# Help with the phase of the solution for a driven oscillator

• etotheipi
In summary, the conversation discusses the equation of motion for an undamped driven harmonic oscillator and the particular solution for a non-homogeneous part. It is noted that the particular solution should have a frequency of ##\omega## and that the phase shift should be zero in the absence of a damping term.
etotheipi
Homework Statement
Obtain the general solution for an undamped harmonic oscillator with a forcing term proportional to a cosine wave, with a maximum of force at t = 0.
Relevant Equations
N/A
My question also applies to the damped driven oscillator, however for simplicity I will first consider an undamped oscillator.

The equation of motion is $$-kx + F_{0} \cos{\omega t} = m \ddot{x}$$ or in a more convenient form $$\ddot{x} + {\omega_{0}} ^{2}x = \frac{F_{0}}{m} \cos{\omega t}$$The auxiliary equation is ${\lambda}^{2} + {\omega_{0}} ^{2} =0$, which has solutions $\lambda = \pm i \omega_{0}$. So the complementary solution is $$x = A\cos{(\omega_{0}t + \phi)}$$Now for the particular solution. For a non-homogenous part relating to sine or cosine, the ansatz is of the form of a sine or cosine of the same argument plus a phase. However, most texts I've read just choose $x = B\cos{\omega t}$, which of course cancels out more nicely.

If we just use $x = B\cos{\omega t}$, the particular solution comes out to be $x = \frac{F_{0} \cos{\omega t}} {m(\omega_{0} ^{2} - \omega ^{2})}$.

However, if we choose the ansatz $x = B\cos{(\omega t + \psi)}$ which is equivalent to $x = Re(Be^{i(\omega t + \psi)})$, we get $$- B \omega ^{2} e^{i \psi} + B \omega_{0} ^{2} e^{i \psi} = \frac{F_{0}}{m}$$ and thus $B = \frac{F_{0} e^{-i \psi}}{m(\omega_{0} ^{2} - \omega ^{2})}$. When we put this together, the $e^{-i \psi}$ will cancel out and we end up with the same particular solution as before when we take the real part.

My question is why are we allowed to ignore the phase of the guess for the particular solution and still obtain the correct answer?

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etotheipi said:
My question also applies to the damped driven oscillator, however for simplicity I will first consider an undamped oscillator.

The equation of motion is

−kx+F0cosωt=m¨x​
This is the equation of motion for the undamped driven harmonic oscillator.

Note that the particular solution should have frequency ##\omega##, not ##\omega_0## and that there is no phase shift in absence of a damping term (if you input a phase shift it will come out to be zero).

etotheipi
Orodruin said:
This is the equation of motion for the undamped driven harmonic oscillator.

Note that the particular solution should have frequency ##\omega##, not ##\omega_0## and that there is no phase shift in absence of a damping term (if you input a phase shift it will come out to be zero).

Yes sorry about the ##\omega_0##, it was a typo.

That makes perfect sense. Thanks!

## 1. What is a driven oscillator?

A driven oscillator is a physical system that experiences oscillations (back-and-forth motion) due to an external force or input. This external force is known as the "driving force" and can be either periodic or non-periodic.

## 2. How do you solve for the phase of a driven oscillator?

The phase of a driven oscillator can be solved for by using the equation: θ = arctan(ω/γ), where θ is the phase, ω is the angular frequency of the driving force, and γ is the damping coefficient of the oscillator. Alternatively, you can also use a graphing calculator or software to plot the phase angle against the frequency of the driving force to find the phase.

## 3. What factors affect the phase of a driven oscillator?

The phase of a driven oscillator is affected by the frequency and amplitude of the driving force, as well as the damping coefficient of the oscillator. Additionally, the initial conditions of the system can also affect the phase.

## 4. Can the phase of a driven oscillator be negative?

Yes, the phase of a driven oscillator can be negative. This indicates that the oscillator is lagging behind the driving force in its oscillations.

## 5. How can understanding the phase of a driven oscillator be useful?

Understanding the phase of a driven oscillator can be useful in various fields such as engineering, physics, and electronics. It allows us to predict the behavior of the system and make adjustments to optimize its performance. Additionally, it can also help us analyze the stability and resonance of the oscillator.

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