FORCED VIBRATION IN SYSTEMS
WITH
ELAST ICALLY STTPPOIRTED DA'PFRS
by
JEROME E.
B.E.S.,
RUZICKA
Johns Hopkins TJniversity
(1955)
STBMITED IN PARTTIAL FULFILLIENT
FOR THTE
OF THE R ETTTREMNYTS
OF
DEREE
MASTER
OF
SCIENCE
at the
TNSTTTTTIE
TASSACHTSETTS
017
TECHNOLOCY
June., 1957
I
Si rnature of Author
De
t
et
of Mechanical Ehcigering
Certified By
Thesis Supervisor
Accepted By
Chairma
_ /Dep
artmental
Committee
on
Graduate
Stud ents
BARRY CONTROLS
OF
-DIVISION
7 0 0
P
L E A S
BARRY
WRIGHT
S T R E
A N T
E
CORPORATION
T
R T O
W A T E
W N
2,
7
M A S S
A C
H U S
ET T S
MARCH 2, 1962
TO WHOM IT MAY CONCERN:
THIS DOCUMENT REPRESENTS OFFICIAL PERMISSION
GIVEN TO PETER C. CALCATERRA TO MAKE CERTAIN CORRECTIONS
ON THE ORIGINAL AND ANY ADDITIONAL COPIES OF THE
UNDERSIGNED MASTERS THESIS ENTITLED:
"FORCED VIBRATIONS
IN SYSTEMS WITH ELASTICALLY SUPPORTED DAMPERS" WHICH
HAVE BEEN RETAINED IN THE M. I. T. LIBRARY.
-O
ROME E. RUZICKA
S
H O
PHONE
C
K,
WA 3-1150
V
-
I
B
R
A T
TELEPRINTER
I O
WTWN
N
-
A
MAss.
N
D
771U
N
0
I S E
WESTERN
UNION
C
0
N
T
R 0
WATERTOWN,
L
MASS.
9
FOPCED VPRATIONI
EM
IN SYST2
1ITH
SU
PPO!
ELASTICALLY
ED DAPRS
JEROME E.
RTZTCKA
TTBM ITTJ)D
-1ICAL TC TE DETPART1ENT OF
ON MAY 20, 1957
IN
PATIAL FTTLFTTLLMN
'F
THE
DEJGREE OF 'MSTEP
ETIMT
OF SCIENCE
ENGINEEP ING
FR
lTE
ABTRA CT
The dynamic response of a Parmonically forced
sinle-de-ree-of-freedom system with the damper
elastically supported is obtained for syvstems containing
Exact analytical
viscous and coulomb type dampers.
expre-ssions for the resDonse of the viscous damped system
are ,-eveloped and a linearized analysis of the coulomb
damped system provides approximate soluti ons for the
dynamic response of that system.
Analog corputor systemrs are devised to provide exact
solutions to both the viscous and coulomb damped systems.
Typical- response curves are given as obtained by numerical
calculation of the derived equations and from the analog
computor analysis. The response characteristics of the
systems are compared to tbe classical sin-le-degree-offreedom systems where the damper is rigidly supported and
a rather complete discussion is riven on the advantages,
disadvantaues, and applications of the systems analysed.
Thesis Supervisor:
Title:
James B. Reswick
Associate Professor of Mechanical Engineering
ACKNOWLEDGEMENTS
The author wishes to thank Professor Stephan H.
Crandall and Mr.
the problem
Charles E. Crede for suggestion
of
and is particularly indebted to Professor
James B. Reswick for his aid and encourage-ment throughout
the course of the investigation.
Additional thanks are also riven to Mr. Richard D.
Cavanaugh for his many helpful suggestions, to
Mr. Neptali Bonifaz for his help in calculation of the
response curves, to Mr. Perley A. Brown for his fine work
done on the drawings,
and to Mrs.
Edith Treggiari for her
effort and patience required in typing the manuscript.
-ii
-
COITTENTS
Page
CHAPTER I
Tntroduction and Nomenclature .
CHAPTER II
Discussion of Viscous Damped System . .
.0
CHAPTER TII
Discussion of Coulomb Damped System . .
.
.
CHAPTER IV
Epilogue
0
.
.
.
.
.
APPENDIX A
.
.
. *
*.. .
.
.
.
7
27
. ..
.
.
.
.
.
.
.
.
Solutions of Viscous System by Analog
Computor
APPENDIX C
.. .
.
Steady State Response of Viscous Damped
System and Approximate Solution of the
Coulomb Damped System
APPENDIX B
.
.
.
.
.
.
.
.
.
.
Analog Computor Solution
Damped
System
.
.
.
..
.
.
.
.
.
.
of Coulomb
.
o.
...
.
.
. . .
REFERENCES
-iii-
96
CTAPTER I - INTRODUCTION AND NOMENCLATURE
The problem to be considered is
that of determining
the steady-state response of a single-degree-of-freedom
system that contains an elastically supported damper.
Two types of damping will be considered,
coulomb or slip damping.
viscous and
Schematic diagrams of these
systems are shown in Figure I-1 and Figure
1-2.
This
type of system is of interest because of applications in
problems of turbine-blade vibration,
structures,
vibration of built-up
and in the 7eneral field of vibration isolation.
Although the viscous damped system, being a linear
syster,
presents no difficulty beyond solving a third order
linear differential eniation with constant coefficients,
rela-
tively very little information on the characteristics of the
s7rstem can be found in the technical literature.
This is prob-
ably a direct result of the fact that the dynamic response of
rn
m
C
JC
77//I-/
Fir. T_- z
F% -. I-i
-1-
this system can be expressed as a two parameter family of
functions (thus presentingT difficulties in showing solutions
graphically) as opposed to the system which has the damper
rigidly supported which is
family of functions.
expressible as aone parameter
Gallagher and Volterra (Reference 1)
have analysed this system (which they refer to as a
"relaxation type of vehicle suspension" system) for both
transient and steady-state type forcing.
However, the
manner in which the results are presented do not show
the real value of such a system.
Reference to this type
of system can also be found in certain books on vibration
(Reference 2),
but again, no convenient and clear response
curves are given.
The coulomb damped system, besides having application
in the design of vibration isolators, can be used to simulate
built-up structures - where the engineer depends on the
energy dissipation of the coulomb or slip damping to
reduce the response displacement and thus the resonant
stresses in the structure.
Investig.ators,
Klumpp and Lazan (Reference 3,
work along these lines.
4, 5)
in Reference
such as Goodman,
have done excellent
4,
Goodman and Klumpp
developed a general theory of slip damping, and investigations
were made into the energy loss per cycle of oscillation.
A study was also made of the damping properties of a beam
made up of two identical beams held toget'er by a clamping
pressure.
Results showed that for zero clamping pressure,
-2-
the energy dissipation is zero as it is for very high
clamping pressure.
It followed that there is always an
intermediate clamping pressure at which the energy
dissipation per cycle is
a maximum,
i.e.,
there is
an
optimum clamping pressure insofar as reduction of resonant
stresses is concerned.
Experiments were run where these
beams were vibrated and t'e displacement response plotted
for various values of uniform clamping pressure.
Double-
valued amplitudes (Jump phenomenon) occurred as would be
expected from a non-linear type vibration.
Studies of the structural damping of built-up beams
have also been made by Pian and Hallowell (Reference 6,7)
who also deal with the energy loss per cycle in such
a system.
All the works cited show the hysteresis type of
load deflection 'and damping that is characteristic of the
slip damping system, and as pointed out in Reference 5,
a complete theory of the dynamic behavior of mechanical
systems subject to hysteresis is
desirable.
In this paper, the author develops solutions to the
viscous damped system and presents typical response curves
for the system.
Both absolute and relative steady-state
response is shown in addition to a comparison of this
system with the conventional sinrle-degree-of-freedom
system where the damper is
ri~gidly supported.
The full
effect of varying damping, from zero to infinity is
snown
and a methiod of obtaining solutions by means of an analog
-3-
computor is developed for ease in obtaining response
curves.
A linearized approximate solution to the coulomb
damped system is also developed which, by itself, is valid
only for a specified frequency range, but when combined with
exact solutions obtained for areas outside of this range,
gives the steady-state response of the system with no
restriction on frequency range.
The effect of varying
damping over a wide range of values is
again shown along
with graphical representations of the steady-state
response.
An analog computor solution is
the non-linear characteristics
developed
which shoows
of the problem and response
curves as obtained from the computor are given.
Again,
damping' is made to take on a wide ranrg~e of values so as
to completely indicate the characteristics
-4-
of the system.
NOMENCLATTRE,
Mass of Supported Body
Spring Rate of Main Support Spring
Spring Rate of Damper Support Spring
Ratio kl/k
C
Viscous Damping Coefficient
jAF =
F
Coulomb DampinrT Force
Normal Force Acting in Coulomb Damper
Kinetic Coefficient of FPriction Between Coulomb
Damper Surfaces
a.
Displacement of Base
Displacement of Damper
Displacement of Mass
(x -0.)
=
Displacement of Mass Relative to Base
(X -Xj)
=
Displacement of Mass Relative to Damper
Maximum Displacement Amplitude of Base
xo
Maximum Absolute Displacement Amplitude of Mass
Maximum Displacement Amplitude of Mass Relative
to Base
Maximum Displacement Amplitude of Mass Relative
to Damper
9
Phase Angle Between Displacements X.
and
Q.,
Phase Angle Between Displacements
and
0k
6
Phase Angle Between Displacements
and CL
Forcing Frequency of Base Excitation
yA,
Ce
Ce
aMW
= Undamped Natural Frequency
ry =
Critical Viscous Damping Coefficient
Eauivalent Viscous Damping Coefficient for Coulomb
Damping
-5;-
A to
=
Coulomb Damping Factor (for Constant
Displacement Excitation)
My1 Ao
=
Coulomb Damping Factor (for Constant
Acceleration Excitation)
A.=
(TA =
Maximum Amplitude of Base Acceleration (for Constant
Acceleration Excitation)
I"
=
Absolute Displacement Transmissibility
1C~I =
Relative Displacement Transmissibility
A
Absolute Acceleration Transmissibility
of Mass
of Mass
=
of Mass
Total ?eal Forces Applied to Mass
C1
=
Viscous Damping Coefficient in Analog System
Mass of Coulomb Damper in Analog System
Time Constant of Integrating Component
Time Constant of Unit Lag Component
*
=Superscript
Referring to Common-Transmissibility
Point
Superscript Refering to Condition Where Damper
is "Locked-in"
Subscript Referring to Values of High Frequency Ratio
L
=
Subscript Referring to "Break Loose"
Points of the Coulomb Damper
or "Lock-in"
CHAPTER
II
- DTSCUSSTON
OF VISCOUS DAMPED SYSTEM
The viscous damped system to be analysed as shown in
Pigure T-1 consists of a mass (constrained to move only in
one direction) supported by a main load carrying spring
which is
The base is
attached to the base.
excited in such
a manner that its displacement varies harmonically with
time.
The motion of the mass is
damped by a viscous damper
which is supported by an elastic member attached to the
base.
Before embarking upon any mathematical analysis,
one
can draw some important conclusions about this system just
by applying some physical reasoning as to the effect of
For example, for zero
elastically supporting the damper.
viscous damping, the system is obviously undamped and
infinite amplitudes would be expected at resonance.
For
this case, resonance occurs at the undamped natural frequency
defined by the relation:
(II-1)
Wh
On the other hand, if the viscous damping is made to approach
infiniity, we see that this corresponds to "locking-in" the
damper which results in having an undamped system whose
natural frequency would be g7iven by:
Ci
where
(4+1
)I
(0
(11-2)
would be the total stiffness for this case.
-7-
Thus, the system would again experience infinite amplitudes
when driven at a frequency given by equation (11-2).
Having established these two limiting conditions by
simple physical reasoning, it then follows that there must
be some value of viscous damping which would give a minimum
This is easily established
peak amplitude at resonance.
if one visualizes increasing the viscous damping to a value
greater than zero which would obviously decrease the
resonant amplitude of the system.
This process could be
repeated to again reduce the maximum amplitude at resonance.
But,
this process is continued
if
as previously verified,
until infinite'damping is enforced, the system again
becom!ies undamped at a higher natural frequency and infinite
Thus, for the peak amplitudes
amplitudes occur at resonance.
to vary in a continuous manner with change in viscous
damping, there must have been some particular value of
viscous damping that produced a minimum resonant amplitude.
This value of viscous damping will be referred to as
"optimum damping".
A rather complete mathematical analysis of this
The expression for the
system is given in Appendix A.
absolute transmissibility of the system was shown to be
it
\N
-8-
-L-
4
I
'
and the relation giving the relative transmissibility
was derived as follows:
t-+0
It is interesting to note that when
these
equations reduce to the equations whicb describe the
undamped res ponse of a single -degree -of -fre edom sys tem.
N-o,
Also, if we let
=ISO
the system reduces to the
classical single-degree-of-freedom system with viscous
FoT-(;;
damping and equations
(II-3) and (II-4) reduce to the
known equations for the absolute and relative transmissibility
Moreover, if the viscous damping is allowed
of that svstem.
to approach zero, the absolute and relative transmissibility
equations become:
~-
TA)
knwneqaton
frA
h
(
(II-5)
asolt
(N
rltie---isiilt
The corresponding equations obtained when the
damping is made infinite are given by:
N
-9-
a
q
o
1(II-7)
__
(II-8)
)z
Thus, equations (TI-5) through (I1-8) Five the
"undamped" response envelopes wh ich must necessarily
bound the damped solutions given by equations (11-3)
and (T-II-).
An interesting observation can be made if one plots
the absolute values of the transmissibilities of the two
undamped conditiots, i.e., zero and infinite damping.
As shown in Figure II-1, the response curve for zero
damping intersects the response curve for infinite damping
at a frecuency ratio which will be deffned as the commonC
C
C
C
C
C
Ftc%.f14 Commvom.
Twtasmtss~bILITJ
-Pol&T
transmissibility frecuency ratio because of the fact that
at this point both of the undamped curves have the same
transmissibility.
The frequency ratio at which the two
response curves intersect can be found by equating the
-10-
O
I
two transmissibility expressions (taking into account
the proper sign of that portion of the response curve
being considered) and solving for the frequency ratio.
Proceeding
in
this manner,
the common-transmissibility
given by:
frequency ratio for absolute motion is
WYn A
(11-9)
-L+
Similarly the common-transmissibility frequency ratio
for relative motion is 7iven bv:
hN+
(IT-10)
To find the trans-missibilities corresponding to these
frequency ratios, one must substitute eauation (II-9) in
equation (TI-3)
and eauation (I1-10)
,hen these substitutions are made,
it
in ecuation (TII-4).
is
found that the
damping terms disappear in each case and the same equation
for transmissibility results for both absolute and
relative motion.
given by:
The common-transmissibility is
TP
thus
1T*
(II -11)
It
is
seen that this expression is
independent of
the damping and thus the conclusions can be drawn that
all damped curves in addition to the undamped curves must
-11-
pass through the common-transmrissibility point.
Note that
even thouph the common-transmissibility frequency ratio
is
different for absolute and relative motion, the value
of the common-transmissibility itself is
equal in each
case.
The commrion-transmissibility frequency ratio is
raphically in Figure 11-2 whi ch is
versus the spring7 ratio.
shown
a plot of thr is quantity
A limiting process applied to
equation (II-9) and (I1-10)
indicates that for values of
spring ratio approaching zero, both the absolute and
relative common-transmissibility frequency ratio tend
toward unity whereas for very large values of spring
ratio the absolute common-transmissibility frequency ratio
approaches\/2-* asymptotically while the relative commontransmissibility frequency ratio increases as the squareroot of one-half the spring ratio.
A graph of the common-transmissibility (for both
absolute and relative motion) is given in Firure 11-3.
As indicated by this graph, for very small spring ratio
(spring ratio approaching zero) i.e.,
a common-transmissibility
frequency ratio approaching unity, the common-transmissibility
approaches infinity.
On the other hand, for very large
spring ratios, the common-transmissibility approaches
unity as an asymptote.
-12-
P
~
I
.
.1
I
*~~
Lt IiLo~L
N+
I
I-
~
4
.....
............
4..
. ........
i
p
....
......
*
~ii
I
. :I
.
-K-
W
WI
z
...
.....
.......
I
I-I-
........
I
....... ...... .
- - -* -I - -
....
+
iwoo00
-
-4... ....
010100
T
...
..
4-N
z
:A-
sum
-Monoti'.
-l
>
0
.~.........
LI = 'V/
L... I. I I
LI
I.
4- - .. . .....
..
....
I'
j-
'
.11
f~
im
--- -- -
20t
Iii
Ii
~
-
N =kI/k -
SPRING
I
RATio
'
Fori
...
0
......
1
___I..~ '
I..."
V.
.~i\
q
q
I
0
.......
.....
P
H
.
0
I
~~i~j.j
MoT1 0NL AT
I
P1
20
I
-4
z
11
z
W
F4
1717
----
0IV
-
4
--. *--
a
-4-
4
4 t
"0%NXDI JS
.......
..
7
...
J7
{
-- -
..
4
- --- . - -
-4
'---*'------'----.*
-4-7
4
....- .
Al SI~ISa
... ....
..
t.
7
-4--
.'-.
4-
.-.....-..-
7 7-
77
10
-
A
7V
-1
-.-.
T
.....
.....
if
---
.--.
7F7
Ah
7-
..
-
+--...-.-...7..----
-----
-
-
C-Ii
.. .
74
....
-4------
---- --+
- ' -7
,mUcI
....
--. ---,-----
t- e
1771
i
-
- -+----.-...
- - ----
:t7-7
47
- -- --~+ - - -*-- -- *
.,++ -- . . -. - -+
:j
-
..
____I___I
AN
1..
ALL ~~-
- --
?
7
47
t-
4 4
-4 77
--+.,. .. -+
7
7-.
4 -4-
- -
..
--- -- ---
4.. +
.'-------.'..*-.7
7-
__7~~~
i
........7
77
-2-
-
......
TRANSMISSIBILTY
-......
.......-.......... ..
....
.~--- r 7-0..
-4
- +. . --- - - - - --- , --- --- ----+--
44
4
7_77 -7,
7 7-i-
.-..-..... ..... . ...4...- ..4....-
777
-
7
7T77
.. .. ..
7 .. .
-- - -4.+
- - --
4.
+
-
TA= TR -COMMON
Thus,
for a riven value of spring ratio, Figure I1-2
and Figure II-3 give information as to the response of
the system at a certain frequency ratio for all values
of viscous damping.
Resorting again to physical reasoning, it seems
evident that of all the different response curves
passing through the common-transmissibility point,
there
must be one which has the common-transmissibility frequency
ratio as its resonant frequency, i.e., its peak transmissibility corresponds to the value of the common-transmissibility.
The value of damping that characterizes this special curve is
then the optimum viscous damping. coefficient and the response
curve is
the optimum damping response curve.
By inspection of equations (II-3) and II-L4), the
transmissibility of the system for large frequency ratio
can readily be determined.
This is done by requiring that
the frequency ratio be very large. and thus eliminate certain
terms on the basis that they are small compared to other
terms.
The resulting expressions are given by:
(
N+I
A
J4-\X.
(IT-12)
(11-13)
The
?
subscript was chosen merely to indicate that
these relations are associated with regions of high
isolation (as opposed to magnification).
with equations
(11-7)
and (II-8),
By comparison
these equations indicate
that, for large frequency ratios, the transmissibility of
the system approaches the undamped response of the system
with the damper "locked-in".
This property is very
important to those involved with problems in vibration
isolation and the value of this particular property will
be made very evident when the response of this system is
compared to the response of the classical single-degreeof-freedom system where the 'viscous damper is rigidly
supported.
Since numerical calculation of equations (11-3) and
(TT-)') required considerable time and care, an analog
computor system was devised to supplement the calculation
of response curves.
A Philbrick type analog computor was
used which is a device that uses voltages to represent all
variables.
The analog computor system is a network of
interconnected operational amplifiers where each amplifier
is designed to perform basic mathematical operations as
well as complex non-linear transformations.
These operational
amplifiers are high gain D. C. vacuum tube amplifiers having
wide bandwidthhigh input impedance for minimum loading,
-16-
and low
output impedance in order that reasonable
driven.
loads may be
A complete description of the analog system
and its associated mathematics can be found in Appendix B.
For further information on the analog computor, the reader
is referred to References 9 and 10.
A typical response curve for this type of system is
shown in Firure II-I4 and Figure 11-5.
Figure II-4 shows
the absolute transmissibility and Figure 11-5 the relative
transmissibility of a system with a spring ratio equal to 3.
A spring ratio of 3 causes the undamped natural frequency
of the "locked-in" system to occur at a frequency ratio
of 2.
These transmissibility curves vividly show the
effect of varying the viscous damping from zero to
infinity.
As reasoned earlier, zero damping corresponds
to infinite amplitudes at a frequency ratio of unity while
infinite damping gives rise to an undamped system with a
natural frequency defined bT equation (11-2).
For values
of damping becoming successively larger than zero, the
resonant amplitude becomes successively smaller while the
resonant frequency ratio becomes greater than unity and
tends toward the common-transmissibility frequency ratio.
Eventually the viscous damping is increased to a critical
value which corresponds to having" the resonant frequency
ratio coincide with the common-transmissibility frequency
ratio.
This value of viscous damping corresponds to "optimum
-17-
TT-1
NS K
Lj
IBILTY
r7
-
-
-.-.
-t
7
--
-7
-------
-~
I
-
---
- --
-
AF
---------
t-
-
I
-
--
--
II.
----
t--
-
-4-
.J
,
-
-
tT
T
T
-
-
-
-
-
--
--
++
+
-
--
--
|
1777K117
4-
4-
T'
44
7N
-41
0--
-2* -K
-
--
-...-----
;.
-l
. ..
--
-
~---
--
.-
-
-
-
- --
- -
-.
- -
-
-
-
-
-
-
-- - --
*-----
*
*
-j~
4
'*
-
-
--
1:
-
-
;i1777M
Hi~j~1~j~~j~
S-, e-
-189-
-..
. +.
-- . -
-
*
-
-n
-I
C
I
2
fit
0
-
.
.~~~~.
.
....
-..
. .-
o-
..
-..
..
4
I
1-
.....----..--..
-m-0~~~~......
VAL.
.
-..
.
-
-
..
bI
.
.
..
-w
.
-...4
--
.
4
77
.
11111
t
-.
--
I-It
-....
*1(I
-
-.
-
-----------
.n
.-
TRANSMISSIBILITY
4.-.-.-.-.-4-'T.
-.......
RELATIVE
-
-
-
- - 4
.
-
-
-
-.-.
-
t.-.T.-t+
-
.......
-
---.--1
T--
.~~~
..--..-..-..-..--
-..-
-..
-
- *.-.
-.
I
damping".
It
is
this value of viscous damping that
minimizes the peak amplitude response for the particular
system under consideration.
davmping is
Further increase in viscous
seen to cause the resonant amplitudes to grow
again while the resonant frequency ratio becomes larger than
the common-transmissibility frequency ratio and tends toward
the frecuency rat o of the undamped "locked-in" system.
Thus,
the effect of varying damping from zero to infinity
on the dynamic response of the system has been demonstrated.
Figure 11-6 is a plot of the absolute transmissibility
of a system whose spring ratio is 8.
This corresponds
to a "locked-in" undamped natural frequency ratio of 3.
This graph naturally has the same general characteristics
as Figure II-)4, but comparison of the two shows the effect
of increasing
the spring ratio.
We see that the increase
of spr:Ing ratio has reduced the isolation at high frequency
ratios and has had a definite effect on the value of
optimum damping.
As predicted,
the transmissibility of
the optimum damped curve has been reduced and the resonant
frequency ratio of the optimum damped curve has been slightly
increased.
It
is
thus apparent that there are some counter-
acting influences that must be considered.
One gains in
isolation by reducing the spring ratio but must then accept
an increase in the peak amplitude at resonance.
On the
other hand, if the system is designed to have a very small
-20-
..........
FTGTTRE TI-6
I II
U ..
,~
I
I
L --J
I
-
~-
q~
U~
;-
~
- - - - - r
;
U
-T
-
8
=
OR
'TRANSMISSIBlILITY
ASO7L
I
~
~
I
I
-
-
-4--
I--
-+
--
t
I
0
U)
4~-t
444--
--
-
-4
+
-
-
-
-
-
-
---
4.+"
-
.
.
.
.
.
.
-
-
-
-
-
. .
-
--
.
.
-
4 - +
.
4. e
..
.=
I
- e ee .
.. .
-21-
--
--
-
-4
4--
--. -.
-
--
4- -4. .
- -
T
---
4
V-
-=4
.
. +--
-e --
--
- -. -
-. -- --
---
--
-
-
resonant amplification (e.g., optimum damping),
the
isolation at high frequency ratio might not be as good
as desired.
The graph shown in Fic-ure 11-7 indicates the manner
in which the maximum absolute transmissibility varies with
the damping ratio for various values of spring ratio.
The optimum damping ratio would correspond to that
value which produces the minimum maximum absolute transmissibility.
As indicated, these curves have a relatively
flat portion in the area of optimum damping.
Thus, it
appears that great accuracy is not required in selecting
Four different values of
a value for optimum damping.
the spring ratio parameter have been plotted to indicate
the effect a variation in spring ratio has on the optimum
damping ratio value.
For purposes of evaluating the vibration isolation
properties of this system, let us
now compare this
system with the conventional viscous damped system with
the damper rigidly supported.
We shall confine our
attention to the absolute motion of the mass during this
comparison.
As mentioned previously, the equation
for the absolute transmissibility of the conventional
viscous damped system is obtainable from equation (11-3)
by letting the spring ratio approach infinity.
When this
is done, the followin7 well known equation is obtained:
-22-
-4
-U
K
~pIc~
C
9
g
---
-
66
4
-
-
2+
-
)..±
-
-
--b
- -
6
-
J
...
- 3 -
-
t4~t
-
.
-F
e e-
-
-
:
-
-
0----
-4
4
-
-
-
T
N
-- -
-
---
-
-
--
--
-44-
.-.-
-
--
.
---------
-i
---
-
.
. . . . . . --
77+
-
+ -
4-
-4 i
-
MAXIMUM ABSOLUTE TRANSMISSIBILITY
-
-
-
.T.
_
H
-
-
-
+
.-
-
L4.
-
- - -
- -
1
0 e
- ----
-
-. . - - - --- -. -- - +-- . - - - -- - - - -
4
st
-
--
----
-
T
=
(TA)TT-1L4
By a limiting process, this equation is seen to reduce
to the following equation for transmissibility at large
frequency ratios:
A(II-15)
Thus, by comparing equation (II-15) with equation
(11-12) we see immediately the value of elastically
supporting the viscous damper.
For large frequency ratios,
the conventional viscous damped isolation system has the
property that the absolute transmissibility diminishes as
the inverse o-f the freauency ratio.
But, the system with
the elastically supported damper is seen to have the
property that the absolute transmissibility diminishes as
the inverse square of the frequency ratio.
Moreover, viscous
damping is still present in the conventional system at
high frequency ratios whereas the expression for transmissibility at high frequency ratio for the system with
the elastically supported damper is
of damping.
seen to be independent
Thus, by elastically supporting the viscous
damper, damping is virtually eliminated at high frequency
ratios which is apparently a good property of a vibration
isolator.
This comparison of performance of the two systems is
All curves on this graph
shown very well by Fig7ure II-8.
are for a value of damping ratio (c/cc) = 0.2 which gives
an absolute transmissibility of approximately 2.5 in the
conventional isolation system.
Several curves are shown,
each for a different value of spring ratio.
The
N=0
curve corresponds to the conventional
isolation system with the damper rigidly supported.
=0
The
curve represents the undamped response curve of
the system.
It
is
seen that for finite values of
spring ratio, the resonant amplitudc'e is slig7htly- increased,
while at hirgh frequency ratios the isolation is enormously
increased.
As a quantitative measure consider, for example,
the curve corresponding to a spring ratio of unity.
Comparinr the response of the two systems, the system with
the elastically supported damper is seen to have a resonant
amplitude of 1.3 times that of the conventional isolation
system.
However, at a frequency ratio of 10, the isolation
has increased by a factor of 2; and at a frequency ratio
of 100,
Thus,
the isolation has increased by a factor of 20.
by accepting7 a relatively small increase in trans-
missibility at resonance, a considerable increase in
isolation at high frequencies can be obtained.
-25-
........
...
- -- -,
- -- --Iz
I
-
-
--
--
-
-
FIGURE II-8
-... .
I
..
. ...................
.. .. .. w
PROPERTIES OF ELASTICALLY SUPPORTED VISCOUS DAMPER SYSTEM
0 4
0
Ir
Q.
t
~
I
t
4 ~
~ .1~
4r
I
*kll-lliPSSINSNVHJL,
31n-toseV A'
I;0_1
I
CHAPTER III - DISCUSSION OF COULOMB DAMPED SYSTEM
The coulomb damped system to be analysed as shown
in Fir'ure I-2 consists of a mass (constrained to move in
one direction) supported by a main load carrying spring
The base is excited in
which is attached to the base.
such
a manner that its motion varies harmonically with time.
The motion of the mass is
danped by a coulomb or dry
friction damper which is supported by an elastic member
attached to the base.
Before getting involved in any mathematical analysis
of the problem, perhaps it would be best to remark about
the characteristics of a coulomb damper and particularly
about the effect of elastically supportin
such an element.
Basically, the force developed in a coulomb damper
arises from the sliding between drr
on the normal force between them.
surfaces and depends
The direction which
the force acts on a rgiven surface is opposite that of the
relative velocity between the surfaces.
Thus,
the damping
force is positive when the relative velocity across the
damper is neg-ative and it is negative when the relative
velocity across the damper is
positive.
It is now necessary to investigate the effect of
elastically supporting a coulomb damper.
This will be
done by constructing the load-deflection and dampingdeflection characteristics of the system.
-27-
Consider applying a load to the mass and observing
the deflection of the mass and the friction force developed
Upon application of the load, the
in the coulomb damper.
mass will displace, but since insufficient force has been
developed in
spring
,
the damper will not move.
Thus,
the motion of the mass is opposed by a stiffness
Note that for this loading the friction force developed
can only equal the force developed in spring
,
i.e.,
the damper friction force increases linearly with the
displacerent,
the constant of proportionality being
.
Thus, for this initial loading, the slope of the load
deflection curve is
(*+*)
damping characteristic
while the slope of the
curve is
.
The load is
now
increased until a force is developed in kwhich equals
,LLF, where
and? is
F
is
the force normal to the friction surfaces
the coefficient
beyond this point,
of friction.
If
the load is
the damper will slide thus reducing7 the
sprinn resistance to th-e motion of the mass too
iP
V
increased
I
1:BI
and producing
a constant friction damping force of value ,AF. Thus,
for this portion of the load cycle,
the slope of the load
deflection curve is A-while the dampingc characteristic
curve has zero slope.
Suppose the load has been increased
to some maximum value.
If
it
now be decreased below this
maximum value the damper will no
longer move since the
force required to move the damper, which was previously
developed in
springR1 , has been diminished.
Therefore,
the motion of the mass will again be resisted by a total
stiffness of
and the daiping force will vary
linearly with the deflection of the mass.
Thus, it
is seen that, if
the load cycle is completed,
the damper will be in motion for part of the cyclQ and
will be at rest for the remaining part.
This means that
elastically supporting a coulomb damper creates hysteresis
type load-deflection and damping characteristics.
These
characteristics for a complete load cycle are shown in
FigTure
I-1
Thus,
and Figure
II-2.
the non-linear manner in which the coulomb
damping enters into the vibration of the system Las been
demonstrated.
It
can be shown (see Appendix C) that the
damping term +B necessarily enters the equations of motion,
and hence,
account must be taken of the non-linearity of
the damning force
by Figure ITT-2.
ith c'hane in displacement as indicated
It
is
fairly obvi ous that obtainin- an exact
-29-
analytic solution of the equations of motion would
undoubtedly be a formidable task (if not an impossible
one)
and thus,
is taken to other means of
recourse
obtaining the dynamic response of the system.
A solution for the dynamic response of the syrstem by
means of an approximate energTy method is
This is
the method of "euivalent
assumption is
7,iven in Appendix A.
viscous damping"i where the
made that equal energ'y is
dissipated by an
equivalent viscous and the coulomb damper in a cycle of
vibration.
Eauatin; the cyclic energy dissipation of a
the following
viscous damper to that of a coulomb damper,
relation is
obtained:
Ce
Tn this expression 15. is
damper and C
is
( III.1)
the relative motion across the
the ecuivalent viscous dampin
coefficient.
By determining- the relative motion across the damper,
using equation (TII-1)
in
and
conjunction with the relations
for transmissibility of the viscous damped svstem previously
derived,
e-pressions for the transmissibility of the
coulomb dam-oed system can be obtained.
qlacement tranissibility
is
The absolute dis-
tTus 7iven by:
( .ti..l
()CC
)
-30-
(III-2)
The eynression for th-e relative displacement
transmissibilitr is
The qu-ant ity
as
follows:
found in both of these expressions is
the coulomb damping factor for constant d3iplcemenit
excitation of the base and is defined hy:
Thus, we see the transmissibility of the coulomb damped
system is
dependent upon the amplitude of cxcitation.
An interestinr' observation can be made by considering
the terms in the numerator of each of the transmissibility
express ions.
term
We see that if
044
vanishes in each case,
independent of damping,
the coefficient of the damrping
the transmissibili ty is
and therefore,
all response curves
have the same transmissibi lity for this condxiition no matter
how - uch damping i s nre sent .
This is
seen to correspond to
the "common-transmssibility" point as defined in Chapter T.
F1or absolute motion, setting' the coefficient of the damping
term ecual to zero e~ives the followine; condition:
Thi s is
the exact same expression derived for the absolute
comwon-transmis s ibility freuency ratio of the viscous
-31-
damped system.
The condition for the relative displacement
transmissibility to be independent
of
daImpi'ng can be
written:
Again, this corresponds to the relative common-tranmissibility
frecuency ratio for the viscous damped system.
The reason
why the same relations hold for both the viscous and
coulomb danped system
.s that the metVod of eauivalent
viscous dampinrg was employed to derive the coulomb damped
ecuations and there will naturalyhbe
some properties common
to hoth systems.
The common-transmissibility for, absolute and relative
motion can be found bT substituting equations
(ITT-5) and
(ITI-6) in the proper eyxpressions for transmissibility.
is found tlh)at the common-transmissibility
both cases and that it
is the same in
also arrees with the expression for
the viscous damped system.
is thus
The common-transmissibility
i1ven byr:
N(III-7)
Therefore, we have determined a point through which
all response curves must pass no matter what value of
coulomb damping is
in
present.
This is
not necessarily true
realityrhut definitelr holds for the linearized system
hein' considered here.
it
One should refer to Chapter I for
-32-
curves of common-transmissibility and common-transmissibility
frecuency ratio.
It
important to realize that the coulomb damped
is
solutions are valid only for a certain frequency range.
It
seems logical that these equations be vplid onily when
relative motion across the damper because the
there is
whole approximate analysis was made on an assumption of
enual enerp
dissipation in the viscous and coulomb
Therefore,
dampers.
coulomb damper,
and thus,
fails.
if
no motion across the
there is
there obviouslv is
no enerry dissipated,
the assumption of equal energy dissipation
not as bad as it
This apparent limitation is
seem at first
glance because if
across the coulomb damper it
t--ere is
might
no relative motion
must necessarily be "locked-in"
undamped with a natural frequency
which means the system is
defined by ecuation (II-2).
Thus,
for frequency ratios
outside the range of validity of the coulomb damped solutions,
the absolute and relative transmissibility of the zystem is
qiven by the undamped solutions expressed by equations
and (TT-8).
(11-7)
The frequency ratios which determine the
extremes of the range of validity of the coulomb damped
solutions were derived fn Aooendix A and are
(III-8)
__r'____N___
L
~
-33-
-,iven by:
The "L" subscript merely indicates that the coulomb
damper has either just "broken loose" or "locked-in"
as
To1ild
be the conditions at the ex:tremes of the range
of validity.
Thus,
by use of the transmissibility
eouations
(III-2) and (T11-3)
enuations
(TIT-8),
and the supplementary
(II-7) and (I-8),
the dynamic response
of the linearized coulomb damped system for all values
of freauency ratio can be obtained.
Several plots of
these eauations will be discussed later.
Before discussing the transmissibility graphs, let
us observe what values of the damping parameter V( are of
practical interest.
and (III-3)
It is evident from equations
(I-2)
that for a frequency ratio of unity the
transmissibility becomes infinite.
Thus,,to prevent such
a condition from arising, the damper must be kept "lockedin" beyond this point.
This obviously puts a limit on
the value of the damping parameter.
can be determined from equation
This limiting value
(I-8)
by requiring
the "break loose" frenuency ratio to be rreater than
unitT and solving for the damping parameter.
restriction on the damping parameter is
The following
obtained:
>
Thus,
the dampin7 parameter must be greater than
to avoid having infinite transmissibility at a frequency
ratio of unity.
(
iiI-9)
The dynamic properties of the coulomb d'amped system
are shown very nicely b
a series of gra hs.
Figure I-3
shows the absolute displacement transmissibility of a
system witb a spring ratio of 3.
As indicated previously,
for values of the damping parameter near
the system
experiences very large transmissibilities in
hood of a frenuency ratio of unity.
parameter above
the neighbor-
Tncrease of th e damping
decreases the resonant transmissibility
V{
until the resonant transmissibility coincides with the
common-transmissibility for the system.
parameter which is
associated with this particular curve
is the "optimum" damping parameter.
in Fiogure ITT-3,
=
1
and
=
The damping
optimum dampingq is
2.
For the system plotted
seen to lie between
For larger values of damping the
resonant transmissibility becomes larger and the resonant
frequency ratio tends toward the undarnped frecuency ratio
of the "locked-in" system.
Tt is
seen that for these
larcrer values of damping, the transmissibility is
gDiven
by the undamped solution for the "locked-in" system until
the damper "breaks loose" at which time the transmissibility
imediately berins to decrease.
As shown,
some of these
damped curves eventually intersect the undamped "lockedin"
curve and proceed along it
frequency ratio.
for higher values of
17A
TUT
T
07T 7-"
1
*7
7
WIT
yq!
'a
*1
71*
7
1
........
....
1
0
II
/
-
4
m in
M.
a
.
.m
i~.
-r
I
V
ri
J
I
~
Ii
~
I
0
b
...
.......
7/j~!
A
4
m,"
47
LA...
Ftt
t e
it
..........
2
- H
. . ..........
.M
rT
,L
I
a
1,
4i
I.
-Jn.
A :Q
il
lip
I
jV
t
Ai ho
-7I
-4
DISPLACEMENT TRANSMISSIBLITY
Q
444L
441
A
n.. 1 ?1
II
ABSOLUTE
-LCRELATIVE
DISPLACEMENT
T RANSMISSIBILITY
0*
rri
0cM
rn
-I
~AJ~
II
-
-
K 1(.
The relative transmissibility for this system is
shown in Ficure III-4.
It is seen that all response
curves do pass through the common-transmissibility point
and remarks about the variation in the damping parameter
made while discussing the graph of Figure 111-3 also
apply here.
The graoh of Figure III-5 shows another example of
absolute displacement transmissibility for the case where
the spring ratio is
8.
Again, very high transmissibilities
exist for damping' ratios approach inj*1/.
The damping is
increased until the resonant transmnissibilitT coincides
with the common-transmissibility
(Optimum damping).
Further increase in damping causes a corresponding increase
in resonant transmissibility as would be expected.
One perhaps might notice that not all the damped
curves eventually become coincident with the "locked-in"
undamped curve.
As a matter of fact,
this is
true and is
explainable by consideration of equation (III-8).
Damping
parameters for wich the denominator becomes nerative
corresponds to those curves which represent a situation
w1 ere the damper never becomes "locked-in".
damper will become "locked-in"
inequality is
satisfied.
4
-38-
Thus,
the
only when the following
0
IzC
"U
01ri
O
0r
1
41
-
r 71- 1
7 0
z
Q
o
! 1111
u
1 1
. ; . - . I-
4
16
%i
O
i
l.A
I"
..I.
1:41~
*1
17Hri
LusL4-0
%
oQo
jij.i
4I.
I
I
I
ii
±
*1
1
I'.
-
All
Wit
1
_______
£
______________
1~
1.
-41---.4--
.i
MI
- I
I
'~~:
K
I
~
ii..
4
p
~1
IL
K........
t..
f
V
#j,.
1114
i2tU Ali 4
i~
[
.
4
Li' i1Lh~L LL*JI <j
4-
.21
TV
.
ABSOLUTE DISPLACEMENT TRANSMISSIBILITY
,
Q.
a
- - - -- -
.
---------
AZ!-
.
'44.+
t
Ii~
'4
I
8
T7
h-VA
1j
KKQ
Thus,
Figures III-3,
),
5 show typical response
curves '.or the coulomb damped system with the damper
elastically supported.
where the damper is
By way of comparison with a system
rigidly supported,
the transmissibility
at hirb frecuency ratios can be determined.
the sprinq ratio approach
By letting
infinity in equation (I11-2),
displacement transmissibility of the coulomb
the absolute
damped system with a ricidly supported coulomb damper is
obtained.
For large frenuency ratios,
this expression
reduces to:
(Ti')N=0
damping
Thus,
is
(Wt2.
(III-11)
always present which tends to decrease
isolation at high frequency ratios.
For the case where the spring ratio has a finfite
value,
the absolute displacement transmissibility at
large frequency ratios is
t
This is
Thus, it
r-iven by the equation:
N+I
IJ
---
1 -)
valid for systems which satisfy ecuation (II7-10).
is
seen that if
the spring ratio and damping
parameter are designed such that equation (1I-10) is
satisfied,
the absolute displace-rent transmissibility at larqe ftecuency
ratios will be smaller than that of the system with infinite
spring ratio by a constant amount given by the ratio
of equation (TII-11)
and (III-12).
An interesting point can be made in favor of
elastically supporting the coulomb damper when one
considers the irratic properties of a rigidly supported
coulomb damper that are sometimes observed at high
freouency ratios.
Althouoh no investigation as to the
reasons for these properties occurring will be made here,
it
seems reasonable that the wear problem in the coulomb
damper may very well contribute to these und esirable effects.
However,
if
one manages to d-esi gn the damper to "lock-in"
at high frequency ratios,
the damper, and thus,
frecuency ratios.
there will be no motion across
no associated wear problem at high
The real value of elastically supporting
the coulomb damper is more effectively shown when acceleration
transmissibility at high frequency ratios is considered.
It is certainly more sensible to talk in terms of acceleration transmissibility for systems operating in regions of
-
K
high frecuency ratios because,
although the displacements
may very well be small, the corresponding accelerations can
be of considerable value.
Also, certain machines operate
under a constant acceleration environment as opposed to
a constant displacement
with the fact that it
inertia force)
is
environment.
TFhese
the acceleration
facts,
coupled
(or its associated
that causes damage to the mounted structure,
-414-
give
good reason to-now consider the effect that
elastically supporting" the coulomb damper has on
the acceleration transmissibility.
The equations riving the acceleration transmissibility
of the system wit* the elastically supported damper can
be obtained from ecouations
(II-2)
and (TI1-3)
by
defininc the coulomb damping parameter for constant
acceleration excitation of the base as follows:
(II I-13)
whereAois the maximum amplitude of the sinusoidal
acceleration being applied to the base.
As indicated
in Appendix A the equations for acceleration transmissibility
are obtained by replacing, in
equation (111-2)
by the
following quantity:
T-).
The expression for acceleration transmissibility is
then c'iven b7:
(4
-.;TA')"
This enuation is
1 .--
W *
--
W+
N+Z(
..-....-
t
N
~
N
naturally valid only for the frequency
ranre for which there is relative motion across the
(TTT-15)
coulomb dariper.
This rancge can be specified by
substituting equation (ITI-1)4)
in equation (III-8)
to obtain:
F+1\4
N
(IE-16)
r
for frequency ratios outside tIe rancre specified
Thus,
by equation (III-16) the s-stem is
acceleration transmissibility is
it
Also,
is
undarmped and the
given by equation (.1-7).
seen b7, inspecting ecuation (II-1-5) that
infinite acceleration transmissibility occurs for a
frenuency ratio of unity.
to prevent such a
Thus,
condition from arisinr the damper must
!e desigred such
tuat it
remains "locked-in" beyond a fremuncy ratio of
unIty,
The limiting, value placed on th: e damping parameter
is
ob;taned by requiring eauation
IIT-16)
than unity for .the lowver freaiu cv ratio.
placed on the coulomb damper is
~kirnA.
to be greater
The limit then
then given by:
4
(111-17)
Inspection of the coefficient of the damping term
in equation (IT-15)
an
reveals that comion-transmi ssibility
the comon-transmissibility frequency rat. o are the
same as for the displacement excited svsterr.
When the
spri ng ratio becomes infinite, ecuation (TI -1)
-4i3-
L_
reduces
to the known app-roximate ecuation. for the acceleration
transmissibility of a coulomb damped system with the
coulomb damper ricidly supported, which is
I
(TA)M=W
Now,
is
+r
&
iven by:
IZ1(.)
_I4
(
for hich freCuency ratios,
ecuation(
"
W
)
( 1I-1 )
seen to reduce to:
AJi,
V'm A.
~ if
But, as previouisly indicated,
(
the svstem with the
elastically supported coulomb damper has a trasmissibility
at
high frequency ratios as
iven by the equation:
(TA)i.
It
much to
immediatelv becomes obvioi s that there is
ain br elastically supoorting thtle coulomb damper.
The conventional system has the proDerty that the absolute
acceleration transmissibility aoproaches a constant value
,iven h
equation (III-19) for large frequency ratios.
Thi s means the isolation property flattens out at high
frecuency -ratios ani a constant amount of acceleration is
transmitted to the supported structure.
notices that bT
However,
one
elastically supporting the coulomb damper,
-L-
1m-19)
the tranissibility at high freouencv ratios no longer
approaches a constant value but varies as the inverse
squar e of the frecuency ratio,
approaches zero as a limit.
or in othe-r words,
Thus,
the introduction of
an elastic suDort for the coulomb damper very effectively
reduces the acceleration,
and thus,
the dama-in7 inertia
forces transmitted to the supported member.
As a means to more exactly ascertain the dynamic
response of the system with the elastically supported
damper,
an analog computor system was devised from which
the absolute acceleration transmissibility was obtained.
The complete analor
computor analysis is
,iven in
Appendix C and includes a full explanation of how the
non-linear damping effects were produced.
The load
deflection and the hysteresis type damping characteristics
were monitored on oscilloscopes during the course of
the coputor analysis to insure that the analo.7 of the
coulomh-damoingr
the sy-Tstem.
in
fact,
elemlent was propoerly introduhced into
Thus,
if
the analogr computor system is,
a cood analog of the mechanical system being
investi cated,
these characteristics being monitored
should correspon'
to Figure III-1 and Figure 111-2.
These characteristics
were very nicely produced
in the analog system and Ficgure III-6 shows a picture of
the typical load deflection and the damping characteristics
as photographed from the monitor oscilloscopes.
Notice
how closely these pictures resemble the characteristics
Fic. 1u-G
as indicated by Figure II1-1 and 111-2.
The top trace
corresponds to the load- deflection characteristic and
the bottom trace corresponds to the hysteresis coulombdamping characteristic.
The curves have sharply defined
corners which would correspond to the characteristics of the
"idealized" mathematical model shown in Figure 1-2.
For purposes of indicating the waveforms of acceleration,
velocity and displacement, a photograph was taken of these
quantities displayed on the monitor oscilloscope and is
shown in Firure III-7.
The bottom curve is
the waveform
of the relative displacement; the middle curve is the
waveform of the relative velocity; and the top curve shows
F .M-7
the absolute acceleration of the mass.
displacement
is
As shown, the
very nearly sinusoidal while the velocity
becomes slightly distorted and the acceleration takes on
the "chopped off" sine wave shape.
FigureIII-8 shows the acceleration transmissibility
of a system with a spring ratio of 3 as obtained from the
analog computor.
The spring ratio of 3 was chosen for
particular study in an attempt to provide some analytical
0
0
"n
_
K LL 11d 2 I II
S SI
OTHO1
jUrT
fiIj
1>
I
"d
ABSOLUTE ACCELERATION TRANSMISStIBLITY
-
5
verification of an experimental curve given in Reference
for a system with an effective spring ratio of 3.
There appears to
be an optimum dampingn curve corresponding- to
~I
.
For
dariingn parameters less than optimum, a slight decrease
in damping is seen to affect a very large change in
transmissibilit.
This sa5n1
greater than optimum.
T he
t
is not true for damping
shape of the response curves
iricate that the approximate linearized analysis gives a
fairly goo(I representation of the solution to the problem.
Ficure ITT-8 also indicates how thc response curves begin
out
on the stiff
undamped curve
at high frequency ratios,
which bears out the results of the
previous simplified analysis.
have a resemblance
"C M
as shown in
computor.
i.e.,
TPhese response curves definitely
to those riven in
5.
Reference
S~ he largce areas of double valued response
this reference
clearly are not shown by thie
The reason for tIs
nay lie in
t"e fact that the
in
static
and kinetic coefficients
the system would have different
of friction,
"break loose
for increasing and decreasing frenruency ratios.
hand,
analog
curves would naturally be affected by the
exoerrimenta.
difference
and eventually return to it
the analo
computor is
just
affected b-
points
On the other
the kinetic
coefficient of friction (because of the "idealized
mathematical -model" chosen) and thus,
would only have
one "break loose" point.
Thus,
it
is
felt that the analog computor solution
rrivenin Appendix C malkes available the non-linear response
of a svste
with a coulomb damper elastically supported.
This solution coupled with the linearized tieory solution
should provide sufficient information to intelligently
desicn such a system.
CTJA PTER IV - EITLOGUJE
The mathematical
analvsis of Appendix A coupled with
the analoc computor analvsis of Appendix P provide means of
obtainin
the exact solution of the dynamic response of the
sy7stem with an elastically supported viscous damiper.
thorouhT
A
discussion and appraisal of the dynamic characteris-
tics of this system along, with graphs of the solutions can
be found in Chapter II.
It
would be convenient if
series of graphs were available, and thus,
a complete
a calculation
program which would emibody many more values of the dinensionless damping and spring ratio parameters is
program for the future.
a suggested
Also, development of an analytical
exDression (riving the optimum damping7 for a g-iven spring ratio
would be an appropriate
addition to the theory developed
herein.
As for the coulomb damped system,
in
Appendix A and, discussed
in
and should be treated as such.
Chapter
the solutions developed
III are approximate
They certainly give much
informatinn about the ch-aracteristics
of the coulomb damped
system, but do not embody the non-linear characteristics
which are known to exist since the solutions are linearized.
It
is
felt
that the computor analysis
gives an accurate description of the dynamic response of
the coulomb damped system.
For this reason,
it
is suggested
that further work be done in obtaining more response curves
from the analog computor in addition to increasing the
number of curves obtained from the approximate solution,
would be interesting to plot craphs of
it
In particular,
the approximate solutions for the acceleration transmissibility
and compare these results with the computor results.
As indicated by the discussions in Chapter II and
Chapter IIf,
these systems should be very useful in problems
of vibration isolation and particularly in problems where a
high degree of isolation is
It
reo ui red at high frequencies.
should be mentioned that solutions for the problem
where the force is
applied to the mass are readily obtained
with a small amount of additionl analysis.
"P=.
For a force
applied to the mass as shown in Figure IV-1
SIN cAt
the analysis is
and Figure IV-2,
carried out in a manner
P. sltacat
P.slcat
t
tm
C
B1
The expression. givin
similar to that in -Appendix A.
the displacement of the viscous damped system is
found
to be:
C \*-/ L 2
4
o
/ak
ap~e
(IV-1)
----
-(
h<
ascAh
fudb
.0
~f2
eazi
thatuth
The force transmissibility, which is defined as the
ratio of the force transmitted to the base to the force
applied to the mass,
that
can be found by realizing
the
force transmitted to the base is riven by the sum of,X.
and
X
The force transmissibility is
F_
thus given by:
(TV-2)
__r
This can be recnnized as the enuation for the absolute
transmissibility for the case where the base is
Thus,
the graphs in Chapter II
beingc excited.
for the absolute transmissibility
of the base ex-cited system also give the force transmissibility
of the force excited system.
vied
-53-
ar expressions can he derived for the coulomb
damped sys term by use0 of the technique of eaui valent viscous
dam-pingT.
For example,
damped syTrs tem is
the displaceent
of the coulomb
given by:
1 (4
B
'
e
(y
-
x-C
By replacing
by
(IV-3)
(TV-3
5
,
the force transmissibility can be
s-own to be the samle as cquation
givingi
(III-15) which is
the equation
the absolute acceleration transmissibility of the
base accelerated excited system
0
to those made in
ChaDter ITTI concerning- ranrre of validity
and 7ood design values of(
Thus,
Statements analo7.ous
can easily be determined.
both analytical expressions and analorr computor
solutions of the forced vibrations of systems with elastically
supported dampers have been developed,
and it
is
hoped that
these solutions will be a significant contribution to
the field of mechanical vibrations.
APPENDIX A
STEADY STATE RESPONSE OF VISCOUS TAPED SYSTEM
and
APPROXIMATE SOLUTION" OF THE COULOMB DAM'PED SYSTEMV
Page
Section A-1
Int rodu ct ion .. .. ..
Section A-2
Derivation of Differential Equations
57
Section A-3
Solutions of the Differential Equations.....
58
...
0009*a@a#*4
.
.
.
.
.
.
.
(1)
Absolute Motion of Mass
(2)
Motion of Mass RelatiVe to Base
(3)
Motion of Mass Relative to Damper
56
...
9*000000000
Section A-1.
Analysis of Undamped System.................
Section A-5
Determination of Common-Transmissibility
63
61
Section A-6
Transmissibility for Large Frequency Ratio.,
66
Section A-7
Equivalent Viscous Damping for Coulomb
Damping...............
67
Transmissibility of Coulomb Damped System...
68
Section A-8
(1) Absolute Displacement Transmissibility
(2)
Relative Displacement Transmissibility
(3)
Limitation of Solutions
Section A-9
Common-Transmissibility Point-Coulomb System..
Section A-10
Large Frequency Ratio Transmissibility Coulomb S
73
Section A-ll Frequency Range for Which Coulomb Damped
Solutions
are
Valid.
*...
75
....................
Section A-12 Summary of Displacement Transmissibility
Expressions. .. . . . . . . . . . .
*.
. . . . . . . . . . . ..
..
*.
Section A-13 Acceleration Transmissibility...............
-55-
77
78
SECTION A-1
INTRODUCTION
The mechanical systems to be analyzed in this appendix
are shown in Figure (A-1) and.Figure (A-2).
Figure (A-1)
shows the viscous damped system with the viscous damper
supported by a spring
a = a
N
,
Figure (A-2)
shows the
a = a0 sin wt
sin wt
FIG. A-I
FIG. A-2
coulomb damped system with the coulomb damper supported
by a spring
.
Conventional methods are used
to solve the viscous damped system and the method of
equivalent viscous damping is
used to solve the coulomb
damped system.
-56-
SECTION A-2 DERIVATION OF DIFFERENTIAL EQUATIONS
Free Body Diagram
of System Shown in
Fi7,ure A-1.
I
c
Applying Newton's First
Law:
c
For Damper: ZF'=O
Substitutinor
(A-2)
- (X-o4
in
(A-1):
Thus we Obtain:
::p,
Differentiating:
IVn
Substitute (A-IL)
in
+jh
~rn
(A-1)
Dc
c)-C(: -)=n
-(--.) -A (0,)-
(A-2)
a4
=0i;
-mw'
+.k (-x- a.) -k, a.)
-,
+(A
I.)
(A-1):
EpY"Ce
But :4
da
Zn
E
For Mass:
(:-k -.i, 1 )
+c
(i+I,)(-
+.A x--)= 0
=
set.
+
(N+I) (i.-&) t
-57-
(ZC-cL)
-. 0C(A-5)
(A-3)
(A-4)
Equation (A-5) is the basic differential equation of
motion for the system of Figure A-1. We shall now
transform this equation into forms which describe
absolute and relative motion of the mass.
From (A-5):
+
..
(A-6)
Equation (A-6) is the differential equation for the
absolute motion of the mass.
Now
S2
Motion of mass relative to base
X
etc.
Substituting these identities into (A-5):
kt
--
yyAN
A =-
E-w
(A-7)
Equation (A-7) is the differential equation for the
motion of the mass relative to the base.
Motion of mass relative to the damper
(x--X)
Now
X
etc.
From (A-2):
.Y
From (A-1):
x
But
DX-
m
+-I
.-
+ k
b
o-- +
GL
-r(-.xI)
Subtractin_
Noting
40.+
.t +C[-
=A
+
*
Y + At(A-8)
Equation (A-8) is the differential equation for the
motion of the mass relative to the damper.SECTTON A-3 SOLUTTONS OF THE DIFFERENTIAL EQUATIONS
The motion of the base is steady state sinusoidal motion.
0. si o-.
o
10.
Cit'
cosu
-.
(A-9,a)
.m
-58-
(A-9,b)
Is cat
SO
(A-9,c)
C
gas &it*
(A-9,d)
SECTION A-3.1 ABSOLUTE MOTION OF MASS
A-
rn
+ CL
.+
.x=C T
.+
(A-6)
Assume the following solution:
:X.
.;
W.
(A-10,a)
+
Da
COS (CAit#+)
(A-10,b )
SV-Xt
c)
(A1(wt+)
cos
.74(
(0at+4)
(A-10,d )
SubstitutinR (A-10,a, b, c, d) and (A-9,a, b) in (A-6):
(-X
) Cos (,t+t)
+ ?YI(-.Xo
-t-(Xe5tNtot*)C
Grouping terms,
a?)
slm (Cut+.) + c
t-)3o)
COS 'A
+ A
C (Wt
Go
sti
Cai.
we obtain:
'
Q. ...
~~-w w~ sinut+)
"W
w+{IC (
COS
(A-ll)
2.leos (u.t4
Now observe the following equation:
U 5Icuat + %I
CoS (t
(A-12)
The absolute value of X is given by:
2.
(A-13)
Thus we may obtain the absolute value of equation (A-ll),
= absolute displaceand noting that (Tz
ment transmissibility:
+ (1c
Note the following definitions and identities:
Undamped angular natural frequency
-59-
(A-15)
C,
CY
E
2=
Critical viscous damping
coefficient
(A-16)
C
101CC ,
= Percent of critical damping
Zb4flJ vqh
Q
-
(A-18)
By using Equations (A-15) and (A-18),
can be shown to take the form:
Equation (A-i.)
ZY
2()
a01
(A-17)
+
(A-19)
TSA-=O'
Thus we have determined the absolute displacement
transmissibility of the mass in terms of the dimensionless
parameters
1 C/c , and W/
SECTION A-3.2 MOTION OF MASS RELATIVE TO BASE
Vy
4-m
64
+ C(
04
-
$=-
(A-7)
Assume the following solution:
& = ;o
(A-20,a)
( Wt +9
$=i5,slt
CoS
(Wt +9)
(A-20,b)
(A-20, c)
-6,ilCos (Odt +Q)
(A-20, d)
b, c, d) and (A-9,c, d,) in (A-7):
Substituting (A-20,a,
4
.Mt,)
Cos (at
+
)
V
(-a. w ) SIN C(,wt + 9)
+0
C.. (Wt
U. W)
31
f*3
cut
Wt
W9
C-A-S
Owt
V"
(Aa
Gr
ouping terms,
we obtain:
Co -
)
we have:
By virtue of Equation (A-13)
2
mcw
-
*
(7~Qo),
Where ( T14
Cs
(+
s
(A-21)
c%
ha
=
=
Ia
Relative displacement trans-
missibility of the mass (Relative to the Base),
By use of Equation (A-15) and (A-l8) this equation can
be shown to take the form:
U
(C0
CS 0
14
C'IN
Thus we have determined the transmissibility of the mass
relative to the base in terms of the dimensionless
parameters
C/Cc
W*W .
-61-
(-2
DAMPER
SECT ION A-3.,3 MOT ION OF MASS R ELA TIVE TO0
SECTION A-3. MOTION OF MASS RELATIVE TO DAMPER
r+c
a
)
O'
±y.
(A-8)
Assume the following solution:
gcs
4
(Wrt
(A-23, b)
+I)
(W ,*.
(A-23,a)
)
:Sig- (Wt+
-
(A-23,c)
r)
0
3
Substituting (A-23,a,
COS
(Wt ty)
(A-23,d)
b, c, d) and (A-9,c) into (A-8):
Mc
~rs
W
) c0s (ut+-fr)
+C ( N(W)
+ vn(9<*)se(t+,-
)ces(Wt
+-g)+A.si
<t+-
-m (-aJw) Sim cut
Grouping terms,
1 0
we obtain:
(MUz)
Sim cut
WbG
+
-62-
t4AcQS(W-L+-jr)
Using equation (A-13)
we have:
By using equation (A-15)
and (A-18):
Thus we have obtained the ratio of the absolute values
of the displacement of the mass (relative to the damper)
to the displacement of the base in terms of the dimensionless parameters
'
cat
.
.
This expression will
be useful when the problem of coulomb damping is
considered (Figure A-2).
SECTION A-4 ANALYSIS OF TJNDAMPED SYSTEM
If the value of
is
in equation (A-19) and equation (A-22)
made to approach zero,
the classical equations for the
undamped response of a single degree of freedom system are
obtained.
The undamped absolute transmissibility is
given by:
(A-25)
-63-
The undamped relative transmissibility is given by:
).
NowasQ
j
IA
26)
is made to approach infinity, the damper becomes
"locked in" and the motion of the mass is resisted by a
total spring rate of (N+*)f,
Thus, it is obvious that this
undamped system has a natural frequency given by:
(--
. (A-27)
I
Substitution of this equation into the two previous equations
gives expressions for the transmissibility of system when
the damper is "locked in".
The absolute transmissibility
of the "locked in" system is given by:
(A-28)
While the relative transmissibility is give by:
(A-29)
SECTION A-5 DETERMINATION
If
OF COMMON-TRANSMISSIBILITY
POINT
one plots the absolute values of the transmissibilities
of the two undamped conditions, it is seen that the response
curve for the zero damping case intersects the response
curve for the infinite damping case.
The frequency at
which the two response curves intersect can be found by
equating the two transmissibility expressions (taking in
-614-
consideration the proper sign of that portion of the
response curve being investigated) and solving for the
frequency ratio.
Thus, for absolute transmissibility:
(A-30)
Solving for the frequency ratio and denoting it
A
by
- the common-transmissibility frequency ratio for absolute
motion:
(A-31)
This expression then gives the frequency ratio at which
the absolute transmissibility of the
undamped cases have
a common value of absolute transmissibility.
transmissibility at this frequency ratio,
To find the
substitute
equation (A-31) in the general expression for absolute
transmissibility as riven by equation (A-19).
substitution is
it
made,
is
When this
found that the damping terms
disappear and the transmissibility reduces to the following
equation.'
*
(A-32)
Thus, since the viscous damping terms do not appear in this
equation, it is deduced that this point is a common-transmissibility point for the system - no matter what value
of viscous damping.
The frequency ratio defining the
-65-
common-transmissibility point for relative motion is
obtained by equating equation (A-26) to equation (A-29),
which results in the following expression:
(A-33)
Substituting this relation into equation (A-22):
(A-34)
1N I+a
Thus, it is seen that the relative transmissibility of
the system at the frequency ratio defined by equation (A-33)
is the same regardless what amount of viscous damping is
present.
Moreover,
it
should be noted that although the
common-transmissibility frequency ratio is different for
absolute and relative motion, the common-transmissibility
itself is equal in each case.
Thus, it has been shown
that there exists a point through which all the response
curves pass no matter what amount of viscous damping is
represented in each curve.
SECTTON A-6
TRANSMISSIBILITY FOR LARGE FREQUENCY RATIO
Of practical importance is
the transmissibility of the
system for large values of frequency ratio.
This trans-
missibility can be obtained by letting the frequency ratio
in equation (A-19) and equation (A-22) become very large
and thus eliminate certain terms on the basis that they are
small compared to other terms.
-66-
Letting
represent the
desired transmissibility ( , subscript merely indicating
that large freoency ratio corresponds to "isolation"
rather than a "magnification"), the following expressions
are obtained:
(Ti)A
P(A-35)
and
(Ti)
(A-36)
By comparison with equation (A-28) and equation (A-29)
it is seen that, for large frequency ratios, the transmissibility of the system approaches the undamped response
of the system with the "locked-in" damper.
Thus, the effect
of viscous damping "cuts-out" at large frequency ratios
thus, rendering the system essentially without damping.
This property is very important to those concerned with
problems in vibration isolation.
SECTION A-7 EQUIVALENT VISCOUS DAMPING FOR COULOMB DAMPING
If one assumes that the energy dissipated per cycle by a
viscous system shown in Figure (A-1) is equal to that
dissipated by a dry friction system shown in Figure (A-2),
one can obtain an expression for the "equivalent viscous
damping" of the dry friction system.
Ce
=N
(See Reference 8
).
(A-37)
Here
must be the maximum displacement amplitude of
By using (A-15)
the mass with respect to the damper.
and (A-] 6)
we obtain:
2.B
(C\
ir
Ce
I W\
. \w )
(A-38)
and collecting terms:
Substituting (, from (A-21), squarin
fz
C
----
(\++ -
tN
(...-----
-
WV.
(A-39)
If we define
Where
=
dimensionless
coulomb damping coefficient,
we obtain:
Cw
U3a
AT
hav
I
(A-10)
Thus we have
obtained an expression for the equivalent
viscous damping ratio in terms of
\
,
the coulomb
damping parameter.
SECTTON A-8 TRANSMITSSIBTLITY OF COULOMB DAMPED SYSTEM
We may now obtain expressions for the transmissibility of
the coulomb damped system shown in Figure (A-2)
equati on (A-)±o),
(A-22), and (A-19).
-68-
by use of
TIRANSNMSS TBTILTTY
"LAENT
SECTITON A-8 .1 ABS OLUTE DIS
To obtain a relation for'the absolute displacemqent
transmissibility, we must replace
by
c
(F)in
equation (A-19)
as r4,ven by equation (A-40O).
44
z
uj41
cc
-69-
~
i~)w
2. .
Let P
Note the following expansions:
=
'.
W
P +Pj+)
(M+)2 (I-z)Z
z
'T-Vi'.
E~s0*0-pa^-
t+ -)-41 lingo
4EceM
-anam Is.4
.. *
07
ftlw%
P
aP'T(M4,"-(m+ld
I)*-10-
2.
[ r'l+13 W
P
zp
L P
4(,M
P7
- z (N+I)]
Substituting, this identity in the relation for
+2)
4M
(T;)A
(TA:'
)2-
Zw;)4
Divide numerator and denominator by
ty [(
A
L
-70-
)
-ii("*)]
Ir2.
(A-41)
gives the absolute displacement
equation (A-)11)
Thus,
transmissibility of the coulomb damped system in
N,
of the dimensionless parameters
and
N-+-0
is of interest to note that when
terms
equation (A-41)
reduces to the known relation for the absolute displacement
transmissibility of a coulomb damped system where the
damper is rigidly supported.
SECTION A-8.2 RELATIVE DISPLACEMENT TRANSMISSIBILITY
To obtain a relation for relative displacement transin equation (A-22)
missibility, we must replace
C
by
as griven by equation (A-140).
4
#W
K z
%( -W-0 6
can)
Ir
N=
-
(2~~ Win)
(±Orr~A
Y(
4
"%\. C
UJ
(7~~j 4
As shown previously:
-14
( AAt N4K\h /t
2.
iI
W
'Z
WVb a
B
WI%
iw M)2
[MZ(
cowl
Ir
ujv
Thus :
4h
(T1) . I&I
(o
+
12
WVS)
-71-
WVIZ
Expanding the terms in the bracket, collecting terms,
dividing numerator and denominator by
(A
Thus,
we obtain:
I
(A-2)
equation (A-h2) gives the relative displacement
transmissibility of the coulomb damped system in terms
of the dimensionless parameters
should be noticed that when
, and
.
It
equation (A-1
4 2)
reduces to the known relation for the relative displacement
transmissibility of a coulomb damped system where the damper
is
rigidly supported.
SECTION A-8.3 LIMITATION OF SOLUTIONS
Having obtained solutions for the transmissibility of the
coulomb damped system under consideration, we must be aware
of the mathematical limitations which are associated with
these solutions.
In the case of the absolute transmissibility,
the solution as given by equation (A-41) is valid only for
real values of the numerator of equation (A-41).
This
condition can be written:
ir
+1
Thus, if equation (A-43) is not satisfied, the solution
for absolute transmissibility becomes imaginary and we are
-72-
now considering a condition which is not in the range of
values of
for which the derived relations are valid.
In the case of the relative transmissibility, the condition
which must be satisfied in
order that we do not obtain
imaginary values of relative transmissibility can be written:
II
Thus, if equation (A-hi') is not satisfied, then
lot)
is
not within the range of values for which equation (A-1
4 2)
V00
Notice that when
is valid.
equation (A-43) and
(A-44) reduce to the limiting conditions for the coulomb
damped system in which the damper is rigidly supported.
SECTION A-9
COMMON-TRANSMISSIBILTTY POINT
-
COULOMB SYSTEM
Since the solutions for transmissibility of the coulomb
damped system were obtained by means of equivalent viscous
damping,
there will exist common-transmissibility points
corresponding to those found in Section A-5 for the viscous
damped system.
The expression for absolute displacement
transmissibility as given by equation (A-41)
be independent of damping if
term vanishes.
is
seen to
the coefficient of the damping
The corresponding value of the common-
transmissibility frequency ratio for absolute motion
is
thus seen to be:
(A-45)
- N
-73
The same reasoning can be applied to obtain the commontransmissibility frequency ratio for relative motion.
Thus from equation (A-L42):
(Aj.
~
(A-L46)
These eqtuations are seen to be exactly the same as those
derived for the viscous damped system.
This is understandable
when one considers the approximate method used in solving
the coulomb damped system.
These freqtuency ratios, when
substituted in the corresponiding transmissibility expressions,
give the value of the common-transmissibility.
As in the
viscous damped system, this value is the same for both
relative and absolute motion and again has the same value
as that determined in the analysis of the viscous damped
system.
Thus the common-transmissibility is given by:
N
(A-47)
SECTION A-10 LARGE FREQUENOY RATIO TRANSMISSIILITYCOULOMB SYSTEM
The transmissibility at large frequency ratios can be found
by letting the frequency ratio in equation (A-41) and
equation (A-4~2) become very large and thus eliminate certain
terms on the basis that they are small compared to other
terms.
Letting
Trepresent
this transmissibility (the
n
symbol representing coulomb damping and displacement
excited system), the following expressions are obtained:
( /A
/A-.\8
-:
and
(A-49)
It should be noted that all of the expressions so far
derived for the coulomb damped s\stem are subject to
Perhaps
conditions such as those stated in Section A-8.3.
more generally it should be stated that the equations for
the coulomb system are valid only when there is
motion in the coulomb damper.
relative
This is a necessary
condition since this whole approximate analysis is based
on the assumption of equal energy dissipation per cycle
in the coulomb and viscous dampers.
Thus, the frequency
range for which there is relative motion in the coulomb
damper must be determined so as to indicate when the
derived solutions are valid.
SECTION A-ll FREQUENCY RANGE FOR WHICH COULOMB DAMPED
SOLUTIONS ARE VALID
Since the energy method used to derive the coulomb damped
expressions assumes relative motion in the damper, the
frequency ranre for which this assumption holds true must
be determined.
The frequency ratios defining the ends
of this frequency range will be designated by
the
"L"
where
subscriptrefers to the point where ei >er the
-75-
coulomb damper "breaks loose" or "locks in".
The relative motion across the damper for the viscous
system is
riven
y equation (A-2).
If the expression
for equivalent viscous damp ing as c-iven by equation (A-410)
is
substituted in equation (A-2L),
the following equation
results:
)2,
(A-50)
Thus, when the numerator of this expression becomes zero,
there is no relative motion across the damper and the
damner has either become "locked in"
or has "broken loose".
ETuating the numerator to zero and solving for frequency
ratios:
N
(A-51)
OWN-
Thus,
the frequency ran--e for which the coulomb damiped
solutions are valid is
defined by:
~~.4)
-76-
SECTION A-l1
SIMMARY OF DISPLACEMENT TRANSMISSIBILITY EXPRESSIONS
viscous DMPE:D STEM EoumoNs
x@10
+ )
+
llo-
(T=I
(-Zc C-)4 ( -zw;)
+ 4
Na
4
'So
(T~
QI
No"
40
coI
(T3iF
. 1
I
z
wv&
C
-4+ NZ. -CC-)
2
4
!D>-. E
I
s 45Tet
EQUi-Tiotsa
(iX)L+
(t4.)
-77-
_
*
SECTION A-13 ACCELERATION TRANSMISSIBILITY
The equations giving the displacement transmissibility of
the viscous damped system also sgive the acceleration
transmissibility of that system.
This is a direct result
of the fact that the viscous damped svstem- is a linear system
in which a ratio of characteristic displacements equals the
ratio of the corresponding accelerations since acceleration
only by a factor of
fCN
Also the viscous damping factor Mc is a function only
differs from'displacement
of the mass and main spring support.
Thus,
the equations
for displacement transmissibility require no chance when
acceleration transmissibility is required for the case when
the base is
excited with constant acceleration oscillation.
However, if the acceleration transmissibility of the
coulomb damped system is
required,
the equations derived
for the displacement transmissibility cannot be used to
represent acceleration transmissibility due to the fashion
in which the coulomb damping factor is
defined.
The
coulomb damping7 factor was defined as:
(A-39)
But, for constant acceleration excitation of the base,
the displacement
excitation.
Thus,
Q,
will vary with the frequency of
this factor is no longer a suitable
"constant" damping factor.
If
the base is
-78.
excited in
a
manner such that
i= Aosa
t
where
Ao
the
the manner in
maximum amplitude of the base acceleration,
which
is
CQO varies with the excitation frequency is diven by:
A.
A.0
(A-53)
Thus we may write:
(A-54)
A
Multiplying the numerator and denominator by
...f ......
NNNW - - - -m Ace A/
irn Ao kv
T :
- -
(A-)5
where we define:
A.
Thus,
A=
(A-56)
is the coulomb damping factor for a system
where the base is being excited by a constant amplitude
acceleration.
Since the equivalent viscous damping technique
essentially assumes harmonic displacement, velocity, and
acceleration functions, ratios of characteristic displacements will again equal ratios of corresponding accelerations.
Thus, the equations describing the displacement transmissibility
of the coulomb damped system may be used if
is replaced
The resulting expressions are written below:
by
(T,A
-
t2(A-57)
)-
-79-
qr
(A-58)
Arain there are mathematical limitations
on these equations
due to the fact that the numerators of these expressions
cannot be allowed to become negative.
These limitatiois
can be obtained by substituting; equation (A-55) in
equation (A-43)
and equation (A-L1.).
Thus,
we have
arrived at expressions for the acceleration transmissibility
of the coulomb damped system by use of the method of
equivalent viscous damping.
the ranee of validity is
The frequency ratios defining
obtained by substituting
equation (A-55) in equation (A-52).
The resulting expression
is:
(A-59)
-80-
APPENDIX B
SOLUTION OF VISCOUS SYSTEM BY ANALOG COMPUTOR
For convenience in obtaining response curves for the
viscous damped system, the technique of obtaining tiese
curves by use of the analog computor will now be investigated.
The equations of motion for the viscous damped system as
derived in Appendix A are given by:
rx
a.)
(B-1)
- 5 )=
(B-2)
and:
C(
From equation (B-2) we obtain:
X
where the letter
C.
(B-3)
is the time-derivative operator.
Subtracting CLfrom both sides of equation (B-3):
Now add and subtract unity in the numerator:
(x,-.)= (x-a.))(B-5)
-.81-
Noting that the second term on the right hand side of
equation (B-6) is the mathematical operaition performed
by the unit lagr computor component,
characteristic time constant
can now be defined as:
which embodies a
this time constant
'
-= ( )(B-6)
Thus equation (B-5) can be written:
(z-o[I+
(B-7)
Substituting equations (R-7) and (B-2) in equation (B-1):
This eauation is used to construct the functional block
diaeram shown in Figure B-1.
This diagram describes the
nature of the physical system.
In contrast to Figure B-1,
a dia7,ram is shown in Fimure B-2 which describes the
nature of the electronic analog system which is used to
represent the physical system.
We see that this system
contains the characteristic time constants
.
The
with the
of the loop.
'1
T,
and
andTZ time constants may he included
coefficient without affecting the gain
Now for the physical system, the undamped
natural frequency is
defined as:
-SE
(B9)
FIG. B- 1
FIG. B-2
PHYSICAL
ANALOG
SYSTEM
SYSTEM
By comparison of the two diagrams, it is seen that the
corresponding auantity in the analog syvstem is
defined as:
C
(B-10)
T, T2.
where the bar indicates analog parameters.
less parameter
spring constants, the dimension-
and
containing the,
N
For the loop
is defined as:
N(B-i
where
N
By considering the
is the spring, ratio.
corresponding loop in the analog system, the following
parameter can be defined:
N
=(B-~2)
By considering the loop containing the unit-lag computor
compnonent, the time constant correspondin- to that f-iven
by equation (B-6) for the physical system is obviously:
Thus for a given
CK,
Cy.
(B..13)
the time constant
value of viscous damping
C
may be varied in
that eauation ('-13) is satisfied.
and the
such a way
Noting that critical
damping is defined by:
Ce
o
The viscous damping factor C
Wr
-84-
is
=2m-d,--(f3-14)
given by:
(BC~
Substituting equation (B-l5) in equation (B-13):
-c
C
-
\
(B-16)
-
Nw
By use of equation (B-10), this becomes:
MIC)
Tj
TT-wmn
C CK
(B-17)
Thus having conveniently defined the time constant of the
unit-laq, component and realizing that
F(cps
it is easily seen that the definition of
),
in the physical
system becomes:
(C
N f,
(B-18)
Since all parameters in the system have been adequately
defined, the response curves for the system are available
by introducing the excitation (L by means of a sine-wave
c-enerator and observing the absolute motion of the mass X ,
and the relative motion of the mass (X-Q.) at proper points
in the analog circuit.
-85-.
APPENDIX C
ANALOG COMPT7OP SOLUTIN O? COULOB DAMPED SYSTEM
Due to cifficulties involved in the solution of
differential equations which involve non-linear functions
(the dampinr force B in this case),
the possibility of
obtaining the response of the system by use of the analog
computor will now be investirated.
In the actual system
under consideration,, the damper is m,,assless thereby
kee-pinp; the system a s inrle-decree-of-freedom
system.
3it
since the non-linear effects of the damrpin7 element must
LICL
be incorporated into the analor
system, it is convenient
to temporarily include the mass of the darper
the analog system.
The inclusion of W)
-86-
I
in
in the analog
system provides a means to produce a feed-back loop in
the analog system which in turn is
used to simulate the
non-linear effects involved in the coulomb damping Dlement.
Also,
it
was found that for reasons of stability of the
motion of the damper, it is desirable to include some
viscous damping
C,
between the main mass and the damper.
The mechanical system that is
actually setup on the analog
computor is shown in Figure C-1.
This system is seen to
reduce to the exact system under consideration when the
values of
and
C,
approach zero.
This is essentially
how the analog computor is used to. solve the problem. The
analog computor system which simulates the mechanical system
shown in Fig7ure C-1 is
constructed and the coefficients M,
and Clare made to approach zero but keeping them as finite
a value as necessary to provide stability of the analog
system.
A functional block diagram is a diagram which shows
the mathematical operations beinr
made on voltage which
is used to represent the variables of the problem.
This
diagram can be constructed by consideration of the equations
of motion for the system shown in Figure C-1.
The free body
diagrams of the mass and damper is shown in Figure C-2 and
Fir'ure C-3.
We see that dynamic equilibrium of mass
gives the following equation:
Dynamic equilibrium of the tdamper requires:
(C-2)
n-'
must be remembered to represent the
r
The damping term
hysteresis dampinm curve previously discussed.
Fic..
It
is
F cC--,3
-
convenient to write the equations of motion in terms of
the relative variables c
where S
and
of the mass relative to the base and
the mass relative to the damper.
is
is the motion
the motion of
In terms of t'hese variables,
the eauations of motion become:
6. -yy
-
(C-3)
(C-b-)
where we def ine:
R.,~~
A: +ae&+ce
.=
-88-
( C-5
as the total real force applied to mass rA.
Equations
(C-3,4,5) are the equations used to construct the
functional block diagram shown in Figure C-).
The functional block diagram indicates the use of
the computor components to perform the mathematical
operations
indicated by the equations of motion.
the most part,
Figure C-,
is
For
self-explanatory and
additional remarks are required only in discussing the
method used to obtain the non-linear characteristics of
the damping element.
The.elements of the analog system which are a part
of the closed loops associated with the dynamic equilibrium
of mass hItare those required to produce the desired
The
non-linear damping force.
coefficient component
and the two integrator components make available the
acceleration, velocity, and displacement of the mass
relative to the damper.' As discussed previously, the
C,
coefficient component is used to provide stability in
this portion of the system.
When the
n
and
C
coefficients
approach zero, we see that the coulomb damping force
+3
is transmitted to the vibrating mass solely by spring-'k
Thus the force
in the analog computor diagram must
represent the non-linear force characteristics of the
coulomb damping element.
The manner in which
Ais
made to have the reauired characteristics will now be
discussed.
-89-
Fics.C-+
FuricTiORaL bLOCK 3)iR
In
Suppose the values of the
and
C,
coefficient
The forces
components have been allowed to approach zero.
applied to the massless damper are then merely is%
and-
These forces must be equal and opposite in order to satisfy
the condition that the massless daiper be in equilibrium.
The type of force produced by the B
components
and C
operating torether on a voltage reprcsenting the velocity
The velocity
is shown in Figure C-5.
of the damper relative to the base.
(-
is the velocity
if this velocity be
Foce
FoRce
I .-
I
Fi
I
SLOPCC
C
multiplied by a viscous damping coefficient
produced with a value
C(-j)
and is
,
a force is
shown in Figure C-5(a).
Now if this force is operated on by a bounding component B
the net result is a force which has the characteristic
shown in Figure C-5(b).
If
the coefficient
C
is now
increased toward infinity, the force produced is indicated
by Fig-ure C-5(c).
-91-
,
Thus,
vibrating in a configuration
the mass is
if
developed in springA
whereby insufficient force is
to
overcome the damper force, there is no relative motion
across the coulomb damper, and the feedback loop originating
A+
and terminating at adder component
to equal force
3
.
Also,
causes the force '
0
since there is no relative
velocity across the coulomb damper, the point representing
the value of the dampinf force 3
in Figure C-5(c) lies
'Therefore,
*
somewhere on the line --
as long as the
displacement of the mass ONl relative to the damper is less
than the value required to cause motion of the damper
relative to the base, the force acting on mass Yi by
spring
,
is
merely.,$
.
When the displacement of
mass "n bec ome s that value required to cause relative motion
between damper and base,
by Point '
or -3
damping force is)V,
the damping force is
in Figure C-5(c).
indicated
The value of this
the maximum obtainable damping force,
which is the hounding value of the
3
component.
value of the damping force will remain at/Al
increase in relative displacement
"r .
Thus,
the
for further
The feedback loop
originating and terminating at adder A4 insures that
-&11 equals)Ffor further increase in ny and thus a
constant damping force is transmitted to mass
spring
for this motion.
M
via
Therefore, it is seen that
the analog computor by use of suitable operational amplifiers
77
has produced the reguired damping characteristics
indicated by Fic'ure C-6.
Fi. C-6
The functional block diag-ram of
igure C-). also
indicates the manner in wliich the net force applied to
the mass and the net damping, force are moni*tored on
oscilloscopes.
The firures displa7Td on these oscilloscopes
are used as a guide to indicate when the
M,
and
C,
parameters should be adjusted so as to more exactly
reproduce the desired force characteristics.
The following is
a discussion of the manner Iin which
the system parameters are set up on the analog' computor.
As shown in Appendix B, the undarped natural frequency is
7iven by:
(C-6)
Cu"At
T, TZ.
-93 -
where the bar placed above COgag.ain indicates that this
T
quantity refers to the analog system.
andT2 are the
time constants of the integrating amplifiers in the feedback loop which originates at adder
adder
A, via adder Ag.
It
is
convenient to collect all parameters into
non dimensional groups.
N=
in
g and terminates at
For example,
merely be defined as
*3/.would
Cr,, and
the analom system were
settincs of the
and k
coefficient
Similarly, ratio of voltaue in
are the dial
components.
the analog circuit can
represent certain dimensionless
system.
the spring ratio
roups in the mechanical
For example, the ratio of the voltapge representing
1lto the voltare input to adder Pilfrom the sine-wave
crenerator represents the ratio of the absolute acceleration
of the mass to the acceleration of the base.
This quantity
is the absolute acceleration transmissibility of the system.
This is true due to the fact that the absolute value of
as
Pvn equals IMC and the sine-wave
generator voltare
represents 1"0.
Since
frequency
= Aostcot,
WJ is
it
is
seen that the forcing
also a parameter whose value will be
-94-
varied.
The conventional dimensionless ratio involving
this quantity is
the ratio of the forcing
which is
frequency to the undamped natural frequency of the system
as defined by equation (C-6).
Finally, if one divides the Cial setting of the B
component (which represents the maximum value of voltage
that the component will -pass) by the sine-wave generator
voltage,
the dimensionless damping parameter
S
is obtained.
Thus,
with measurements made from the analog computor,
the absolute acceleration response of the system is
obtained in terms of the dimensionless parameters
~0 I
c:~'~
~-'
U
REFERENCES
1.
Gallagher, J.,
Volterra, E.,
"A Mathematical Analyrsis
of the Relaxation Type of Vehicle Suspension", Journal
of Applied Mechanics, September, 1952, pp. 389-396.
2.
Myklestad, N. 0., "Fundamentals of Vibration Anailysis",
McGraw-Hill Book Company, New York, 1956.
3.
Klmpp, J. H., Lazan, B. J., "Frictional Damping and
Resonant Vibration Characteristics of an Axial Slip
Lap Joint",
WADC Technical Report No.
54-64
March,
19b
r. Goodman, L. E., Klumpp, J. H., "Analysis of Slip
Damping With Reference to Turbine-BladeVibration",
Jovrnal of Applied Mechanics, September, 1956,
pp.
h21-L29.
5.
Lazan, B. J., Goodman, L. E., "Effect of Material and
Slip Damping on Resonance Behavior", from "Shock and
Vibration Instrumentation", ASME Publishing Department,
New York.
6.
Pian, T. H., Hallowell, F. C. Jr., "Structural Damping
in a Simple Built-Up Beam", Proceedings of the First
U. S. National Congress of Applied Mechanics, 1952,
pp. 97-102.
7.
Pian, T. H., "Structural Damping of a Simple Built-Up
Beam With Riveted Joints in Bending", Journal of Applied
Mechanics, Paper No. 56-A-2.
8.
Timoshenko, S., "Vibration Problems in Engineering"
(3rd Edition), D. Van Nostrand Company, pp. 77-78, 93.
9.
Korn, G. A., Korn, M. K., "Electronic Analog Computors"
(2nd Edition), McGraw-Hill Book Company, New York, 1956.
10.
"Applications Manual for Philbrick Computing Amplifiers",
George A. Philbrick Researches, Inc., Boston, Massachusetts.
-96-