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:
()
()
11
iii
pp
iii i i ii x i y i z i
mxx yy zz F x F y F zδ δδ δ δ δ
==
++= + +
∑∑
&& && &&
? Define virtual displacements
1
N
i
ij
j
j
x
xq
q
=
??
?
=
??
?
??
∑
δ δ
? Substitute in and noting the independence of the
j
qδ , for each
DOF we get one Lagrange equation:
11
iii
pp
iii i i i
iiir x y z
rrr r r r
xyz x y z
mx y z q F F F q
qqq q q q
==
???
??? ? ? ?
++ = + +
???
??? ? ? ?
???
∑∑
&& && &&
r
?
?
?
δ δ
? Applying lots of calculus on LHS and noting independence of
the
i
qδ , for each DOF we get a Lagrange equation:
1
iii
p
ii
xyz
i
rr r r
x
i
r
y zdT T
FFF
dt q q q q q
=
?? ? ?
? ????
?= + +
?? ?
?? ? ? ?
?? ?
∑
&
?
?
? Further, we “moved” the conservative forces (those derivable
from a potential function to the LHS:
1
iii
p
ii
xyz
i
rr r r
x
i
r
y zdL L
FFF
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
riii
p
ii
qxyz
i
rr
x
i
r
y z
QFFF
qq
=
??
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
? DOF = 3 – 2 = 1.
? Constraint equations: y = z = 0.
? Generalized coordinate: q
? Kinetic Energy:
2
1
2
= &Tmq
? Potential Energy:
2
1
sin
2
qmgq=?Vk θ
Massachusetts Institute of Technology ? How, Deyst 2003 (Based on Notes by Blair 2002) 3
16.61 Aerospace Dynamics Spring 2003
? Lagrangian:
22
11
sin
22
LTV mq kq mgq=?= ? +& θ
? Derivatives:
,,
LdL L
mq mq kq mg
qdtq q
?????
===?+
??
??
&&&
&&
sinθ
? Lagrange’s Equation:
sin
r
q
dL L
mq kq mg Q
dt q q
????
?=+? =
??
??
??
&&
&
θ
? To handle friction force in the generalized force term, need to
know the normal force ? 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: cosNmg θ=
Massachusetts Institute of Technology ? How, Deyst 2003 (Based on Notes by Blair 2002) 4
16.61 Aerospace Dynamics Spring 2003
? Recall Wδ δ=?Fs
q
. Two nonconservative components, look
at each component in turn:
o Damper: Wcqδ δ=? &
o Friction Force:
sgn( )
sgn( ) cos
WqNq
qmg q
=?
=?
δ μδ
μ θδ
? Total Virtual Work:
( )
sgn( ) cosWcq qmg q=? ?
&
δ μθδ
? The generalized force is thus:
()
sgn( ) cos
r
q
r
W
Qcqqmg
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 μθ
θ μθ
?
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
riii
p
ii
qxyz
i
rr
x
i
r
y z
QFFF
qq
=
??
q
? ??
=++
??
???
??
∑
o
o For example, the gravity force:
,sin,
sin
i
r
i
yi
q
y
Fmgyq
q
Qmg
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
rr
dL L
Q
dt q q
??
??
?=
??
??
??
&
with L=T-V and
r
N
qx y z
rr
x
r
y z
QF F F
qqq
? ??
=++
? ??
? For non-conservative forces that are a function of , there is
an alternative approach. Consider generalized forces
q&
1
(,)
n
N
iij
j
Qcq
=
=?
∑
&
j
tq
where the are the damping coefficients, which are dissipative
in nature ? result in a loss of energy
ij
c
? Now define the Rayleigh dissipation function
11
1
2
nn
ij i j
ij
Fc
==
=
∑∑
&&
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
rr
n
N
qj j q
j
r
F
cq Q
q
=
?
==?
?
∑
&
&
? So that we can rewrite Lagrange's equations in the slightly
cleaner form
0
rrr
dL L F
dt q q q
??
???
? +=
??
???
??
&&
? In the example of the block moving on the wedge,
2
1
2
F cq=
&
sin
r
q
dL LF
mq kq mg cq Q
dt q q q
?????
′
?+=+? +=
??
???
??
&& &
&&
θ
where
r
q
Q
now only accounts for the friction force. ′
