeneous equation,
$$x_h(t)=C_1 \exp(-\mathrm{i} \omega t)+C_2 \exp(\mathrm{i} \omega t),$$
plus any particular solution of the inhomogeneous equation. Physically it's clear that the ansatz
$$x(t)=C \exp(-\mathrm{i} \Omega t)$$
should lead to a solution. Now it's easy for you to find the solution yourself.

Also the particular solution of the singular case of resonance, i.e., $\Omega=\omega$ can be found by taking the limit $\Omega \rightarrow \omega$ at fixed initial conditions (e.g., choosing as initial conditions $x(0)=\dot{x}(0)=0$).

I really appreciate the effort you put into your response but solving differential equations like this may be a bit out of my reach as I'm still in high school. Is there an intuitive way to approach this?

 

[Philip Wood]

We learned in class that for a simple harmonic damped forced oscillation, the amplitude decreased exponentially over time.

I'm afraid you're a bit confused here. The displacement of the system is the sum of (1) a steady simple harmonic oscillation at the frequency of the driver, and (2) a damped oscillation at the system's natural frequency. It's only (2) that dies away, leaving (1) as the 'steady state', constant amplitude response. In vanhees71's reply, x_h(t) is (2) and x(t) is (1).

 

[anonymous24]

I'm afraid you're a bit confused here. The displacement of the system is the sum of (1) a steady simple harmonic oscillation at the frequency of the driver, and (2) a damped oscillation at the system's natural frequency. It's only (2) that dies away, leaving (1) as the 'steady state', constant amplitude response. In vanhees71's reply, x_h(t) is (2) and x(t) is (1).

Thank you for your response, it does clear up the confusion. Would undamped oscillation also return to steady state over time?

 

[Philip Wood]

No, but do bear in mind that real systems are damped to some degree. An undamped system is an idealisation, and one that may not be very helpful when trying to understand forced oscillations.

 

[tech99]

Thank you for your reply but sorry I don't understand what do you mean by circuit. I meant in a simple harmonic system. Perhaps I should edit my post.

Sorry, I now realize you are studying mechanical systems, but the same happens with an electrical circuit having an inductor and a capacitor.
Intuitively, you may know that a wine glass will resonate and shatter if a singer hits the exact resonant frequency. But because the damping of the wine glass is small (efficiency high, bandwidth narrow) the frequency of the note must be exact.