Coulomb Damping
Today’s Objectives:
Students will be able to:
6
b) Identify system parameters when
it has Coulomb damping
4
2
Displacement
a) Understand the difference
between Coulomb and viscous
damping
0
-2
-4
-6
Trivia of the day
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Mechanical Engineering
SDOF concept map
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Rose-Hulman Institute of Technology
Mechanical Engineering
Coulomb Damping (dry friction) – EOM
Damping Force: Fd=µkN (independent of velocity)
(SEP)
x
k
m
FBD
a) Motion to the right
(SEP)
µk, µs
b) Motion to the left
(SEP)
x
mg
x
mg
kx
kx
Fd
Fd
N
N
Find EOM using conservation of linear momentum rate (Newton’s 2nd Law) in the x-direction
↓
r
dPsys
dt
r
= ∑F
→
m&x& = − kx − Fd sgn (x& )
123
sign of x&
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Rose-Hulman Institute of Technology
Mechanical Engineering
Coulomb Damping (dry friction) – EOM
We can write this as two EOM (one for each direction)
m&x& + kx = − Fd
m&x& + kx = Fd
x& > 0 (moving right)
x& > 0 (moving left)
(1)
x& > 0 (moving right)
(3)
x& > 0 (moving left)
(4)
(2)
Or in standard from
&x&
+x=−
&x&
+x=
ω
2
n
ωn2
Fd
k
Fd
k
where
ωn =
k
m
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Rose-Hulman Institute of Technology
Mechanical Engineering
Coulomb Damping (dry friction) – Solution
We can solve Eqns. 3 and 4
F
xR (t ) = A1cosωn t + A2sinωn t − d
k
F
xR (t ) = A3cosωn t + A4sinωn t + d
k
Assume IC’s
x(0 ) = x0 ⎫⎪
⎬ starts moving to the left
x& (0 ) = 0 ⎪⎭
(moving right)
(5)
(moving left)
(6)
Procedure (see book for details)
1.
2.
3.
4.
5.
Eq. 1 and 2 are valid for only ½ a cycle
Solve Eq. 6 for A3 and A4 using IC’s
Endpoint of this ½ cycle becomes the IC for the “moving right” phase
governed by Eq. 5
Solve Eq. 5 for A1 and A2. Endpoint of this ½ cycle is IC for next ½ cycle
Go to step 2
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Coulomb damping – Free response
Observations
6
x0
4
x0 −
4 Fd
k
Offset
Displacement
2
0
-2
-4
-6
2F
x0 − d
k
2π
ωn
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Rose-Hulman Institute of Technology
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Hysteretic Damping (structural or solid damping)
• Experimental results indicate that
for most structural materials such
as steel or aluminum the energy
loss is:
– Independent of the frequency
– Proportional to the amplitude
squared
• Notes:
– Rarely is only one form of damping
present
– A semilog plot can provide an
indication of the damping present
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Rose-Hulman Institute of Technology
Mechanical Engineering
Damping records of automobile motion (Figures taken from
An Introduction to Mechanical Vibrations by Steidel)
Well function shock
absorber
Shock absorber action is
poor and damping is
primarily structural
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Rose-Hulman Institute of Technology
Mechanical Engineering
Damping records of automobile motion (Figures taken from
An Introduction to Mechanical Vibrations by Steidel)
Dry friction or rubbing
structural parts is apparent
Shock absorbers are
ineffective. A loose fitting
plunger makes the damping
different in each direction
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Rose-Hulman Institute of Technology
Mechanical Engineering
We will use Simulink
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Generic Simulink Model for SDOF System
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One way to get the envelope is to use the
Hilbert Transform
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Rose-Hulman Institute of Technology
Mechanical Engineering