aho01_DHM.

# aho01_DHM. - Damped Harmonic Motion Closing Doors and Bumpy...

This preview shows pages 1–3. Sign up to view the full content.

Damped Harmonic Motion Closing Doors and Bumpy Rides Andrew Forrester May 4, 2010 Prerequisites and Goal Assuming you are familiar with simple harmonic motion, its equation of motion, and its solutions, we will now proceed to damped harmonic motion. In this analysis we’ll ﬁnd that among all the possible levels of damping, from zero to inﬁnity, that there is a special “sweet spot” level of damping called critical damping, that will allow you to automatically shut a door most quickly (and smoothly) without slamming it, for example. This principle can apply to shocks in vehicle suspension as well, to quickly absorb impacts from bumps in the road. We’ll examine these topics again near the end of this paper. Qualitative Analysis of Motion Before getting mathematical, we should examine some real systems and get an intuitive feel for them. Well, on paper, we’ll just examine some graphs representing damped harmonic motion with diﬀerent levels damping and discuss why they make sense. In this section we’ll pretend that the graphs are created using data from a real oscillator, but we should realize that real data would actually be a bit more complicated. You can ignore the graphs’ Keys for now and refer to them once you’ve examined the mathematics later in this paper. - x 0 0 x 0 0 2 π / ϖ n 4 / n x t Damped Harmonic Motion: initial conditions 1 Key r = 0 r = 0.2 n r = 0.4 n r = 0.6 n r = 0.8 n r = 1.0 n r = 2 n r = 7 n r = 20 n r = 99 n r = 1000 n Figure 1: Damped harmonic motion with diﬀerent levels of damping, with initial conditions 1. We’ll just examine one-dimensional motion in this paper. For concreteness, we’ll imagine the oscillator is an object with mass sitting on a frictionless table, attached to a spring (which is attached to a wall) and a damper (which contains some viscous ﬂuid). The variable x will denote its displacement from its equilibrium position. We’ll watch the motion of the object under several conditions, where the spring and mass are kept the same but the damping ﬂuid in the damper is made progressively more viscous, starting 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
from zero viscosity. We’ll examine two sets of initial conditions; 1) one where the object is released from rest at x 0 (see Figure 1), and 2) one where the object starts at the equilibrium position with a velocity v 0 , after being smacked with the appropriate impulse (see Figure 2). The ﬁrst noticeable fact from Figure 1 is that there are two distinct kinds of motion: oscillations with some rate of decay and “decay” or relaxation with no oscillations. Explaining the extreme cases in Figure 1 should be most obvious. With no damping, we simply have simple harmonic motion, a sinusoidal motion which is given by an unshifted cosine because of the initial conditions. With inﬁnite damping, the object does not move at all, being “frozen” by an inﬁnitely viscous ﬂuid (that is, a solid) at its initial position, and so its position-versus-time curve is a straight horizontal line. What about the intermediate cases? We
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 11/12/2010 for the course PHYSICS 1B 318007241 taught by Professor Corbin during the Spring '10 term at UCLA.

### Page1 / 7

aho01_DHM. - Damped Harmonic Motion Closing Doors and Bumpy...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online