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.
y
ω
x
Massachusetts Institute of Technology © How-Deyst 2003 (Based on Notes by Blair 2002)
1
16.61 Aerospace Dynamics
Spring 2003
Derivatives:
∂T
= mrD,
∂rD
∂T
d  ∂T 
D
D
=
mr
= mrω 2
,
 D
dt  ∂r 
∂r
External forces: None
Lagrange’s equation gives the equation of motion as rCC − rω 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
θ
r
g
ω
Spherical coordinate set (2-11)
Two holonomic constraints
• r = constant
• φ = ωt+φ0 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
h
s1
y1
m1
y2
l
m2
s2
m3
s3
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 k3
k2
g
k1
s
m1
ω = const.
Motor
m2
a
Elevator
Massachusetts Institute of Technology © How-Deyst 2003 (Based on Notes by Blair 2002)
1