m,L

1
ME6441, Dr. Ferri
Example, Constrained Dynamics
Slender rod of mass m and length L slides without
friction on inclined plane
B
m,L
C
A
q
b
b
Free-Body Diagram
q
C
b-q
NB
mg
NA
2
Force/Moment Balance
SFx

SFy

SMc

Kinematics
3
Combine
Differentiate
or
and
4
Finally, get differential equation for q
5
Lagranges Equation Approach
Recall
6
Potential energy
Virtual Work
Calculate necessary derivatives
7
Lagranges Equation
where
8
B
Alternate Formulation
C
NB
q
A
yc
NA
b
O
xc
9
Virtual displacements
Virtual work
10
Lagranges equations
j1,2,3
xc
yc
q
11
B
Alternate Formulation
C
q
yc
A
b
O
xc
In this formulation, we assume that the rod is
free to move in a 2-D plane. Other than
(conservative) gravity forces, no other forces
are modeled.
12
Constraint at A
C
A
O
13
Constraint at B
B
C
O
configuration constraint
14
Constraint summary
Constraint at A
Thus,
Constraint at B
Thus,
15
Lagranges Equations
j1,2,3
xc
yc
q
Comparing with Newton-Euler equations