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

0 1 2 3

-1

0

1

x, v

, a

time

x v a

2

-1.0 -0.5 0.0 0.5 1.0

-0.5

0.0

0.5

v

x

Phase space plot (v vs. x)

k

m

Non-ideal spring (mass)

How should we consider the effect of finite spring mass?

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

Damped harmonic oscillation (underdamping)

1costx Ce t

0 5 10 15

-1.0

-0.5

0.0

0.5

1.0

x

time

x v a

0 5 10 15 20-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

x

time

x v a

Approaching the ideal limit2 2

0

-1.0 -0.5 0.0 0.5 1.0-1.0

-0.5

0.0

0.5

1.0

v

x

Phase space plot (v vs. x)

Detection: ultrasonic motion detector

2r v t