3498 -1 -F ANALOG COMPUTER INVESTIGATION OF A BOUNDARY - LAYER- CONTROL SYSTEM MARGARET M. SPENCER PAUL S. FANCHER January 1960 ANALOG COMPUTER LABORATORY THE UNIVERSITY OF MICHIGAN Ann Arbor, Michigan

The work described herein was done for Continental Aviation and Engineering Corporation under Project No. 026-03457. This project was established under an Air Force prime contract, AF 33(600)38666. This report was written under Continental Aviation and Engineering Corporation Project No. 045-03498.

The University of Michigan Willow Run Laboratories 3498-1 -F CONTENTS List of Figures and Tables.................... v List of Symbols.......................... vii A bstract..................... 1 1. Introduction and Conclusions.................. 1 1. 1. Introduction 1 1. 2. Conclusions 2 2. The BLC System.......................... 3 2. 1. Description of the BLC System 3 2. 2. Derivation of Equations 6 2. 2. 1. Compressor 6 2. 2. 2. Rotor Dynamics 7 2. 2.3. Surge Sensor 8 2.2.4. Blow-Off Valve 10 2, 2. 5. Duct 14 2.2. 6. Manifold 14 3. Analog Computer Mechanization of Linearized Equations........ 15 3. 1. Equations Mechanized for Computer 15 3. 2. Definitions of Coefficients 16 3. 3. Computer Mechanization 19 4. Computer Data.................22 4. 1. Case 1 Without Damping 22 4. 2. Case 2 Without Damping 28 4. 3. Case 3 Without Damping 28 4. 4. Case 1 With Damping 28 4. 5. Case 2 With Damping 28 4. 6. Case 2 Without Damping, Spring Removed from Surge Sensor 30 4. 7. Cases 1 and 2 With Piston 30 5. Analog Computer Mechanization of Nonlinear Equations.......43 Appendix A: Stability of the Blow-Off Valve............... 44 Appendix B: Digital Computation of Operating Points........... 48 Appendix C: Simulation of a Long Manifold............. 50 R eferences.............................. 54 Distribution List.......................... 55 iii

The University of Michigan Willow Run Laboratories 3498-1 - F FIGURES 1. BLC System.................................. 4 2. Single-Compressor BLC System................... 4 3. Schematic Diagram of Anti-Surge Control.............. 5 4. Coulomb Friction....................... 13 5. Linear Simulation......................... 20 6. Computer Mechanization of the Linear Equations......... 21 7. Case 1 Without Damping...................... 23 8. Cases 2 and 3 Without Damping............... 29 9. Case 1 With Damping....................... 31 10. Case 2 With Damping................. 33 11. Effects of Springs on System Operation for Case 2 Without Damping... 34 12. Case 1 With Piston................. 36 13. Case 1 With Piston.......................... 37 14. Case 2 With Piston..................... 38 15. Case 2 With Piston........................ 39 16. Case 2 With Piston.................. 40 17. Computer Mechanization of the Nonlinear Equations......... 41 18. Development of a Network Analog for the Manifold........ 52 TABLES I. Steady-State Values for Parameters................. 24 II. Coefficient Values for Parameters........... 25 III. Computer Runs of the System With Piston.............. 35 IV. List of Analog Equipment............... 43 V. Operating Points Computed by Digital Computer.......... 50 V

The University of Michigan Willow Run Laboratories 3498-1 - F SYMBOLS Symbol Definition Units A Effective area of the leakage path in. 2 L A Combined area of all exhaust orifices in. 2 M A Cross-sectional area of the duct in. 2 3 B Magnitude of coulomb friction lb F Spring force when x is zero lb H Spring force when y is zero lb HPT Horsepower from the power turbine to drive the compressor Hp Kf Viscous damping coefficient lb-sec/in. N Compressor rotor speed RPM P Total compressor inlet pressure, pounds per square psia inch, absolute IP The compressor inlet static pressure and the total psia pressure in chamber 2 P Total compressor outlet pressure psia PA Total pressure in chamber A psia P Total pressure in chamber A psia A P Total pressure in chamber B psia P^ Total pressure in chamber C psia P Total pressure in the manifold psia T Temperature of the atmosphere ~K 1 mT3 Temperature of the compressor outlet air OK T Temperature in chamber A OK TB T Temperature in chamber B OK B T Temperature in chamber C OK C T Temperature in the manifold OK V Volume of chamber A in. 3 V Volume of chamber B in. 3 B V Volume of chamber C in. 3 C W Air-flow rate out of the compressor lb/sec Wbi Air-flow rate into chamber B lb/sec hbi W Air-flow rate out of chamber B to the atmosphere lb/sec W.i Air-flow rate through the needle valve into chamber C lb/sec W Air-flow rate out of chamber C lb/sec co W Air-flow rate past the piston lb/sec vii

The University of Michigan Willow Run Laboratories 3498-1 -F SYMBOLS (Continued) Symbol Definition Units W. Air-flow rate into chamber A lb/sec W Air-flow rate through exhaust nozzles in the lb/sec manifold to the atmosphere WL Air-flow rate through the duct to the manifold lb/sec WV Air-flow rate through the blow-off valve lb/sec x Displacement of the surge sensor poppet ball from its seat in. y Displacement of the blow-off valve cover or the opening of in. the needle valve T 0 Referred temperature, 5184 P 6 Referred total pressure, 4 7 14. 7 CONSTANTS Symbol Definition Value C Flow coefficient for above critical flow 0. 8 c Specific heat at constant pressure 0. 2417 Btu/lb~ - R f 3l f32 f3v Flow parameters for above critical flow V/O0R f, f sec 1' 3p g Gravitational constant 386.4 in. /sec2 J Mechanical equivalent of heat 778 ft-lb/Btu K Experimental constant for the air-flow rate ( 343)10-3 lb/sec a (0. 343)103/lee relation for chamber A -n 2 6 Ilb-in. K6 Experimental constant for the relation (4 08) (10-4) s2/lb2 between P and, (4.08) (10 ) sec /lb y6 between P2 and W a 2t a R Gas constant 639. 6 in. /OR 7T 3. 1416 DIMENSIONS Symbol Definition Value A Area of the diaphragm between chamber A and al 9 chamber 2 7.06 in. 2 A Area of the diaphragm between chamber B and chamber 2 0. 601 in. 2 viii

The University of Michigan Willow Run Laboratories 3498- 1 - F SYMBOLS (Continued) Symbol Definition Value A Area of the diaphragm at the bottom of chamber C 40. 72 in. 2 cl A Area of the cover of the blow-off valve 38.48 in. c2 A Cross-sectional area of the duct 44. 18 in. 2 D A. Area of the inlet orifice in chamber B 0. 00196 in. 2 1 A Area of the outlet orifices in chamber B 0. 003236 in. 2 o A Area of the poppet exposed to pressure from chamber C 0. 0768 in. 2 A Area for flow from chamber C to the atmosphere 0. 694x in. 2 A Area of the opening of the needle valve to chamber C 0. 0387y in. 2 D Diameter of blow-off valve cover 7. 0 in. c2 I Moment of inertia of rotor 0. 1805 ft-lb-sec2 K Spring constant in surge sensor 6 lb/in. x K Spring constant in blow-off valve 32.8 lb/in. m Mass of moving part of surge sensor (3. 675)(10 ) lb-se X in. I lb-sec2 m Mass of moving part of blow-off valve (7. 169)(10-3) lby in. v Volume of chamber A with poppet valve closed 1. 83 in. 3 AO V Volume of chamber B with poppet valve closed 0. 196 in. 3 BO V Volume of chamber C with the blow-off valve closed 58.43 in. V Volume of the manifold 15321. 15 in..-..... S' +100 y XY H 10_0 X nput A / / / / -100 J Servo Multiplier ix

The University of Michigan Willow Run Laboratories 3498-1 - F X Y- W -(X + 4Y + 10Z) Z Summing Amplifier Amplifier with Feedback Removed Y- 4 > -...... W: - / (X + 4Y + 4Z)dt Integrator X a....Y Y: aX Scale-Factor Potentiometer Function Generator X P X Input 1 Axis l W_ 2 Y Input 2 Y Axis T ~_~~ ~2 T5 - T Z Probe 3 -2 Z T2 Output Dual-Input Function Generator Z = f(X, Y) X

3498-1 - F Analog Computer Investigation of a Boundary- Layer- Control System ABSTRACT A limited study using an analog computer was performed to investigate the stability of a pneumatic system consisting of a gas-turbine-driven compressor operating into a manifold from which air was escaping through a number of orifices. Simulation of linearized equations which describe the transient operation of the system for small departures from steady-state operating conditions was completed, and results of this simulation are presented. The results from this incomplete study show that the system has strong tendencies toward instability. A circuit for simulating the general nonlinear equations for the system is also presented. The linear and nonlinear simulations are compared. Methods of simulating a system consisting of a long manifold with one compressor at each end are discussed. 1 INTRODUCTION and CONCLUSIONS 1.1. INTRODUCTION This is a final report on a study conducted by the Analog Computer Laboratory of Willow Run Laboratories of The University of Michigan for CAE (Continental Aviation and Engineering Corporation) under contract to the Air Force. The purpose of the study was to investigate, with an analog computer, the stability of a BLC (boundary-layer-control) system which was designed to change the flight characteristics of an aircraft by exhausting air through small holes distributed along the surface of the wing, thus disrupting the boundary layer of air at the wing's surface. The system consisted of a cylindrical manifold (90 feet long and 8. 5 inches in diameter) which received air from two gas-turbine-driven compressors at each end and exhausted air through holes uniformly distributed along its length. Each of the compressors was fitted with an anti-surge control which opened a blow-off valve and thus increased the air flow from that particular compressor when its operation approached surge conditions. The ultimate objective of the study was to simulate the entire system consisting of the manifold and four compressors, and to include the effects of pressure variations along the length of the manifold. The study was not, however, completed. 1

The University of Michigan Willow Run Laboratories 3498-1 - F Initially, the analog computer was used to solve the linearized equations for one compressor, its anti-surge control, the duct, and the manifold. This preliminary investigation included the dynamics of the compressor and the inertial effects due to the mass of the moving elements of the controls. Since this resulted in a ninth-order system, a theoretical investigation would be quite difficult. This simulation was valid for small variations of variables from a steady-state operating point. The computer results are included in the report. The digital program for obtaining steady-state points and the computer potentiometer settings is included in Appendix B. The second phase of the study was to have been to simulate the nonlinear equations for the system. Termination of the prime contract by the Air Force resulted in cancellation of this project; hence ultimate objectives of the contract were not fulfilled. A new contract was obtained from Continental Aviation and Engineering Corporation to write and publish this report. 1 2. CONCLUSIONS Since the computer circuit used linearized equations which were valid for only very small changes of variables from a specific steady-state operating point, the results from the computer cannot give a complete description of the system's operation. Instead, the computer results merely indicate whether or not the system is stable at each operating point which was examined. When the system itself is in operation, an instability at some point causes the operation to shift to another point, which may be either more stable or less stable than the previous point. The nonlinear system could be regarded as a linear system in which the coefficients of the equations are continually changing and are functions of the instantaneous values of certain variables of the system. The linearized equations-are quite useful for a preliminary analog computer investigation of a system, since their mechanization on a computer requires a minimum of equipment; none of the less-accurate nonlinear computer components, such as multipliers and function generators, is used; and the linearized equations are less difficult to manipulate if it becomes desirable to verify analytically the validity of some unexpected result from the computer. Changing of the steady-state operating point for the computer mechanization requires a considerable amount of tedious hand computation to find the new steady-state values of variables and the potentiometer settings, although this part of the work may be performed quite rapidly on a digital computer. A simulation of the general equations would be needed in order to completely explore the behavior of the system. This would permit the entire operating range of the system to be ex2

The University of Michigan Willow Run Laboratories 3498-1 - F plored for regions of unsatisfactory operation, and would thus provide the system designer with a rather complete picture of its characteristics. The following conclusions concerning operating characteristics of the system are based upon results obtained from simulating the linearized equations. The simplified system, consisting of one compressor and anti-surge control operating into a manifold, is unstable when the blow-off valve is well open. This instability seems to be caused by similarities in magnitudes of time constants of the compressor, rotor, the manifold volume, and the control rate between the surge sensor and blow-off valve. If the manifold volume is increased the tendency toward instability is reduced. Also, the system becomes slightly more stable when the blow-off valve is nearly closed, probably because the time constant between the surge sensor and blow-off valve is increased under these conditions. The blow-off valve itself has a tendency to oscillate at a frequency much higher than the natural system frequency. This tendency is strongest when the valve is nearly closed. This instability of the blow-off valve could be eliminated by installing a suitable viscous damping device, such as a dash pot, on the moving part of the blow-off valve. Possibilities for unstable operation would be enormously increased if four compressors, each with an anti-surge control, were connected to a common manifold, because such an arrangement has many more feedback loops than the simplified system. Although the simulation indicated that the system as described in this report is somewhat unstable, it is felt that the system instability may be eliminated by changing one of the three time constants mentioned earlier. Further computer investigation would probably show what changes should be made to make the system operation completely satisfactory. 2 THE BLC SYSTEM 2. 1. DESCRIPTION OF THE BLC SYSTEM Figure 1, which is reproduced from a drawing supplied by CAE, is a schematic diagram of the BLC system. The system consists of four gas-turbine-driven compressors which are forcing air into a single long cylindrical manifold. Air escapes from the manifold through a number of orifices distributed along its length. The duct from each compressor to the manifold contains a check valve to prevent air from flowing from the manifold to the compressor. Each compressor is equipped with an anti-surge system (not shown in Figure 1). The system which was simulated is shown schematically in Figure 2. It consists of a single compressor and anti-surge control, working into a manifold having one-fourth of the volume of the manifold shown in Figure 1. 3

The University of Michigan Willow Run Laboratories 3498-1 - F l~ —-------— 90, — o" —.O-..... —90 0 Orifices 4t I ^ V - __ /__ Manifold Check Valves in Duct O \ *>~~~~~ / ^^''^ yCompressor with..' J ( # L^^^ 2 Compressors 2 Compressors FIGURE 1. BLC SYSTEM =12^ —------ 2 —22' 6" r- Orifices Manifold _ Check Valve'~y~ Compressor, >/,Sr Surge Blow-offf I Sensor Valve-7 I Anti-Surge Control FIGURE 2. SINGLE-COMPRESSOR BLC SYSTEM 4

The University of Michigan Willow Run Laboratories 3498-1 -F The anti-surge control consists of the surge sensor and the blow-off valve. The surge sensor is a pneumatic device which monitors the compressor air-flow rate and outlet pressure. When the compressor air flow decreases or compressor outlet pressure increases sufficiently to cause the compressor operation to approach surge conditions, the surge sensor causes the blow-off valve to open. This permits air from the compressor to escape through the blow-off valve, thus increasing the total air flow and removing the danger of surge. A mechanical schematic diagram of the surge sensor and blow-off valve is shown in Figure 3. The difference between total pressure (P1) and static pressure (P ) at the compressor outlet is a function of air-flow rate from the compressor. This pressure difference applied across the diaphragm between chamber A and chamber 2 of the surge sensor causes a downward force to be exerted on the poppet valve at the bottom of the surge sensor. The pressure (^ -q_^ Ai4i LOW- OFF VVE FIGURE 3. SCHEMATIC DIAGRAM OF ANTI-SURGE CONTROL 5 FIGURE 3. SCHEMATIC DIAGRAM/ OF ANTI-SURGE CONTROL

The University of Michigan Willow Run Laboratories 3498-1 - F which is in chamber B is proportional to the compressor outlet pressure, and exerts an upward force on the valve. Whenever a stall condition is approached, the air flow from the compressor decreases, thereby causing a decrease in downward force on the poppet valve, and the compressor outlet pressure increases, thus causing an increase in upward force. This change of forces causes the valve to open, thus permitting air to escape from chamber C of the blow-off valve. The resulting drop in PC permits the upward force on the valve due to P3 to open the valve and permit air to escape. The upward motion of the blow-off valve also opens the needle valve at its base, and thus permits air to flow into chamber C. For a given opening of the surge-sensor poppet valve, the blow-off valve always assumes a steady-state position such that the air-flow rate into chamber C through the needle valve just equals that out through the poppet valve. When the poppet valve closes, the pressure in chamber C rises sufficiently to close the blow-off valve and needle valve. Air passing through small orifices in chambers A and B of the surge sensor restricts the flow rates and thus improves the stability of the surge sensor by introducing damping effects. 2.2. DERIVATION OF EQUATIONS Mathematical equations which describe the operation of the system are developed in this section. For each major component of the system, first the general nonlinear equations are written, and then the linearized equations are derived. The linearized equations are specialized equations which describe the relationships between changes in system variables from their steadystate values. These equations are valid only for very small changes of variables, and are obtained by the process illustrated in this section. 2.2. 1. COMPRESSOR. In functional notation, the compressor pressure ratio (P /P1) may be expressed as a function of two variables, the corrected rotor speed (N/1f) and the corrected air-weight flow rate (W V1//6): a 3 f N a P 1 6 P. ( N We) (1NL) T3 T1 = f2' 6' tion of the same two variables: 3T l =2(1/ W6 ) (2NL) 6

The University of Michigan Willow Run Laboratories 3498-1 - F P T -T 3 3 3 1 Writing the total differential for - in Equation 1NL, and for T in Equation 2NL gives 1 1 P P a_3 3 3 X W= }N + A(WN N+a N (1L) T3-T T-T T T T 3 1 1 z W. (2L) A — = AN + AW (2L) T1 N a a a The operating characteristics of the centrifugal compressor which is used in the system were obtained from a "map" which had P3/P1 for its ordinate, W O//6 for its abscissa, and contained lines of constant N/J/1, and lines of constant efficiency. Using the Keenan and Kaye gas tables (Reference 1), the air temperature rise due to compression was computed from pressure ratio and compressor efficiency, as obtained from the map. The resulting data was then cross -plotted in a form which permitted graphical determination of the values of the partial derivatives in Equations 1L and 2L. 2. 2. 2. ROTOR DYNAMICS. The rate of kinetic energy storage in the compressor rotor is equal to the difference between power supplied by the turbine and power required by the compressor. In equation form 550 HpT - JcpWa(T3 T1) = INN (3NL) where HpT is the horsepower supplied by the turbine (assumed constant) J is the mechanical equivalent of heat (778 ft-lb/Btu) c is the specific heat of air I is the moment of inertia of the combined compressor turbine rotor N is the rotor speed (revolutions per minute) The linearized form of Equation 3NL is -Jc (T - T ) AW JcW AT3 I (NAN + NAN) p 3 1 a p a 3 \60/ Since for steady-state conditions N = 0, the term NAN may be eliminated, giving 27rn2 INAN =-Jc (T - T)AW - Jc W AT (3L) (27r)P 3 1 a p a 3' 7

The University of Michigan Willow Run Laboratories 3498-1 - F 2. 2. 3. SURGE SENSOR.1 The anti-surge control uses the compressor air-weight flow and outlet pressure to control the position of the blow-off valve. The blow-off valve may be operated directly by compressor-outlet pressure, or it may be controlled by the surge sensor, which is controlled by compressor air-flow rate and compressor-outlet pressure. Referring to Figure 3, the surge sensor contains chambers 2, A, and B, diaphragms Aal and A 2, orifices A, A., and A, a spring, and a single poppet valve. P2 is the compressora2' 1 0 O inlet static pressure. P is the compressor-inlet total pressure. The pressure in chamber 2 of the surge sensor is P2, the compressor-inlet static pressure, and thus is dependent on the rate of the air-weight flow through the compressor. If the air-weight flow decreases, the pressure in chamber 2 increases. Since diaphragm A is larger than diaphragm A 2, the force resulting from the pressure in chamber 2 is such that it tries to open the poppet valve. The inlet and outlet orifices of chamber B are adjusted in the cross-sectional area so that the pressure in chamber B is proportional to P, the compressor outlet pressure. The resulting force on diaphragm Aa2 tends to open the poppet ball. Also, the force from the pressure in chamber C of the blow-off valve is pushing on the bottom of the poppet ball. There are two forces which tend to close the poppet valve. One force is proportional to the amount the spring is displaced from its equilibrium position. The other downward force results from the pressure in chamber A (the compressor-inlet total pressure, P ) exerted on diaphragm A1 The rates of pressure change in chambers A and B are restricted by placing small orifices in the paths of air passing into and out of these chambers. This introduces damping and thus reduces the tendency of the surge sensor to oscillate. When the poppet is open, air flows from chamber C in the blow-off valve through the poppet to the atmosphere by way of an opening in the side of the surge sensor. If the probes which sense static and total pressures at the compressor inlet are properly adjusted, P 1- P W va0 1- 2 K (K a ). (4NL) Assuming laminar flow through orifice A, W = K (P - PA). Combining this with the ai a 1 A general gas-law equation for chamber A gives RT = W aidt= Ka(P1 - PA)dt. (5NL) Most of the equations and parameters which are used in this section were obtained from a Cosmodyne Corporation report (Reference 2). 8

The University of Michigan Willow Run Laboratories 3498-1 - F For the volume of chamber A, VA = VAO- Aal (6NL) where VAO is the volume of chamber A with the poppet valve closed, and x is the displacement of the poppet ball off its seat. The flow from the compressor outlet into chamber B through orifice A. is given by the equation Cf31AiP3 TW bi -1 i 3 (7NL) B where fg is the flow parameter C is the flow coefficient A. is the cross-sectional area of orifice A. 1i 1 T is the temperature of the air in chamber B B The flow from chamber B to the atmosphere is through two orifices labeled A. The combined cross-sectional area of these two orifices is A, and the flow is given by Cf APB w = 320 B (8NL) boyr The volume of chamber B is V-, = VBO + A a2x (9NL) B BO a2 (9NL) where V is the volume of chamber B with the poppet valve closed, and x is the same as for BuO Equation 6NL. The gas-law equation for chamber B is P V PBVB = f (10NL) RT BB bi- Wbo )dt. (1NL) The force equation for the surge sensor, for x > 0, is written by setting the sum of the upward forces equal to the reaction force due to acceleration: -F +A (P - P ) +Aa P +(A - A 2)P2 - A P -K x= m x, (1NL) o p C 1 a2B al a2 2 alA x x where F is the spring force when x is zero A is effective area of the poppet valve p K is the spring constant m is the mass of the moving part of the surge sensor x 9

The University of Michigan Willow Run Laboratories 3498-1 - F Equations 4NL through 11NL may be linearized by writing the total differentials as was done for the compressor. These linearized equations are listed below. AP 2KW 2 a (4L) - OAW (4L) P 2 a 1 6 VAAPA PAAV RTA RT A a d (5L) A A AV -A = Ax (6L) A al AW = 31 AP + i iAf. (7L) hi TAr 31 b B v B Cf A CA P AW - A B- + Af3 (8L) bo F AB 3 2 AVB = A a2Ax. (9L) V AP PBAV T + RT = (AWbi - AWbo)dt. (10L) B RTB bi /Wbo)dt. (1OL) Bo B A PC +A AP + (Aa -A )AP - A AP -K Axx = m Ax. (11L) p C a2 B al a2 2 al A x x 2. 2. 4. BLOW-OFF VALVE. Functional components of the blow-off valve include a spring, chamber C, diaphragm A' a needle valve which allows air at P to flow into chamber C, and a large valve of diameter D 2 and area A2 which can release compressor-outlet air c2 c2 to the atmosphere. The downward force exerted by the pressure in chamber C acting on diaphragm Al tends to close the blow-off valve. The spring applies a downward force which is directly proportional to its displacement from its equilibrium position. Thus, the spring force also tends to close the blow-off valve. The upward forces on the valve are caused by P3, the compressoroutlet pressure, acting on the bottom of A 2' and P acting on the bottom of diaphragm A 1 c2' 1 c1 If the pressure in chamber C is reduced sufficiently due to air flow out of this chamber through the poppet valve in the surge sensor, the forces holding the blow-off valve closed will be decreased, and the valve will open. Excess compressor air flow is then bled through the valve, permitting the compressor air-flow rate to increase, and preventing the compressor's 10

The University of Michigan Willow Run Laboratories 3498-1 - F operating point from crossing the surge line on the compressor map. The opening of the blowoff valve also opens a needle valve which permits air from the compressor outlet (at pressure P3) to flow into chamber C. This permits the air pressure in C to build up and the blow-off valve to close whenever the poppet valve in the surge sensor closes. With the poppet valve and the blow-off valve closed, the three factors which determine when the anti-surge control begins to operate are the forces from the springs, the calibration of the sensing probes, and the adjustment of orifice A in chamber B. The bleed air-weight flow to the atmosphere through the blow-off valve is given by the equation CP 3f3vTD y W = 3v (12NL) v VT3 where y is the displacement of the large valve from its seat rD c2y is the area of the opening of the large valve f3v is the flow parameter If the flow through the needle valve from the compressor to chamber C is assumed to be incompressible, then the flow rate is given by dA /2gP w=Ci- 3(PS-P), (13NL) ci dY (y RT3 (P3 C) (13NL) dA where A = Y is the area of the needle valve opening, as a function of y. Y dy The air flow out of chamber C enters the surge sensor through an area A in the poppet x valve. The flow rate is given by Cf33PCA Wc- C- (14NL) where f33 is the flow parameter. An approximation of the area is A = 0. 694x. (15NL) x The volume of chamber C is given by VC = CO Ac (16NL) where V is the volume of chamber C with the blow-off valve closed. ~~~~~~~~~~~~CO ~~ 1

The University of Michigan Willow Run Laboratories 3498-1 - F Applying the gas-law equation to the air in chamber C gives PcVC = (Wci )dt, (1 7NL) RT c co where Wci is the rate of air flow into the chamber through the needle valve W is the rate of air flow out through the poppet valve in the surge sensor co The expression for a force balance in the blow-off valve for y > 0 is -H+ (P3 - P)Ac2 -(PC - P)Ac -Ky = my (18NL) where H is the spring force when y is zero K is the spring constant m is the mass of the moving part of the blow-off valve y If a piston is used instead of diaphragm A 1 three changes in the equations take place. First, mass m is increased; second, there is leakage from chamber C to the atmosphere around the piston; and third the force-balance equation must include a term to represent the friction between the piston and the cylinder wall. (See the next two sections for Equations 19NL through 22NL. ) The leakage air-flow rate for the piston is given by Cf P A W c 3pC L (23NL) cp fT~C where fp is the flow parameter AL is the effective area of the leakage path The gas-law equation for chamber C is modified by adding the leakage term, W: cp PCV C (W - W - W ) dt. (24NL) RT J ci co cp The friction force between the piston and cylinder wall is a constant which changes sign when the piston reverses its direction of motion, as shown in Figure 4. This coulomb friction force of magnitude B may be represented mathematically by the expression -B, and adding this force to the force-balance equation for the blow-off valve gives -H 2 +( P-)Ac - = y + IIB (25NL) 12 B. (25NL) 12

The University of Michigan Willow Run Laboratories 34981- -F +B -B FIGURE 4. COULOMB FRICTION Equations 12NL through 18NL and Equations 23NL and 24NL can be linearized to give the following equations. /f3v 7rD AWv = ((YPP3 + P3Ay). (12L) dA / Y(2P 3 - P)AP3 P 2 A W - /C2 + - P P y (13 A C dy 2g 2 C 3 3 C -2P\ P 23 P -P2C1 PC 3 3 - 3 C Cf3 CPcAX Ao V(AXPC + P + A Af3 (14L) V C C x T 33' C VTC AA = 0. 694Ax. (15L) x AVC= - AclY. (16L) APcVC P CC + C AV =r(AW i.AW )dt. (17L) RTC CRT C Rci co c2AP3 - Ac AP - Kyy = myAy. (18L) Cf3 AAPC 3p L C (23L) Tcp APCVC PcVc RT —+ = (AW - W - AW ). (24L) KRTC R J ci co cp 4L) 13

The University of Michigan Willow Run Laboratories 3498-1 -F 2. 2. 5. DUCT. Air from the compressor outlet has four possible exits. Most of the air flow is through the duct into the manifold and through the blow-off valve, but some also flows into chamber B of the surge sensor and chamber C of the blow-off valve. Thus, the total airflow rate (W ) is the sum of these four separate air-flow rates. In equation form, a W = W + WV +W +W.. (19NL) a L V bi ci Since the pressure drop across the duct is quite small, the incompressible-flow equation may be used: WL =CA3 RT V 3(3 M (20NL) 3 and WL = 0 for (P3 - PM) < 0, (check valve closed) where A is the cross-sectional area of the duct M is the manifold pressure Equations 19NL and 20NL may be linearized to give AW = AWLA + A W AWbi + AW.. (19L) a L V bi ci CA 23 AWL 2 /P ( P3 [(2P3 -PM)P3 -P3AP (20L) 2. 2. 6. MANIFOLD. The manifold pressure, PM, is always very much greater than atmospheric pressure, P1, so the flow parameter used in the equation which describes air flow from the manifold through the exhaust orifices is a constant. The flow equation is flCA PM 1W = M (21NL) o TM where f is the flow parameter A is the combined area of all the exhaust nozzles The gas-law equation for the manifold is PMVM = (WL - W)dt. (22NL) RTM 14

The University of Michigan Willow Run Laboratories 3498-1 - F The volume of the manifold for the case of one compressor was taken to be one-fourth the volume of the manifold shown in Figure 1. The manifold equations can be linearized to give f CA AW AP (21L) and APM V R M: (AWL - AW ) dt. (22L) RTM J L 0 3 ANALOG COMPUTER MECHANIZATION of LINEARIZED EQUATIONS Linearized Equations 1Lthrough 22L, Section 2.2. 1 through 2. 2. 6, are, in principle, the equations solved by the analog computer to obtain a linearized simulation of the system. The actual equations which were mechanized are listed in this section. The assumptions which are applicable to each equation are also stated. The constant coefficients are designated as a's with subscripts; their definitions are given in Section 3. 2. 3. 1. EQUATIONS MECHANIZED FOR COMPUTER The number or numbers in parentheses to the left of each equation indicate which equations, from Sections 2. 1 through 2. 6, are used. Compressor and Rotor Dynamics (1L) AP3 = CAN + c2AWa (1C) where a > 0 and a2 > 0. (2L, 3L) AN =-a - AW - AN. (2C) 3 a 42 Surge Sensor In chamber A, assume TA = T1. (5L, 6L) AP A = AX - AP (3C) 15

The University of Michigan Willow Run Laboratories 3498-1 - F In chamber B, assume that T = T3 and that f31 and f32 are constants (choke flow through orifices A. and A ). 1 o (7L, 8L, 9L, 10L) APB 7 7AP3 8 P B - 9 (4C) The force-balance equation for the surge sensor is (llL, 4L) m Ax = A APC + A aAP - 10AW - A AP - K Ax. (5C) x p C a2 B 10 a al A x Blow-Off Value In chamber C, assume that TC = T3 and that f33 is a constant. (13L, 14L, 15L, 16L, 17L) APC =11 3 + a12 - 13AP - "14A + 15 Y (6C) The force-balance equation is (18L) mAy= =A A PcP - A P - KAy. (7C) y c2 3 ci C y Duct and Manifold Assume T = T3. (7L, 12L, 13L, 19L, 20L) a 16AP3 17 PC +18 Y -al9 PM (8C) (20L, 21L, 22L) APM = a20AP - 21 AP (9C) 3. 2. DEFINITIONS OF COEFFICIENTS Each a corresponds to a potentiometer in the computer mechanization, and the value of each a at a steady-state operating point determines the setting of the corresponding potentiometer. 16

The University of Michigan Willow Run Laboratories 3498-1 -F al ~ )N W P3 aPa 1 8W aN1 a3 Ta - T1 Jc aJ 3 - T1 + T W 7 N a \60/ "' a T3 - T1 JcW T T p aP L\ i 1 4 7r2 N wa PAAal AA al 5 VA K RT a 1 6 VA Cf 31AiRVT3 7 V Cf32 AoRV 3 8= VB A P a2 B 9 V 2KP W 0 10 = 2 (Aal- Aa2) 17

The University of Michigan Willow Run Laboratories 3498- - F dA Cy - y2 gRT (2P - P dA c dy VgT3P3( PC) 1 2 VC ~~~dA dy 3 33 C Cf33AxRV Cy dy 32gRT3 a13 Vc 2VcVP3(P - ~Cf 33 33PC C -J VC dA Cy 31 CrDc2f3 dyRT3 -a = ++ (2P -P c26 3+3 3 c'i CA3 2/ (2P3 - P 2 P3 (P3 -PM) dA y 2Yg dy RT3 P c17 ~1PP 3 23 3 C C 7rDf P dA c2 3 3 _ p 3 CA3 2RTg a19 P(P -P P 2y3p3 M3 18

The University of Michigan Willow Run Laboratories 3498- - F CA3 VgRT3 a20 2VMVP3(P3 P) (23 PM CAP3/2gRT Rl3 flCAM 21 + ~21 2VM33(P3 -3 PM) V M 33 M M 3.3. COMPUTER MECHANIZATION Figure 5 is a block diagram of the linear simulation. Each block represents a separate component of the system, as indicated, and contains equations which show the mathematical relationships between its input and output variables. The information flow shows exchange of AP3 and AW information between the compressor and the duct. The surge sensor receives 3 a AP3 from the compressor, AW from the duct, and APC from the blow-off valve. It sends Ax 3 a C information to the latter. The blow-off valve also receives AP3 information and transmits AP, and Ay to the duct and manifold block. C An unscaled diagram of the computer mechanization which solves the equations in each block is shown in Figure 6. The symbols representing analog computer components are defined at the front of this report. The computer diagram is labeled to correspond to the blocks. One equation is not mechanized as specified, it is Equation 8C in the duct and manifold. The first term of this equation is a16AP3, which is obtained by substituting Equation 1C from the compressor and rotor dynamics. Therefore, AN is fed to a potentiometer set at (a 16)(a1) and, likewise, AW is fed to a potentiometer set at (a2)(a 16). a 216 The mechanization uses 9 integrators, 9 summing amplifiers, and 31 coefficient potentiometers. Amplifier A and integrator B with potentiometers a1, -a2, a3, and a4 solve equations which describe the compressor and rotor dynamics. Note that one potentiometer multiplies by -a2 instead of a because a2 is physically a negative number in this case, and potentiometers can have only positive settings. Integrators C and D solve the gas-law equations for chambers B and A in the surge sensor, while integrators E and F with amplifier G solve the dynamics of the moving parts of the surge sensor. The blow-off valve equations are solved in the lower right of the diagram by integrators H, I, and J, and amplifier K. Integrator L's output is the pressure in the manifold, and amplifier MTs output is the rate of air-weight flow from the compressor. Because of the high response speed of certain parts of the system, it was impractical to use a "real-time" computer simulation (one in which computer operating time exactly corre19

The University of Michigan Willow Run Laboratories 3498-1 -F Compressor and Rotor Dynamics Duct and Manifold Zw. 53' 1. N + c% z6a z a = ~6 z53 - "Ik aC + a"8 z~ - a zM -ca ~ - M PM -g b0 5 "-0 - 21 aM Surge Sensor =A - a6PA'6P3, |o 6PB 7a3 aSaB - Cx9 m a = ALPc + A2zP - o, - A- A - K A aPc Blow-Off Valve PC = 0ll?P3 12 6 3C - + > ^C +' myr Ac 3 - A cC - Ky FIGURE 5. LINEAR SIMULATION sponds to the operating time for the system). The solution on the computer was slowed down by a factor of 100, thus making 100 seconds of computer time correspond to 1 second of operating time for the system. This time-scale change affects the scaling of the problem by multiplying the gain of the integrators in the computer mechanization by the ratio of computer time to problem time (in this case, 100). Therefore, if the quantity APM is used as an input to an integrator with a nominal gain of unity (1 megohm input resistor and a 1 mfd feedback capacitor), the output of the integrator is 100 APM The dotted lines immediately below integrator I in the computer diagram, Figure 6, show the mechanization of the coulomb friction of the piston, which was added to chamber C in the blow-off valve. The term Y B of Equation 25NL is generated by a high-gain amplifier with its olt output limited to ~100 volts, and a potentiometer which converts this output to a voltage corresponding to the friction force B which, in turn, is added to the other forces entering integrator I. The term in Equation 23L which represents leakage air-flow rate for the piston has a 20

The University of Michigan Willow Run Laboratories 3498-1 -F ACP 1 sec. real time = 100 sec. of computer time Surge Sensor aL 1 -Kx _x k "loRo Dyn-amics v - Chamber B P[f V le Cf A I lfolOJ 1<8 A d I Compressor O -- coeffic t of L, I m b aA Chamber A Rotor Dynamics 4 - - 3 MY poT. e cmpuer dagrm alo sows he oteniomter hic is dde toimuateheper 21 R~otorDynamih r Blow-Off Valve + q _a, r -U- > \ h-i'' - \`I Cfg A^ coefficient of —-3, which must be added to a 13 The setting of potentiometer a13 in the feedback loop of integrator H will contain this added quantity when the piston is added. Several runs on the computer were made with the piston and are presented in Section 4. ation of a dash pot placed on the blow-off valve. This potentiometer, which is in the feedback Kf loop of integrator I, is set tO~y and produces at its output the friction force due to the dash pot. 21

The University of Michigan Willow Run Laboratories 3498-1 -F 4 COMPUTER DATA The data presented in this section was recorded from the computer mechanization of the linear equations. Transient data was obtained in terms of the departures of variables from their steady-state values. Three different cases of steady-state operating conditions were explored. The steady-state solutions for each case were found by hand computation, and parameters from the steady-state solutions were used to compute the potentiometer settings. The first case was for both valves in the surge sensor and blow-off valve just closed (y = 0, x = 0). In the second case, the steady-state condition was for y = 0. 4 inch, x = 0. 01043 inch. For the third case, the steady-state condition was for y = 0. 8 inch, x = 0, 01193 inch. For all three cases, the same operating point on the compressor map was used; therefore only one set of values for the compressor partial derivatives was required. The steadystate operating point was changed from one case to the next by changing the total area of the exhaust orifices, AM, in the manifold, and thus changing W, and consequently WL. The manifold volume remained constant at one-fourth the volume of the manifold shown in Figure 1. The adjustable valve (A ) in chamber B was also changed. 0 Table I shows the steady-state values of the parameters used for each of the three cases, and Table II shows the values of the coefficients of the linearized equations for all three cases. 4. 1. CASE 1 WITHOUT DAMPING A computer run made for case 1 (valves just closed) is shown in Figure 7. This figure shows eight variables of the system plotted as functions of time by a Sanborn recorder. The time marks along the lower edge of the record mark off one-second intervals in computer time, which are equivalent to 0. 01-second intervals for the problem. A time scale, in terms of problem time, is also shown along the lower edge of the figure. The eight variables which are plotted as functions of time are, from top to bottom on the figure, AW, Ax, Ay, AN, /APC APM, APB, and AP3. The A's represent incremental departures from steady-state values of M' B' 3 the variables. The scaling shown for the ranges of A's in the figure is such that the curves show the response of the linearized equations to a step decrease of WL of 1 pound per second at time t = 0. Since the equations are linear, all of the scales at the left side of the figure could be reduced by a factor of 1/K, and the curves would be correct for a step decrease of 1/K pound per second of WL at time t = 0. Thus, by using the scales shown, the curves may be interpreted in terms of the change of each variable per pound per second step decrease in WL. Since the linearized equations are 22

The University of Michigan Willow Run Laboratories 3498- 1 - F (W =19.25) 0 (x = O) 0 Ay (N = 0).... (P,=53.60) 00 K APB: 100(PB = 33..59) 1~ -.:.... (P3 -=55.86)..0' i4.....'. o. —'.. i. l l.....l -.FIGURE 7. CASE 1 WITHOUT DAMPING;El r- 5 3.6 0 ) O I ~I ~ i:i-l ~:i- f — II ~i -i-ii ~ ~-i iii tH 1!...'i....:.i..:......... 1/.....0. -....:. 0............ 1.0................. ":...... (PB = 18.59) -1.0 (P3 = 53.86) _ -1.0 AP - O........1..2 0. FIGURE 7. CASE 1 WTITHOUT DAM.. PING

The University of Michigan Willow Run Laboratories 3498-1 - F TABLE I. STEADY-STATE VALUES FOR PARAMETERS W = 19.25, P = 55. 86, T = 846. 7, P = 12. 48, W a 3 3 2 bi -3 N = 1.589x10, = 18,940, andPA = P1 14. 7 Case 1 Case 2 Case 3 x 0 0.01042 0.02224 y 0 0.4 0.8 p 33.59 33.74 33.79 B p 53.60 53.27 52.95 C p 52.14 54.40 55.62 M W. 0 0.005627 0.01193 ci W_ 19.25 12.08 4.90 WV 0 7.173 14. 346 A 3.241 x 10-3 3.226 x 10-3 3.221 x 10-3 o A 25.29 15.21 6.035 M applicable to the nonlinear system for only very small changes of the variables, the curves should not be interpreted in terms of absolute magnitudes of the A's, but instead in terms of relative magnitudes. The steady-state value of each variable is indicated near the zero mark for the A scale. For case 1, both valves were closed, so in the actual system a negative value of Ax or Ay would be impossible. These restrictions on Ax and Ay could have been introduced into the computer circuit by the use of diode-limiting circuits, but it was considered preferable to first examine the operation with no nonlinearities in the equations. If the A's were extremely small and the valves were just barely open, the curves for case 1 would be applicable and negative values of Ax and Ay would be permissible. The step decrease in WL may be interpreted as an equivalent step increase in the manifold pressure (caused, perhaps, by a sudden increase of air flow from one of the other compressors in a four-compressor BLC system). It does not cause a step on the APM curve in Figure 7, because the recorder stylus for this curve was connected to a point in the analog circuit at which the step would not appear. Instead, the APM curve shows the variations of PM about the value which resulted from adding the step. 24

The University of Michigan Willow Run Laboratories 3498-1 - F TABLE II. COEFFICIENT VALUES FOR PARAMETERS Coefficients Case 1 Case 2 Case 3 Compressor and Rotor ca 0.0062 0.0062 0.0062 a -1. 176 -1. 176 -1. 176 2 a3 1841.4 1841.4 1841.4 a4 2.699 2.699 2.699 Surge Sensor a 56.71 58.97 62.15 a 62.34 64.82 68.31 a 78.62 76.28 73. 73 7 ca 130.7 126.3 121.9 8 a9 103.0 100.4 97.17 9 a1 1.491 1.491 1.491 Blow-Off Valve a11 0 14.62 45.16 12 121.8 180.7 312.3 a 0 15.31 47.64 13 a14 5034 6935 11235.2 15 37.35 51.46 83.38 Duct and Manifold 16 2.761 4.368 10.487 16 a 0 1.086 x 10-3 2.0 x 10-3 17 1a 17.94 17.94 17.95 18 ac 2.587 4. 128 10.18 19 ac 97.54 149.8 361.57 20 a21 104.55 153.8 363. 13 25

The University of Michigan Willow Run Laboratories 3498-1 - F The curves of Figure 7 show that the system tends to be oscillatory at three separate and distinct frequencies. The lowest-frequency oscillation, with a period of about 0. 3 second, appears on each curve. This oscillation indicates the.degree of stability of the entire system, including the manifold, compressor, rotor dynamics, surge sensor, and blow-off valve. The higher-frequency oscillation, which appears also on all of the curves but is most predominant on the curve for APC, is caused by an oscillation of the moving element in the blow-off valve. The existence of the unstable conditions which produce this oscillation is shown mathematically in Appendix A. Its period is about 0. 013 second, corresponding to a frequency of about 80 cycles per second, The third oscillation, which is even higher in frequency, may be seen on the first part of the curve for APB. It is caused by vibration of the valve in the surge sensor, and it dies out after about 0. 1 second because of the damping action of orifices A and A. Although the system response after about one-half second is masked by the oscillation of the blow-off valve, the response for about the first 0. 2 second is valid in Figure 7. Since AWL is not one of the variables recorded, the step decrease in WL does not appear in the figure, but its effect in decreasing W appears as a step decrease of about 0. 25 pound per a second in W at the beginning of the top curve. The decrease in W is not as great as the dea a crease in WL because the increased compressor outlet pressure causes the air flow through the duct to increase immediately, and thus partly cancels out the effect of the step decrease. The action of the surge sensor in controlling the blow-off valve is governed by various time constants, so AW is not returned to zero until about 0. 1 second after application of the step. a Because of the time constants the control causes AW to overshoot the zero value and to reach a a positive peak at about 0. 17 second. Then, AW decreases again, and the curve for W would a a show a gradual convergence to zero, if the high frequency oscillation of the blow-off valve were not present. A better understanding of the curves and of the system may be gained by examining the status of the variables at various points along the time scale in Figure 7. When t = 0, the step decrease of W has caused a momentary increase of P of about L 3 0. 3 psia. This produces excessive force on the blow-off valve. Because of the valve's inertia, it does not open immediately, but the time constant is quite short and the valve is open about 0. 03 inch after 0. 04 second. Superimposed on this plot of y is the oscillation of the valve, which was started by the initial disturbance and is growing in amplitude exponentially. The compressor air flow decreases by about 0. 25 pound per second, as shown on the first curve of Figure 7. If all time constants which actually exist in the system had been simulated, the compressor air flow would have momentarily decreased by 1 pound per second, thus cor26

The University of Michigan Willow Run Laboratories 3498-1 -F responding exactly to the decrease in WL, and then it would have rapidly changed to the value shown in the curve, as the increased P3 forced more air through the duct and effectively cancelled out part of the step decrease of WL. This small pulse in the AW does not appear, a because the very small time constants which would produce it were not included in the equations. The step change in air-flow rate through the compressor initiates a high-frequency transient oscillation of the surge-sensor valve (variables Ax and AP ). The pressure in chamber C of the blow-off valve starts to rise because of the rise of ALy. Superimposed on this pressure is the waveform due to the oscillation of y. The surge sensor valve displacement (ALx), the rotor-speed change (AN), and the manifoldpressure change (AP ) start to rise. The pressure in chamber B of the surge sensor also starts to increase due to the increasing of Ax. When t = 0. 1 second, starting with the first curve, the change in compressor air flow, AW, has returned to zero because of the corrective action of the surge control, but AW is still increasing because of interaction of lags in the system. The surge-sensor valve has reached a peak in its opening, as shown by the maximum for Ax. The blow-off valve opening, as shown by the curve for Ay, is still increasing because the air flows into and out of chamber C have not yet reached equilibrium. The engine speed has increased because of the decreased load on the compressor, which resulted from the initial decrease in W. a The average pressure in chamber C reached a peak at about 0. 05 second and is now decreasing because of air being exhausted through the surge sensor's poppet valve. The manifold pressure, P M, is decreasing, after having just reached a maximum. The cause of this behavior may be explained in terms of the behavior of P, since the manifold 3 pressure variations follow the P3 variations, with a slight phase lag. The pressure in chamber B is behaving in a manner similar to that of the manifold pressure, since it also follows the P3 variations, with variations due to changes in x superimposed. The transient oscillation because of vibration of the valve at the beginning of the run has disappeared, but the influence of the vibrations of the blow-off valve is beginning to appear in the curve for AP. The compressor outlet pressure, P3, has reached a peak and is decreasing because the blow-off valve opening is increasing. At t = 0. 3 second, the oscillation of the blow-off valve has grown to a sufficient magnitude to make the remaining parts of the curves invalid. 27

The University of Michigan Willow Run Laboratories 3498-1 -F 4.2. CASE 2 WITHOUT DAMPING A set of curves obtained for case 2 (valves partly open) from the Sanborn recorder is shown in Figure 8(a). For this run, the paper drive on the recorder was operated at one-half of the speed used for Figure 7, but each time mark still equals 1/100 second, so the time scale is different. The total system operating time covered by this set of data is about 1. 5 seconds. The initial behavior of each variable is similar to that for case 1, but the system is now stable. The blow-off valve oscillation, or chatter, still occurs, but it dies out after about a second. The system itself is stable, but the stability is somewhat marginal, since several cycles of oscillation occur before the transients die out, and control systems are usually considered to operate satisfactorily if they produce only one small overshoot when they are subjected to a step change of a controlled variable. 4.3. CASE 3 WITHOUT DAMPING results obtained from the simulation of case 3 (valves well open) are shown in Figure 8(b). Here, the oscillation of the blow-off valve is so small and of such short duration that it is no longer objectionable, but the system has become unstable. The low-frequency oscillations are growing rapidly in amplitude. 4, 4. CASE 1 WITH DAMPING In order to learn more about the nature of the blow-off valve oscillation, the computer circuit was modified to include viscous damping on the motion of the valve. The results obtained when the viscous damping coefficient, Kf, was 0. 0717 pound-second/ inch are shown in Figure 9(a). The curves show that the damping retarded the rate of growth of the oscillation, but the oscillation still grew in amplitude. Hence, the damping was not of sufficient magnitude. For Figure 9(b), the damping coefficient was raised to 0. 1742 pound-second/inch. The curves show that with this amount of damping, the oscillation exponentially decays, but at a rather low rate. In Figure 9(c) the Kf is 0. 3584 pound-second/inch, and for Figure 9(d) it is 1. 4336 poundsecond/inch. In Figure 9(d), the oscillation is completely under control, and the transient dies out in a few cycles. Comparison of Figure 9(d) with other results for case 1 shows that addition of the damping had negligible effect on the system response, although it completely eliminated the valve chatter. 4.5. CASE 2 WITH DAMPING The results of operation with values of Kf of 0. 3584 pound-second/inch, 0. 7168 poundsecond/inch, and 1.4336 pound-second/inch are shown in Figure 10. Comparison of these 28

The University of Michigan Willow Run Laboratories 3498- - F W 0.... I..........A W a xW.i':;':.:'T:::I::::::"' r:: q A x.. (W, = 19.25) 0 -- - — (W= 19.25) 0:;.;! i0.2::i -,0.2 _.._...._.. -:': (x=0.74) 0 —..-.- (C (=0.0.95) 0- -j - FL -1.0 -';......................... -1.0,o o ^.......100. O0'N::::_7 z —:::': j.. I......: -::....':...........:..0............ — —.-}.....-. —......' (N = 18,940) 0 i~.Ij I(N = 18,940) 0 —.[-.l — ~~oo=-jiAP~ii~=Z' i~ ij~ c j4j1%=^i~j-........0......... (P= - 1 04.2) o- ( 52...9) I^i —— i~ ^ E^ ^^^ ^E-5.0'/ __ (P3= 33.74) 0-(PB = 33.79) 0 -. i 1.0 -1 -— _.__.iI 0- i'..:'":''___ -1.'- - (P3 - 55.86) 0:...,::' (P3 = 55.86) 0-.:,.l.. -i r-i...................... _ Ti~ ~~ ~~~~~.........,.....:...,....i.... ——.."-~ t= 05 10 1.51 t00 05 TIME (sec) TIME (sec) j~ =';:i~~-.. —i — ~~-::;';'.. 1._.j. i;.'::,T_.j/.:................ P!.0',.:, i' ji - T- -' —-~-!-.... — 1-I -_- _. o ----- (P 53.27(a) Case 2 (b) Case 3 I.- I' FIGURE 8. CASES 2 AND 3 WITHOUT DAMPING 29 (P, = 54.40. x0.:'-':.:_...... -:-. 6 —" (P 255.6) 0 -'~ -1.0 —'-..- -- 1.0 1.5 t 0. (P, = 33.74) 0Ca s e 2 (b) C 33.79) 0

The University of Michigan Willow Run Laboratories 3498-1 - F runs with Figure 8(a) shows that the system response under "valve-open" conditions is not affected by the addition of viscous damping to suppress the high-frequency blow-off valve oscillations. 4.6. CASE 2 WITHOUT DAMPING, SPRING REMOVED FROM SURGE SENSOR The results of an investigation of possible stabilizing effects of the springs in the surge sensor and blow-off valve are shown in Figure 11. Four sets of curves are shown in this figure. All four sets were obtained for case 2, and for zero damping (Kf = 0). For Figure 11(a), the surge-sensor spring constant, K, was set to zero. For Figure ll(b), normal values for both K and K were included. For Figure ll(c), the blow-off valve spring constant, K, x y y was set equal to zero, and for Figure ll(d), both K and K were zero. Since the runs seem x y to be identical, it appears that eliminating the specified spring constants does not appreciably affect the operation of the system. 4.7. CASES 1 AND 2 WITH PISTON A number of computer runs were made for a piston substituted for the diaphragm in the blow-off valve. Two changes in the analog computer diagram were made in order to include the piston simulation. First, an additional air-flow term, corresponding to the leakage between the piston and cylinder wall, was required in the gas-law equation for chamber C of the blow-off valve. Second, the coulomb friction force between the piston and cylinder walls became a term in the force-balance equation of the blow-off valve. The actual method of adding these features to the computer diagram is described in Section 3. 3 of this report. Since the magnitude of the coulomb friction force depends only upon the direction of motion of the piston and not upon its velocity, the introduction of the coulomb friction term made the simulator's operation nonlinear. Thus, the earlier statements concerning interpretation of magnitudes of variables on the computer data for the linear system do not apply for the system with coulomb friction. Thus, if the initial step in WL is specified as 1 pound per second, the coulomb friction force (say 90 pounds) must be scaled to conform to this limitation, and the resulting curves are for a 1 pound per second change in WL. With proper precautions, however, certain interpretations can be made. If the same curves were regarded as results for a step of 0. 1 pound per second in WL, the ranges would decrease by a factor of 10, and the coulomb friction would increase by the same factor. The leakage area would remain the same, so the results would then be for a 0. 1 pound air-weight flow decrease and coulomb friction of ~900 lbs, with ranges of 1/10 of those specified on the scales for the data. 30

(W. 19.25)........... -1.0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~l i~~t~t~t~- ~ ~ ~ ~ i~tiiiiill:WIT 0.1~~~~~~~~~~~~i..~rrt~~t iiiii-irliijiijiijij:.iiiiilii~ljiiililii jl (X= 0 O ~~.......................' iiii i ii- ii i................... 1:-1-7................................~ ~ ~ ~ ~ ~ ~ ~~~~~i~i....:'";;';;..............it~ ~ ~rirr ~~~ -— ~; ~~-; I..;... -0. I 0.2..............~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~iilljlillil~iliilliiliili:i;li (N=8 90) ~~c~ —t,~:iiiiiii~,r 1100 - --- --- - ~ -100~~~~~~~~~~~~~~~~~~~~~~ (Pc 53.60 0~~t~~f~ ~li:i~r~l~l l'i~i~7~ii~~~~~ I.0-~- -iI _~ A PA......... 7:i~' M: M.: - -....................................................................................... (P A = -- 4.7) 0:;.-.........:;:::::::;: -.:....,................................ iiii iiii I i -_Cl~_ I I i ~ i;i I i -i Iiiiiiti -ii i i1.................. iiiiiiiiii::: 1 ~I:::: 1:::: 1::::::............................................... 0 ~ ~ ~ ~~~~~~.............. liliill!lii~iliiiil:::':::...........:::::::::::~~~~~~~......... (P, 5 5.8 6 ) 0..:iiiiiii..::::iiii ~~~~~~~~~~~~~~~~~~..........:.......................................j i i: i i...................................................:; ~ ~. ~ ~ ~:: i: I c- I:' ] I ":..:: 1 --.0...............................`I I' A P ~ ~ I I~ I t=O 0.5 1.0 1.5 t=................. HH l:i. H.. H:!;:;1111Hll..............~~~~TI E (ec................... 1n8 (d K 1. 433 -1.0~~~~~~~~~~~~~~~~~~~~~~~~3

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The University of Michigan Willow Run Laboratories 3498-1 - F AWa' |'.... -:::'.....: -0 211 I Z (Wa = 19.25) ~0 i E _':1>:' -1.0 0 - -:!:_-..,...'._ "........ ~ (x = 0.01042). -.. j... _..... 1g.. O'.............,,.'... oy 1i —.:::i. —.......... i....... O w _..........._................... _...... 2AN ^ISBIljl tt)fflBjilll iiyli|IIIB (P y =50.4) _ -_ - _ 1~00-. --: X..... -1.0 -^f^i^,.^- ~ 1......,- 1. ---- -- -- 34n(PB=_ —- ^-.1 3; -7. if - -" -,;.................... t= 0.2 0.4 t 0.2 0.4 0....t..... (P ( 53 =27) Both_ Sptr ig { _ _ _: _ i _ =0................. -0.2 e T_; ~- -_ —i — - - i=11 1- 1 —-1 —1 iii=O _,_, ii i iiii I:ii::=::I................-i..l..iiil iI....;!:........- _ A N_ 1........-.... N8ITHOT....DAMPI 3100 -_ -- _ _...::............................. o..... -........................ (P B= 33.74)........... ii~!:~.1..~?.......................................................i: (P, 53 27) -........~::t't.........'"' i ~-~r"T"F ~i:,~'~i'.i~.....~l...~....i.~.,..~ "I~"""..... -1.0 (a)K = luded -::::: (d) Kx:: -:::: K:::: O t 00. 04 =O0. 04 t= 02.4 t= 02. ~"'~:': ~~~IM.... (a)~~~~~~~~'-...'~ =. 0 b ohSrng c. d)Kx=0 Included~~~~~~~~~~~~~~~:i:: FIGURE~~~~~. 1.. ~ EFFECT O.. SPRNG... --- SYTE OEATONFRAS WIHU DAMPING..,.~....,.....

The University of Michigan Willow Run Laboratories 3498-1 -F Figures 12 through 16 are Sanborn recorder curves from the computer for cases 1 and 2 and pistons having various combinations of friction and air leakage. The parameters for the runs are listed in Table III. TABLE III. COMPUTER RUNS OF THE SYSTEM WITH PISTON Figure Case Friction Leakage WL (lb/sec) 12 1 90 0.009 1 13(a) 1 1 0.09 1 13(b) 1 1 0.360 1 13(c) 1 9 0.360 1 14 2 90 0.009 1 15 2 90 0.009 0. 1 16 2 90 0.360 0. 1 The runs show, in general, that the addition of the piston suppresses the chatter of the blow-off valve but is detrimental to the stability of the system itself. In fact, for large coulomb friction, the variables show a continuous nonsinusoidal oscillation after the original transient has died out. 35

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-......:.................. _.. _-..._ —_.-:... __.:~ _ _ - _^..:::.._ _ _.:::!:..:. _ _-:.'............'' I.:. 4- - - - --.. -oI 0k E I EH EEE Illiil l- i - - ~ ~ I v~0:::::=::::I:I:I:I:I:I:I I I I I I nW W|ffilE W-W HE X iiffii -t WS~t i F:::i:;:i:: -- ~ ~ 3;:!. \, ____ _ r _______ _ 1,,lEl, _______ _,,,_ l_,______ _ -_ ___^__ _ _______ _ l ___ ______.;- ___:___::::.,:::::::::::::::::':::'. _ ^ _...!...... L.'!...... - = - -:: -- —. - - - -- = -. -:::.: i.. — 1:1 1:..'i:.- -:1 " -' ~: 1.1 1:1 1:' —: ~ ~1 - ~1 11:1 —,............. - 1..........1-. j^~ ~~ I F i:::: EI 1 - ----- ___ __l l E............:....:''..:.i..1- 1.:.1 l:'.:'.".: 1 ~..4. $' __ __ __ __ __ -_ -_ _ _ __ - -; __ __ __ __ _ __ _ __ _ __ __ __ __ L.............. — ___ _4^ ^:_;___ _ - [- -,"-::......... -;: ii::f:::!:O:ii:: "i~::::fi'.":i:2:.::f:::f:::f:>::^:! ~::::E ||g|||g:|||||' |:|z:||l||" ||| ||B| |E|| |||:E..'"...... ~ = I E-::E:EEEEE::1:EE:E 0ii|!EEI lEEH I X I I:-F — 7~ ^ _ _.-~ -'f "'!" [..... - -,. -. - - ^ s - - ^ i - i- | K i a f ^ _".':::::::::,::'::::;:: ~.......'.....::, ~::, ~:::.;: ~::...:........ I I' _ I I I I I II I I I I 1 1:; 1:::: _:::;::1 _:: I:l^l.:l':.l."l''"l"'l'"'i:"'"''l:1::::;:_ S' -::....::.::::.. ".:............-............ 1 —- -— 1 0 1 I I! 1 1 1 1 1 1 1 II 1 1 1' 1::'.:: f::f::::::: ~ I I:........... 11 1 11111 111. 1 ~..................:^EfEEE:,::EJ::,E:::E.::,:EEE EEE,:E.:::,:,::iEI!,:l:i!H: IB:!:,!..:.. -................:::::.:::::::::::.::::::::::::..::....::: ~ 4-..........:.[:::;...:;:;:....:::..'.......................- H:::::::::I::::::::::^:;[~ I:l- - ~~~~~~~~~~~~~~~~~~~~~~o 0EEEEEEEEEEEE^^ElE^Ellll~~~~l~llH~ttl~~lll 0 0 - 0'- ~CN 0 N0 0,00 000 00 00 0 00 0"0 I I I I 7 I I W.....I.I.I..I I:I:II.z. ItI 1 1 1I.11; I:.11.:t1::1"1.1H:::::::::::::::::::::::::11':-.::1::1: 1::1'.-I'I''..1...:........:....:....F..:.....:.:..:..::::..::...:.:.1:1.:'1:. - 1 "' 1' mI1:'1:1:.:-1.'' i.'.:,-' i I! 8 I | | | |;; | 10 1 1" F 1 1 11 I z I'i t. L.. I.':'.::;| x |:,: |, |:: I'::: |::::::::::::::::: | |::;: |: 1.....::::. 4:........::..........._ I L I I I- I l I 1:1 I.1:::1.1.:.l. 1:1:::..I..:l::::l::l'.:..:.1:: I::.'1: 1I':::.1- I::l:.i:l l::ll':::::::: l l:::: l: 1:::.::::: i.........:.:...:.:.:. A:.:: J:::..... ~ 1 1...-. > I = H z -'- I.... L._/ It I:;I.'1t I _ _, I I I ~ - X-,.- o o o d~ o, _ ~ ~,_.:,_:..... I _. o.. O I... o,,.,

The University of Michigan Willow Run Laboratories 3498-1 - F a^w, 5 *0 ~~'.T-T.!-!'-!-! 1 "]-''TT_[.T!-'.-!......l.T..T....... "::.T.TT.-........ ~.. —-I —..'' i i i (Wa =19.25) 0-.: -5.0 =::::::s:: _.... g...,., ='1 11-''... Ax 0' (x= 0.01042) 0 -_ F -0,1 1.0 --:...............:...:......~ AP F5G0. 1......::::.::W:::.:::::: l:/::s.:e:c:. - "..:~:::. ( = 53.7) O t3- |W8HC|| ||_et - 1.0........................... (N 18,940) 0.0::....:-..i::.::::..: 5.0 t 00 51.0.......2 FIGURE 14. CASE 2 WITH PISTON. B = 901b, A = 0.009 in., W = 1 b/sec.,P,:: 5 ) il:!;;.C.-.l./i....l....!-.:. 1-. —.................. P5.0.:............ Pi ll.i I I. I:.. I I J' I -5.0':;:::;! TIME (sec) FIGURE 14. CASE 2 WITH PISTON. B = 90lb, A = 0. 009 in., W = 1 lb/sec. 38

6C'oas/qT T'0 = M''UT 600'0 = V'qT 06 = a'NOJ~SIc HJLIM g [SVD'S T[a HfI O (Das) aWIJ _cT 0 8g';0~~l 0'' =t 1O' c9Z OZ 9' LO' L9,1 I1. I 1- I 1 I 1 I l..................!~...(.................... I I I I I I I I I I I I I I I I I I I".I I.I I.t -I, -IO' L-..... _.I -: _ _ _ --. - - 7 -......-!!.;.: - _ _ -. — __ - - -. -. - -. _...................:... w.. ( 9 8' = c-d ) CdV I" T-1.-.I!........-_._.'1.L.....:]::..;:i.'.:..:.:..: -..:' __ h-:_;:;::..'::::.;z;:.;:::::.::::/:::..h./:::!:.L 0'.-. -- 1.::::d.. ~...:...:................... I I,+...-...... -.....':......:-~::.:. -: -..::~,~ L,'i::,::~ Z::::: 1!;: iliii.......... --:.. - r............:":~::::: 1,.;.~ ~:.:::~::r:::.::..-. I0;9';':........'( d)::. 88PLE = Bd [::~~i..:::....::JI!: [[: [~;~,'i'/,[i'[:~',:~;;i,- ~[,[[ [~......... " ~~~~.:..:::.. -..................:..:............................I. I.....::I ] ~. [::?:~::: f:::H J:::::]]:::~ ~~~...:........... ~..:i~.....................:.......................................... 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The University of Michigan Willow Run Laboratories 3498-1 - F 5 ANALOG COMPUTER MECHANIZATION of NONLINEAR EQUATIONS Figure 17 is a scaled diagram of a computer circuit for simulating the nonlinear equations of one compressor, its anti-surge control, the duct, and the manifold. It solves the nonlinear Equations 1NL through 22NL, which are listed in Section 2. The information flow diagram would be basically the same as Figure 5, except that the blocks would contain appropriate nonlinear equations, and the information flow lines would be labeled in terms of magnitudes of variables instead of increments. Table IV gives the amount of analog computer equipment required. TABLE IV. LIST OF ANALOG EQUIPMENT No. of Units Type of Analog Equipment 2 Function Generators of 2 Variables 13 Servo Multipliers (electromechanical, capable of 5 multiplications by the shaft variable) 9 Integrators 32 Amplifiers 37 Potentiometers oFor this estimate, it was assumed that all flows through orifices were choked flows, and thus the flow parameters would be constants and therefore would not need to be generated as variables. It might be necessary to add some function generators to allow the simulation to operate for the complete range of variables. For example, the computation of air-weight flows in and out of chamber B of the surge control may require corrected air-weight flows as functions of the pressure ratios across this chamber's inlet and outlet orifices, and thus would require two function generators. In Figure 17, the circuit which represents the compressor uses two function generators of functions of two variables in order to incorporate the compressor characteristics. These P3 TT - T functions are - as a function of N/ r/ and WJ//6, and - as another function of the same P T 1 1 two variables. The rotor dynamics computer uses two servo multipliers, with the shaft of one positioned by Wa and the shaft of the other positioned by (27r/60 IN. In the diagram servodriven multiplier potentiometers are labeled by the shaft variable written along with a curved arrow beside the potentiometer symbol. The multiplier potentiometers are represented by larger circles than those used as symbols for fixed coefficient potentiometers. 43

The University of Michigan Willow Run Laboratories 3498-1 - F Appendix A STABILITY of the BLOW-OFF VALVE The computer results show the presence of exponentially-increasing oscillations of Ay for the case where the blow-off valve is nearly closed. If the simulation is changed to include a sufficiently large linear damping term in the force-balance equation for the blow-off valve, the resulting operation is stable. This damping term could be produced in the actual system by a dash pot attached to the moving part of the blow-off valve. The following mathematical development verifies the instability of the system and demonstrates the effect of adding the dash pot. If the dash pot is included in the system, the equations for the blow-off valve are: (P - 1)A c2 -(PC - P)Acl -Ky =my +Kfy RTC =(W - W )dt. dAy 2gP3(P3 -PC Wci = C y -j RT3 co 3 7V3 V = VCO -Ac Y C CO cl1 The first equation, in linearized form, is Ac2P3 -AclAP -Ky = myAy + KAy. (1) Substituting the last three equations into the second, differentiating with respect to time, and linearizing the resulting equation gives 44

The University of Michigan Willow Run Laboratories 3498-1 - F (2) AC 11 3 + 12y a 13 PC -14 Ax "15 (2) dA Cy Y2gRT(2P - P) where a d = 3 g 3 -C 11 2IP3P (P -PC) dA C 2gRT3P3(P3 - PC) 12 VC dA Cf33AxVTiR C y 23 c13 V 2V VP3(P3 -P) Cf3 3PCV3R 14 = V C PA C cl 15 V, Assuming that the effects of AA and AP3 do not appreciably contribute to the oscillatory x 3 behavior of the blow-off valve, the terms containing these quantities may be eliminated from Equations 1 and 2 to leave, from Equation 1: -AclAPC - KyA y = + KfAy, (3) and, from Equation 2: ApC =' 12AY - 13APC+ a15AY. (4) Solving Equation 3 for -APC, and differentiating with respect to time, K K m y f Y -AP y A + y C l cl c cl Substituting this expression for -APC into Equation 4 gives 45

The University of Michigan Willow Run Laboratories 3498-1 - F K K m K K m ny a y "12y + a13 Lyf rnf +. YA~ y- Ay' A y =a ~1Ay+a 1 Ay +-Ay + a A + Ay. A A 12 A A A 15 cl cl c cl 1 cl cl 1 A cl Rearranging and multiplying by — gives y (113 m Y+ A15 m y1y + +12C1 A Y+ 1 +m Ay + + m m m 0. (5) Y/ Y Y Y Y y The characteristic equation for Equation 5 is of the form 3 2 p + ap +bp + c = 0. Next, we apply a test to determine whether this equation has roots with positive real parts (the Hurwitz criterion) (Reference 3). Application of the mathematical algorithm for determination of the Hurwitz properties of polynominals leads to the following array. 1 -P a 2 p3 ap + + bp 3 c a p P +-P (b - ) (b -c)/ap2 + c (b - )p 2 c ap c c c lb- C p (b ca For stability, (b - 0> (References 3 and 4), or b > - \ ~~a / ~a In Equation 5, Kf a = 13 + f; 13 m y a-K 15Ac K b = 13 f + 1 c m mm Y Y Y 46

The University of Michigan Willow Run Laboratories 3498-1 - F c13Ky 12 Ac c + m m Y Y then the condition for stability is a K aA 13 y + 12 cl a 13Kf 15Acl K m m 13_f 15c1 _ +__________. (6) m m m K Y Y Y f a13 + — 13 m y If Kf is zero corresponding to no dash pot, the condition for stability is: a13K 12Ac1 a A K m m 15 cl y y y m m a y y 13 which simplifies, through algebraic manipulation, to: 12 15 a13 13 Substituting for a15, a12, and a13 gives dA 2 _C I ~P (P PC PCcl dy RT3 3 -3 3 > dA (7) c Cy / P Cf33 Ax T 3 2 P3(P3 - PC) VT3 2 where A = 0. 694 x. x For case 1, x = 0 and y = 0, thus making the denominator of the right side of Inequality 7 zero. Since the numerator is not zero, the right side of this inequality is infinite, while 47

The University of Michigan Willow Run Laboratories 3498-1 - F the left side is finite. This means that without the damping action of the dash pot, it is impossible for the inequality to be satisfied and consequently for this part of the system to be stable for case 1. If the valves are open, however, it is possible for the inequality to be satisfied. This is verified by the computer runs for case 3 in Section 4 of this report. In order to determine the magnitude of Kf needed to provide stability for case 1, we examine Inequality 6 under the condition that x = 0 and y = 0. Rewriting Inequality 6, with a13 = 0, 15Ac 1 + K a A 15 cl y 12 cl n > m K y f or, 12 cl Kf >m A +K (8) f y a15 cl + y Thus, the blow-off valve is stable for case 1 for the values of Kf specified for Inequality 8, and since opening of the valve tends to increase the stability, it will be stable for cases 2 and 3 also. Appendix B DIGITAL COMPUTATION of OPERATING POINTS The use of a digital computer for computing steady-state operating points is highly recommended for either a nonlinear or linearized analog computer study of a system such as the one described in this report. Use of a digital computer in this manner permits many more steady-state points to be investigated in a linearized simulation, since the digital computer may be programmed to not only obtain the steady-state points but also to compute the coefficients of the linearized equations from the steady-state parameters. Hand computation would then be necessary only to obtain the partial derivatives from the compressor map. In the nonlinear simulation the steady-state operating points would be useful as check points on the analog computer. Input data to the digital computer was P1, T1, 6, 0 (based on the altitude specified) and P3' T3' W, N obtained from the compressor map. TM was also an input. The digital program was written in GAT (Generalized Algebraic Translator) language (Reference 5), and the 48

The University of Michigan Willow Run Laboratories 3498-1 - F digital computer was an IBM 650 computer. The digital computer program was written to incorporate the following procedure of computation and also to compute the a coefficients as defined in Section 3, except that it is assumed here that T is not equal to T3. 1. The following quantities are specified: P3 Wa, N. T TM," P1P 6, 0, and T. 3 a 3 2. Calculate P2 from P2 = P 1 - (4. 08)10- 4 ] 3. Calculate PB from P = (0. 5907)P3 B B 3 4. Calculate PC x, and y by trial and error from these 3 equations: x = -(0 0768P + 0. 601P= + 6. 459P - 7. 1368P ); 6 C B 2 1 8.6629079xP C Y P ~ (P - P ) U 3 3 C 1 P = 4072(38 48P + 2. 24P1 - 32. 8y). 9.3418399P3y 5. Calculate WV from WV = V V VT3 6. Calculate W from W = W - W L L a V (0. 00066253671)W 2T3 7. Calculate PM from PM = P - 22. 55688 P 8. Calculate W from = T M 9. Calculate VA from VA = 1. 83 - (7.06)x. 10. Calculate VB from VB = 0. 196 + (0. 601)x. 11. Calculate V from V = 58. 43464 - 40. 72y. The values of the parameters for six operating points are shown in Table V. 49

The University of Michigan Willow Run Laboratories 3498-1 -F TABLE V. OPERATING POINTS COMPUTED BY DIGITAL COMPUTER Altitude (ft) 20, 000 30, 000 Quantity Point 1 Point 2 Point 3 Point 4 Point 5 Point 6 P 6.753 6.753 4.362 4.362 4.362 4.362 6 0. 4594 0. 4594 0. 2967 0. 2967 0. 2967 0. 2967 0 0. 8604 0. 8604 0.7917 0.7917 0.7917 0.7917 T 447.4 447.4 411.70 411.70 411.70 411.70 P 27.01 26.94 17.45 17.71 17.40 17.58 3 W 9.708 9.713 6. 535 6.485 6. 538 6.501 a N 18385 18366 17636 17689 17618 17680 T3, 746.38 746.14 685.63 687.08 685.27 686.50 TM 741. 28 741. 16 681.73 682.46 681.55 682. 17 2? 5.694 5.693 3.678 3.689 3.678 3.686 p- B15.95 15.91 10.31 10.46 10.28 10.38 P 25.36 25.40 16.37 16. 17 16.40 16.23 x 0.0203 0.0155 0.0135 0.0374 0.0102 0.0269 y 0.6669 0.5305 0.4417 1.003 0.3483 0.7766 W V 6. 160 4.888 2.750 6.333 2.163 4.868 WL 3.549 4.825 3.785 0. 1518 4.375 1. 633 P 26.78 26.51 17.08 17.71 16.90 17.51 -7W 22. 19 21.97 14.75 15.29 14.60 15. 12 VA 1.687 1.720 1.735 1.566 1.758 1.640 VB 0.2082 0.2053 0.2041 0.2185 0.2021 0.2122 VC 31.28 36.83 40.45 17.58 44.25 26.81 Appendix C SIMULATION of a LONG MANIFOLD One of the ultimate objectives of the original project was a simulation of a manifold driven by more than one compressor. A simulation of a manifold driven by four compressors would have been impractical because of the large amount of analog equipment which it would have required. In the complete BLC system, which has two compressors driving each end of the manifold, there are two basic types of interaction between the compressors. One type is the interaction between compressors at the same end of the manifold. Here, changes in pressure 50

The University of Michigan Willow Run Laboratories 34981 - F due to a change in output air-flow rate from one compressor immediately affect the other compressor. The other type of interaction is between compressors at opposite ends of the manifold. It is felt that the system's operation may be explored adequately by first simulating two compressors operating into a manifold at stations which are adjacent to each other, and then simulating a system which has a compressor at each end of the manifold. When compressors are passing air into opposite ends of the manifold there are time-delay effects involved. For example, a sudden increase of air-flow rate from the compressor at one end causes the pressure in the manifold at that end to rise, but this change in pressure does not appear at the other end of the manifold instantaneously. Instead it propagates down the length of the manifold at the speed of sound. Since the manifold length is 90 feet, this delay is on the order of 1/10 of a second. In addition, the rising pressure is somewhat attenuated in passing down the manifold because of losses due to friction of the air against the manifold walls, and pressure drops due to air lost through the exhaust orifices. One possible method of simulating the action of the pressure wave in traveling down the manifold would be to simulate a separate smaller manifold for each of the two compressors, and then to couple the pressures of the two manifolds together by suitable analog dead time or time-lag simulators. Further consideration shows, however, that this approach might not be feasible because analog circuits which simulate transport delays have certain limitations which might cause instability of an arrangement such as the one described above, while the manifold itself would be quite stable. The ideas described in the following paragraphs might be developed further to give a simple method of simulating a manifold with a compressor at either end. First, consider the analogy between pneumatic systems and electrical systems: pressure may be considered as voltage, air-flow rate as current, T as capacitance for a chamber PV RT where - = JWdt, the viscous drag of the air as resistance, and the inertia (mass of the air) as inductance (Reference 6). Figure 18 shows how an electrical transmission line whose operation is analogous to that of the manifold might be developed. Figure 18(a) shows a schematic representation of a manifold with five exhaust orifices and with air entering each end. Limiting the number of exhaust orifices simplifies the explanation, but the principles developed in Figure 18 apply to a manifold with any number of exhaust orifices. If the air is considered to have no inertia and is incompressible, the electrical network of Figure 18(b) is the electrical analog of the manifold. The pressure at either end of the manifold is represented by voltages at either end of the network. Current through the series resistors is analogous to longitudinal air flow in the manifold, and current through the shunt conductances corresponds to air flow through the orifices. 51

The University of Michigan Willow Run Laboratories 3498-1 - F W2 W3 W4 W5 W6 w1-P P P P W 1P P2 P3 P4 P5 P6 P7 7 T T (a) Manifold with five orifices lp1 R P2 PR R P4 R P6 R P Ij vv- T vv vr T vv T uv f V~ V Wl I I! 1, W7 G W2 G5W3 G W4 GW GWGW6 (b) Electrical network which is analogous to the manifold i-f inertial and compressibility effects of air are not considered PI P P3 P4 P5 P6 P7 $y- -wTP —- Vy2P —— s —Yyvir — L LL LL L LL L L L cC C C CC C (c) Electrical network analogy for a manifold without air friction and without orifices P R P2 R P3 R P R P R P6 R P7 L L L L L L L L LL G G C (d) Network analog for the manifold FIGURE 18. DEVELOPMENT OF A NETWORK ANALOG FOR THE MANIFOLD Voltage drops due to current in the series resistors correspond to pressure drops between adjacent orifices in the manifold resulting from friction of the air with the manifold walls. Now consider the effects of air compressibility and inertia. Then examine the electrical analog of the manifold where these effects are present, but the orifices are closed, and friction of the air against the manifold walls is not considered. If the manifold is considered to be divided into six equal cylindrical volumes, each separate volume of air may be considered to have a certain mass and also to act as a spring. The electrical analog of a mass is an inductance, and the electrical analog of a spring is a capacitor. Thus, electrically, each of these 52

The University of Michigan Willow Run Laboratories 3498-1 - F six sections may be represented by a series inductance and a shunt capacitance; and when the sections are combined, the electrical network shown in Figure 18(c) is the result. This network corresponds to a lumped-parameter electrical transmission line with no losses. Actually, since the air in the manifold could be divided into an infinite number of sections and each section would have mass and compressibility, the correct electrical analogy for the manifold would be a distributed parameter transmission line instead of the network shown in Figure 18(c). It is probable, however, that the lumped-parameter transmission-line analogy would be adequate. Further investigation would be needed to determine the number of stations required. Figure 18(d) results from combining Figures 18(b) and 18(c). It may be recognized as a general lumped-parameter transmission line. The behavior of networks of this type has been extensively investigated, and complete mathematical descriptions of this behavior are available. The transmission line may be simulated on the analog omputer by one of two methods. First, it may be possible to develop a transfer function for the generalized transmission line in terms of relationships between variables at its ends, and to use an analog computer to simulate this transfer function. Such a transfer function would involve transport delay, but would also include attenuation, and it could probably be mechanized so that tendencies toward instability in the computer circuit would be eliminated. The second method would involve writing an ordinary differential equation for each station in the transmission line, and setting up a computer circuit which would solve this resulting set of simultaneous linear ordinary differential equations. The advantage of the second method is that it permits measurement of air pressures and. air-flow changes at intervals along the transmission line. Simulation by this second method would require a rather large computer set-up if further investigation showed that it was necessary to use a large number of stations in the simulation. The first method (the use of transfer functions) would not require a lumpedparameter transmission-line analogy and thus might be simpler and more accurate, providing the transfer function itself is not too difficult to simulate. 53

The University of Michigan Willow Run Laboratories 3498-1 - F REFERENCES 1. Joseph H. Keenan and Joseph Kaye, Gas Tables, Wiley, New York, N. Y., 1948. 2. John K. Jackson, Transfer Functions of Anti-Surge Control, Report No. R-1088-1, Cosmodyne Corporation, Los Angeles, Calif., 31 August 1959. 3. Ernest A Guillemin, Mathematics of Circuit Analysis, Wiley, New York, N. Y., 1949, pp. 395-409. 4. Eric B. Pearson, Technology of Instrumentation, Van Nostrand, New York, N. Y., 1957, pp. 118-122. 5. R. Graham and B. Arden, The Generalized Algebraic Translator, Statistical and Computing Laboratory, The University of Michigan, Ann Arbor, Mich., April 1959. 6. Allan R. Catheron and John F. Taplin, "Pneumatic Components, " in John G. Truxal (ed. ), Control Engineers' Handbook, McGraw-Hill, New York, N. Y., 1958, Sec. 16, pp. 7-11. 54

The University of Michigan Willow Run Laboratories 3498-1 - F DISTRIBUTION 30 copies Continental Aviation and Engineering Corporation (1 repro) 12700 Kercheval Avenue Detroit 15, Michigan ATTN: J. E. O'Shea, Purchasing Agent 55

+ HAD Div. 30/2 UNCLASSIFIED AD Div. 30/2UNCLASSIFIED Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Boundary-layer control Willow Run Laboratories, U. of Michigan, Ann Arbor1. Boundary-layer control ANALOG COMPUTER INVESTIGATION OF A BOUNDARY-LAYER systems -Design ANALOG COMPUTER INVESTIGATION OF A BOUNDARY-LAYERsystems -Design CONTROL SYSTEM by Margaret M.Spencer and Paul S. Fancher. 2. Boundary-layer control CONTROL SYSTEM by Margaret M. Spencer and Paul S. Fancher. 2.Boundary-layer control Jan 60. 55 p. incl. illus. 5 tables, 6 refs. systems - Effectiveness Jan 60. 55 p. incl. illus. 5 tables, 6 refs. systems -Effectiveness (Rept. no. 3498-1-F) 3. Boundary-layer control (Rept. no. 3498-1-F) 3. Boundary-layer control (Contract AF 33(600)-38666) Unclassified report systems -Performance (Contract AF 33(600)-38666) Unclassified reportsystems Performance A limited study using an analog computer was performed to inves- Boundary-layer control A limited study using an analog computer was performed to inves- 4. Boundary l to 4.tm Smlto systems - Simulation tigate the stability of a pneumatic system consisting of a gas-systems - Simulation tigate the stability of a pneumatic system consisting of a gas- B-e turbine-driven compressor operating into a manifold from which air 5. Boundary-layer control turbine-driven compressor operating into a manifold from which air 5. Boundary- trl systems - Test results air systems - Test results was escaping through a number of orifices. Simulation of linearized M eacal ompters -was escaping through a number of orifices. Simulation of linearized 6. Mathematical computers - equations which describe the transient operation of the system for co mputersequations which describe the transient operation of the system for Analog computers small departures from steady-state operating conditions was corn- I Scc Me.asmall departures from steady-state operating conditions was co- Spencern pleted, and results of this simulation are presented. The results pencer, Margaret M. anpleted, and results of this simulation are presented. The results Fancher Paul S from this incomplete study show that the system has strong tend- ancher, au S from this incomplete study show that the system has strong tend- Continental Aviation and encies toward instability. A circuit for simulating the general non- Continental Avia ion anencies toward instability. A circuit for simulating the general non- Engineering Corporation linear equations for the system is also presented. The linear and Engineering Corporation linear equations for the system is also presented. The linear and P ie III. Prime Contract AF 33(199)-II.PieCnrcAF3(9) nonlinear simulations are compared. Methods of simulating a sys- II.P imeCnrcfF3i0)-nonlinear simulations are compared. Methods of simulating a sys-38666nrog nonlinear simulations are compared. ~~~~~~~~~~~~~~~~~~~~~~Methods of simulating a sys-386te costigfalngm iodwthneoprsratahed311 tem consisting of a long manifold with one compressor at each end386 te consisting of a long manifold with one compressor at each end are discussed. Armed Services are discussed. Armed Services (over) Technical Information Agency (over) Technical Information Agency UNCLASSIFIED UNCLASSIFIED -1 -1 — - AD Div. 30/2 UNCLASSIFIED AD Div. 30/2 UNCLASSIFIED Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Boundary-layer control Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Boundary-layer control ANALOG COMPUTER INVESTIGATION OF A BOUNDARY-LAYER systems - Design ANALOG COMPUTER INVESTIGATION OF A BOUNDARY-LAYER systems - Design CONTROL SYSTEM by Margaret M. Spencer and Paul S. Fancher. 2. Boundary-layer control CONTROL SYSTEM by Margaret M.Spencer and Paul S. Fancher. 2. Boundary-layer control Jan 60. 55 p. incl. illus. 5 tables, 6 refs. systems - Effectiveness Jan 60. 55 p. incl. illus. 5 tables, 6 refs. systems -Effectiveness (Rept. no. 3498-1-F) 3. Boundary-layer control (Rept. no. 3498-1-F) 3. Boundary-layer control (Contract AF 33(600)-38666) Unclassified report systems - Performance (Contract AF 33(600)-38666) Unclassified report systems -Performance A limited study using an analog computer was performed to inves- 4. Boundary-layer control A limited study using an analog computer was performed to inves- 4. Boundary-layer control tigate the stability of a pneumatic system consisting of a gas- systems - Simulationtigate the stability of a pneumatic system consisting of a gas- n system mulon turbine-driven compressor operating into a manifold from which air 5. Boundary-layer controlturbine-driven compressor operating into a manifold from which air 5. Boundary-y control was escaping through a number of orifices Simulation of linearized systems - Test results was escaping through a number of orifices. Simulation of linearized systems - Test result s ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~was escaping through a number of orifices. Simulation of linearized Mteaia optr.Mteaia optr equations which describe the transient operation of the system for 6. Mathematical computers -equations which describe the transient oputeration of the system for -m small departures from steady-state operating conditions was corn- Analog computers small departures from steady-state operating conditions was com- na og c e pleted, and results of this simulation are presented. The results I. Spencer, Margaret M. and pleted, and results of this simulation are presented. The results Fpncer, P S' Fanchef, Paul S. FanchefPulS from this incomplete study show that the system has strong tend- Fncheral S from this incomplete study show that the system has strong tend- ne II. Continental Aviation and II. Continental Aviation and encies toward instability. A circuit for simulating the general non- I enciesnen twr it toward instability. A circuit for simulating the general non-Ee ntanene C oio linear equations for the system is also presented. The linear andEngineering Corporation linear equations for the system is also presented. The linear and I. Pie system ~~~~~~~~III. Prime Contract AF 33(600)nonlinear simulations are compared. Methods of simulating a sys- III. Prime Contract AF 33(600)-nonlinear simulations are compared. Methods of simulating a sysnonlinear simulations are compared. ~~~~~~~~~~~~~~~ ~ ~ ~ ~ ~ ~~Methods of simulating a sys - 366tmcnitn faln aiod ihoecmrso tec n 81 tem consisting of a long manifold with one compressor at each endtem consisting of a long manifold with one compressor at each end are discussed. Armed Services are discussed. Armed Services (over) Technical Information Agency (over) Technical Information Agency UNCLASSIFIED UNCLASSIFIED H- -1- -H

AD UNCLASSIFIED AD UNCLASSIFIED UNITERMS UNITERMS Analog computer Analog computer Stability Stability Pneumatic system Pneumatic system Compressor Compressor Manifold Manifold Orifices Orifices Simulation Simulation Linearized equations Linearized equations Steady-state Steady-state Instability Instability Nonlinear equations Nonlinear equations System System UNCLASSIFIED UNCLASSIFIED + AD UNCLASSIFIED AD UNCLASSIFIED UNITERMS UNITERMS Analog computer Analog computer Stability Stability Pneumatic system Pneumatic system Compressor Compressor Manifold Manifold Orifices Orifices Simulation Simulation Linearized equations Linearized equations Steady-state Steady-state Instability Instability Nonlinear equations Nonlinear equations System System UNCLASSIFIED UNCLASSIFIED

+ -1 —-H AD Div. 30/2 UNCLASSIFIED AD Div. 30/2 UNCLASSIFIED Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Boundary-layer control Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Boundary-layer control ANALOG COMPUTER INVESTIGATION OF A BOUNDARY-LAYER systems - Design ANALOG COMPUTER INVESTIGATION OF A BOUNDARY-LAYER systems - Design CONTROL SYSTEM by Margaret M.Spencer and Paul S. Fancher. 2. Boundary-layer control CONTROL SYSTEM by Margaret M.Spencer and Paul S. Fancher. 2. Boundary-layer control Jan 60. 55 p. incl. illus. 5 tables, 6 refs. systems - Effectiveness Jan 60. 55 p. incl. illus. 5 tables, 6 refs. systems-Effectiveness (Rept. no. 3498-1-F) 3. Boundary-layer control (Rept. no. 3498-1-F) 3. Boundary-layer control (Contract AF 33(600)-38666) Unclassified report systems - Performance (Contract AF 33(600)-38666) Unclassified report systems-Performance A limited study using an analog computer was performed to inves- Boundary-layer control A limited study using an analog computer was performed to inves- 4 Boundary-l r coto tigate the stability of a pneumatic system consisting of a gas-systems - Simulationtgat t stability of a pneumatic system consisting of a gas-S iutltion turbine-driven compressor operating into a manifold from which air Boundry-layer controlturbine-driven compressor operating into a manifold from which air. Boundaryayer contro was escaping through a number of orifices. Simulation of linearized systems - Test results was escaping through a number of orifices. Simulation of linearized systems - Tes resuls equations which describe the transient operation of the system for 6 Mathematical computers -equations which describe the transient operation of the system for 6. Aalo co small departures from steady-state operating conditions was cor- Analog computers small departures from steady-state operating conditions was com- I nacer M e Mo. d pleted, and results of this simulation are presented. The results. Spencer, Margaret M. and pleted, and results of this simulation are presented. The results Fanher PauS from this incomplete study show that the system has strong tend- Fancher, Paul S. from this incomplete study show that the system has strong tend- FanchE encies toward instability. A circuit for simulating the general non- Continental Aviation andcies toward instability. A circuit for simulating the general non- EngConteerng Croation linear equations for the system is also presented. The linear and Engineering Corporationlinear equations for the system is also presented. The linear and Egine nonlinear simulations are compared. Methods of simulating a sys- III. Prime Contract AF 33(600)- nonlinear simulations are compared. Methods of simulating a sys-tIIIg a ss ten consisting of a lons are fomd Me compressor at each end 38666 tem consisting of a long manifold with one compressor at each end 38666 are discussed. Armed Services are discussed. Armed Services (over) Technical Information Agency (over) Technical Information Agency UNCLASSIFIED UNCLASSIFIED -t -1 —-H AD Div. 30/2 UNCLASSIFIED AD Div. 30/2 UNCLASSIFIED Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Boundary-layer control Willow Run Laboratories, U. of Michigan, Ann Arbor 1. Boundary-layer control ANALOG COMPUTER INVESTIGATION OF A BOUNDARY-LAYER systems - Design ANALOG COMPUTER INVESTIGATION OF A BOUNDARY-LAYER systems -Design CONTROL SYSTEM by Margaret M. Spencer and Paul S. Fancher. 2. Boundary-layer control CONTROL SYSTEM by Margaret M. Spencer and Paul S. Fancher. 2. Boundary-layer control Jan 60. 55 p. incl. illus. 5 tables, 6 refs. systems - Effectiveness Jan 60. 55 p. incl. illus. 5 tables, 6 refs. systems - Effectiveness (Rept. no. 3498-1-F) 3. Boundary-layer control (Rept. no. 3498-1-F) 3. Boundary-layer control (Contract AF 33(600)-38666) Unclassified report systems - Performance (Contract AF 33(600)-38666) Unclassified report systems-Performance A limited study using an analog computer was performed to inves- 4 Boundary-layer control A limited study using an analog computer was performed to inves- 4 Boundary-laye control tigate the stability of a pneumatic system consisting of a gas- systems -aSimulation tigate the stability of a pneumatic system consisting of a gas- syster - to turbine-driven compressor operating into a manifold from which air Boundary-layer controlturbine-driven compressor operating into a manifold from which air 5 Boyndct was escaping through a number of orifices. Simulation of linearized systems - Test results was escaping through a number of orifices. Simulation of linearized systems - Test results equations which describe the transient operation of the system for 6. Mathematical computers - equations which describe the transient operation of the system for 6. Mathem cal computers - small departures from steady-state operating conditions was com- Analog computers small departures from steady-state operating conditions was com- Analog computers pleted, and results of this simulation are presented. The results. Spencer, Margaret M. and pleted, and results of this simulation are presented. The results I Spencer, Margaret M. and from this incomplete study show that the system has strong tend- Fancher, Paul S. from this incomplete study show that the system has strong tend-Fanche encies toward instability. A circuit for simulating the general non- I Continental Aviation and encies toward instability. A circuit for simulating the general non-. Continental Aviation and linear equations for the system is also presented. The linear and Engineering Corporation linear equations for the system is also presented. The linear and Engineering Corporation nonlinear simulations are compared. Methods of simulating a sys- III Prm Contract AF 33(600)-nonlinear simulations are compared. Methods of simulating a sys- III Prime C tem consisting of a lons are fomd Me compressor at each end 38666 tem consisting of a long manifold with one compressor at each end 38666 are discussed. Armed Services are discussed. Armed Services (over) Technical Information Agency (over) Technical Information Agency UNCLASSIFIED UNCLASSIFIED F ^t

AD UNCLASSIFIED AD UNCLASSIFIED UNITERMS UNITERMS Analog computer Analog computer Stability Stability Pneumatic system Pneumatic system Compressor Compressor Manifold Manifold Orifices Orifices Simulation Simulation Linearized equations Linearized equations Steady-state Steady-state Instability Instability Nonlinear equations Nonlinear equations System System ___.. (i- C UNCLASSIFIED UNCLASSIFIED,1'a1-O -HT ~ ~ ~ ~ ~ ~ r __ AD UNCLASSIFIED AD UNCLASSIFIED UNITERMS UNITERMS Analog computer Analog computer Stability Stability Pneumatic system Pneumatic system Compressor Compressor Manifold Manifold Orifices Orifices Simulation Simulation Linearized equations Linearized equations Steady-state Steady-state Instability Instability Nonlinear equations Nonlinear equations System System UNCLASSIFIED UNLASSIF