Lecture 5 - Coulomb Damping
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Transcript of Lecture 5 - Coulomb Damping
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8/13/2019 Lecture 5 - Coulomb Damping
1/12
Rose-Hulman Institute of TechnologyMechanical Engineering
Vibrations
Todays Objectives:
Students will be able to:
a) Understand the difference
between Coulomb and viscous
damping
b) Identify system parameters whenit has Coulomb damping
Trivia of the day
Coulomb Damping
-6
-4
-2
0
2
4
6
Displacement
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8/13/2019 Lecture 5 - Coulomb Damping
2/12
Rose-Hulman Institute of TechnologyMechanical Engineering
Vibrations
SDOF concept map
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8/13/2019 Lecture 5 - Coulomb Damping
3/12
Rose-Hulman Institute of TechnologyMechanical Engineering
Vibrations
Coulomb Damping (dry friction) EOM
= Fdt
Pd sys rr
( )321 &&&
&xofsign
xsgnFkxxm d=
Find EOM using conservation of linear momentum rate (Newtons 2nd Law) in the x-direction
Damping Force: Fd=kN(independent of velocity)(SEP)
x
k
mk, s
FBD
(SEP)
x
kx
mg
Fd N
(SEP)
x
kx
mg
Fd N
a) Motion to the right b) Motion to the left
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8/13/2019 Lecture 5 - Coulomb Damping
4/12
Rose-Hulman Institute of TechnologyMechanical Engineering
Vibrations
Coulomb Damping (dry friction) EOM
(2)left)(moving0
(1)right)(moving0
>=+
>=+
xFkxxm
xFkxxm
d
d
&&&
&&&
(4)left)(moving0
(3)right)(moving0
2
2
>=+
>=+
xkFxx
xk
Fx
x
d
n
d
n
&&&
&&&
We can write this as two EOM (one for each direction)
Or in standard from
where
m
kn=
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8/13/2019 Lecture 5 - Coulomb Damping
5/12
Rose-Hulman Institute of TechnologyMechanical Engineering
Vibrations
Coulomb Damping (dry friction) Solution
( )
( ) (6)left)(movingscos
(5)right)(movingscos
43R
21R
k
FtinAtAtx
k
FtinAtAtx
dnn
dnn
++=
+=
( )
( )leftthetomovingstarts
00
0 0
=
=
x
xx
&
We can solve Eqns. 3 and 4
Assume ICs
Procedure (see book for details)
1. Eq. 1 and 2 are valid for only a cycle
2. Solve Eq. 6 forA3 andA4 using ICs
3. Endpoint of this cycle becomes the IC for the moving right phase
governed by Eq. 5
4. Solve Eq. 5 forA1 andA2. Endpoint of this cycle is IC for next cycle5. Go to step 2
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8/13/2019 Lecture 5 - Coulomb Damping
6/12
Rose-Hulman Institute of TechnologyMechanical Engineering
Vibrations
-6
-4
-2
0
2
4
6
Displacement
Coulomb damping Free response
k
Fx d
20
k
Fx d
40
Observations
x0
n
2
Offset
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8/13/2019 Lecture 5 - Coulomb Damping
7/12
Rose-Hulman Institute of TechnologyMechanical Engineering
Vibrations
Hysteretic Damping (structural or solid damping)
Experimental results indicate that
for most structural materials such
as steel or aluminum the energyloss is:
Independent of the frequency
Proportional to the amplitude
squared Notes:
Rarely is only one form of damping
present
A semilog plot can provide anindication of the damping present
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8/13/2019 Lecture 5 - Coulomb Damping
8/12
Rose-Hulman Institute of TechnologyMechanical Engineering
Vibrations
Damping records of automobile motion (Figures taken from
An Introduction to Mechanical Vibrationsby Steidel)
Well function shock
absorber
Shock absorber action is
poor and damping is
primarily structural
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8/13/2019 Lecture 5 - Coulomb Damping
9/12
Rose-Hulman Institute of TechnologyMechanical Engineering
Vibrations
Damping records of automobile motion (Figures taken from
An Introduction to Mechanical Vibrationsby 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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8/13/2019 Lecture 5 - Coulomb Damping
10/12
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8/13/2019 Lecture 5 - Coulomb Damping
11/12
Rose-Hulman Institute of TechnologyMechanical Engineering
Vibrations
Generic Simulink Model for SDOF System
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8/13/2019 Lecture 5 - Coulomb Damping
12/12
Rose-Hulman Institute of TechnologyMechanical Engineering
Vibrations
One way to get the envelope is to use the
Hilbert Transform