lecture 5 - coulomb damping

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


2/12


Vibrations

SDOF concept map


3/12


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


4/12


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=


5/12


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


6/12


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


7/12


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


8/12


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


9/12


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


10/12


11/12


Vibrations

Generic Simulink Model for SDOF System


12/12


Vibrations

One way to get the envelope is to use the

Hilbert Transform

lecture 5 - coulomb damping

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