# Forced oscillations: solution of Differential Equation at resonance

• jellicorse
In summary, the conversation discusses the steps taken to find a solution for the equation \frac{d^2u}{dt^2} + \frac{k}{m}u = Fcos\omega t. It is mentioned that at resonance, F cos\omega t is a solution to the homogeneous equation and the full solution is given by Acos\omega_0t +Bsin\omega_0t+\frac{F}{2m\omega_o^2}tsin\omega_0t. The steps for finding the complementary function and particular integral are also discussed. However, the solution obtained is not correct as it introduces units of time due to the term t in \frac{F}{m\omega_0
jellicorse

## Homework Statement

Following a worked example in my book, I have been trying to get a solution for the equation

$\frac{d^2u}{dt^2} + \frac{k}{m}u = Fcos\omega t$

The book says that at resonance, i.e. when $\omega_0$ (the natural frequency) = $\omega$ (the forcing frequency), the term $F cos\omega t$ is a solution to the homogenous equation and the solution to the differential equation above is$$Acos\omega_0t +Bsin\omega_0t+\frac{F}{2m\omega_o^2}tsin\omega_0t$$

## The Attempt at a Solution

To get the full solution:

Complementary Function:

$$Acos\omega_0t+Bsin\omega_0t$$, where $$\omega_0=\sqrt{\frac{k}{m}}$$To get the Particular Integral:
Assume $u= Ctcos\omega t +Dtsin\omega t$

Then $$\frac{du}{dt}= C cos\omega t - Ct\omega sin \omega t + D sin\omega t +Dt \omega cos \omega t$$

$$\frac{du}{dt}= (Dt\omega +C ) cos\omega t + (D-Ct\omega) sin\omega t$$

And $$\frac{d^2u}{dt^2} = -C \omega sin \omega t -C\omega sin \omega t - C \omega^2 t cos \omega t + D\omega cos \omega t + D\omega cos \omega t - D \omega^2 t sin \omega t$$

$$\frac{d^2u}{dt^"}=(2D\omega -c \omega^2t)cos\omega t +(-2C\omega -D\omega^2t)sin \omega t$$Back substituting these into the original differential equation:

$$(2D\omega -C\omega^2t + \frac{k Ct}{m}) cos\omega t + (\frac{KDt}{m}-2C\omega -D\omega^2t) sin \omega t = \frac{F}{m} cos\omega t$$
Equating coefficients:

$$(2D\omega -C\omega^2t + \frac{k Ct}{m}) =\frac{F}{m}$$

and $$(\frac{kDt}{m} -2 C \omega - D\omega^2 t) = 0$$

After this, I have tried solving for D and C but it seems to end up in a pretty intractable mess.

I know somehow D should be equal to $\frac{F}{2m\omega_0^2}$ (and I assume since we are talking about a situation in which $\omega_0=\omega$, D= $$\frac{F}{2m\omega_0^2} = \frac{F}{2m\omega}$$) but can not see how to get there. Can anyone tell me if I have been going in the right direction so far?

Last edited by a moderator:
From the form of the particular solution, you're apparently assuming that ##\omega=\omega_0 = \sqrt{k/m}##, so the two equations reduce to
\begin{align*}
2\omega D &= \frac Fm \\
-2\omega C &= 0.
\end{align*} The solution you're trying to obtain is not correct: ##\frac{F}{m\omega_0^2}t\sin\omega_0 t## does not have units of length.

1 person
Thanks a lot Vela. I should have noticed that when equating coefficients.

I don't quite understand this though:

vela said:
The solution you're trying to obtain is not correct: ##\frac{F}{m\omega_0^2}t\sin\omega_0 t## does not have units of length.

I can't see anything to do with units of length in the equation...

The denominator is ##m\omega^2 = k## where ##k## has units of force/length, so ##\frac{F}{m\omega^2}## will have units of length. That would be fine if the coefficient multiplied only ##\sin \omega t##, but it doesn't. It multiplies ##t \sin\omega t##. The factor of ##t## introduces a unit of time.

Oh, I see... That makes sense, thanks!

## 1. What is the concept of forced oscillations?

Forced oscillations refer to the oscillations of a system that is subjected to a periodic external force. This external force can be in the form of a constant force or a varying force, such as a sinusoidal force.

## 2. What is resonance in forced oscillations?

Resonance occurs in forced oscillations when the frequency of the external force matches the natural frequency of the system. This results in a significant increase in the amplitude of oscillations.

## 3. How is the solution of a differential equation obtained for forced oscillations at resonance?

The solution of a differential equation for forced oscillations at resonance involves finding the particular solution by using the method of undetermined coefficients. This solution is then substituted into the original equation to find the coefficients and obtain the final solution.

## 4. What is the significance of resonance in forced oscillations?

Resonance is significant in forced oscillations because it can cause a system to vibrate with a large amplitude, which can lead to damage or failure of the system. It is also useful in many applications, such as in musical instruments and radio receivers.

## 5. How can resonance in forced oscillations be avoided?

Resonance in forced oscillations can be avoided by changing the frequency of the external force, adjusting the damping in the system, or modifying the system's natural frequency. This can help prevent excessive vibrations and potential damage to the system.

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