Damped SHM w/ forced motion

In summary, the homework statement is that two functions, xs(t) and xp(t), are related through a differential equation. The problem asks for the coefficients of the function xh(t), which is found by solving for x(t) and collecting terms. However, the last term on the LHS is missing an x(t). This causes problems when trying to solve for C1 and C2, as they must be determined from the coefficients. After correcting the mistake, the problem can be solved for xh(t).
  • #1
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Homework Statement



[PLAIN]http://img848.imageshack.us/img848/1150/damped.png [Broken]

Homework Equations





The Attempt at a Solution


Part i)

x(t) = C1coswt + C2sinwt
dx/dt = -C1wsinwt +C2wcoswt

using IC's:
xo = C1
dxo/dt = C2w

therefore x(t) = xocoswt + (dxo/dt)/w sinwt

but for part ii I do the same thing and get A=0 and B=0. I know I'm doing something wrong here what is it?
 
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  • #2
First off, the notation in the problem seems a bit sloppy to me. The two functions xs(t) and xp(t) refer to the same function. Also, the form of the homogeneous solution isn't correct if you take A and B to simply be constants.

Anyway, what you did is not how you solve for C1 and C2. You want to plug the particular solution into the LHS of the differential equation and then collect terms. You'll get (something)cos ωt + (something else)sin ωt, where the two coefficients will depend on C1, C2, and the various parameters in the equation. When you compare that to the RHS, you can see that you must have (something)=f0 and (something else)=0. You can solve these equations for C1 and C2.

(If you've already taken differential equations, you're just solving for the particular solution using the method of undetermined coefficients.)

Once you have that, we can go on to tackle finding the homogeneous solution xh(t).
 
  • #3
Ok, thanks for your help, I followed through with it and got this, which seems kind of insane but I checked it a lot and it seems right.

[PLAIN]http://img859.imageshack.us/img859/1029/scan0008k.jpg [Broken]
 
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  • #4
That can't be correct. Both coefficients should be proportional to f0, and the units don't appear to work out either.

I just noticed there's a typo in the differential equation. The last term on the LHS is missing an x(t). It should be

[tex]\ddot{x}(t) + 2\zeta\omega_n\dot{x}(t)+\omega_n^2 x(t)=f_0\cos\omega t[/tex]

The coefficients will be

[tex]\begin{align*}
C_1 &= -\frac{f_0(\omega^2-\omega_n^2)}{(\omega^2-\omega_n^2)^2+(2 \omega\omega_n\zeta)^2} \\
C_2 &= \frac{f_0 (2 \omega\omega_n\zeta)}{(\omega^2-\omega_n^2)^2+(2 \omega\omega_n\zeta)^2}
\end{align*}[/tex]
 
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  • #5
Wow, nice find, I just double checked and you're right about the missing x term. Interesting since the question was straight out of the textbook and the teacher hasn't mentioned anything about it when he assigned the questions. I've emailed him to post a note on it so hopefully other student's don't solve it the same way too.
 
  • #6
Just redid the whole problem and got the same coefficients, thanks for all your help!
 
  • #7
Were you able to work out the rest of the problem?
 

1. What is damped simple harmonic motion with forced motion?

Damped simple harmonic motion with forced motion is a type of oscillatory motion where an external force is applied to a damped harmonic oscillator. This force can either be a constant force or a periodic force, causing the oscillator to move with a specific frequency.

2. How does damping affect the motion of a forced harmonic oscillator?

Damping refers to the gradual decrease in the amplitude of oscillation in a forced harmonic oscillator. This means that the amplitude of the oscillations decreases with time, eventually reaching a state of equilibrium where the amplitude remains constant. The amount of damping present in the oscillator affects the amplitude and frequency of the motion.

3. What is the equation for damped simple harmonic motion with forced motion?

The equation for damped simple harmonic motion with forced motion is given by: x(t) = Ae^(-bt)cos(ωt+φ) + F/((ω0^2-ω^2)^2 + (bω)^2), where x(t) is the position of the oscillator at time t, A is the initial amplitude, b is the damping coefficient, ω0 is the natural frequency of the oscillator, ω is the frequency of the external force, φ is the phase angle, and F is the amplitude of the external force.

4. How does the amplitude of the external force affect the motion of a damped forced oscillator?

The amplitude of the external force directly affects the amplitude of the motion of a damped forced oscillator. A larger amplitude of the external force will result in a larger amplitude of the oscillations, while a smaller amplitude of the external force will result in smaller oscillations. However, regardless of the amplitude of the external force, the damping coefficient and the natural frequency of the oscillator also play a significant role in determining the amplitude of the motion.

5. What is the significance of resonance in damped forced oscillations?

Resonance occurs in a damped forced oscillator when the frequency of the external force matches the natural frequency of the oscillator. This results in a large amplitude of oscillations and can potentially lead to damage or failure of the system if the amplitude becomes too large. Therefore, it is important to consider resonance when studying damped forced oscillations.

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