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)
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16.61 Aerospace Dynamics
Spring 2003
Generalized Forces Revisited
• Derived Lagrange’s Equation from D’Alembert’s equation:
p
∑ m ( &&x δ x + &&y δ y
i =1
i
i
i
i
i
p
(
+ &&
ziδ zi ) = ∑ Fxi δ xi + Fyi δ yi + Fzi δ zi
i =1
)
 ∂x 
• Define virtual displacements δ xi = ∑  i δ q j


j =1  ∂q j 
• Substitute in and noting the independence of the δ q j , for each
DOF we get one Lagrange equation:
N
 ∂xi
∂yi
∂zi
&&
&&
&&
+
+
m
x
y
z
∑
 i
i
i
∂qr
∂qr
i =1
 ∂qr
p
p


∂xi
∂yi
∂zi 
+ Fyi
+ Fzi
δ qr = ∑  Fxi
δ qr
∂
∂
∂
q
q
q
i =1 

r
r
r 
• Applying lots of calculus on LHS and noting independence of
the δ qi , for each DOF we get a Lagrange equation:
d  ∂T

dt  ∂q&r
p
 ∂T

∂xi
∂yi
∂zi 
−
=
F
+
F
+
F
∑  xi


yi
zi
∂qr
∂qr 
 ∂qr i =1  ∂qr
• Further, we “moved” the conservative forces (those derivable
from a potential function to the LHS:
d  ∂L

dt  ∂q&r
p
 ∂L

∂xi
∂yi
∂zi 
−
=
F
+
F
+
F
∑  xi


yi
zi
∂qr
∂qr 
 ∂qr i =1  ∂qr
Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002)
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16.61 Aerospace Dynamics
Spring 2003
• Define Generalized Force:

∂x
∂y
∂z 
Qqr = ∑  Fxi i + Fyi i + Fzi i 
∂qr
∂qr
∂qr 
i =1 
p
• Recall that the RHS was derived from the virtual work:
Qqr =
δW
δ qr
• 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.
• DOF = 3 – 2 = 1.
• Constraint equations: y = z = 0.
• Generalized coordinate: q
g
k
q(t)
c
1
• Kinetic Energy: T = mq& 2
2
m
θ
µ
1
• Potential Energy: V = kq 2 − mgq sin θ
2
Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002)
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16.61 Aerospace Dynamics
Spring 2003
1
1
• Lagrangian: L = T − V = mq& 2 − kq 2 + mgq sin θ
2
2
• Derivatives:
∂L
= mq& ,
&
∂q
d  ∂L 
= mq&&,


&
dt  ∂q 
∂L
= − kq + mg sin θ
∂q
• Lagrange’s Equation:
d  ∂L  ∂L
−
= mq&& + kq − mg sin θ = Qqr


&
dt  ∂q  ∂q
• 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.
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
mq&&
Fs
mg
Fd
N
Ff
o Sum forces perpendicular to the motion: N = mg cosθ
Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002)
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16.61 Aerospace Dynamics
Spring 2003
• Recall δ W = F ⋅ δ s . Two nonconservative components, look
at each component in turn:
o Damper: δ W = −cq&δ q
o Friction Force:
δ W = − sgn(q ) µ N δ q
= − sgn(q ) µ mg cosθδ q
• Total Virtual Work:
δ W = ( −cq& − sgn(q) µ mg cosθ ) δ q
• The generalized force is thus:
Qqr =
δW
= ( −cq& − sgn(q ) µ mg cosθ )
δ qr
• And the EOM is:
mq&& + kq − mg sin θ = −cq& − sgn( q ) µ mg cosθ
⇒ mq&& + cq& + kq = mg ( sin θ − sgn(q ) µ cos θ )
Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002)
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16.61 Aerospace Dynamics
Spring 2003
• Note: Could have found the generalized forces using the
coordinate system mapping:

∂xi
∂yi
∂zi 
Qqr = ∑  Fxi
+ Fyi
+ Fzi

∂
q
∂
q
∂
q
i =1 
r
r
r 
p
o
o For example, the gravity force:
Fyi = − mg ,
yi = − q sin θ ,
∂yi
= − sin θ
∂q
⇒ Qqr = mg sin θ
Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002)
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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
d  ∂L

dt  ∂q&r
 ∂L
N
−
=
Q

qr
∂
q
r

with L=T-V and
Q N qr = Fx
∂x
∂y
∂z
+ Fy
+ Fz
∂qr
∂qr
∂qr
• For non-conservative forces that are a function of q& , there is
an alternative approach. Consider generalized forces
n
Q N i = − ∑ cij (q, t )q& j
j =1
where the cij are the damping coefficients, which are dissipative
in nature Î result in a loss of energy
• Now define the Rayleigh dissipation function
1 n n
F = ∑∑ cij q&i q& j
2 i =1 j =1
Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002)
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16.61 Aerospace Dynamics
Spring 2003
• Then we can show that
n
∂F
= ∑ cqr j q& j = −Q N qr
∂q&r j =1
• So that we can rewrite Lagrange's equations in the slightly
cleaner form
d  ∂L

dt  ∂q&r
 ∂L ∂F
+
=0
−
 ∂qr ∂q&r
• In the example of the block moving on the wedge,
1 2
F = cq&
2
d  ∂L  ∂L ∂F
−
+
= mq&& + kq − mg sin θ + cq& = Qq′r


dt  ∂q&  ∂q ∂q&
where Qq′r now only accounts for the friction force.
Massachusetts Institute of Technology © How, Deyst 2003 (Based on Notes by Blair 2002)
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