Summary 3.11 non-linear models (Telephone wires, rocket motion, pulled rope)

3.11 Non-linear models
(telephone wires, rocket
motion, pulled rope)

Table of Contents
Nonlinear Springs
Hard and Soft Springs
Nonlinear Pendulum
Telephone Wires
Rocket Motion
Variable Mass


Nonlinear Springs
for a linear function where F (x) = kx

d2
m 2 + F (x) = 0
dy

for a nonlinear function F (x) = kx + k1 x3

d2 x 2 d2 x
m 2
+ kx = 0 or m 2
+ ks + k1 x3 = 0
dt dt
Hard and Soft Springs
The restoring force is given by

F (x) = kx + k1 x3 , k > 0

The spring is hard if k1 >0
The spring is soft if k1 <0



Nonlinear Pendulum



3.11 Non-linear models (telephone wires, rocket motion, pulled rope) 1

, a simple pendulum is a special case of the physical pendulum and consists
of a rod of length l to which a mass m is attached at one end\

the angular acceleration is

d2 s d2 θ
a= 2 =l 2
dt dt
For Newton's second law

d2 θ
F = ma = ml 2
dt
Equate the the two different versions of the tangential force to obtain

d2 θ g
+ sin θ = 0
dt2 l
If we assume that the displacements θ are small enough to justify using the
replacement sin θ ≈ θ then

d2 θ g
+ θ=0
dt2 d



Telephone Wires
the firs order differential equation gives

dy W
=
dx T1

Since W = ρs

dy ρs
=
dx T1

Since the arc length between points P1 and P2 is given by




3.11 Non-linear models (telephone wires, rocket motion, pulled rope) 2