# Forced Mass-Spring System - Diff Eq

1. Feb 20, 2008

### DualCortex

1. The problem statement, all variables and given/known data

You have a forced, mass-spring system, without damping.
Spring constant = 4 N/m
weight of mass = 9.8 N
mass = 1 Kg

Find the motion X(t) of the mass if ω = 1.5 (Hz) and deduce the
maximum elongation of the spring. Sketch the vibrations X(t). Do same for ω = 1.9, 3.

Find the range of all safe frequencies ω. That is, frequencies that do
not destroy the system (spring breaks if elongated more than 0.06 m.

At t = 0, system is at equilibrium/rest.
Therefore X(0) = 2.45 and X'(0) = 0.

2. Relevant equations
External force: 0.04*cos(ω*t)
" ... the external frequency ω can be adjusted ... "

3. The attempt at a solution
Diff eq should be

$$X\text{''}+4*X=0.04*\cos (\omega *t)$$

Therefore,
$$X(t) = \left(2.45-\frac{0.04}{4-\omega ^2}\right) \text{Cos}[2*t]+\frac{0.04 *\text{Cos}[t *\omega ]}{4-\omega ^2}$$

So, plugging in ω isn't that bad for the first part. However, I'm not sure how I am supposed to find the maximum elongation at each of the frequencies.

Thanks for any hints/help.

FORGET ABOUT THIS PART: Manually (the dirty method of plugging in values), using Matlab I found ω = (19.97653, 20.0560) to be the range where the amplitude went over 0.06 cm.

Turns out the professor made a mistake on one of his given variables, so the spring constant is actually 4 instead of 400, which makes a lot more sense. Just noticed the problem now is that either the spring should be able to be stretched longer or the mass weighs less ... will have to email my professor.

Last edited: Feb 20, 2008
2. Feb 20, 2008

### blochwave

I'm assuming you got the solution correctly

If you just plug in the frequencies like you've done you get X(t)=some time varying function

The maximum elongation will occur when the velocity is 0, right?

3. Feb 20, 2008

### DualCortex

Well yeah, problem is that then you'll have something like
-Sin(1.5t) = Sin(20t)
if we let ω = 1.5 and not all waves have the same amplitude. Or, well, the amplitude will be pretty much constant unless ω gets close to 20.

Last edited: Feb 20, 2008
4. Feb 21, 2008

### blochwave

Again, assuming you did it right(took dx/dt, set = 0)you'd solve for t to find the time that the velocity equals 0, and then you can use that time in your position equation to find the corresponding value of x

5. Feb 21, 2008

### DualCortex

Hey blochwave:

if you could please explain to me how to solve:

$$\frac{0.04*\omega}{4 - \omega^{2}}*sin(\omega * t) = 2*\left(\frac{0.04*\omega}{4 - \omega^{2}}-2.45\right)*sin(2*t)$$

$$\omega = 1.5, 1.9, 3$$, as the problem asks me.

Last edited: Feb 21, 2008
6. Feb 21, 2008

### blochwave

Ok to backtrack

I'm looking at the part where you're asked to find the maximum elongation. This occurs when the velocity of the spring is 0, as you can imagine(it's changing direction when the velocity is 0 so it's as far as it goes)

You had found an expression for its displacement from equilibrium in terms of time

by differentiation with respect to time, you find the velocity as a function of time. You set that velocity equal to 0, then you can solve for TIME(you're given the values of omega to use, just plug them in one by one), this will be the time(s) that the maximum elongation occurs, and then you can plug that into the X(t) equation to find what the actual elongation is

7. Feb 21, 2008

### DualCortex

blochwave, I fully understand what you are telling me, and it is the method I tried at first right after I solved the diff. eq.

The equations above is X'(t) set equal to 0.
So I plug in, say, 1.9. Then what?

I don't know how to solve it mathematically, but apparently the max is 2.45.
As w grows or decreases, terms go to zero and the only one that is left is 2.45*cos(2t)

On the other hand as w approaches 2, terms cancel out and again the only one left is 2.45*cos(2t)

ACTUALLY, forget about this problem for now. There's various issues with the given conditions. After he told us of the mistake, the spring constant is way too low (4, compared to 400 previously), first the weight on the system is too high (which I contacted my professor about) stretching the string too much, and the external force is way too low

Last edited: Feb 21, 2008