Mass with a Spring and a Rubber Band

This system consists of a mass attached to both a spring and a rubber band subject to an external force proportional to

sin(ωt)

, where

ω

is the angular frequency. The other controls are the damping factor

b

and the restoring forces

k

1

and

k

2

. The spring obeys Hooke's law, as does the rubber band when it stretches, but not when it contracts. Systems of this type have been suggested as possible causes of the collapse of the Tacoma Narrows Bridge.

Details

The equation of motion is

y''(t)+by'(t)+

k

1

y(t)+

k

2

h(y)=10+.01sin(ωt)

with

h(y)=

y	y≥0
0	y<0

.

The gravitational acceleration

g

is approximated as 10

m/

2

s

.

[1] P. Blanchard, R. L. Devaney, and G. R. Hall, Differential Equations, 4th ed., Boston: Brooks/Cole, 2012, pp. 454–456.

[2] Tacoma Narrows Bridge Collapse "Gallopin' Gertie."[Video]. (Dec 9, 2006) www.youtube.com/watch?v=j-zczJXSxnw.

External Links

Gravitational Acceleration (ScienceWorld)

Angular Frequency (ScienceWorld)

Spring (ScienceWorld)

Hooke's Law (ScienceWorld)

Hooke's Law

Phase Portrait (Wolfram MathWorld)

Permanent Citation

Enrique Zeleny

"Mass with a Spring and a Rubber Band"
http://demonstrations.wolfram.com/MassWithASpringAndARubberBand/
Wolfram Demonstrations Project
Published: January 2, 2013