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ENGINEERING REPRINT SERIES
Reprint Number 8
Engineering Experiment Station
Columbia, Missouri
USE OF^ THE^ CENTRIFUGAL^ GOVERNOR
MECHANISM AS A^ T©RSrONAL
VIBRATilON ABSORBER
0.A. PRINGLE .o, Assistant Professor of Mechanical Engineering
Reprinted from the Transactions^ of^ the
American Society^ of^ Mechanical
Engineers, Vol. 75, 1953
COLLEGE OFENGINEERING THE ENGINEERING EXPERIMENTSTATION
The Engineering Experiment Station was organized in 1909 as a partofthe College of Engineering.Thestaff oftheStation includes all members ofthe FacultyoftheCollege of Engi- neering, together with Research Assistants supportedby the StationFunds. The Station is primarily an engineering researchinstitutionengaged intheinvestigation of fundamental engineering problems of general interest, intheimprovement of engineering design, and inthedevelopment of new industrial processes. The Station desires particularly to co-operate (^) with industries of Missouri inthesolution of such problems.For thispurpose, there is availablenotonlythespecial equipment belongingto the Stationbutall ofthe equipmentandfacilities oftheCollege of Engineeringnotin immediate use for class instruction. Inquiries regarding thesemattersshould be addressedto
T~E
TheDirector, Engineering Experiment Station University of Missouri Columbia, Missouri
UNIVERSITY OF MISSOURI BULLETIN
VOL. 55. NO. 25 ENGINEERING EXPERIMENT STATION REPRINT SERIES, NO. 8
Published by the Universityof Missouri at Room102,BuildingT-3, Columbia, Missouri. Entered as second-classmatter, January2,1914,atpost office at C-:,lumbia, Missouri, under ActofCongressof August24, 1912 .Issuedfour time • monthly October (^) through May,three times monthly June through September.
July 10, 1954
60 TRANSACTIONS OF THE ASME JANUARY, 1953
In Equation [5] the average angular velocity n is used in-
stead of the instantaneous velocity n + 8. w cos wt. The in-
clusion of the lat ter term results in an expression similar to the
last term of Equation [l ] and is not important in the present
discussion.
OPERATION OF ABSORBER
The action of the absorber is shown diagrammatically in Fig. 2.
In Fig. 2 (a), the instantaneous magnitude of a quantity is the
projection of its vector upon the Y-axis. At resonance, T 0 will
be in phase with the velocity of 80 w, and will lead the amplitude
80 by 90 deg. Let the natural frequency of the absorber system
equal w. Then the absorber amplitude r 0 lags 90 deg behind its
exciting force F', which is due to and in phase with the velocity
8.w. The absorber radial velocity r 0 w = (dr/dt) is then in phase
with T 0 • The inertia torque T; is of opposite sign to r 0 w, and
therefore joins with the damping torque Td in opposing T 0 •
This is shown more clearly in Fig. 2(b ), which omits the displace-
ments and velocities.
EQUATIONS OF MOTION
Applying Newton's law to the disk and neglecting damping
'2;T = la
(a) y
(b)
X
FIG. 2 VECTOR DIAGRAM OF ABSORBER ACTION
dr d^28 T 0 sinwt - 2mR ij- -K,8 = I ~d dt t•
.. [ti]
The corresponding equation for the absorber i~
'2;F = ma 0
d8 d'r 2m Rn dt - Kr = m dtl... .... [7 /
Inspection of these equations shows that the following will be a
solution
8 = 80 sin wt r = r 0 cos wt
Also
Making the foregoing substitutions results in
(2mRnw) K, K
. [8)
Equation [8] shows that 8. becomes zero if the natural fre-
quency of the absorber equals the applied frequency.
Owing to the effect of centrifugal force, the absorber-system
spring constant and natural frequency vary with the angular
velocity as follows
K = k - mfl• ............. ....... [10)
w,. = - 122 • .•••• ••• •.•••••• [BJ
The action of the absorber is due to the fact that it adds an-
other degree of freedom to the original system. An additional
natural frequency is added, making a total of two for the case
under discussion, neither of which corresponds to the original
natural frequency. These new frequencies are found by setting
the denominator of Equation [9] equal to zero. A convenient
form of the resulting equation, in which the order of vibration N
has been introduced and R is the original desired radius, is
( w2)^ [^ (^ k^ wt^ )-']^ (4m^ R^ (^2) w•) 1- - l-w 2 --- - n,.• m N• INtD,. 2
X ( 1 m n"^ 2)^ 1 - mw - - = 0 ... (12}
(^2) ( 2) -2( k 2 )- k kN 2 m N 2
Solving Equation (12] for w gives the two natural frequencies.
In a practical design, the undesirable effect of the two new
natural frequencies is eliminated by means of stops such as
shown in Fig. 1. Initial spring force holds the mass against the
inner stop until the speed at which the absorber .is to operate is
approached, at which time it is pulled away from the stop by
centrifugal force. At some higher speed the mass is forced against
PRINGLE-CENTRIFUGALGOVERNOR^ MECHANISM-^ TORSIONAL^ VIBRATION^ ABSORBER^61
theouter stop. Thus the absorber doesnotoperate at^ speeds
which might alJow resonanceatthetwo^ natural^ frequencies.
DESIGN^ OF^ .ABSORBER
An equation useful in proportioningtheabsorber is obtained
from Equation [9]when w = Wn 2m R Or 0 wn = T•. ........... ........(13]
Theangular velocity 11, the natural^ frequency^ Wn,^ and^ the
torque T 0 willbe known Themass^ m^ and^ radius^ R^ must^ be
chosen sothatthe amplitude^ r 0 will be reasonable. Thestops will
be locatedtoeliminate the natural frequencies of^ Equation^ (12].
Thespring constant is found from Equation (11]. Theinitial
force in the spring must balance the centrifugal force onthe
mass atthespeedatwhichtheabsorber is to begin operating.
The details of constructionwilJvary withtheapplication,but a
possible design is shown in Fig. 3. Themassismounted on a
cantilever spring, which transmits the absorber torque to the
&s,o,
ABSORBERMASS
STOP
FIG. 3 A Po88IBLE ABSORBER^ CoNSTRUCTioN
vibrating body. An alternative construction would have the mass
pivot on antifriction bearings. Adjustable stops are above and
below the mass. The spring is loaded in compression, the initial
load being adjustable. The distance between the center of
gravity of the mass and the line of action of the spring force is also
adjustable, allowing the spring constant to be varied slightly.
Thetype of mounting depends on the shape of the^ vibrating
body. For greater effectiveness, two or more absorbers may be
mounted radially about the center of rotation. •
COll.lPARISONS
Fig. 3 emphasizes the factthat,if desired,^ the^ absorber^ may^ be
constructed so thatits characteristics are somewhat variable.
This would facilitate exact tuning^ to^ correct for manufacturing
tolerances, approximations in calculations,^ or^ wear.^ Other types
of dynamic absorbers,^ particularly^ the^ rotating-pendulum^ ab-
sorber, are often difficult to tune exactly. 3
An important^ comparison^ may^ be made^ by^ writing^ the^ equation
for either the rotating-pendulum absorber or the Frahm absorber
which corre.,ponds to Equation (13]. This equation is
m R r 0 wnt = To' ...... ............. (14]
Equation^ (14]^ states^ that^ a.^ rotating-pendulum absorber of^ a
certain mass, radius, amplitude, and frequency wilJ balance
- "Practical Solution of Torsional Vibration Problems," by W. K. Wilson, John Wiley & Sons, Inc., New York, N. Y., second edition, vol. 2. 1941, p. 570.
3 \
2
"'
N FIG. 4 ABSORBER COKPAUD^ WITH^ ROTATING-PENDULUM^ ABSORBEB
a certain exciting torque T .'. Equation (13] states that theabsorber
under discussion ofthe same mass, radius, amplitude, and fre-
quencywilJbalance a different exciting torque T •· Forthe sake
of comparison, let the theoretical relative effectiveness E, be the
ratio of T. to T.'. Then
2m ROr 0 w. 20 2
E, = ---- = - = - ........... [15J
m R r 0 w;.^1 Wn N
This result is visualized in Fig. 4. For second-order vibration
both absorbers are equally effective; in other words, for^ a^ given
application, the absorbers would be about the same size.^ For
higher orders, the rotating-pendulum absorber would be preferred
on the basis of Equation^ (15].^ On^ the^ other hand, for orders
of vibration lowerthanthe second, the absorber^ under^ discus-
sion would bethemost effective.
Discussion
R. J. HARKER.• This paper on the use of a radially oscillating
mass tosuppress torsional vibrations in rotating systems is novel
and appears to present some interesting possibilities, particularly
with 11espect tolow orders of excitation. The author istobe con-
gratulated for his ingenious proposal and for his contribution to
the theory of the tuned dynamic vibration absorber.
Equations[8]and[9]of the paper may be simplifiedbydefining
the main system as a single mass at a radius of gyration equal to
the radius R, or I = MR 1, and by using the foIJowing notation
f = (wn/On) = absorber tuning factor
g "" (w/On) = forced-frequency ratio
μ, = (m/M) = mass ratio
Then Equations(8]and [9] become
8. =
(T0 /K,)^ (1^ -^ g•]
To=
- Associat.e Professor of Mechanical Engineering, University of Wisconsin, Madison, Wis. Mem. ASME.
Reprint
No.
PUBLICATIONS OF THE: E:NGINE:E:RING RE:PRINT SE:RIE:S
*1. Response of Circuits to Steady-State Pulses, by Dr. L. Waidelich, Professor of Electrical Engineer-
ing. Reprinted from the Proceedings of the I. R. E., Vol. 37, No. 12, December 1949.
*2. Heat Transfer to Water Boiling Under Pressure, by E. A. Farber, Graduate Student, now Assistant
Professor of Mechanical Engineering University of Wisconsin, and R. L. Scorah, Professor of Mech-
anical Engineering. Reprinted from the Transactions of The A. S. M. E., May 1948.
*3. steady-state Waves on Transmission Lines by D. L. Waidelich, Professor of Electrical Engineer-
ing, (1950).
4. Theory of the Adiabatic Bubble, by Ralph Scorah. Reprinted from the Proceedings of the Mid-
western Conference on Fluid Dynamics, J. W. Edwards, Ann Arber, 1951.
5. Equivalent Load Method for Analyzing Prestressed Concrete Structures, by Robert B. B. Moreman,
Professor of Civil Engineering. Reprinted from Journal of the American Concrete Institute,
Vol. 23, January, 1952.
6. Design of Low Frequency Constant Time Delay Lines, by C. M. Wallis, Reprinted from Transactions
American Institute of Electrical Engineers, Vol. 71, 1952.
7. The Engineer Becomes A Professional Manager, by Harry Reebey, Reprinted from Journal of Engineer-
ing Education, Vol. 43, 1953.
8. Use of the Centrifugal Governor Mechanism as a Torsional Vibration Absorber, by 0. A. Pringle,
Assistant Professor of Mechanical Engineering, Reprinted from the Transactions of the American
Society of Mechanical Engineers, Vol. 75, 1953
