Upload
octavia-cannon
View
263
Download
4
Tags:
Embed Size (px)
Citation preview
Lab 5: Damped simple harmonic motion
• Simple harmonic oscillation
• Damped harmonic oscillation
381 Mechanics
Ideal case: no friction
Simple harmonic oscillation
Hooke's law:
Newton's 2nd law:
F kx
F mx
kx mx
0mx kx 2 0x x
k
m2
2m
Tk
solution: cosx A t : Amplitude
: phase
A
2 k
m
Simple harmonic oscillation (cont.)
displacement: cosx A t
velocity: sinv x A t
= cos2
v A t
cos sin
2
2accelaration: = cosa x A t
2= cosa A t cos cos
2Force: = cosF ma Am t kx
Friction: retarding motion (energy dissipation)
Damped simple harmonic oscillation
1
2
Hooke's law:
Damping force:
Newton's 2nd law:
F kx
F Rx kx Rx mx
F mx
0mx Rx kx 22 0x x x
, 2
k R
m m
Assume a solution: tx e
2 22 0te
Damped simple harmonic oscillation (cont.)2 22 0
2 2 2 2*
* *t t tx e Ae Be
2 20
2 20
2 20
underdamping: <
critical damping: =
overdamping: >
2 2* i
* 0 2 2*
Damped harmonic oscillation (underdamping)
1costx t Ce t
2 21let's define
1 1*i t i ttx t e Ae A e B is replaced by A* because x is a real function. let
2iC
A e
1 1
2i t i i t itC
x t e e e
using Eular's formula cos2
ix ixe ex
Damped harmonic oscillation (underdamping)
1costx t Ce t 2 21
0 10 20 30 40-1.0
-0.5
0.0
0.5
1.0
x
time (arb. unit)
0 1; 0 0x v
0 10 20 30 40 50-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2x
Time
=10 = =0.1
e
Damped harmonic oscillation