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V I B R A T I O N P R O B L E M S IN H Y D R A U L I C S T R U C T U R E S

A r m y E n g i n e e r W a t e r w a y s E x p e r i m e n t S t a t i o n V i c k s b u r g , M i s s i s s i p p i

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DISTRIBUTED BY:KimNational Technical Information Service Ü . S . D EP A R T M EN T O F COM M ERCE5285 Port Royal Road, Springfield Va. 22151

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}' VIBRATION PROBLEMS

IN HYDRAULIC STRUCTURES

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ß/ Miscellaneous Paper No. 2-414

V December 1960 y

U. S. Army Engineer Waterways Experiment Station

CORPS OF ENGINEERSVicksburg, Aiississippi

ARMY-MRC VICKSBURG. MfS3.

?r ? x ? -x ^ 7Tri.'zzs~+ir ''-jkXit'. * -.>■■ .H i&jta1«^£

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Preface

I This paper vas presented by Mr. i'rtnk 3. Campbell, Chief, Hydraulic

j Analysis 3ranch, Hydraulics Division, U. 3. Array Engineer Waterways Experi-

' ment 3tation> at the New Orleans meeting of the American Society of Civil

• Engineers on 8 March 1$60 under the sponsorship of the Committee on Hydro-

! mechanics. It has been submitted for publication in the Journal of the| Hydraulics Division. ASCE.

• The paper is based on data gathered in tests of prototype structures

• sponsored by the Corps cf Engineers Civil Works Investigation program. The

; author wishes to acknowledge the supervision of the field work discussed

; herein by Mr.. Ellis B. Pickett, Chief of the Prototype Section, Hydraulic

a • ^^Tysis Branch, and by his predecessor, Mr. Benson Guyton; most of the

• electronic instrumentation used in the tests was done by Messrs. Lei land M.

Duke and George C. Downing of the Instrumentation Branch, Waterways Experi­

ment Station. It was only with the cooperation of many engineers in the

Divisions and Districts of the Army Corps of Engineers that the field tests

i were possible. Engineers of the Waterways Experiment Station too numerous

; to mention also participated in this work. Mr. E. P. Fcrtson, Jr., is

j Chief of the Hydraulics Division and Mr. J. 3. Tiffany is Technical Direc-

; tor, Waterways Experiment Station. The field work was performed over a

I period when Colonels C. H. Dunn, CE, A. P. Rollins, Jr., CE, and Edmund H.

Lang, CE, were successively Directors of the Station.

Preceding page blankiii

t 22265

Contents

Preface.................... . . . . . . . . . . . . ..................iii

Summary . . . . .......... . . . . . • • • • • • • . . . . . . . . . vii

Introduction . . . . .• . . . . . . . . . • . . . . . • . • • . . . . 1

Scope . • . • • • .............. 1

Applicability of Classical Theory . . . . . . . . . . . . . . . . . . 2

Definitions of symbols . . . • . . . . . . . . . . . . . . . . 2Basic equation ......................... .......... ........ 2Damped free vibrations . . . . . . . . . • • . . . .... . . . . 3Forced vibration......................... . • « . . . . . . 5Constant friction . . ... . . . . . ................. 6Hydraulic gate friction . . . . . .... . . . . . . . . . . . . 7

Gates on Elastic Suspensions . . . . . • • ... . . . . . . . . . . . 8Model studies . ................................................ 8Prototj^e tests.............. .................... .. . • . . . 10

Sluice Gates . . . . • . . . . . . . . . . . . . . . . . . . . . . . 13

Flexural Vibration of Elascic Beams . . . . • • • • • • . . . • • • . Ih

Plates and Shells . . . . . . . . . . . . . . . . . . . • . . . . . . It

Hcvell-Bunger valves . . . . . . . • . . . . ... . . . . • • . . ItRayleigh-Ritz method . . . . . . . . . . . . . . . . . . . . . 15Gate skin plates . . . . . . . . . . . . . . . . . . . . . . . 16

Exciting Forces . . . . . . . . . • . . . . . . . . . . . . . . . . . l6Self-excited oscillations . . . . . . . « . . . . . . . . . . . 16Free water surface phenomena . . • . . . . . . . . . . . . . . 17Channel waves . . . . . . • • . . . . . . . . . . . . . . . 17Other exciting forces . . . . . . . . . . . . . . . . . . . ..... 18

Summary cf Conclusions . . . . . . . . . . . . . . . . . . . . . . . 18

References . . . . . . . . . . . . . . . . . . . . . . . . . , . . . 19

Preceding page blankv

Summary

This Paper is concerned p r in c ip a lly w ith the v ib ra tio n o f c o n tro l ••.tes end regu la ting valves commonly used in hydraulic s t ru c tu re s , such =s

p |e P f1“®1?31 find ings of twelve y ea rs of v ib ra tio n in v e s tig a tio n s vy the Arnty Engineers c iv i l works a c t iv i t ie s a re reported .

The c la s s ic a l theory of v ib ra tion i s reviewed to s e le c t those o c r- tions applicable to tne su b jec t. Evidence of Coulomb damping i s demon- .traced , and tne re su lts of f ie ld te s t s a re analyzed to determine the con-

l r ic t io n fo rce . Both laborato ry and f ie ld te s t s are reported .Empnasis i s placed or. ex c itin g forces expected to be found in hydrau-

1 ic s tru c tu re s. The Von Karman vortex t r a i l , s e lf -e x c ita t io n involving f l e e t i n g pressure waves, end other hydraulic p u lsa ting phenomena a re° c*i.,cussed. Ceroam p ra c t ic a l conclusions are drawn.

Preceding page blank■vit

A

VIBRATION PROBLEMS IN HYDRAULIC STRUCTURES

Introduction

1. The study of vibration theory has had an interesting history.Ire physical phenomena involved with musical instruments was of interest to

7*the scientist of the eighteenth century. Lagrange1 solved the physical ,rcblem of the vibrating string in 1T59*

2. A great advance in the physics of vibration was contributed moreIVthan a century later by Lord Rayleigh upon the publication of his theory cf sound. Many refinements to Rayleigh1s work have since been accomplished. In this century, the mechanical engineers have written and are continuing to write extensively on the subject of mechanical vibrations.

3* It is the purpose of this paper to emphasize to civil engineers the importance of certain types of mechanical vibrations in hydraulic structures. These particular vibration problems are not adequately covered even in the mechanical engineering literature.

Scope

U. The title of this paper could have been "Mechanical Vibrations in high Head Hydraulic Structures"; however, such a title would be toe restric­tive for the purposes of this paper. The author wishes to place special emphasis upon the problems of exciting forces which arise from certain hy­draulic phenomena.

5* Ibis paper is principally a discussion of the vibration problems ‘rising in connection with the operation o gates and valves commonly em­ployed to control and regulate the flow of water through or over dams. The vend "operation" is used because much research must be done before a rydraulic structure can actually be designed with confidence that the me­chanical elements involved will, not vibrate under certain conditions of '-Iteration.

Raised numbers refer to similarly numbered items in the list cf refer­ences at the end cf the text.

T V *

2

6. The author is associated withprototype investigations to deter­mine causes of vibrating gates or gate elements• Therex'ore, this paper will constitute, in a sense, a summary report of those experiences of the U. S. nrmy Corps of Engineers with undesirable vibration which have been brought to the attention of the Waterways Experiment Station.

7* The problems cf vibrati; in hydraulic machinery will not be treated in this paper, nor will the problems of water hammer and surge tank design. Extensive information on these subjects may be found elsewhere.

Applicability of Classical Theory

8. It is important tc examine the classical theory of vibration to determine which portions of the science are useful to the solution of the specific problems that might be encountered in hydraulic structures. The theory may be found in various textbooks such as these of Freberg and Kemler, Den Hartog, Timoshenko and Young, and Jacobsen and Ayre. ° Nu­merous other good texts are avai3 able.Definitions of symbols

9* A definition of symbols can sometimes be given more clearly by asketch, such as shown in fig. 1. Thecase of undamped free vibrations serveswell for the purpose of definition ofbasic physical quantities. The deviceshown is a mass, defined as w/g , withan elastic suspension which has aspring constant k expressed inpounds per inch. In this case, letus imagine a cube of steel supported by

- tension spring. There will be an initial static ej.ongation called 6 .sBasic, eouation

1C. If the cube is forceably pulled downward and released., a vertical vibration will ensue. The basic differential equation is shown in fig. 1. ihe symbol x signifies the second derivative of displacement with respect to time. This derivative is recognized as acceleration. The first term is there';ere - the product of mess and acceleration, which represents the

PERIOC* T * ~

FREQUENCY: T = 2 i f ^

' mitiens- -undamped ribraticns

3

inertial force of the vibrating body. The second term is the product of the spring constant and the displacement, which is referred to as the restoring force.

11. The useful solution to the basic equation is shown in fig. 1.One period of the cosine curve is 2* or o^t . The subscript "n" is used to denote the natural..frequency of the elastic system. The symbol o> is commonly called the angular frequency, although various authors employ dif­ferent symbols. The natural period is then 2it/co . Since the frequency in cycles per second is the reciprocal of the period we can also write:

fnCDat

The following commonly used equation can also be written:

It may readily be oeen that the natural frequency is inversely proportional to the square root of the initial static elongation . For the elastic system described this value is expressed simply as = w/k . The spring constant is thus seen to be an important characteristic of the vibratory system. Some textbooks give tabulations of the spring constant for various types of elastic systems.

12. The phase angle circle is included to demonstrate the initial displacement position. It should also be mentioned that certain authors, notably Jacobsen and £yre,^ have developed graphical phase plane solutions which appear to be very useful in certain types of damped vibrations.Damned free vibrations

13. When a restraining force is im­posed on a vibrating system the type of damping must be known before analysis.The basic differential equations for the three common types of damping are showTi in fig. 2. Their applicability to the prob­lem of vibration in hydraulic structures is important to this discussion. It Fig. 2. Damped free vibrations

CONSTANT FRCTCN *♦ V» ♦ CcC sgn *) * 0

(*0 SINGl : COMPLETE SOLUTION)

VISCOUS damping J V + Cí * tk i * 0

«•e^* (Acosu'nt)

TURBULENT DAVP.NGy V Ct CÜ* ♦ fc*» 0 ( no direct simple solution)

k

should be noted that each of the differential equations contains inertial

force imclvinj mass and acceleration plus restoring force involving the

spring constant and displacement. To each cf the equations is now added a friction or damping force.

IV. The constant friction force has the sign of the velocity. The

force always opposes the motion and thus changes sign at each half cycle,

.therefore, no single complete solution can be written. Each part of the

solution must be considered between the largest positive and negative dis­

placements of each half cycle.

15. The second term of the viscous damping equation contains the

damping coefficient which is multiplied by the first power of the velocity.

The general equation has a relatively simple and complete solution. The

first factor on the right contains an exponential function which involves

the damping constant Cy and mass. The second factor contains a cosine

function of the natural frequency. A large part of the classical theory of

vibration is based upon viscous damping. Unfortunately, the viscous damp­

ing solution dees not seem to be generally applicable to problemsencountered in high head hydraulic structures.

16. Turbulent damping is the third case to be considered. It may

be noted that the turbulent damping force is a function of the square of

the velocity. This is the common type of force with which hydraulic engi­

neers ordinarily deal in their problems cf pipe friction loss. Unfortu­

nately, the equation is a second order, second degree differencial equa­

tion and has no direct simple solution. Prof. Milne°,1C' has obtained an

elaborate exact solution. He has also prepored tables for the evaluation

cf the solution which have been four useful in surge tank problems for

hydroelectric power plant design. The technique involved is beyond the scope of this paper.

17* We now turn our attention to the problem of attenuation of dis- .

placement in damped free vibrations. In the case of viscous damping, the

attenuation for each cycle is called the logarithmic decrement and can be

evuluated when the viscous damping coefficient, the mass, and the frequency

ore known. It may be seen that the attenuation is affected by an exponer.-

"icl lav; so that the displacement is damped rather rapidly in the begir.nin--.

• -.iter the damping factor becomes negligible, we consider that a steady

*- > - i, » m U m

late exists. Reference will be mede to steady state vibrations later in this paper.

18. Constant friction is cnaracterized by an arithmetic decrement as indicated on the top sketch of fig. 2. It is important to note that only the constant friction force and the spring constant are involved in the de­termination of the arithmetic decrement. This is an important relationship, as will be shown later. ’

Forced vibration

19- We have considered the aspect of a damping force which decreases the energy cf the vibrations. It is now appropriate to consider an excit­ing force which can increase the energy and hence the amplitude of the vi­brations. The basic equation for forced vibrations with the classical vis­cous damping is shown at the top of fig.3* Tne solution shown is for the case of steady state vibration. In other words, the damping effect of the expo­nential term has become negligible.The exciting force F is considered to be applied with a cosine variation.The solution for the steady state vibra­tion is shown in fig. 3. Ttie first fac­tor of the solution is the ratio F/k , which represents the ’’static elongation” which would be produced by the maximum exciting force with the existing spring constant. If the cosine function in the numerator Is Placed equal to 1, we then have the factor one divided by the radical, which is ordinarily known as the Magnification Factor. It will be noted that the Kagnilicaticn Factor involves ratios of forcing frequency to natu­ral frequency for the various damping ratios Cy/Cvc . Tne denominator Cvc of thls ratio is the critical damping coefficient which can be evalu­ated according to the equation in fig. 3. .She critical damping coefficient invokes only the natural frequency cf the vibrating system and its macs.

20. In considering the subject of damping friction, it would be ex­propriate to quote from Jacobsen and lyre6 as follows: 'Friction is there­fore, a priori, a more complicated property to deal with than either inertia or restoration/’

Fig. 3* Forced vibrations— viscous damping

6

21. A graph can be constructed of the'Magnification Factor as a function of the frequency ratios. The various damping ratios are repre­sented by a family of curves. It should be noted that for a damping ratio of zero, the curve represents an undamped system. Near the condition of resonance where the ratio of frequencies is equal to one, the Magnification Factor, and hence the amplitude, approaches infinity. One might surmise that the system is safe against wildly fluctuating vibrations in the resonant range if the damping ratios are substantially greater than 0.3. Constant friction

22. As mentioned previously, the differential equation for the caseof constant friction has no simple continuous solution. Nevertheless, Prof. Den Hartog1 has developed a remarkable solution for maximum dis­placement in the case of steady state vibration with constant friction. His equation for maximum displacement is shown in fig. it. The solution again contains a ratio of exciting force to

spring constant. The variables A and B under the radical are two different functions of the ratio of forcing frequency to natural frequency of the system. The second term under the radical contains a square of the ratio of the constant friction force to the exciting force. The magnification can be expressed as the ratio of the maximum displacement to the ’static elongation attributable to the exciting force.

23. It may be seen that when the second term under the radical be­comes larger than the first, the evaluation of maximum displacement in­volves the square root of a negative number. The Dc-n Hartog equation shown is only of value in the determination of magnification above the dashed line shown. The family of curves represents various ratios of the constsn:1 notion force to the maximum exciting force. It can be seen that for a *_tio of ici'ces greater than n A , the magnification approaches infinity.

2*i. By a very complex procedure, Prof. Den Hartog1 was also success- iul ir. evaluating single points underneath the dashed line and thus he con­structed curves representing high force ratios. The friction is sufficieuv

Fig. it. Forced vibrations— constant friction

*¿*¿*»4 ... WU

l

in the res!on of high force ratios to cause a stoppage in the vibratory mo­tion. It is interesting to note that the two lines representing force ra­tios of 0.85 and 0.95 from the Pen Hartog investigation have maxima at fre­quency ratios substantially smaller than the resonant frequency. One might expect that, in the case of constant friction, if the force ra.tos are con­siderably greater than 0.8 there would be little likelihood of developing very large .vibratory- amplitudes. In the case of viscous damping, we sur­mised that the situation would be fairly safe if the damping ratios exceeded 0.3.

2 5. ihe engineer is then faced with the problem of determining the constant friction force and the exciting force. 3cm.e test data are avail­able on friction factors, which should be applicable to hydraulic gates.On the other hand, experimental data on the magnitude of exciting forces in the various hydraulic phenomena are practically nonexistent. The problem of excitation ir. hydraulic structures '.rill be discussed later.Hydraulic rate friction

¿6. A substantial amount of research has been conducted on sliding 1 fiction and rolling friction in the mechanical engineering laboratories. However, nuch of the recent expeidr.enxaticn has been concerned .with the ef­fect of various types cf lubrication upon journal friction. It is believedthat ordinary lubricants are not effective for very long periods cf time when used cn submerged high head hydraulic sates.

27. The U. 3. Bureau of Reclamation11’12 has conducted tests cn both sliding and rolling friction in connection with their studies of gates for hydraulic structures. These laboratory tests were conducted in the dr ■r ither than with water surrounding the element, as would be expected in -n outlet works, for example. It appears that the coefficient for oliding friction is in the neighoerheod of 0.1, whereas the coefficient cf rolling friction may be approximately C.C01 depending upon the type cf «etals- -employed. In the field cf design hoists for high head gates, pro­vision must be mc.we for adequate capacity to bread; the seizure of metals 1 the gate has been closed under a high head for a. substznti 1 neriod

cf time. Therefore/with application cf a judiciously chosen factor'of - °tyy the coefficients mentioned wcuId probably be entirely adequate * r purposes of heist capacity design. Cn the other hand, it. is believed

8

Z £T ir™u — * * — or f r ic t io n ^ ln th e Broto

, L 1m 2 " ” “ " " ' to V41“ f ” « " t b . problems of g ate

fe.tes on E la s tic Suspensions

Model stu d ies

. . . 26’ ? * * Pl8Xe ntm~ 8j,<lriiulic "»del s tu d ie s of a mass 0„ an e las« uspensron have b e » found to be p r a c t ic a l and useiU i in a s t u * T.teaha W s C C r t e ^ L T r i s ^ 1’1“ 1 S ta t l0 n StU illeS ^

~ r desi s „ as ^ “

An e J L T “ ” “ ^ ^ ” “s s l“ ^ t e d in th e „cd,« te n s io n spring vas se le e te d »hich produced a s t a t i c e lo n g -tio n s i - i l -

LLTT ln the P r0 t° t W - C“ s i t o .b l e care »as tahen to reduce’ in th e model by the use of m iniature r o l i e r b e a r i n g Both th

v ib ra to ry notion and the fo rces on the g ates »ere m e a s u r T ^ t

described in a W a t e r y Experiment « . „ o n te c h n ic a l r e p o r t . « '

30. Cue of the o b jectiv es of th e t e s t s »as to determ ine „h eth er a***

i

i......... r i . 2,i -¡ 5 i y¡2 2 '

1 J i r

»■0.0126 F T

^ ■ ¿ r i l ^ C P S

IMAGE FOR 1 7 - fT GATE O P E N IN G .Ln -0 .3 3 3 FT■j -------- MOOE L VELOCITY Vm --1845 FT/3 EC

S T R O U H A IN Q VflRPOR NORM AL P LA T E:

V - f = 0143

~ L ~ a 0°9K- 8.0 °-9

h 1192

vortex t r a i l from a p a r t ly open,

fla t-b o tto m g ate could produce v lt

t io n s . The b asic th e o ry of t h i s i

v e s tig a tio n i s dem onstrated in f ig

The i n i t i a l s t a t i c e lo n g atio n o f t!

model g ate on i t s e l a s t ic suspense

was 0 .0126 f t . The n a tu ra l freque:

o f th is system according to th e

theory p rev io u sly p resen ted i s

Fig. 5 . F la t bottom g ate— vortex t r a i l resonance

sim ulated to be approxim ately 8 cycles p er sec.

Von K c i n ^ ***** — * « rcu m , therte x t r d l is shed from each edre o f n i m .

r f ^ . cu&e 01 ^ne p la te . The freaup*

involves t h e ^ u ^ t T r “ !. ^ ^ 3* ' " “ 1 nUBber ' ,hlcif l »14 , \ “ le "3th ° f the P l“t e - - d the v e lo c ity of th e

nre ^ Ur i n T ™ thB l 0 f t Sh°“S f lS t P l“ * - i t . 1 . ;'lent,.: : t V ^ ^ « - - t i e l opening OfA X liC : iJ'il/i o*p r \ f ' - f t . .— , -

o - ta e ¿ l a t p la te above th e rn->f* __ , ..

l/

PH*\gptwgpp m

o✓

completes the normal concept of the vortex trail. During the early phases

of the tests, it vas not known whether the vertex trail can be shed from

half of a plate as shown in this situation. However, it was soon learned

that this phenomena can exist for the protrusion of a gate leaf into a

conduit* .32. Various experimenters have determined the Strcuhal number for a

flat plate normal to the flow to be approximately one-seventh.^ If the

vortex trail can produce an exciting force on the bottom of the gate which

would come into resonance with the natural frequency of the elastic system,

it would be expected that the gate would produce vibrations of large ampli­

tudes. -For a 17-ft prototype gate opening, end a model velocity of18.5 ft per sec, the theory indicated that there would be essential reso­

nance. This computation indicates that the ratio of forcing frequency to

natural frequency is 0.$9. It was actually determined that when the model

gate was set to simulate a 17-ft prototype gate opening, violent vertical

vibrations were produced. It was therefore obvious that resonance between

the frequency of the vortex trails from a gate projection acting as an

exciting force on an elastic system could actually be produced in the

laboratory.33. Split-leaf gates. In connection with the subject of the vortex

trail as an exciting force, mention should be made of split-leaf spillway

gates. With this system, one leaf rests on top of the other. In times of

flood flow, both leaves are removed.

3 . After the top leaf has been removed and while the bottom leaf is

being raised, water can flow over and under the partially raised leaf. The

full vortex trail is then effective in producing periodic pulses alter­

nately on the top and bottom of the gate leaf.

35* The Waterways Experiment Station studied this problem with a

model of the Old River Control Structure gates. [36. Improvement of flate lips. It was learned from the studies of

Fort Randall gate vibration that a flat-bottom gate had very high down-

pull. The tests at the Waterways Experiment Station further showed that '

a flat-bottcm gate also is more susceptible to vibrations. Various lip

extensions from the downstream edge were tested. The flat-bottcm gate

jnd the lip extension schemes are shown in the top two sketches of

1C

fti.

iII

Iili

il

FLAT BOTTOM GATE

t :UP EXTENSION

■ 'CSTANDARD 45* L ip

H f ~ “ ADVANTAGES\ l I. LOW DOWNPUll- ECONOMICAL HOIST

2 VORTEX TRAIL PROM DOWNSTREAM EDGE

Fig. 6. Development'of standard ^5-degree lip

essentially from- the lip extension

fig. 6. Typical oscillograms trans­lated in terms of vertical displace­ment from the accelerometer records are shewn. A fairly regular motion of 8 cycles per sec may be seen on the top oscillogram, whereas no such

• regularity is shown for the case of a lip extension.

37* It was reasoned that the vortex trail was probably springing

rather than from the upstream edge ofthe ilat-bottom gate. The engineers of the Waterways Experiment Stationtried the next logical step, which was a gate with a bottom surface slcoed at 1*5 degrees. It was found that the vortex trail would then soring from the downstream edge and minimize pulses on the bottom of the gate.

38. He tendency for vibration was found for the ^-degree gate lip. Furthermore, the dovnpuli or rather the reduction of pressure cn the bottom of the gate was much less for the sloping gate bottom than for the flam bottom;.• Therefore, the 1+5-degree gate lip has become standard cn Irmy En­gineer gates for the reason that reduced dovnpuli is indicated and that there is less tendency toward vibration of the gate.Prototype tests

3 '* Kptthtville Par, tests. Before proceeding with a report of the prototype tests, it is appropriate to discuss briefly the important nrcb- len of the spring constant for cable-supported control gates. Tests were made by waterways Experiment Station engineers at Knightvllle Dam in New England in the fall of 1955* In the analysis of the data it was of course desirable to calculate the spring constant of the cable supports.If the total metallic cross-sectional area of the cable and the modulus of elasticity are known, the spring constant can be determined. A fairly accurate value of metallic cross-sectional area can be furnished by the uionui'acturers of wire rope.

^0. The only information cn modulus of elasticity of wire rope at time was a value of 12 million lb per sa in. for. unstressed cables

iurnished many years- ago b- one of the wire rope manufacturers. The use of

tsr^

.

rb

f

I

\}}

¿¿turn* • * * --~-- • . i ^ J l ^ '

11

this value did not produce a theoretical frequency vhich agreed v/ith the measured frequency at Khightville Dam on tests at three different gate openings ranging from 10-l/2 to 11 ft.

k l . The natural frequency measured at Knightville was about 6-1/2 cycles per sec. With the other physical data pertaining to the vibrating system- known, the modulus of elasticity was estimated to be approximately 21 million lb per so in. Soon thereafter information became available from tests on prestressed wire rope. These results indicated that the modulus of elasticity should be in the neighborhood of 20 million lb per sq in.This value is now considered to be a fairly reliable modulus of elasticity for wire rope which has been stressed repeatedly in operation.

h 2 . It was clearly evident that the mass of the water contained within the gate must be added to the mass ci\ the gate metal for a proper vibration analysis.

^3* Fort Randall Dam tests. The Waterways Experiment Station con­ducted a series of three tests cn the control gates for Fort Randall Dam in 195 and 1955 for the Omaha District, while the reservoir was filling. The results of these tests are contained in a technical report1^ prepared by the Waterways Experiment Station. The information shown in fig. 7 pertains to the test made with a head of 110 ft. The spring constant for the ca­ble suspension was l.j x 10^ lb per in.•md the gate weighed approximately ‘<7 tons. Gate roller tracks and side ■guides are shown schematically. The 'ride guides restrain any large motion of the gate in the upstream direction.Half of the guides were spring- mounted and the other half were not.

h k . The oscillograph records show that for large gate openings, t here were occasional vertical vibrations of approximately A cycles per sec . *e°e vibrations continued for several cycles and were then damped cut. v. m an examination of the record, it seemed apparent that the frictional

•^ree involved in the rollers was rather, high cut that occasionally a surge gainst the., downstream face of the gate moved the gate upstream so that

rpn^-HOlST♦0.36 ¡Ml

cabled rl -031 n H

k«UXiOT LBS/IN

z y !

.025SEC.

DAMPING RECORD

S’:E ! /ROLLER GwOSSn ' , track

I f* y

CAMPING FORCE Cc5^ r

GATEOPENING

20 FT

19-21

CcPOUNDS

27402800 (AVG CF 4

OBSERVATIONS)

Fig. 7. Elastic suspension-- Coulomb damping

m

* t -t c w s

1 - ■ ■

nxmenturily the rollers were not in contact with the track. During these intervals the gate was permitted to vibrate vertically. As the rollers ag in cau»e in contact, the vibration was damped. Further examination of the damping periods of the record indicated that the damping was of a Coulomb character. One of the damping records is shown in the upper right- hand corner of fig. 7. It may be noted that straight lines connect the peaks and troughs of the curve.

1*5. This was e significant finding, for although experimental evi­dence was ?.t hcnd to shew that the damping was predominantly caused by aconstant friction force, the theoretical problem was known to be complex.6It is pertinent to quote again from Jacobsen and Ayre as follows:'It is therefore clear that the presence of constant friction greatly complicates the analysis cf steady state response to an alternating force/ especially if the friction is large enough to enforce stops in the motion.”

•’;6. The arithmetic decrement discussed above was then calculated for several damping periods. Knowing the spring constant and the arithmetic decrement, the damping force can be computed. The section of the record shewn is for 3 20-ft gate craning and the damping force was computed tc be 2~kC lb. An.average of four observations indicated a damping force cf about 25CC lb. It should be mentioned that in two cases studied for the same opening,' one dumping force appeared to be double the dumpingforce for another period. It is possible that the lower force could have bean ■c-vuscd by only one roller track coming in contact with the guide. In the case cf . the'higher force, it appears that both roller tracks came in contact with the guides simultaneously.

Uj. 'We then realised that although the theory oi" Coulomb damping is complex/ the simple determination of the arithmeticaldecrementaffords a means of measuring friction. It is believed that a study of damping rec­ords cun furnish valuable information cn the total damping force for any particular system under observation. 'There has not been an opportunity to utterf.pt to compare the toual damping force measured in the prototype with a lcul ted foice based on friction factors as measured in the laboratory.

..r: estimation of the frictional forces of the individual components, of a roller train r.vy not be a simple matter.

13

Sluice Gates

**8. The hydj.Tulj.caUy operated sluice gate is a device commonly used

for controlling the flow-of water through concrete dams. Prototype tests

were made on the sluice gates of Pine Flat Pan for the Sacramento District

— HYDRAULIC Q« HOIST oi

I n ?? BONNET

-02,5 . ...., 1 SEC— 1 ■ 1

8 02p r < oi 3

GATE OPEN 75FT MAX X *628 X10*« IN

in 1952. Measurement of the vibrations cf the gate leaf vas a part cf these tests. The gate size vas 5 by

9 ft. Fig. 8 shows a schematic sec­

tion of the slide gates.

The 6-tcn sluice gate was

supported by a 6-l/2-in.-diameter

steel stem, h typical oscillogram is

shown in the upper right-hcnd corner

cf fig. 8. The maximum amplitude cf

vibration was of the order of six-

Tiillicnths cf an inch. This empha-

~5‘X9‘CATE -o.ü Í * * ; * i -r r r i' ! ter 1 .

HEAD* 110 FT

Fig. 8.

.JAECg

MOVING GATE f - 35 CPS

MAX X = 1.46 XI0*1 IN

Hydraulically operated sluice gate

sizes the fact that we are dealing with a relatively lightweight gate on stiff suspension.

50. The oscillogrsm in the lower right-hand corner cf the figure

represents a significant observation on the Pine Flat sluice gate. This

record represents the vibration existing when the gate was being moved from

~ 3"*ft to a 4-ft opening. For such a situation, the amplitude exceeded one-

thousandth of an inch or a thousand times that observed for the stationary

position. It seems reasonable to believe that this larger vibration ampli­

tude was caused by alternate seizure and relief of the metals on the slide

bearing. This is sometimes called “chatter.

51- although the hydraulic head for the west just described was

11C ft, this gate was subsequently operated under a head in excess of

jCC ft. No detectable damage to the gate was evident. A cursory study

was made of the fatigue aspect in the gate stem. The stresses in the

stem proper were so low as to eliminate any further consideration of

fatigue. No study has been made of the possible fatigue aspects of

the supporting elements at either end of the stem.

li*

Flexural Vibration of Elastic Beatr.s

FLOW

j-ne noiizontaJL beams of aDEFLECTION* 6 XK f4IN.(WET) 5

l I| frEQuencymnterior dry tractor gate are normally very stiff.fhe i?ort Randall gate beam is shown inSECTION A»

rtf p SECTION B-B

FREQUENCY-INTER'CR WET E. • '40 CPS

Fig. 9- Control gate— lateral vibration

fig. 9 as an example.53* For a • oncept of the stiff­

ness it may be noted that the initial static deflection based on the mass of the gate and contained water is only c X 1C in. A comparison is shown

between the theoretical frequency of the beam with and without contained water.

54* £he frequency for the dry condition may be seen to be 2C5 cvcles per sec, and that for the condition of inclosed water lfcC- cycles per sec. This emphasises the importance of accounting for the mass of the contained water for gates of this type.

Plates and Shells

Hovell-Furirer valves55- Seme years ago it was learned that weld seams on 4-vane Hcvell-

Bunger valves at two different projects had failed. Fig. 10 is a sche­matic representation of the vibration characteristic of both a 4-vane and a o-vane valve shell. Prof. Timoshenko treats the subject of vibration ofthin-wailed elastic cylinders.1^

56. Fig. 1C shews schemata- colly a 4-vane valve shell vibrat- ’ Tig with the fundamental mode and a 6-vane valve shell vibrating with ~ secondary mode • The equation at the top of the figure gives the nat- t:r:i frequency for the fundamental r.ode end any of the higher modes.-t rr/y be seen that the f: ctor in

-■-k / O// \\ //1 v1 V

SOLUTION FOR SHELL ONLY FOUR VANE VALVE SIX VANE VALV?

fundamental MODE.-=2 SECONDARY MODE?* =iM2 = 2 S 8 _ M j * 7.59

f2 :US- \(ICT - , 759* In U * r * k =282 h-f; V f c

Fig. 10. Hove11 -Bunger valv^ vi.tration

15

the first equation, involving the radical, is Concerned with the mess and

elasticity of the shell. The last factor is here designated <? mode factor.

For the fundamental mode, i = 2 , and the resulting mode factor is 2 .6 8 .For the secondary mode, i = 3 , end the value of the mode factor is ?.59»

fir.ee the natural frequencies of the shells ere proportional to ratio of

the mode factors, it may readily fee seen that the ratio of the frequency

of the secondary mode to that of the fundamental mode is 2.82.

57- This ignores the existence of vanes within the shell. However,

it appears that by using 6 vanes and producing a vibration of the secondary

mode type, the frequency could be expected to be much higher than for the

4 vanes and the fundamental mode, ’.ihen the frequency is substantially

higher, we would normally expect that the amplitude of vibrations would

be lower and that for each cycle the total amount of energy is less for

the higher mcc.es. For this reason, it is believed that a 6-vane valve

is less liable to shew -failure of the seams which connect the vanes to the shell.

58. To the author's knowledge, the problem of the Kowell-Bunger

valves considering the mass and elastic characteristics of both shell and

var.es has not been treated. It would be interesting indeed to see a

thorough analysis of the 4-vane valve compared "with the analysis of the

6-vane valve. Furthermore, it is conceivable that a 5-vane valve would

cause the vibration to go to a still higher mode end hence would have

less likelihood of failure of weld seams.

Rayleimh-Ritz method

59• It is appropriate to mention the Kayleigh-F.ito method by which

problems of this nature car. be solved. Lord Rayleigh^“ ir his classical

theory of sound treated only the case of the funds-ental mode. He devised

the method of writing the basic equations for both potential energy and

kinetic energy. By equating these two energies and by proper manipulation'

of the equation, he determined the frequencies of the various types of

clastic systems. Seme years later - extended Lord Rayleigh’s method

to treat the problems of the vibration characteristics with, modes higher

than the fundamental. Since that time numerous contributions hove been

•Vide to the theory of vibration of elastic elements using the Rayleigh-Rits method.

lo

CV; te skin plates60. The vibration of the skin plate on a submergible rate at

Cheatham Dam on the Cumberland River offers an example of this type c:‘vibratory motion. Hie operator:

1L . - HOIST - CHAIN

2»

06 0!

SECOND

V ' V - K m o tLA ':£ ACCELERATION RECORD 23’ \ A V * ‘ HEAD ON GATE 3 F T

6A.L E A F ' S . * . 2 • THEORETICAL f 59 CPS

~ A C C E L E R O V E T E R I5 0 -2 0 0 CP5 S I L L \ * SOUND I I S * IS O CPS

Fig. 11. Skin-plate vibration— submergible gate

reported a very loud noise i z su: from the Cheathun* rate when the overflow head was about 3 ft. sketch of the gate is shown in fig. 11.

6l. The Waterways Experi­ment Station investigated the cause of the noise in cccperati; with the Nashville District.

uelercmeters were placed to measure the principal freedoms of notion of r gate itself as veil as those of the upstream and downstream skin plate. The specific source of the noise was net known at that time.

62. It vas found that the frequency cf the downstream skin piste 0 preximated the. frequency of the sound. The frequencies of ether freedom: of motion were substantially different. It was therefore concluded that the downstream skin pi are was vibrating and causing the sound.

63. The resonant air chamber inside the gate was analyzed but feu:: to have a very low frequency in comparison. Similarly, the vortex trr.il from the strut arms possessed a very lew frequency. As the exciting for: of the vibratory motion could net be readily determined, it has been rec: mended that the skin plate be restrained by a support in the middle of t* panel. It was reasoned that this would greatly increase the natural fre­quency of the skin plate and eliminate the vibration which emitted the noise.

Exciting Forces

felf-excited oscillationsoh. The action of the bow on a violin string is the classic examg

which is often cited. A geed example of self-excitation from the hydrov engineering viewpoint is the interaction of a reflecting pressure wave i

17

:''<weÜSS=»

c

*:f--

é r

CONQUÎT profile

-T.*— </-Pr

WAVE FRONT POSITION—

4=ïT‘ii o I o i

TIME-PRESSURE CURVE AT GATE

G-GATE i • INTAKE

Fig. 12. Self-excitation with pressure wave

conduit and the vex^tical vibration of a gate. This is shovn schematically in fig* 12*

65* A small vertical displacement of the gate would cause a pressure wave to travel upstream in the tunnel.Vhen the v;ave reaches the reservoir, it is reflected and changes from a positive pressure wave to a negative pressure wave or vice versa.

66. If the speed of the wave is *’a" and the distance from gate to res­ervoir is ' L, '* the period of the pulsa­tion against the gate bottom would be -L/a sec. The pressure pulses on the gate bottom are shown in fig. 12 as a square wave for the sake of simplicity. Y/hen the natural period of verti­cal vibration of the gate is close to the natural period of the pressure vave pulse, a condition of self-excitation could exist.

67. It is interesting to note that the Vcn Karman vortex trail is a self-excited oscillation in itself.Free water surface phenomena

68. Two free water surface phenomena are shown in fig. 13* Eithercould constitute an exciting force on anelastic structural element.

69. The fluctuating nappe for a small, head on a sloping weir has been noted by various observers. This phenome­non -was extensively observed ana analysed 8by Bruno Leo.

FLUCTUATING NAPPE

Fig. 13. Exciting forces— free water surface

The phenomenon was also observed by the Bureau of Reclamation'1' cn the drum gates at Black Canyon Ccm. In this case, the vibration was eliminated by aerating the space under the nappe.Channel- waves

70. The binoaal wave which has a clapetis type action from each side wall was observed in the fish ladders at KcNsry Dam. A trincdal wave was Iso observed in the Bonneville fish ladders. These waves can constitute n exciting force cn adjoining structures.

18

Other e:\citin~ forces71. Various-.ether phenomena can possibly constitute exciting forces

in vibratory notion. The successive foliation and collapse of vapor cavi­ties in the phenomenon- cf cavitation seex to have e periodicity. Very little experimental information is available for the multitude of possible geometrical boundsry situations. Some ■ experimentors have suggested a Strcuhal type number which defines the frequency of shedding of cavities.

72. The fixed cavity which forms on the trailing side of an obstruc­tion in high-velocity.flow has an intermittent pressure pulse. However, very little research'has been conducted on this problem.

73. The toe of the hydraulic jump is known to pulse in an upstresn- dcwnstreain direction. The possible effect of this is unknown.

Summery of Conclusions

7t. Tne following conclusions are derived from this study:a. The modulus of elasticity of wire rope suspensions cun be

determined by measurement cf vertical vibrations.b. The type ox" friction which affects the damping of hydraulic

gates has been shown to be constant friction or Coulomb damping.

c. The magnitude of the constant friction force can be evalu­ated by an analysis of the oscillograms.

d. The Von Karr.an vertex trail has been observed to be the exciting force which can cause vibration of gates.

e. A gate leaf with a sloping bottom sheds the vortex-trsi 1 from the downstream edge and therefore minimizes vibratory motion.

f. Extensive research is needed on the magnitude of the excit­ing forces before gates can be designed to be free of vibration..Considerable research is needed on the character of viora-

~ tion of both fixed end traveling cavities and other hydrau­lic pulsating phenomena.

References

Den Hartog, J. P., 'Forced vibrations with combined Coulomb ^nd vis­cous friction.’ Transactions,, fcsericsn Society of Mech-nic..! Er. tri - neers, Faper tPM 53-51 presented at National rppl. Mech. Meeting, Puidue University, June 1931.

» Mechanical Vibrations. Uth ed., McGraw-Hill Book Co., Inc., lieu Yora., 1956. ~Freberg, C. R., and Kemler, E. N., Elements of Mechanical Vibrations. 2d ed., John Wiley and Sons, Inc., New York, 1$^9.Glover, 3. E., Thomas, C. W., and Hammett, T. F., Report cn Vibration Studies Made at Black Canyon £vm. U. 3. Bureau, of Reel-.in tion Hydraulic Laboratory Report No. 58, Denver, Colo., 2k July 1939.Goldstein, 3., Editor, Modern Developments in Fluid Dynamics■1st ed., Oxford, Clarendon Press, 1938. (See vol II, p 571.)Jacobsen, L. S., and :'.yre, R. S., Engineering Vibrations. 1st ed., McGraw-Hill Book Co., Inc., New York, 1958. ~Lagrange, J. L., ‘Recherches our la Nature et la Propagation du Son.' (Miscellanea Taurinensia, t. 1 , 1759)> vol 1, Oeuvres de Lagrange, Paris, (.1867), p 39. ~Leo, Drvr.o, Self Excited Vibrations at Overflew Weirs (Selbstgeste- r.erte Schwenkungen an Uber der orten '.ehren;. U. S. ; .my Engineer W&terweys Experiment Station Translation No. 35-6, Vicksburg- Miss., June 1955*Milne, U. E., Darroeft Vibrations. University of Oregon publication, vol 2, No. 2; 1923.

. Table of Damped Vibrations. University of Oregon, Mathematical Series, voll, No. 1 , March 1925.Noonan, N. G., and Strenge, W. Ir.., Report of Tests cn Coefficients of friction. U. bureau cf Réclina.tion Technic;.1 Kemcrmclum }Io~. ‘'¿2,30 January 193^_______ __, Tests on Rollers. U. S. Bureau of Reclamation TechnicalMemorandum Ko. 399> Separate No. 26, 26 September 1936.Rayleigh, J. W. 3., The Theory of Sound (1st ed., 1877)* 1st American ed., New York end Dover, 159-5.Ritz, Wither, Gesaimelte Werk a. Paris, I9II.Timoshenko, o., end Yeung, D. Hr, Vibre tien Fretiens in Enrineerim.3d ed., Van Nest3:wild Co., Inc., New York, 1555.U. S. .-any Engineer Waterways Experiment Station, CE, Vibration and Pressure-Cell Tc-c-ts FIrod-Control Intake G ates, Fort H_r.i'.li Mis .-curl River -outh D.kot a. Technical Report 2-9 35, Vicksburg, Mias., June I5-56.

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1 7 . UV S. /\nry Engineer W atervjys E ^ e r ir .e n t S ta tio n , CE, Old River Lev- S i l l Control S tru cture, Hydraulic V-odel In v e stig a tio n , Report 1 , Dctnrsull Fcicer. cn V e it ic ? 1 -L i f t ^ c e s . Technical Report 2 -V*;-7, Vicksburg, H is s ., Deceirfoer 1S56.

18. _______ , Sul llv sy and C u tlet Vorks, Fort Psndall Pair., M issouriRiver, South Di.kcta; Hydraulic H-cdei InvestigationV Technical Report 2-528, Vicksburg, H is s ., October 1S5S •

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ISReferences

Ii i. Den Hartog; J. P., 'Forced vibrations with combined Coulomb and vis­

cous friction." Transactions, .tr.erican Society of Mechanic..1 Engi­neers , Paper /.PM 53-9, presented at National appl. Meek. Meeting, Puidue University, June 1931*

| -• » Mechanical Vibrations, l*th ed., McGraw-Hill Book Co.,I Inc., New Yor*., 1956- ~ ~| 3. Freberg, C. R., and Kemler, E. N., Elements of Mechanical Vibrations.j 2d ed., John Vfiley ana Sons, Inc., New York, 151*9.I U. Glover, S. E., Thomas, C. W., and Hammett, T. F., Penort cn Vibration I Studies Made at Black Canyon Lam. U. 3. Bureau of Reclamation| Hydraulic laboratory Report No. 58, Denver, Colo., 2b July 1939.$ 5. Goldstein, 3., Editor, Modern Developments in Fluid Dynamics.

1st ed., Oxford, Clarendon Press, 1>38. (See vol II, p 571.J6 . Jacobsen, L. S., and .-lyre, R. 5., Engineering Vibrations. 1st ed.,

McGraw-Hill Book Co., Inc., New York, I95ST7. Lagrange, J. L., ‘Recherches cur la Nature et la' Propagation du Son.'

(Kiscellsnea Taurinensia, t. 1, 175S'), vol 1, Oeuvres de Lagrange, Paris, (.1867), p 39*

i!iti

*5

A

10

il

Leo, Bruno, Self Excited Vibrations at Overflow Heirs (Selbstgeste- nerte Schwengur.gen an Über Strömten '.’ehren,. U. 3. .\rmy Engineer Waterways £:?eriuent Station Translation No. 55-6, Vicksburg, Miss., June 1955-Milne, U. E., Parsed Vibrations. University of Oregon oublication, vol 2, No. 2; 1923.

._____ > Table of Draped Violations. University of Oregon,Mathematical Series, vol 1, No. 1, March 1929.Noonan, N. G., and Strange, U. II., Report of Tests cn Coefficients of Friction. 1!. bureau cf Reclamation Technical memorandum No. i-32, 30 January 193:,

I

CJ..j\IlIi

_______ > Tests on Rollers, U. S. Bureau of Reclamation TechnicalMemorandum Ko. 3$9> Separate Mo. 26, 26 September 1936.Rayleigh, J. 17. S. / The Theory of Sound (1st ed., 1877)* 1st American ed., Kev York and Rover, I9Â-5.Ritz, V. lther, GesciiVielte Uerhe. Paris, 1911.Tiii.oshenkc, o., and Yeung, D. K., Vibration Problems in EnTineerirr:.3d ed., Van Nostr^nd Co., Inc., îlexr York, 1>55.U. S. i-ar?y Engineer v/ateways Experiment Station, C B , Vibration . nd Pressir/e-Cell Tests ii ?o<i-Control Intake G vtes, Fcrt ll.nl a 11 Dr-iri, i-lis souri River :outh P-kata. Technical Report 2-s35^ Vicksburg, Mies., June ^ 56.

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18.

U. S. Anry Engineer Vkterwuys Experiment Station, CE, Old R iver Lcv- 3111 C en tra l c t r r c tu r e , H vdr.n lic ?-Todel In v e s tig a tio n , Report 1, DeATrr.’. l l Foieer. cn V c itic c l - i . i f 't - - ¿e s . Teciinieo.1 Deport Vicksburg, H is s ., Decerfoer IS 56•

, S n illv sy -md C u tle t .'ork s , F c rt Rsndy.ll ■Dar.. M issouri R iver; South Ik .k ttd ; H ydrenlic Model. In v e s tig a tio n . T echn ica l Report 2-528, Vicksourg, M iss ., October 1>5S*

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