THE UNIVERSITY OF MICHIGAN INDUSTRY PROAM OF THE COLLEGE OF E LLEGE OENGINEERING RESONANCE IN GOVERNED EYDRO PIPING SYSTEMS V, L. St;eetcr E. EB Wylie July, 1965 IPN711

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS.........E...................*.... ii LIST OF FIGURES..............1oV~X~c iv NOMENCLATURE......................................................... v I INTRODUCTION.................................................... 1 II RESONANT FREQUENCIES OF VISCOUS FLUID PIPING SYSTEMS........... 2 III EQUATIONS FOR IMPULSE TURBINE AND GOVERNOR SYSTEM.............. 8 A, Turbine.................... *................******** 8 B. Governor Equations......................................... 10 IV ENERGY RELATIONSHIPS............................ *..... 15 V PIPING SYSTEM EXAMPLES...................**..........***.*..... 17 A, Waves on Reservoir................................. *...O 17 B. Governed Turbine........................................... 18 C, Other Excitations.................................... 19 VI CONCLUSIONS........................... 21 REFERENCES............................................... o,**.~ 22 iii

LIST OF FIGURES Figure Page 1. Schematic View of Governor System (Borel3).................. 23 2. Flyball Governor......................................... 24 3o Spool-Valve Linkage to Flyball and Dashpot................ 24 4o Dashpot Assembly................................ 25 5. Series Piping System......................................... 26 6. Pressure Oscillations at Closed Valve....................... 27 7o Transfer Function, Closed Series System Connected to a Reservoir.......................................... 28 8. Impedance Diagram for Stability Analysis.................... 29 9. Pressure Head and Discharge Oscillations for Unstable Governor................................................... 530 10. Pressure Head and Discharge Oscillations for Stable Governor..................................................... 1 iv

NOMENCLIATURE A = cross sectional area of pipe; constant A = nozzle area n a = wave speed in pipe B = constant C = constant Cd = discharge coefficient of nozzle D = diameter of pipe e = turbine efficiency f = Darcy-Weisbach friction factor g = acceleration due to gravity H = complex function of x only HR = complex head at upstream end H = complex head at downstream end h = elevation of hydraulic gradeline h = steady-state elevation of hydraulic gradeline h' = fluctuation of hydraulic gradeline elevation I = moment of inertia of rotating masses i - v K = number of harmonics k = integer - = length of pipe Q1,l2 = linkage lengths in governor m = ic v

N = wheel speed Na = displacement due to permanent droop a ANc = displacement of dashpot cylinder ANm = manual speed setting Np. = displacement of dashpot piston n = exponent on friction term P = power of turbine PG = power absorbed by generator p = pressure in dashpot Q = complex function of x only Q, = complex discharge at upstream end QS = complex discharge at downstream end q = discharge q = steady-state discharge q' = discharge fluctuation R = resistance coefficient; governor constant R1,,Rj,RRg = governor constants r = radius to centerline of buckets r2,r4r,rg = governor constants t = time T -- function of t only; torque; period u = peripheral speed of centerline of buckets V = fluid velocity leaving nozzle X = function of x only vi

x = distance along pipe Y = displacement of spool valve z = hydraulic impedance z = characteristic impedance ZR = impedance, upstream end zS = impedance, downstream end = real part of 7 = pure imaginary part of y = vane angle, exit to buckets = governor arm angle y = complex number; unit weight of liquid p = fluid density cA = phase angle, nozzle area to head c = phase angle, speed to head N CDq = phase angle, discharge to head ) = circular frequency vii

Io INTRODUCTION In calculating the stability of penstock systems under the control of a governor, the resonant periods of the complex piping systems have not been adequately taken into account. Gaden(l) and Rich(2) assumed the penstock system could be modeled by a simple pipe having the same theoretical period Z 4v/a as the actual penstock with its usual reductions in diameter. Borel(3) takes the pressure pipe and surge tank into account, but considers incompressible surge onlyo Wylie(4,5) has developed methods for finding the fundamental (apparent) period of a piping system and its various harmonics, for inviscid flow conditionso In this paper equations for determination of impedance phase and magnitude are developed for analysis of complex piping systems with real fluids. Then equations for an impulse wheel and governor are developed, linearized, and programmed to determine phase and magnitude of impedance as well as other pertinent variables, A criteria for stability analysis is presented based on energy concepts and the real fluid impedance calculations. Questions of stability are examined for two situations, waves on the reservoir upstream from the penstock when the unit is not operating; and series systems under control of a governor. -1

II, RESONANT FREQUENCIES OF VISCOUS FLUID PIPING SYSTEMS The water hammer equations for continuity and equation of motion, in terms of discharge q and elevation of hydraulic gradeline h are, respectively x +,gA ht 0 qx a+ aand h + 1 t + f qn = 0 (2) x agA n 2gDA in which a is the speed of the pressure pulse wave, A is the pipe cross-sectional area; f is the Darcy-Weisbach friction factor for steady state conditions, D is the pipe diameter, n is the exponent of the velocity in the loss term, g is the acceleration of gravity, x is the distance from upstream end of pipe, and t is the time. The subscripts x and t denote partial differentiation, These equations are solved for steady-oscillatory flow, following methods of transmission line theory.(6,7) The variables h and q are considered as h =h + h' q = q + q1 (3) with h, q average steady-state values, and h', q' variations from the average. By making these substitutions into Equations (1) and (2) and simplifying q' + A h' = 0(4) x 2 t a -2

-3and h' + 1 q' + Rq' =0 (5) x gA t in which nfq n-1 R = n (6) 2gDAn is used as the linearized friction coefficient for turbulent flow. Equations (4) and (5), by differentiation, may be written in terms of a single dependent variable, q', or h', q I-= 1 q' + Eq Rq (7) xx 2 tt 2 t or hi =. h' + gA Rh (8) xx 2 tt 2 t a a To solve Equation (8), let h' = X(x) T(t), in which X is a function of x only and T is a function of t onlyo After substitution and rearrangement X = T + R'- Y (9 x a2 T a2 T in which Y = a + iP must be a complex constant. The equation for X in terms of y yields X = AeY + Be-x (l0)

-4with A and B constants of integration to be determined by conditions at inlet and outlet of pipe. By restricting the solution for T to the steady-oscillatory case, T = C1 emt with m = i); i = V-, and c the angular frequency, substitution into Equation (9) yields 2 (a + ip)2 -= g A(iR - L) (11 a2 gA After solving for a and p (D)2 R2 sin (I a t e \( -f) + R2]1 sin tan1 gAR) (12) Va2 ~ gA 2 V )2 + R21 cos (1 tan gR) (13) Va2 gA J 2 2 Since R and X are always positive, the angle is in the first quadrant and a and p are real, positive numbers. These equations are easily solved by computer for each pipe at the frequency w being considered. The equation for h' is hi, eit (AeYX + Be-yX) (14) After combining this equation with Equation (4) to eliminate h, q = gA eiot (AeX - Be-YX) (15) ia y The hydraulic impedance z(x) is the ratio of head fluctuation to discharge fluctuation h'/q' z(x) = ya2 AeYX + Be-7 (16 gA AeYX - Be-Yx

-5The characteristic impedance z of a pipe is a function of u, defined by Yz = a2 (7 - io) (17) C iJcgA AgA so Aeyx + Be-7x z(x) = - z A + (18) u yx -yx Aex - Be-Y A and B must now be expressed in terms of conditions at inlet and outlet of the pipe. Equations (14) and (15) may be written iUt h'(x, t) = H(x)e (19) q'(x, t) = Q(x)e (20) in which H and Q are complex functions of x only. For the inlet, let x = o, H(O) = HR, Q(0) = QR (21) After substitution in Equations (14) and (15), eliminating A and B, H(x) = HR cosh yx - QR ZC sinh yx (22) -HR Q(x) = -- sinh yx + QR cosh yx (23) zC ZC and z(x) = H(x) ZR - ZC tanh yx (24) Q(x) 1 - (R/ZC ) tanh yx

-6in which zR = HR/QR For known conditions at the pipe outlet, x = ~, let x1 = - x, H() = HS, Q() = Q, zS HS/Q (25) hence H(x) = Hs cosh yx1 + zC QS sinh yx1 (26) HS Q(x) - sinh yxl + QS cosh yx1 (27) ZC and zS + zC tanh yx! z(x) = C 1 (28) 1 + (zS/zC) tanh yx1 Since zR, zS ZC, and y are complex, z(x) is a complex number having a magnitude |z(x)| and a phase angle p, z(x) = [z(x)| e iq o The discharge lags the head by cpq. By analyzing a piping system, starting with known impedances, say, at reservoirs, deadends, or orifices, the impedance at some desired point in the system may be calculated for a given u, when R, A, a, f, Q, n and q are known for each pipeo As a simple example, if a known sine wave variation of head is superposed on a steady flow at a point, and the discharge variation is desired at this point, the calculation of z = |zI e q yields the magnitude of discharge fluctuation. h' = Ah sin owt, q' A= q sin (ct - pq) (29) Ah Aq = IZI For more complex periodic head variations, the head is expressed by a Fourier series by use of harmonic analysis~ The impedance for each of

-7the frequencies is calculated and q' is then found by adding the harmonics taking phase into account, K q' = Z Aq sin (kt - Pk) (30) k=l in which there are K harmonics and X is the frequency of the fundamental period, Those frequencies yielding a high impedance are the resonant frequencies for the system.

IIIo EQUATIONS FOR IMPULSE TURBINE AND GOVERNOR SYSTEM Since the principal aim of this paper is to correctly handle the pipeline transients, the turbine equations are limited to the impulse impeller, and variations in electrical transients are neglected. Ao Turbine The power P produced by an impulse wheel is given by P = pqe (V - u) u(l - cos:) (31) in which p is the density, q the discharge, e the turbine efficiency, V the fluid velocity leaving the nozzle, u the peripheral speed of the centerline of the buckets and ~ the vane angle of the bucketso If the power absorbed by the generator is PG, then the equation for determination of speed change of the wheel is (from TN = INN, with N the wheel speed in radians per second, T the net torque, and I the moment of inertia of rotating masses) P - PG = INN (32) N is the angular acceleration. From nozzle dynamics V = CdJ, q = CdAn F, u = N r (33) in which Cd is the discharge coefficient of the nozzle, An is the nozzle area, h is the head at the base of the nozzle, and r is the radius of centerline of the buckets. By use of Equations (31) to (33) -8

-9P CdAn V2gh e(l - cos 5) N r (Cd 2gh - Nr) - PG INN (34) By assuming constant e for small discharge and speed changes (ioe. constant windage, bearing, splitter and friction losses), Equation (34) for steady operation becomes p CdAno 2gh e(l - cos 0) Nor (Cd Vgh - Nr) - PGo =0 (35) After dividing Equation (34) by Equation (35) An ~ N Cd 2gh - Nr P I 2 N N __ __ ______ = - - N0 -.. nho No -Cd Oa P Go PG o 0 o (36) Let the superscript indicate the dimensionless variable, eogo, A = An/An N* = N/No, N* = N/No For a small fluctuation AN = N - N, and AN* = N* - 1, etc. Equation (36) may be reduced to AA* + AN* 1 - + - Ah 1 + C -- APo L C V o - Nor JL CVgo -Nr o 37) I N2 N* N PGo Since N* remains close to unity, let N* = 1, and define Nor Cd V o C 1 = - - = 1 + Cd 2gh 0 - or2 Cd 2gho - N (38

-10then A* + C1 A* + C2 Ah* = C AN + APG (39) is the linearized equation for the impulse turbine, When changes in generator loading are not under consideration AP* = 1 G Bo Governor Equations The governing system of Borel,(3) shown schematically in Figure 1 is analyzed as a typical system with temporary droop provided by a dashpoto In Figure 2 the flyballs of the governor are assumed to pivot as shown, and friction and gravity loads, except on the flyballs, are neglected. Let N be the angular velocity of flyballs, then N = Cg N After taking moments about 0 2 N 2 m C R sin 9 cos = mg Rg sin Q (4) and cos = I (41) 2 2 gCg -N For steady-state conditions cos = 2 (42) gg 0 The displacement of governor collar from steady state position is 2gr (N+ No) 04 AN = 2rg (cos Q - cos 0) = 2 (N - N ) (4) RgCg 0N'

-11After linearizing C4 --- = --- cos 9 44) RgCg No N In Figure 3, let Y = displacement of spool valve from closed position (neglecting overlap); then Y c4 AN+ ( A - C4 AN) (45) P e1 +'~2 in which ANp is the displacement of dashpot piston from steady-state position, and i1,.e2 are the linkage lengths shown. To nondimensionalize this equation, first divide by RN, with R to be determined, Y C4,2 a ga^P y ~x Y - M A+. — RNO R (~ + Y2) (Q + 2?)Ro (46) To simplify the arithmetic, let 2C i2 g 1, N = No C4,AN R(^2^ Po -1 N (7) then Y** AN* + AN ( 48) p To analyze the dashpot behavior, Figure 4, the force F in the piston rod is given by F = Ks = (P2 Pl ) Ap (49

-12K is the spring constant, p1 and p2 the pressures in the cylinder, and A the piston areao The flow through the piston of the dashpot Qp is assumed to be laminar C (P2 - P )0t D4 Qp- - Ap (50) 128 -to D is the diameter of opening through the piston, io its length, 4 the viscosity, and C a correction for end effects due to the short tube flowo T is the piston position; then C K D4 N -N = s - (51) C P 128 () (51) p using Equations (49) and (50) and N = p - N Let p a 1.28 v4 A2, _ P (52) 3 CV K it D4 yielding ZNp - ANa= R3 (TC ~ p) (53) After dividing through by N to nondimensionalize Po AN* -AN -- R1 (LN N) (54) p 3 C p The displacement of spool valve is taken proportional to the flow through it, which is proportional to the rate of change of position X of the area control of the nozzle, X -X 1 Y* 0 d:* = - - Y- (55) dt X I R1

-13 - The nozzle area change is proportional to change in X for small displacements from steady state &A = r2 AX* (56) n The position of the dashpot cylinder is proportional to XM Figure 4i CN' r4 AX (57?) and the position Na of the permanent droop control, Figure 4, is given by AN = r4 AX*- AN* (58) a r4 m5 * in which ANm is the manual speed control. By elimination of Y ** A XNp, X N and in a u Equations(48). and (55) to (58) AnR+ R3r4) = r4 - A r.AN *+ r2 AN - R r2 A* + R3 A An 1 3 4 12'2 m 35 23 (59) In practice R1 << R3 r4, so the \A* term may be dropped. By defining 1 1 + R3 4 o, r2 r4 * n 2 n - [ r2 An + AN* + R3 AN - AN* (60) Equations (39) and (60) for turbine and governor may be solved by assuming a steady-oscillatory sine wave variation of all variables at the selected frequency c for the analysiso The following four equations

-14define the magnitude and phase relations among the variables: Ah* AH sin Dt (61) Aq* - AQ sin ((ot - cpq) (62) ZAN = a sin (at - cN) (63) AAn - aA sin (ct (PA ) (64) The substitution of Equations (61), (63), and (64) into Equations (39) and (60) permits the solution for AN/MAn,H/AAncpAand cppoThen by use of the nozzle equation from q = Cd An 2gh Aq* = MA* + 1 Ah* 2 permits AQ/nA and cp to be determined, A computer program has been writen to effect these solutions written to effect these solutions.

IV. ENERGY RELATIONSHIPS Whenever oscillatory flows and pressures exist in a system the energy input and output may differ from the steady state flow condition, If a net energy input exists due to the oscillations, that is, the energy in exceeds the energy out plus losses, the amplitudes will be magnified until an energy balance is reached or a failure occurs. This is the situation when resonance develops. Assuming the oscillatory motion superposed upon a steady state condition or upon a zero flow condition in a system, the energy being added during each wave period, T, is T f y hT q' dt 0 If sinusoidal waves are assumed the evaluation of this integral yields the energy in foot-pounds entering the system during each wave period. y h' ql T E = y ^ cos pq (65) 2 qO sji For any given magnitude of oscillation, h' and q', the phase angle, (pq can be seen to be the important controlling parameter At the input end of a system a phase angle in the first or fourth quadrant permits the addition of energy over the original steady flow condition. At the outflow terminal of a system a phase angle in the second or third quadrant allows a lower outflow of energy than the original steady flow condition. In each of these cases the energy level will be increasing (less losses) within the system if the other terminal of the system -15

-16maintains the same energy relationship as in the steady flow condition, as is commonly the situationo Using the impedance methods, phase angles can be evaluated for any particular system over a wide range of frequencies, The phase angle at the fundamental period of the exciting mechanism (or at the period of one of the predominate harmonics of the wave form) provides the necessary information to decide if an oscillating condition will develop or attenuate,

Vo PIPING SYSTEM EXAMPLES A. Waves on Reservoir A series penstock system leading from a reservoir to a turbine is shown in Figure 5. If the turbine valve is closed, it is possible for surface waves on the reservoir to set up pressure fluctuations in the pipe. Should the surface wave period be such that the corresponding phase angle from the piping system is at or near zero degrees, only a few consecutive cycles would be required to add enough energy to cause a severe oscillation, The development of pressure fluctuations at the closed valve in the system identified as system b in Figure 5 are shown in Figure 60 These results were obtained using the characteristics solution(8) of the equations for water hammer, Equations (1) and (2), on the digital computero A two-foot sinusoidal wave form was assumed at the reservoir surface. The wave period was selected to match the third harmonic of the piping system at which point the phase angle is zero, Figure 7 shows a plot of the pressure head transfer function, calculated using impedance methods, vso frequency, This transfer function is the ratio of the oscillatory head developed at the closed valve end to the oscillat ory head applied at the reservoir, The fundam-ataL, third, fifth and seventh harmonics are easily identified at the frequent cies associated with the large transfer function, -17

-18B. Governed Turbine With the turbine operating in the example shown in Figure 5, the piping system can be analyzed over a wide range of frequencies using the impedance concepts, yielding phase angle and impedance magnitudes at the valveo The governor can also be analyzed in order to evaluate the same parameters at the valveo If the impedance magnitudes are matched at a frequency where the phase angles of the two system components are in the second or third quadrant, an unstable operation will resulto In analyzing a given system with a particular governor setting, a graphical representation of phase angle and impedance modulus is helpfulo For the example under consideration, these are shown in Figure 8, The upper curves represent, pq, the phase angle between the pressure and discharge at the valve, plotted against frequencyo The lower curves show impedance modulus vso frequencyo In each section the solid line is information obtained from an analysis of the governor aloneo The dashed lines represent the results of the impedance analysis of the piping system only. Piping system b (Figure 5) represents a situation wherein the governor and complex piping system are matched in a stable zone of operation, That is, at the matched condition of impedance the governor phase angle is 285~ which indicates stability. On the other hand, at the matched impedance magnitude point of system a, the governor phase angle is 2570 indicating an instability. Figure 9 displays the unstable operation of system a, Figure 5o These data were obtained from a digital computer analysis which simultaneously

-19solved the linearized governor differential equations by Runge-Kutta Methods and the hydraulic transients in the piping system by the method of characteristics. This same turbine, valve and governor setting on piping system b, Figure 5, is shown to be stable as exhibited in Figure 10o The initial disturbance introduced to the system is shown to die out quickly on the computer analysis. The phase angle relationships at the valve can be observed in Figures 9 and 10. The unstable condition in Figure 9 shows Pq = 256~, while the stable condition in Figure 10 shows cp = 280~o Co Other Excitations A reciprocating pump produces a periodic pulse superposed upon the mean flow through the pump, This periodic motion can be analyzed by harmonic analysis and cpq can be determined from the system for each of the harmonics of the pump periodo Energy considerations show that, on the suction side of the pump, pq in the vicinity of 180~ at the predominate frequencies of the pump should be avoided. On the discharge side, pq in the vicinity of 0~ at these same frequencies should be avoidedo A comprehensive analysis of the entire system using impedance methods can be easily accomplished on the digital computero Any mechanism on the outflow end of a pipe line which will operate in a manner such that the head and discharge are approximately 180~ out of phase can cause an instabilityo This fact has been well illustrated by a leaky seal on a large penstock valve at a Power Planto(5)

-20In this case, on the high pressure side of the oscillation the leakage was shut off, at low pressure the seal opened allowing a higher flowo This provided an "ideal" exciter for a severe resonating conditiono

VIo CONCLUSIONS A method of stability analysis is presented for complex systems which are connected to various forms of exciters. The primary basis for the analysis is the impedance theory which is developed for viscous systems. Equations for the governed impulse wheel are also developed and placed in a form suitable for solution on the digital computero The solution of these equations over a wide frequency range along with the impedance equations from the piping system yields the necessary information for the stability criteriao Confirmation of the analysis is provided using the characteristic solution on the digital computero Energy considerations are discussed and related to the impedance calculations as they apply to the development of resonance in practical pipe line systems. The impedance theory enables the analyzer to take into account the complete detail of the piping system. Simplifying assumptions which approximate the system are no longer necessary. -21

REFERENCES lo Gaden, Daniel, Considerations sur le probleme de la stabilite des regulateurs de vitesse, Editions La Concorde, Lausanne, 1945o 2o Rich, Go Ro, Hydraulic Transients, McGraw-Hill Book Co., New York, 1951, ppo 47=92. 30 Borel, Lucien, Stabilite de reglage des installations hydroelectriques_ Pay of Lausanne, Dunod Paris, 1960. 4, Wylie, Eo B., "Resonance in Pressurized Piping Systems", Journal of Basic Engineering, Trans. ASME, Paper No. 65 - FE - 6, Applied Mechanics/Fluids Eng. Conf., Washington, D. Co, June 1965. 5o Wylie, E. Bo and Streeter, V. L., "Resonance in Bersimis No, 2 Piping System", Journal of Basic Engineering, Trans. ASME, Paper No. 65 - FE -10, Applied Mechanics/Fluids Eng. Confo, Washington, Do C., June 1965o 60 Moore, Ro Ko, Traveling-Wave Engineering, McGraw-Hill Book Co, New York, 1960, ppo 13-36. 7~ Waller, Eo Jo, Prediction of Pressure Surges in Pipeline by Theoretical and Experimental Methods, Publication No. 101, Oklahoma State University, Stillwater, June 1958. 8. Streeter, Vo L., "Water Hammer Analysis of Pipelines", Journal of the Hydraulics Division, ASCE, 90, No. HY4 (July 1964), 151-172. -22

LIST OF ILLUSTRATIONS Figure 1 Schematic View of Governor System. (Borel3) Figure 2 Flyball Governor Figure 3 Spool-Valve Linkage to Flyball and Dashpot Figure 4 Dashpot Assembly Figure 5 Series Piping System Figure 6 Pressure Oscillations at Closed Valve Figure 7 Transfer Function, Closed Series System Connected to a Reservoir Figure 8 Impedance Diagram for Stability Analysis Figure 9 Pressure Head and Discharge Oscillations for Unstable Governor Figure 10 Pressure Head and Discharge Oscillations for Stable Governor

0 FLYBALL HEAD SPOOL VALVE l _ -- F ~ DASH POT MANUAL SPEED SETTING o TO OPEN SERVO MOTOR Figure 1. Schematic View of Governor System. (Borel3)

-24e 22 rn_ mCg N2Rg sine mg C4AN Figure 2. Flyball Governor C4AN Y ANp I,'. I2 HIGH PRESSURE-'Fgure. Spool-Valve Linkage to Fl Figure 5. Spool-Valve Linkage to F~Sball and Dashpot,

-25Nc Fig 4r ~

RESERVOIR VALVE 4 2 " TURBINE Turbine & Governor Constants Pipe 4 3 2 1 a Diameter 5,. 4.5 4. 3.5 R1.06 C1 =.077 Wave Speed(ft/secj 3600. 3800. 4000. 4200. Ho-2590 ft. r21. C2 = 1.462 System a Length(ft) 1080. 1520. 1600. 1260. Qo=107.5 cfs r4=.02 C3 5.03 System b Length(ft) 900. 950. 1200. 1050. Speed: 600 rpm_2 r4=.2 I-19500 slug-ft R3 2. Figure 5. Series Piping System

140 " -- 80120 w w LLJ Lt. DP 60 0 2 4 6 8 10 12 14 TIME (SEC) Figure 6. Pressure Oscillations at Closed Valve

50 ___ --- -- z LJ w L, 30 CD z 4 Cr w I0 Cn Cn w 0 I 2 3 4 5 6 7 8 9 I I w (RADIANS/ SEC) Figure 7. Transfer Function, Closed Series System Connected to a Reservoir

-29-- 360 c GOVERNOR i 270 NS ABLE' /Zp^^F/, -J Z 180 4<~. PIPING SYSTEM a u so L j PIPING S YSTEM a4 90 I I (n 1i / tOVSRNOR 5v-. l il i i o 30 z FREQUENCY, r (RAD/SEC) Figure 8. pedance Diagram for Stability Analysis LLJ o I / 101 0ERIES PIPING SYSTEM b 0.0 0. 1 15 20 2.5 3.0 nKFREQUENCY I w (RAD/SEC), Figure 8. Impedance Diagram 1or Stability Analysis

25.0 10 #q q256* q'' / - 12.5 /,'. / r \ I -, O 0.L s' 0 o Fw ~ ~ ~~~~~~~~~~~~~~~~~~~- 0 Z \ t I IJ Co -12.5 -5 | j) h -25.0 ________ ____________________ ____________________-10 0 5 10 15 20 25 30 35 TIME (SEC) Figure 9. Pressure Head and Discharge Oscillations for Unstable Governor

100 40 50I I 2 I 0 0' 0 0w ~~~~~~~~~~~~0 IC wH 50 —-— 0 ___2 -100i ur _ _ _ _ __ur H ad an _ _ _ _ _ _rg Os i40ton 0 5 10 Gov 20no,- v, eo~ r\ ~ ~ IM (SEC)0 c-~~~~~~o Stbl Goero s\' I -1 0______________________ 4 D ~ 0 10 152 TIME (SEC)~~ Fiur 10 rsueHa.ad DshreOclai fo Stbe oero