# Investigating damped Harmonic Motion in a Spring?

I'm going to conduct an investigation into the dampening of a spring.

Essentially, what specific factors could be investigated? Currently I'm planning to investigate the effect of changing the mass on the end of the spring, the effect of using different spring constants, and the effect of changing the initial extension of the spring on the amount of dampening.

-
These two questions are obviously linked: physics.stackexchange.com/questions/8402/… – Roy Simpson Apr 12 '11 at 11:43
I don't understand your accusatory tone, are users not permitted to ask two questions on the same topic? – MathsStudent Apr 13 '11 at 6:39
that's not an accusation directed at you but simply a statement for everyone interested that these questions are related and their answers might thus complement – Tobias Kienzler May 3 '11 at 10:27
Can anyone point me towards some information on whether or not mass affects the damping constant? I can't find any reliable information. – MathsStudent May 5 '11 at 7:21

and welcome to the site. I suppose that you have to do this as some kind of homework so I would not try to invent new things and just look what mathematical description(s) of such a system exists. The governing equations will have some constants in it and it would be already a nice thing to look if the given framework can explain the dynamics and if you can determine those constants.

Before we start, you can be assured that the harmonic oscillator is most likely the physical model with the highest impact on life in general - applications from molecule spectra to airplane design.

## Mathematical description: Damped harmonic oscillator

Wikipedia has a really nice collection of articles for these kind of things. There, we can see that a damped harmonic oscillator can be described by the equation of motion

$$m\ddot{x} + c\dot{x} = -k x$$

where now $x$ is the position of the mass depending on time, dots denote derivative(s) with respect to it, $k$ is the spring "constant" and $c$ is the damping coefficient. All you have to do is to solve this equation by an exponential ansatz of the form $$x(t) = x_0 e^{\lambda t}$$ and discuss the several cases as given in the mentioned article

The motion in the underdamped case will look like an exponentially decaying oscillation $$x(t) = x_0 e^{-\frac{c}{2m}}\cos (\omega t + \phi_0)\,:$$

The picture was taken from the very good spanish wikipedia site concerning the topic, $c \equiv b$.

## Experiment

For a damped oscillation, you will have to determine two constants, $k$ and $c$. They are linked to the angular frequency of the system by $$\omega = \sqrt{\frac{k}{m}-\left( \frac{c}{2m}\right) ^2}\, .$$

Now, can you tell me, with respect to the given form of the movement, how to measure the constants just with a stopwatch and a ruler?

After you figured everything out, you can go into a lot of different directions like how the spring constant depends on $m$ (nonlinear motion), aperiodicity, Fourier space (Lorentzian) etc.

Sincerely

-
Thanks Robert! This is exactly what I'm looking for. I would just measure 'k' using F=kx and altering the masses on a spring and measuring the extensions (more accurate than a stopwatch) to find k. I'm not sure how to find 'c' though, what is 'c' anyway? The 'different directions' is something I'm also very interested, do you have any additional suggestions? Thanks! – MathsStudent Apr 12 '11 at 10:37
Have a look at Mathsstudents two questions within one day. I doubt he will understand essential parts Your answer. – Georg Apr 12 '11 at 10:38
I do understand most of it Georg :) – MathsStudent Apr 12 '11 at 10:41
So to what use this question today? This is not a chat forum. – Georg Apr 12 '11 at 10:45
@MathStudent: Your way of finding $k$ is just fine. Now, can you imagine a way to measure $\omega$ or the damping directly? Greets – Robert Filter Apr 12 '11 at 11:01

Is this a real spring? If it is, then damping is affected by a) The number of active coils (mass that is in motion), b) The cross section of the wire, c) Any rubbing or clashing between the coils, and d) the rate at which loads are applied on the coil (frequency dependency).

I suggest putting strain gauges on each 1/4 turn and hitting it with a hammer. You will see the response of the spring clearly and how the dynamics of the coils really affect the overall behavior.

I suggest find a good engineering book an spring design and analysis and you will be amazed at the complexity that springs exhibit in real life.

-