16.61 Aerospace Dynamics Spring 2003
Lecture #8
Examples Using Lagrange's Equations
Massachusetts Institute of Technology ? How, Deyst 2003 (Based on notes by Blair 2002) 2
16.61 Aerospace Dynamics Spring 2003
Example
Given: Catapult rotating at a constant rate (frictionless, in the
horizontal plane)
Find the EOM of the particle as it leaves the tube.
ω
x
y
Massachusetts Institute of Technology ? How-Deyst 2003 (Based on Notes by Blair 2002) 1
r r
16.61 Aerospace Dynamics Spring 2003
Derivatives:
?T
= mr
D
,
d
?
?
?T
?
?
= mr
DD
,
?T
= mrω
2
? D dt
?
? D
?
?r
External forces: None
CC ?Lagrange’s equation gives the equation of motion as rrω
2
= 0
What do we get if we solve this via Newton’s method?
Massachusetts Institute of Technology ? How-Deyst 2003 (Based on Notes by Blair 2002) 3
16.61 Aerospace Dynamics Spring 2003
Example
Mass particle in a frictionless spinning ring.
Ring spins at constant rate ω
m
θθ
g
r
ω
m
Spherical coordinate set (2-11)
Two holonomic constraints
? r = constant
? φ = ωt+φ
0
which gives the spin rate of the tube
So only 1 DOF ? use θ as the generalized coordinate
Massachusetts Institute of Technology ? How-Deyst 2003 (Based on Notes by Blair 2002) 1
16.61 Aerospace Dynamics Spring 2003
Example
System of 3 “particles” suspended by pulleys.
(Neglect mass of pulleys.)
g
m
1
m
2
m
3
l
h
y
1
y
2
s
1
s
2
s
3
Massachusetts Institute of Technology ? How-Deyst 2003 (Based on Notes by Blair 2002) 1
16.61 Aerospace Dynamics Spring 2003
Example
2 particles in a frictionless tube held by springs. Assume that
s = 0 and a = 0
Elevator
ω = const.
a
g
s
k
1
k
2
k
3
m
1
m
2
Motor
Massachusetts Institute of Technology ? How-Deyst 2003 (Based on Notes by Blair 2002) 1
