16.61 Aerospace Dynamics
Spring 2003
Lecture #10
Friction in Lagrange’s Formulation
Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002)
1
16.61 Aerospace Dynamics
Spring 2003
Lecture #10
Friction in Lagrange’s Formulation
Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002)
1
16.61 Aerospace Dynamics
Spring 2003
Generalized Forces Revisited
•
Derived Lagrange’s Equation from D’Alembert’s equation:
(
)
(
)
1
1
i
i
i
p
p
i
i
i
i
i
i
i
x
i
y
i
z
i
i
i
m x x
y y
z z
F x
F
y
F z
δ
δ
δ
δ
δ
δ
=
=
+
+
=
+
+
∑
∑
&&
&&
&&
•
Define virtual displacements
1
N
i
i
j
j
j
x
x
q
q
=
∂
=
∂
∑
δ
δ
•
Substitute in and noting the independence of the
j
q
δ
, for each
DOF we get one Lagrange equation:
1
1
i
i
i
p
p
i
i
i
i
i
i
i
i
i
r
x
y
z
i
i
r
r
r
r
r
r
x
y
z
x
y
z
m x
y
z
q
F
F
F
q
q
q
q
q
q
q
=
=
∂
∂
∂
∂
∂
∂
+
+
=
+
+
∂
∂
∂
∂
∂
∂
∑
∑
&&
&&
&&
r
δ
δ
•
Applying lots of calculus on LHS and noting independence of
the
i
q
δ
, for each DOF we get a Lagrange equation:
1
i
i
i
p
i
i
x
y
z
i
r
r
r
r
x
i
r
y
z
d
T
T
F
F
F
dt
q
q
q
q
q
=
∂
∂
∂
∂
∂
−
=
+
+
∂
∂
∂
∂
∂
∑
&
•
Further, we “moved” the conservative forces (those derivable
from a potential function to the LHS:
1
i
i
i
p
i
i
x
y
z
i
r
r
r
r
x
i
r
y
z
d
L
L
F
F
F
dt
q
q
q
q
q
=
∂
∂
∂
∂
∂
−
=
+
+
∂
∂
∂
∂
∂
∑
&
Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002)
2
16.61 Aerospace Dynamics
Spring 2003
•
Define Generalized Force:
1
r
i
i
i
p
i
i
q
x
y
z
i
r
r
x
i
r
y
z
Q
F
F
F
q
q
=
q
∂
∂
∂
=
+
+
∂
∂
∂
∑
•
Recall that the RHS was derived from the virtual work:
r
q
r
W
Q
q
=
δ
δ
•
Note, we can also find the effect of conservative forces using
virtual work techniques as well.
Example
•
Mass suspended from linear spring and velocity proportional
damper slides on a plane with friction.
•
Find the equation of motion of the mass.
g
c
k
m
q(t)
µ
θ
g
c
k
m
q(t)
µ
θ
•
DOF = 3 – 2 = 1.
•
Constraint equations:
y = z
= 0.
•
Generalized coordinate:
q
•
Kinetic Energy:
2
1
2
=
&
T
mq
•
Potential Energy:
2
1
sin
2
q
mgq
=
−
V
k
θ
Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002)
3
16.61 Aerospace Dynamics
Spring 2003
•
Lagrangian:
2
2
1
1
sin
2
2
L T V
mq
kq
mgq
= − =
−
+
&
θ
•
Derivatives:
,
,
L
d
L
L
mq
mq
kq mg
q
dt
q
q
∂
∂
∂
=
=
= − +
∂
∂
∂
&
&&
&
&
sin
θ
•
Lagrange’s Equation:
sin
r
q
d
L
L
mq kq mg
Q
dt
q
q
∂
∂
−
=
+
−
=
∂
∂
&&
&
θ
•
To handle friction force in the generalized force term, need to
know the normal force
Æ
Lagrange approach does not
indicate the value of this force.
mg
F
s
F
d
F
f
N
mq
&&
o
Look at the free body diagram.
o
Since body in motion at the time
of the virtual displacement, use
the d’Alembert principle and
include the inertia forces as well
as the real external forces
o
Sum forces perpendicular to the motion:
cos
N mg
θ
=
Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002)
4
16.61 Aerospace Dynamics
Spring 2003
•
Recall
W
δ
δ
= ⋅
F
s
q
. Two nonconservative components, look
at each component in turn:
o
Damper:
W
cq
δ
δ
= −
&
o
Friction Force:
sgn( )
sgn( )
cos
W
q N q
q mg
q
= −
= −
δ
µ δ
µ
θδ
•
Total Virtual Work:
(
)
sgn( )
cos
W
cq
q mg
q
= − −
&
δ
µ
θ δ
•
The generalized force is thus:
(
)
sgn( )
cos
r
q
r
W
Q
cq
q mg
q
=
= − −
&
δ
µ
θ
δ
•
And the EOM is:
(
)
sin
sgn( )
cos
sin
sgn( ) cos
mq kq mg
cq
q mg
mq cq kq mg
q
+
−
= − −
+
+
=
−
&&
&
&&
&
θ
µ
θ
θ
µ
θ
⇒
Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002)
5
16.61 Aerospace Dynamics
Spring 2003
•
Note:
Could have found the generalized forces using the
coordinate system mapping:
1
r
i
i
i
p
i
i
q
x
y
z
i
r
r
x
i
r
y
z
Q
F
F
F
q
q
=
q
∂
∂
∂
=
+
+
∂
∂
∂
∑
o
o
For example, the gravity force:
,
sin ,
sin
i
r
i
y
i
q
y
F
mg
y
q
q
Q
mg
sin
∂
= −
=
∂
−
=
=
θ
θ
θ
−
⇒
Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002)
6
16.61 Aerospace Dynamics
Spring 2003
Rayleigh's Dissipation Function
•
For systems with conservative and non-conservative forces,
we developed the general form of Lagrange's equation
N
qr
r
r
d
L
L
Q
dt
q
q
∂
∂
−
=
∂
∂
&
with L=T-V and
r
N
q
x
y
z
r
r
x
r
y
z
Q
F
F
F
q
q
q
∂
∂
∂
=
+
+
∂
∂
∂
•
For non-conservative forces that are a function of , there is
an alternative approach. Consider generalized forces
q
&
1
( , )
n
N
i
ij
j
Q
c q
=
= −
∑
&
j
t q
where the are the damping coefficients, which are dissipative
in nature
Î
result in a loss of energy
ij
c
•
Now define the Rayleigh dissipation function
1
1
1
2
n
n
ij i
j
i
j
F
c
=
=
=
∑∑
& &
q q
Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002)
7
16.61 Aerospace Dynamics
Spring 2003
Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002)
8
•
Then we can show that
1
r
r
n
N
q j
j
q
j
r
F
c q
Q
q
=
∂
=
= −
∂
∑
&
&
•
So that we can rewrite Lagrange's equations in the slightly
cleaner form
0
r
r
r
d
L
L
F
dt
q
q
q
∂
∂
∂
−
+
=
∂
∂
∂
&
&
•
In the example of the block moving on the wedge,
2
1
2
F
cq
=
&
sin
r
q
d
L
L
F
mq kq mg
cq Q
dt
q
q
q
∂
∂
∂
′
−
+
=
+
−
+
=
∂
∂
∂
&&
&
&
&
θ
where
r
q
Q
now only accounts for the friction force.
′