THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING DYNAMIC CHARACTERISTICS OF ELECTRO-MAGNETIC COMPRESSORS Wen-Jei Yang Harng-Sen Huang February, 1968 IP-807

ACKNOWLEDGMENT Part of the work reported was initiated at the Tecumseh Products Research Laboratory, Ann Arbor while the first author was associated with the laboratory as a consultant. He wishes to express his appreciation for providing the opportunity. ii

TABLE OF CONTENTS Page ACKNOWLEDGMENTS................................................... ii LIST OF FIGURES............................................... iv NOMENCLATURE.................................................. o v INTRODUCTION............................................... 1 ANALYSIS................................................... 3 RESULTS AND DISCUSSION....................................... 13 CONCLUDING REMARKS............................................ 23 PROGRAM FOR UNSTEADY BEHAVIORS OF MOVING-COIL TYPE COMPRESSOR..................................... 24 PROGRAM FOR UNSTEADY BEHAVIORS OF MOVING-IRON TYPE COMPRESSOR..oo............................... 30 REFERENCES..........................................o............ 37 iii

LIST OF FIGURES Figure Page 1 Schematic Diagram of Electromagnetic Compressors...o..... 4 2 Displacement-Time and Pressure-Displacement Diagrams..... 8 3 Dynamic Characteristics of "Moving-Coil" Type Electromagnetic Compressor for n = 1.41................o 15 4 Dynamic Characteristics of "Moving-Iron" Type Electromagnetic Compressor with M = 0.812 ibm, kI = 550 lbf/in., kII = 0, and n = 1 0........................... 16 5 Dynamic Characteristics of "Moving-Iron" Type Electromagnetic Compressor with M = 0.812 Ibm, kI = 330 lbf/in., kII = 0 and n = 1.0............................ 17 6 Dynamic Characteristics of "Moving-Iron" Type Electromagnetic Compressor with M = 0.812 Ibm, kI = 550 lbf/in., kII = 0 and n = 1.41..............18 7 Dynamic Characteristics of "Moving-Iron" Type Electromagnetic Compressor with M = 0.696 Ibm, kI = 550 lbf/in., kII = 0 and n = 1.41......................... 19 8 Dynamic Characteristics of "Moving-Iron" Type Electromagnetic Compressor with M = 0.812 Ibm, kI = 300 lbf/in., kII = 0 and n = 1.41.............. o 20 iv

NOMENCLATURE A Area of the flux cross section of air gap effective in pro22 ducing tractive force, ino; A1, for inner yoke, A2, for outer voke. Ab Cross-sectional area of buffer space, in. 2 Ap Cross-sectional area of piston, in. B Flux density of the magnetic field for the "moving-coil" type, maxwell/in. C Capacitance, f. d Diameter of coil turn, in. E Root-mean-square coil voltage, volt. Em Maximum coil voltage, volt. F Instantaneous mechanical force for the "moving-coil" type or instantaneous magnetic-tractive force for the "movingiron" type, lbf. g Gravitational acceleration, in./sec. i Instantaneous coil current, ampere. I Root-mean-square coil current, ampere. J Integer used for numerical computationo k Spring constant, lbf/in., kI, for spring I; kii, for spring II. i Piston position measured from the cylinder head for zero spring force, in; RI, for spring I; AII, for spring II M Mass of the moving parts including the piston, end frame and part of springs, lbm (or divided by 386.088, lbf-sec.2/in.) N Number of coil turns, turns. n Ratio of specific heats. Pb Gas pressure in buffer space, psia. PC Gas pressure inside cylinder, psia. v

NOMENCLATURE (Continued) Pd Discharge pressure, psiao PS Suction pressure, psia. R Resistance of coil, ohms. t Time, sec. At Time interval used for numerical computation, sec. x Instantaneous piston position measured from the cylinder head, in.; xo at the beginning of compression stroke, xl, at the end of compression stroke; x2, at the beginning of expansion stroke; x3, at the end of expansion stroke; xa for zero buffer space; xb, for maximum buffer space. Ya ~ Compression ratio. () ~ Instantaneous magnetic flux per gap, maxwell. w Angular frequency of imposed voltage, rado/sec., = 2r (cps)o Superscript ~ ~' Time derivatives. vi

INTRODUCTION In the design and control of a compressor, the knowledge of dynamic performance is of prime importance. This paper is devoted to an examination of the transient behavior of electromagnetic compressors, specifically for small horsepower~ A comprehensive review of various types of compressors for refrigeration and compressed air and gases has appeared in references 1 and 2. In this paper, attention is focused on two different types of electromagnetic compressors: "moving-coil" type or the so called "swing motor" and "moving-iron" type. The former is analogous to the common electromagnetic speaker found in radios and high-fidelity equipment, while the latter operates on the same principle as a high-speed magnet which is designed to move an external load having mass against (3) the action of a constant load force, friction, and a spring. The resonance phenomenon is used to advantage in both typeso In a reciprocating-type compressor, the rotary motion of an induction motor is converted mechanically to reciprocating motion to compress the gas. However an electromagnetic compressor produces this reciprocating motion directly. Therefore the electromagnetic compressors have the following major advantages. (i) Low friction losses: In a conventional compressor converting the rotary motion to reciprocating motion involves friction at several contacting parts, while in the electromagnetic compressor the only point of contact is between piston and cylinder. (ii) Small size, light weight and low cost: The mechanism -1 -

-2 - of the electromagnetic compressor is basically simple and has few partsO (iii) High efficiency and low power consumption: Because the resonance phenomenon is used to advantage in the electromagnetic compressor, the power factor is higher than that for the induction motor. Due to the high efficiency of the electromagnetic system and the low friction losses, these electromagnetic compressorshave, in general, a low power consumptiono The dynamic behavior of the electromagnetic compressors is simulated by means of a digital computer in the present paper~ The results for the "moving-coil" and "moving-iron" types are obtained and compared

ANALYSIS The electromagnetic compressors to be investigated are shown in Figure o1 A schematic view of the "moving-coil" type is illustrated in Figure 1-a. The coil is suspended by two springs in the ring gap formed between the yoke, pole piece and the permanent magnet. A piston is connected to the coilo This piston is inserted in a cylinder which allows the piston and coil to move back and forth without contacting the yoke and pole piece. When the coil is connected to an AC power line, the compressor experiences a starting transient for a certain period of time. But as soon as the starting transient period is elapsed the piston and coil will move back and forth at the given power frequency (50 cps or 60 cps, and so forth)o The "moving-iron" type compressor is schematically shown in Figure l-bo The coil is wound between the stationary inner and outer yokeso The piston is connected to the end frame with a concentricring shape, flat-faced armature. The piston and end frame are suspended by two springs. This piston is inserted in a cylinder which allows the piston and end frame to move back and forth. When the coil is connected to an AC power line, an instantaneous magnetic force is generated across the air gaps to move the end frame and piston. Under a steady periodic operation, the end frame and piston will vibrate at twice the power frequency. For both type compressors, an intake valve and an exhaust valve permit the cylinder and piston to act as a simple pump to compress the gaso By utilizing the principle of resonance, it is possible to -3 -

END FRAME ARMATURE PISTON BUFFER SPACES EXHAUST VALVES (a) MOVING-COIL TYPE (b) MOVING- IRON TYPE Figure 1. Schematic Diagram of Electromagnetic Compressors.

-5 - produce a vibration with a large amplitude at the given power frequency for the "moving-coil" type or at twice the given power frequency for the "moving-iron" type. The mass of the vibrating parts and the strength of the springs maybe calculated to place the mechanical resonance of the system at a certain frequency to achieve high efficiencyo The following assumptions are made in the formulation of the problem (i) Friction, part and valve losses are negligibleo (ii) The gas leakage from the cylinder and buffer space is negligibleo (iii) The gas behaves ideally and undergoes reversible polytropic processes in the cylinder and buffer spaceo (iv) The resistance voltage (IR) in the magnet is negligible, This means that the reactance voltage is equal to the supplied voltage~ (v) There exists a linear relationship between the magnetic flux (or flux-linkage NO) and the current i e (vi) The piston is subjected to the forces exerted by the gas on both front and rear surfaces of the piston and by the gas in the buffer spaceo With these assumptions, the application of force balance produces M(x-g) + ki(x-i) + Ap Pc(x) + Ab Pb(x) = F + Ap Ps + Ab Ps + kI(x-l (1) where x(t) represents the instantaneous location of the piston measured from the cylinder heado The initial conditions are x(o)- xo and x(o) = 0 For an ideal gas undergoing a reversible polytropic process, x(o) = 0 o For an ideal gas undergoing a reversible polytropic process,

the gas pressure in the cylinder space Pc(x) may be related to the suction pressure Ps as follows. During compression stroke when x1 < x < xo P (x) P ( 2-a During discharge stroke when x2 < x < xl P (x) = 1 P = discharge pressure (2-b) c x1i S During reexpansion stroke when x2 x < x3 Pc(x) x 2 P n (2-ec xi X j During suction stroke when x3 < x < xo Pc(x) = Ps (2-d) Similarly) for the gas inside the buffer space, one can write Pb(x) = 0 when x > xb (3-a) and Pb(X) x= x- Ps when x < xb (3-b) For the "moving-coil" type compressor, the instantaneous (1) mechanical force on the conductor in the direction of motion is F = 8086 X 10- TdNB(Em/R)sinot (4) where B is the flux density of the magnetic field, For the "movingiron" type compressor, the instantaneous magnetic-tractive force is

-7 -F [(t)/v]2 1 + (5) 72 X l06 Al A2 where v is the leakage coefficient which is a function of the air gap length. Figure 2 shows the displacement-time characteristics of the compressors during starting transient. It is easy to realize that the locations xo, x1, x2 and x3 are all time-dependent. xb is the location of the piston for the maximum buffer space and is predetermined by design. x and x2 indicate the ends of the suction and discharge strokes of the compressor cycle, respectively. They correspond to the location of the piston at the moment x becomes zero during the suction and discharge strokes, respectively. The current values of xl and x3 are determined by xl = xo/1l/n and x3 = x2 1/n, respectively. Only after the steady-periodic operation of the compressor is established, then the locations xo, xl, x2 and x3 would become time-independent The compression stroke of the compressor cycle may consist of processes 1 and 20 Process 1 represents the interval xo X> x > xb while process 2 is for xb > x > x3. Process 1R or 2R takes place only when the compression stroke fails to yield the specified discharge pressure, that is, when the current value of (Xo/xl)n is less than the specified compression ratio a. The resulting return stroke may be either process 2R or 1R or 2R followed by 1R depending upon the current position of the piston x. If the return stroke begins at x > xb, the process is called 1R, while if it begins at x < xb, the process is called 2R. During process 1R if x = 0 occurs before the piston

STARTING TRANSIENT STEADY PERIODIC OPERATION.~ 6 00 -.f x 10/ IR 4i Ae R 6 X.b __ /-_ ^ \^\ / - rd __ _ _IJ__ __ \ \ xl I -> \ ^/ y \^,\ — _X2 Pd xX2 X3Xl xb X0 Xa TIME, t Figure 2. Displacement-Time and Pressure-Displacement Diagrams.

-9 - reaches the current xo, the location x becomes a new xo o However, process 1R may continue beyond the current xo. For such cases, process 1R ends at the current xo and is then followed by process 6 until the piston turns back at the location where x = 0 o That location becomes a new x. Process 2R maybe followed by process 1R when the piston moves beyond xb. Or, it maybe succeeded by process 2 when the piston turns back before it reaches the location xb o Process 3 refers to the discharge stroke which begins at x x3 and ends at x = x2. It is then followed by process 4 which represents the reexpansion process of the residual gas in the compressor cylinder after the discharge stroke is completed. Process 4 begins at x = x2 and ends at x = xl when P < Ps and the suction valve is forced to open. The suction stroke is represented by process 5 for x1 _ x < xb and by process 6 for x > xb. The location x = xb has a physical significanceo When the piston moves beyond xb, the gas in the buffer space would exert no net force on the piston. However, when the piston moves within x < xb, the gas in the buffer space becomes under compression and thus would exert a pressure force on the piston. In summary, Equation (1) may be rewritten for the eight processes as follows. Process 1, from x = xo (or x = 0 following process 1R or 6) to x = xb: M(x-g) = k(i-x) + AP Ps [ ) - ] - F, (6-a) subject to the initial conditions x(O) = xo and x(O) = 0 Process 2, for x < xb and x > 0 following process 1.

-10 - M(x-g) = k(e-x) + P - 1 - F. x (6-b) Process 2, from x = xb (or x = 0 following process 2R, 4 or 5) to x = 1 = Xo/C / M(x-g) = k(,-x) + Process 2R, for xl < x *0 M(x-g) = k(Q-x) + Process 3, from x = xl M(x-g) = k(Q-x) + Process 4, from x = x2 M(x-g) = k(i-x) + A P [(-)n - 1+ Ab Ps[(b -xa 1 6-)F %\ ) - a (6-c) < Xb and x > 0 following process 2 o 1/n ~ /n Fio n l1 APFIXbXaln_ l1 A~P5 i-)1 +IJ Ab L/XXa - (6-d) to x = 0 at which x = x = x3/aln -X n 1fxb-xa n AP Ps +_ Ps n - 1 F - P J b X-Xa j (6-e) 1/n or x = 0 following process 3 to x = x3 = x2uc Ap Ps 1 + Ab P [- - F LI X~l I J L ~ a 4 6-f) Process 5, from x = x3 to x = xb: M () b [(x Xa) n M(x-g) = k(i-x) + Ab Ps x -- -i - F L. aJ (6-g) Process 6, from x = xb to x = 0 at which x= x: M(x-g) = k(Q-x) - F (6-h) For the "moving-coil" type compressor, the impressed voltage is known as E = Em sinwt. However, for the "moving-iron" type compressor, the impressed voltage equation for a magnet circuit shunted by a condensor is 1 t E = No + IR + C f Idt o

-11 - or E sinwt = 10-8 N i + iR + 106 t i dt (7) C o where C is the capacitance in microfarads. When the force required to establish the flux in the iron part of the circuit is ignored, the total magnetic force across the air gaps can be written as iAI A2 where i is the permeability of air ( = 3.192 maxwell/ampo-turn in an inch cube). Equations (7) and (8) are combined to yield the magnetic flux equation 8 N.8 xA1l 0.628X106 A1 t E sinwt = 10-8 No + 0.628 A 1 + A ( 1 + -xdt m' NA1 A CNA1 A2 0 (9) Its approrriate initial condition is 4(0) = 0. Em sinwt in Equation (9) is the impressed voltage or the forcing function of the physical system. In case of the "moving-coil" type compressor, the nonlinear differential Equation (1) together with its appropriate initial conditions provide a complete statement of the problem. However, for the "moving-iron" type compressor, Equation (1) is coupled with the integrodifferential Equations (9). They have to be solved simultaneously for x(t) and f(t). Numerical reductions for both cases are performed by means of the finite difference technique. Equation (1) may be rewritten in finite difference form as

-12 - x(J) = {Mg + k[e-x(J-l)] + Ap Pc(J-l) - Ap PS + Ab Pb(J-l) - Ab PS _ C ]A2 I A1 + 1 -} M(At ) 72X10A1 A2 I ' where J = 1 corresponds to t = 0, t = (J-l)At, and time interval used in numerical reduction. Equation (9) duced to the differential equation by a differentiation. equation may then be expressed in finite difference form (10) At is the is first reThe resulting as )(J)= {EmW cos [w)(J-l)At] + [ 2X108N + 0.628 1 + Al 2lO-8 N + 0.628R 1 + -1( (At)2 NA1 A2 x(J-2) 0.628 X106 l (1 +_ x(J-) (J-l) - 08N (J-2)} At NCA1 A2 (At)2 10-N + 0.628R x(J-l) [0 + _ 1+ l J-( 2 NA A At'A2 t (At) N1 N 2I The appropriate initial conditions for the last two equations are x(l) = x(2) = xo and 0(1) = 0(2) = 0.

RESULTS AND DISCUSSION An examination of Equation (1) reveals that the displacementtime characteristics of the compressor is functions of the mass of the vibrating parts M, the spring constants kI and kII, the initial spring forces as represented by iI and eII, the geometrical configuration of the piston as described by A and Ab, the initial location of the piston xo, the suction pressure PS, the polytropic exponent n, the piston position at the maximum buffer space xb, and the magnetic force F. As shown by Equation (4), the magnetic force for the "moving-coil" type compressor depends upon the number of coil turns N, the diameter of coil turns d, the flux density of the magnetic field B, the resistance of the coil R and the impressed voltage Em and frequency c. On the other hand the magnetic force of the "movingiron" type compressor depends upon the magnetic flux ( and the areas of the flux cross section of the inner and outer air gaps Al and A2 This magnetic flux is coupled with the displacement governed by Equation (9). Therefore, the magnetic flux as well as the displacement are functions of the impressed voltage Em and frequency w, the number of coil turns N, the coil resistance R, the capacity of the condenser C and the areas of the flux cross section of the air gaps. The numerical computation was performed by means of an IBM 7090 digital computer for both compressors. A set of input data were selected: Ap = 1.0 in.2, Ab = 0.25 in.2, PS = 19.2 psia, Pd =192 psia, kII = 0 lbf/in., xa = 0 in., xb = 0.05 in., xo = 0.05 in., i = 0.0454 in., R = 40 ohms, c = 60 cycles/sec. N = 725 turns, C=OIf, -13 -

-14 - Em = 110 volts, d = 1 in., B = 80,000 maxwell/in.2, A1 = A2 = 14 in.2 and v = 1.41. Two values each of the important quantities M, kI and n were selected to investigate the role they play in the dynamic characteristics of the compressors: M = 0.696 and 0.812 lbm, kI = 300 and 550 lbf/in., and n = 1.0 for isothermal process and 1.4 for isentropic process (in case of air). The results are presented in graphical form in Figure 3 for the "moving-coil" type compressor and in Figures 4 through 8 for the "moving-iron" type compressor. It is seen in Figure 3 that the magnetic force of the "movingcoil" type varies sinusoidally like the impressed voltage or current. Immediately following the starting, the piston moves toward the cylinder head. The compression stroke fails because it yields a low compression ratio. A full force cycle following the first positive maximum value of the force results in the first full compressor cycle. Only the compressor having M = 0.812 lbm (0.0021 lbf-sec. /in.) and kI = 300 lbf/ in. gives the desired compression ratio of ten. More importantly, this compressor has almost attained the steady periodic operation after the second force cycle. The period of each compressor cycle is identical with that of the impressed voltage. The other two compressors having M = 0.812 Ibm, kI = 500 lbf/in. and M = 0.696 lbm, kI = 550 lbf/in. produce the compression ratio of less than ten with irregular vibrating periods. In addition, the piston undergoes two compression strokes with entirely different compression ratios in every compressor cycle. One of these compression ratios is too small to be of any use. Therefore the compressor having M = 0.812 lbm and kI = 300 lbf/ino is better than the other two.

0.30 0.25 0.20 0.15 - | - M = 0.812 LBM, K}= 300 LBF/IN. CID 0 20 30 40 0 70 80 -005 550 X 0110 SECONDS -0.15 M0200.812 LBM, K1= 300 LBF/IN. -0.25 x - 0812 550., --- 0.696 550 -0,30 Figure 3. Dynamic Characteristics of "Moving-Coil" Type Electromagnetic Compressor for n = 1.41. I I

3.0. 2,5 0 F -I 1.5 I0 I I I i I I -0i,5 MI - I.5 I x CDl I090 Compressor or M = o.81 lbmy k, = 50 l II =.1and n - Iili,~u..' I -1 -2.5 I II -3,0 Figure 4. Dynamic Characteristics of Moving-Iron Type Electromagnetic Compressor for M = 0.812 ibm, kI 550 lbf/in., kii = 0, and n = 1.0. I I

An I'a 3.0 i- 2.5 ~1g5 I I -3:.5 Figure 5. Dynamic Characteristics of "Moving-Iron" Type Electromagnetic Compressor with M = 0.812 ibm, k1 = 300 lbf/in., k11 = 0, and n = 1.0. 'x-1, - C114~~~~ n = 1.0.~ ~

6.0 5.0 0 --.0 " 32,0 ' LL 1 0 to 10 20 2 30 L40 50 80 90 -1X0 90 -5.o0 -6.0 Figure 6. Dynamic Characteristics of "Moving-Iron" Type Electromagnetic Compressor with M = 0.812 ibm, kI = 550 lbf/in., kII = 0 and n = 1.41. H Oo i

5.0 400 CL 4.0 ^ -5.0 _ -6.0.,0,1 If! LA -10 ~r I h 0 01069 5 l / a 90 ~-1, O t CD /010 MILISECONDS '4-2.0 -;-3.0 -4,0 -5.0 -6,0 Figure 7- Dynamic Characteristics of "Moving-Iron" Type Electromagnetic Compressor with m =.696 lbm.,k 5 b/n., k = and n - 1.41. I \O I

6.0 ^r(, -5.0 -.0 I 20 -I 3: 0 with M = 0.812 lbm, k, = 3~00 bf/in., kII = 0 and n = 1.490 t MILISECONDS ~-2.0 - x -5.0 -6.0 Figure 8. Dynamic Characteristics of "Moving-Iron" Type Electromagnetic Compressor with M = 0.812 ibm, kI = 300 lbf/in., kII= 0 and n =1.41. I r) 0 1

-21 - The displacement-time, force-time, current-time and magnetic flux-time characteristics of the "moving-iron" type compressor are graphically illustrated in Figures 4 through 8. It is disclosed by examining these figures that in the same time interval of 83.3 miliseconds, this compressor may perform twice as many compression cycles as the previous type since two cycles of the magnetic force are generated in each cycle of the impressed voltage. This is the principal advantage of the "moving-iron" type compressor. However it is rather difficult or time consuming to find out an appropriate set of the system variables for a desired performance of the compressor. Only five sets of the system variables are studied in the text. The results are presented in Figures 4 through 8. It is disclosed that the patterns and magnitudes of the variations in the magnetic flux, magnetic force, current and displacement in the first 10 miliseconds are about the same for the five cases. During the 10 milisecond time interval, both the magnetic flux and force have attained their highest maxima, while the current vibrates twice and the piston performs two compression strokes. The effect of the polytropic exponent n may be found by comparing Figures 4 and 6. After 50 miliseconds has elapsed, the compressor undergoing an isothermal process has come closer to an almost steadyperiodic operation than the compressor undergoing an adiabatic process, although in practice an adiabatic process is more likely to occur. It is interesting to note that the current-time variations in Figures 4 and 6 are of the "M" shape for positive current and of the "W" shape for negative current.

-22 - As was revealed by comparing Figures 6 and 7, the increase in the mass of the moving part from 0.696 Ibm to 0.812 Ibm results in an increase in the swing displacement of the piston and consequently the compression ratio. A comparison of Figures 6 and 8 for adiabatic processes shows that the reduction in the spring constant kI from 550 lbf/in. to 300 lbf/in. results in a higher compression ratio and a smoother operating condition. However, a comparison of Figures 4 and 5 for isothermal processes indicated that the effect of the spring constant on the compression ratio is insignificant.

CONCLUDING REMARKS It is rather hard to draw a conclusion on the effects of each system variable on the dynamic characteristics of the compressors from the limited numerical results. However, it is obviously feasible to find out an appropriate set of the system variables to design a compressor with the desired performance. The "moving-iron" type compressor has an undefeatable advantage over the "moving-coil" type, that is, twice the pumping rates. The penalty the former type has to pay is more computations. -23 -

PROGRAM FOR UNSTEADY BEHAVIORS OF MOVING-COIL TYPE COMPRESSOR -24 -

-25 - $COvMPILE FASTRAN,EXECUTE, I/) DUiljP,PRIiNT IBJECT,PlJUNCH UBJECT C PR(.IGRAMI FOR LW\ISTEADY BEHA\VIORS OF IMOVIING-C(.IIL TYPE CUiJiPREkSSUkDIiMENSIOiN X(5), PHI( 5)_ __ READ INPUT TAPE 7,2003,A,BSO,XL,XKAP,AB-,PD,PS,Xl'iASSw!t,V __60 READ INPUT TAPE 7,2005,CAPACtRESISTWIIID\,S,DELTIiv,LIVmIT,)IA, GFX,__ 1V OL T WRITE OUTPUT TAPE 6,3002 WRlRITE (UTPUT TAPE 6,3003,A,B, SOXL,XKAP,AHPl), PS,V WRITE OUTPUT TAPE 6,3004,CAPAC,RESIST, W I NiDS,O EL I MLI, L I IT,X,,ASS,IMi WRITE UlT PUT TAPE 6,3005?,IIA, GFX,VOLT S I G=PD/PS S1=S 0 / SIG ( 1./V) ABPS= A B:PS APPSSO=AP 'PS SO: *V APPSS1=APPSSO/S1*V _ ABPSA=B ABPS P-A) P S APPS=APPS ____ _ DELTS = )ELT I vt i,* 2 SO 1 =S ) PC 1 = 170.0 - PC2=1.OE-8 INDS ___ ___ _ PC3=.468-' RES ST/7 I S NI DS _ ___C_. _0______ __ _______ _.___ ___ _ ____________________________ PC5=DELTS. P C 5 = O) E L T SO E I'lFC OlN1 = 1. (4E -8 100 D Mvi = DELTS ) / X l ASS X( 1 )=SO __ _ ___ _ __ T =.) E L T I M HE iN F = 0 X ( 2 ) =.5 ( ( X K (X L- X( 1 ) ) - F )= I;i + 2. X( 1 ) ) L=3 C --— PHASE 2 OF CYCLE --- IA =3_ __ _ 70C) )DO 808 I=IAHLIlnI" X I I -1__ _ _ _____ ___ _ ______ _____ ____ -_______ -T=X I M' DELT I M E Mi F=. 278 W I \l D2)S l) I A G F X V.1L T /RES I S-r S I i\', ( WX VT ) ': 1. (O* E -6 X( L)=(XK ( XL- X( L-1 ) ) +ABPSB/ ( X(L-1 )-A) 44V+AP SSO/X ( L-1 ); v-AtS1 A P P S-E Iv F ) "D19 +2. X ( L- )-X ( L-2 ) WRITE O:.ITPT TrAPE 6,500(2,X(L), T, ErIF X(1)=X(L-2) ___ X(2)=X( L-1) X (3)=X(L) L=4 IKEEP=I _ S 2=S 1/S I ':/; ( 1. / V ) IF (X(3)-S02) 151,800,800 _ ___ __ __ 800 RATE=X ( 3 )-X ( ) IF (RATE) 808,802,803)3 808 C 11NT r, I.J LE G_ _ T C] T) ____ _ ____ _ 8 C03 RATENi =X( 3)-X( 1 ) IF (RA_ -E+RA EN) 802,805,805_ _ _____________ 805 IKEEP=IKEEP-1 8(0)2 XIM= IKEEP 4 (') T A L = X I, [) L T I i.' TU'TAL=XIu9'v4OLELTI 'l W.RITE OLJTUIT PUT T1APE 6,O()1, X ( ), 7,.TOTL, E F C --— PHASE 2R OF CYCLEH(01 IAA=IKEEP ^ A P I A + - - " " ' -r - - - - - -- - - - - - - - - —.- - -

-26-.D0 806 I=IAAP,LIlMvl XI I I - 1 T = X I M DELI_ I 1__ EMvF=.278*W INDS'D IA*' GFX*VOLT/RES I ST*S IN (W*T ). OE-6 __X (L)= XK", (XL-X (L-1) )+ABPSBA/( X(L-1 )-A) **V+APPSSO/X ( L-1) **V-ABPS-AP 1 P S- E vi F ) V+2, *X ( L- 1 ) - X ( L-2 ) W RITE OUTPUT T1 APE 6,5002,X(L),T,EM-.F X(1)=X(L-2) X( 2)=X(L-1) X(3)=X(L) L=4 _ IKEEP=I IF (X(3)-B) 807,807,810 807 RATE=X(3)-X( 1 ) IF (RATE) 809,806,806 809 IAB=IKEEP * X I i M= I KEE P TOTAL=X I I'*DELT I i WRITE OUTPUT TAPE 6,5001,X(3),TOTALEMF GO TO 700() 806 CONT I NUE _ ___ _ GO TO 60() C --— PHASE 1 ROF CYCLE ---810 IAC=IKEEP IACP=IAC+1 DO 811 I=IACP,LIIiIT X I Mr = I-_ ___ ______ T = X I Iv:'[ E L T I i E MiF = 278.W I NDS) I A* GFX*VOLT/RES I ST*SI N ( WI T ) *1.OE-6 X ( L )=( XK*( XL-X ( L-1 ) )+APPSSO/X ( L-1 )**V-APPS-EMF );viDM+2. 'X ( L-l )-X( L1 2) WRITE )OUTPUT TAPE 6,5002,X(L),T,EMF X( 1)=X(L-2) X(2)=X(L-1) X(3)=X(L) L=4 IKEEP=I IF (X(3)-SO1) 814,814,812 814 RATE=X( 3)-X()_ _ IF (RATE) 840,840,811 840 IABP=IKEEP IAB=IABP+1 - XI M=IKEEP T10 TAL=X I lM*DELT I Mv _ _WRITE OUTPUT TAPE 6,5001,X(3),TOTAL,EMF IF (X(3)-B) 700,700,1001 1001 SO1R=X(3)_ _ S01=SO1R I END=IKEEP GO TO 1002 811 CONTIN UE GO TO 60 C --— PHASE 6 OF CYCLE ---- 812 IAE=IKEEP I AEP= I AE+1 DO 813 I=IAEP,LIMIT X I M= I-1 T=X I M;:)ELT I M\ EMF=.278*WIN D S *IA G FX. VOLT/RES IS T SI! ( W*T ) *1.OE-6 X(L)=(XK*(XL-X(L-1))-EMF)*DM+2.* X(L-1)-X(L-2)

-27 - WRITE OUTPUT TAPE 6,5002,X(L),T,EMF X( 1)X(L-2) X(2)=X(L-1) X(3)=X(L) L=4 IKEEP=I RATE=X(3)-X(1) IF (RATE) 816,815,813 813 CONTINUE ___ _____ GO TO 60 816 RATEN=X(3)-X( 1 )__ IF (RATE+RATEN) 815,817,817 817 IKEEP=IKEEP-1 815 IEND=IKEEP X I M = I K E E P_ TOTAL=X IM *DELTI M WRITE OUTPUT TAPE 6,5001,X(3),TOTAL,E4MF C --— PHASE 1 OF CYCLE ---SOLAST=X(3),SO1=SOLAST 1002 A P P S S O = A P - P S l SO (1: ', V_ _ I AF= I EiN IAFP=IAF+1 _ DO 830 I=IAFP,LIMIT X I ~ = I -1 T = X I M:' EL T I M EIvF=. 278 ' -l' I t\l -)S' DIA * (;F X ' VLT/RESIST'S I Nl ( W *TP T 1).OE -6 - -x(L)=( XK*(XL-X( L-1 ))+ APPSSO/X (L-1) --- V-APPS-E'IF)lvl+2.,X( L-I )-X( L1 2) iWRITE OUTPUT TAPE 6,5002,X(L),T,EMF X(1)=X(L-2) X(2)=X (L-1) X(3)=X (L)_ _ L=4 IKEEP=I_ _ _ _ _ IF (X(3) —) 831-,832.T832 832 RATE=X(3)-X(). IF (RATE) 830,835,835 835 X IM=I KEEP __ ____ I. )T AL=X I i;: OELT I Mi VWRITE OUTPU-' TA APE 6,50 0l1,X(3),TtTAL,E IVIF__..) -rT 8].() 830 CO T I N\U LE G([ T O 60 831 IAB=IKEEP+1 GO TO 700 C --— PHASE 3 OF CYCLE ---151 IC=IKEEP ICP=IC+1 S02=SO1/SIG (./V) -APPSS1=AP*PS* SOl 01 V __ __ _ _ __ __ __ ____________ _______ A P P S S 0 = A P;: P S ': S (01 1-'- V APPSS1=APPSSO/SO2':-' *V DO 170 I=ICP,LIi'I'T ___ _ X I i= I- 1 T=X II,': E LT I v E F =. 278 1 ^I Si)S I /A;; F X: V ) L T/ k E S I S' S I i ( I. ' T ): 1, OE-6 X ( L ) ( XK ( XL-X ( L-1 ) )+APS,/( X ( L-1 ) - ) >::+APPSS1-A tPS-AP PS-EiF ); IDM 1+2. -~ X(L-1)-X(L-2) - 'iR ITE.I..) T P 1APE 6,b ()2,X ( L ), T l, I.-___ X(1)=X(L-2)

-28 - X(2)=X(Il -l) X( 3)=X(L) I_ =_ _ _ __= 4 ____ ________ _______ ________ __ ______ _____ IKEEP=I _RA AE=X( 3)-X( 1 __ _____________ IF (RATE) 170,185,180 170 C I.,) T I I UE_ GO T (.J S 1.0 R ATEN..=X _ 3) -X(1 _______ IF (RATE+RATEN'l) 185,181,181 181 IKEEP=IKEEP-1 185 ID=IKEEP S2=X(2)-(X(2)-X(1) ) =,-RATE/(RATE-RATEN) S3=S2''SI, '- ( 1 l./V) X I IiM = I__ __ A P PS S0 = A P: P SS'3 ':: V T_ UTAL =X I Il:) EL T I _ __iv S01=S3 tjR.ITE OUT PU TAPE 6,5001,X(3),TOTAL,EMF IKEEP=IKEEP+1 "IF ( B-S3)801,8(01,900 _ __ C --— PHASE 4 IOF CYCLE ---900( I K E =I I _ __ _ ___ _ _ P = EPI A()= IKE E P I A O P = I A. o + 1 00 902 I= IA(P,LIi. IT X I i = I-___ T= X I lU-: 0 E LT Ii _EvM F =. 2 78 ' W I * l.) S; ) I ' F X 'V. L / R ES S I S T S I ixU ( _^", T ) -_____ 1.t__ '__ X ( L ) = ( XK ( X L- ( L - S L-1 T aX - ) )+ / + A P P S SSO/X ( L-1 ) \!-A, P S -AP 1 PS-E iF ) ':. +?.;: X ( L-1 ) -X ( L -2 )!RITE, I TE I.J U TiPE 6, 5002, X ( ) T, T,E F X (1) =X (L-2) X(2)=X(L-1) X(3)=X( L) L=4 I KEE P= I IF (X(3)-S3) 904,t904,9()6 90 )4 R A Ti-E=X ( 3 ) -X ( 1 _ _ ).._ __ _ _ __ IF (RATE) 908, 908(,902 90()2 C O T I,i\!E _ _ _E _ __ _ _ __ () TO 60 (908 IAI =IKEEP X I i = I K ' E P T OTl A E-L X L_ I'_ _ _I..__,_ -. _L_ ________'___ 1R ITE t ITP T APE 6, 5()()1, X (3 ), TOT AL, tlF GCU TO ( 70 ( C --— PHASE 5 iF CYCLE --- '906) I'aP=IKEEP IA PP = I AP + 1)ti 910 I=IAPP,LIiiiIT X I I, = -1 1 'T = X I *oi' - '.I E L ' 1" ____ I - ivF=. 278-: OS, F, i 7XILT/RS:I 'S S I ( X 'T -) 1. ) E-6 X ( L ) = ( XK* ( XL-X( L-1 ) ) +AiHPSSA/ ( X ( L-1 ) -A) **V-AKPS- iiF ) *+2 * X ( L-1 ) -X 1 (L-~) Ik RITE O l TPOT T l.iP-_6 _ X (,5 )_,, u2, X _ (_ __ ____ X(.1)=X(L-2 ) X ( 2) =X ( - ) X( ) =X ( L )

-29 - L=4 IKEEP=I IF (X(3)-B) 912,912,812___ ___. 912 RATE=X(3)-X(1) IF (RATE) 914,914,910 910 CONT INUE GO:) TO 60 914 X IM = I KEEP T (JTAL=X I V, 'l)DEL I l l_ WRITE OUTPUT TAPE 6, 5001,X(3) TOTAL, EIvF SO1=X() A P P S S = A P 'P S S 1 -': V IAB=IKEEP+1 GO TO 70( 2003 F[ORMAT (10F6.O,Fl(. 0,F10l.1) 2005 FlRPI AT (3F6.1,F8.6, 14, 3F10.2 ) 3002 FORMAT (52H1UNISTEADY-STATE DYNAMIC ANALYSIS OF AXIAL COMPRESSOR) 3003 FORMAT (3H A=,F10.5, 3H 3t=,F10.5,4H SO=,F10.5,4H X =,tF10.5', 14H XK=,F1O.5,4H AP=,F1O.5,4H AB=,F1O.5,4H PD=,FlO.5,4H PS=,F1O.5,. 23H V=,F5.1) 3004 FORMAT (7H CAPAC=,F6.1,8H RESIST=,F6.1 7H WINDS=,F6. 1,H DELT I =, 1FB.6,7H LIMIIT= I4,7H XMIASS=,F1.5,3H!=,F10.6) 3005 FORMAT (5H DIA=,F10.2,6H vIGFX=,F10.2,6H VOLT=,F10.2) 5001 FORMVAT (6H XEND=,F9.5,6H TIME=,F7.5,7H FORCE=, 1F10.5) 5002 FORM1AT (3H X=, F9.5,6H TIlvlE=,F7.5,7H FORCE=, 1F10.5) END $DATA

PROGRAM FOR UNSTEADY BEHAVIORS OF MOVING-IRON TYPE COMPRESSOR -30 -

-31 - DIMENSION X 5 ),PHI(5) READ INPUT TAPE 7,200 3,A,B,SO,XL, XK,AP, AB, PD,PSXMASSW,V 60 READ INPUT TAPE 7,2005,CAPAC,RESI ST,WINDS,DELTIM, LI MIT WRITE OUTPUT TAPE 6,3002 WRITE OUTPUT TAPE 6,3003,A, 6,SO0,XL,XK,AP,AB,PD,PS,V WRITE OUTPUT TAPE 6,3004,CAPAC,RESISTWINDS,DELTIM,LIMIT,XMASSW SIG=P D/PS S1=SO/SIG **(./V) ABP S= AB*P S APPSSO=AP*PS*SO **V APPSS l=APPSSO/Sl**V ABP SBA=AB*PS* (B-A ) **V APPS=AP*PS DEL TSQ=DELT IM**2 SO1=SO PC1=1 7i.0 *W PC2= 1.OF- 8* I ND S PC3=.468*RES I ST IW INDS PCt=0.0 PC5=DELTSO EMFCON=1.04E-8 100 DM=DELTSQ/XMASS X( 1)=SO T=DELTIM PHI ( 1 )=C. PHI(2)=PHI( 1) EMF=EMFCON* PHI( 2)** 2 X(2)=.5*( (XK*(XL-X( 1) )-EMF)*DM+2. X( 1)) L=3 C -— PHASE 2 OF CYCLE ---IAB=3 700 Dn 808 I=IAB,LIMIT X IM= I-I T=XIM*DEL TIM PHIA=PC 1*CUS(W*T) + ( 2. *PC2/PC5+PC3*X( L-2)/DELT I M 1 -PC4*X(L-1 )*PHI L-l-PC2*PHIL-I )-(L-2)/PC5 PHI =tPC2/PC5+PC'3*X(L-1)/DELTIM PHI (L )=PHIA/PHIB EMF=EMFCON*PHI( L)**2 PHI (1)=PHI(L-2) PHI (2)=PHI (L-) PHI ( )=PHI (L) X(L)=(XKW(XL- X(L-1 ) +AtPS[A/( X(L-1)-A)**V+APPSSO/X(L-1)**V-ABPS1 APPS-EM F )*DM+2.*X(L-1)-X(L-2) CUR T=.468X( L )*PHI( L)/WINDS WRITE OUTPUT TAPE 6,502,tX L) Tt PH I(L),EMFtCURT X ( ) = X L L-2) X(2)=X(L-1)

-32 - X(3)=X(L) L =4 IKEEP=I S02=S01/S IG** ( 1./V) IF (X(3)-502) 151,800,800 800 RATE=X(3)-X 1) IF (RATE) 808,802,803 808 CONTINUE GO TO 60 803 RATEN=X(3)-X(1) IF (RATF+RATEN) 802,805,.805 805 IKEEP=IKFEP-1 802 XIM= IKEFP TOTAL =X IM*DELT IM CURT=.468*X (3 )*PHI( 3)/WINDS WRITE OUTPUT TAPE 6 5 001,X( 3) TOTAL,PHI 3) EMFCURT C --— PHASE 2R OF CYCLE ---801 IAA=IKEEP I AAP= IAA+l DO 806 I=IAAPLIMIT "XIM=I-1 T=XI M*DEL TIM PHIA=PC *COS(W*T)+(2.*PC2/PC5+PC3*X(L-2)/DELTIM-PC4*X(L-1))*PHI(L1 1)-PC2*PHI(L-2)/PC5 PHI B=PC2/PC5+PC 3*X(L- 1)/DEL TIM PHI (L )=PHIA/PHI B EMF=EMFCUN*PH ( L)* 2 PHI (1)=PHI(L-2) PHI (2)=PHI(L-1) PHI( 3)=PHI(L) X(L)=(XK*(XL-X(L-1))+ABPSBA/(X(L-1)-A)**V+APPSSO/X(L-1)**V-ABPS-AP 1PS-EMF) *DM+2.*X ( L- )-X(L-2) CURT=.468*X( L )*PHI( L) /WINDS WRITF OUTPUT TAPE 6,500 2 X( L),T, PHI (L),EMF CURT X(1)=X(L-2) X(2)=X(L-1) X(3)=X(L) L=4 IKEEP=I IF (X(3)-B) 807,807,810 807 RATE=X(3)-X( 1 ) IF (RATE) 809,806,806 809 IAB=IKEFP X IM=I KEP TOTAL=X IM*DELTIM CURT=.468X( 3)*PHI(3)/WINDS WRITE OUTPUT TAPE 5,500 1,X( 3),TOTAL,PHI (3 ),EMF,CURT GO TO 7C,0 806 C ONT I NUE GO1 TO 60 C --— PHASE 1P OF CYCLF --- 810 IAC=IKEFP I ACP= 4A+1 00 811 I= IACP,L IMIT XIM=I-1 T=X 1 M*3F LTI M P I A=PC l1*COS( W',T)+(2.*PC2/PC5+PC3*X(L-2)/DELTIM

-33 - 1 -PC4*X(L-1 )*PH I L-1)-PC2*PHI (L-2)/PC5 PHI B= PC2/PC5+PC3*X(L-1 )/DELTIM PHI(L)=PHIA/PHI B EMF=EMFC ON* PHI (L) **2 PHI (1 )=PHI(L-2) PHI (2)=PHI(L-1) PHI (3)=PHI(L) X(L)=(XK(XL-X(L-1) )+APPSSO/X(L- 1)**V-APPS-EMF)*DM+2.*XL-1)-X(L1 2) CURT=.468*X(L)*PHI (L)/WINDS WRITE OUTPUT TAPE 6,5002,X(L),T,PHI(L),EMF,CURT X(1)=X(L-2) X(2)=X(L-1) X(3)=X(L) L=4 IKEEP=I IF (X(3)-S01) 814,814,812 814 RATE=X(3)-X(1) IF (RATE) 840,840,811 840 JABP=IKEEP I AB= I A3P+ 1 XIM=IKFEP T OTAL=XI M*DELTIM CURT=.468*X(3)*PHI(3)/WINDS WRITF OUTPUT TAPE 6,5001,X(3),TOTAL,PHI(3),EMF,CURT IF (X(3)-B8 700,700,1001 1001 S01R=X(3) S01=S01R IEN= IKEE P GO TO 1002 811 CONTINUE GO TO 60 C --— PHASE 6 OF CYCLE --- — 812 IAF=IKEEP I AE P= IAE+1 DO 813 I=IAEP,LIMIT X IM=I-1 T=X IM*OELTIM PHI A=PCl! *C3S(W*T)+( 2.*PC 2/PC5+PC3*X(L-2) DELT IM-PC4*X (L-1) )*PHI (L1 ) -PC2*PHI (L-2 1 /PC5 PHI B= PC2/PC5+ PC3* X (L-1 /DELTIM PHI (L )=PHIA/PHI B EMF=EMFCON*PHI ( L)**2 PHI ( 1 )=PHI (L- 2) PHI2 2)=PHI( L- 1) PHI(3)=PHI(L) X(L)= (XK* (XL-X(L-) )-EMF)*DOM+2.*X(L-1 )-X(L-2) CURT=.468*X(L )PHI (L) /WINDS WRITE OUTPUtT TAPE 6,5002,X(L),T,PHI(L),EMF,CURT X( 1 )=X( L- 2) X(2)=X(L-1) X(3)=X(L) L=4 IKEEP=I RATE=X( ) -X( 1 ) IF (RATF) 816,815,813 813 CONTIN JF

-34 - GO TO 60 816 R.ATEN=X( 3)-X( 1) IF (RATF+RATEN) 815,817,817 817 IKEEP=IKEEP-1 815 I ENO=IKEEP XIM=IKEEP TOTAL=XIM*DEL TIM CURT=.468*X (3)*PHI (3 /WINDS WRITE OUTPUT TAPE 6,5001,X( 3),TOTAL,PHI ( 3),EMF,CURT C --— PHASE 1 OF CYCLE ---- SOLAST=X( 3) SO1= SOLAST 1002 APPSSO=AP*PS*SO 1**V IAF=IEND IAFP=IAF+1 DO 830 I=IAFP,LIMIT X IM=I-1 T=X IM*DEL TIM PHIA=PCCOS(W*T) + ( 2.*PC2/PC5+PC3*X(L-2)/DELTIM-PC4*X(L-1) )*PHI(L1 1)-PC 2*PHI (L-2 )/PC5 PRHIB= PC2/PC5+PC3*X(L- 1 )/DELTIM PHI ( L )=PHA/PHI B EMF=F MFCON* PH I (L) ** 2 PHI (1 )=PHI( L- 2) PHI(2)=PHI(L-1) PHI (3)=PHI(L) X(L)=(XK*(XL-X( L-l) )+APPSSO/X(L-1 ) **V-APPS-EMF)*DM+2.*X(L-1)-X(L1 2) CURT=.468 *X( L)*PHI ( L) /WINDS WRITE OUTPUT TAPE 6,5002,X(L),TtPHI(L),EMF,CURT X(1)=X(L-2) X(2)=X(L-1) X(3)=X(L) L=4 IKEEP=I IF (X(3)-R) 831,832,832 832 RATE=X(3)-X( 1 ) IF (RATE) 830,835,835 835 XIM=IKEEP TOTAL=XIM*DEL TIM CURT=.468*X 3 )PHI (3) /WINDS WRITE J-UTPUT TAPE 6,5001,X(3),TOTTAL,PHI(3),EMF,CURT GO TO 810 830 CONTINUE GO TO 6C 831 IAB=IKEEP+1 GO TO 700 C --— PHASE 3 OF CYCLE ---151 IC=IKEEP ICP=IC+1 S02=S01/SIG **(./V) APP SSO=AP*P SSO I**V APPSS 1=APPSSO/S02**V DO 170 I=ICP, LIMIT XIM=I-1 T=X IM*D[LT IM PHIA=PC 1 C S( W* T )+(2.*PC2/PC5+PC3* X(L-2) /DEL TI M-PC4*X (L- 1) )*PHI (L

-35 - 1 1 )-PC2*PHI (L-2) /PC5 PHI B=PC 2/PC 5+PC3* X(L-1) /DEL TIM PHI (L )=PHIA/PHIB EMF= EMFCON*PH I( L)**2 PHI (1)=PHI(L-2) PHI (2)=PHI L-1) PHI ( 3)=PHI (L) X(L)=(XK*(XL-X(L-1))+ABPSBA/(X(L-1)-A)**V+APPSSI-ABPS-APPS-EMF)*DM 1+2. *X L-1)-X L-2) CURT=.468*X(L )*PHI(L)/WINDS WRITE OUTPUT TAPE 6,5002,X( L)T, PHI (L) EMF,CURT X(1)=X(L-2) X(2)=X(L-1) X(3)=X(L) L=4 IKEEP=I RATE=X(3)-X( 1 ) IF (RATE) 170,185,180 _JL7O CONT I NUE GO TO 60 180 RATEN=X(3)-X(1) IF (RATE+RATEN) 185,181,181 181 IKEEP=IKEEP-1 185 ID= IKEEP S2=X(2) -( X(2 ) -X(1 ) )*RA RAT( E-RATEN) S3=S2*SIG **(1./V) XIM=ID A PP S SO=AP*P S *S 3**V TOTAL=X IM*DELT I M S01=S3 CURT=.468*X(3 ) *PHI( 3)/WI NDS WRITE OUTPUT TAPE 6,5001,X(3),TOTAL,PHI(3),EMF,CURT IKEEP=IKEFP+1 IF (B-S3)801,801,900 C --— PHASE 4 OF CYCLE ---900 IKEEP=IKEEP- 1 I AQ=IKEEP I AQP=I AQ+1 DO 902 I=IAQP,LIMIT X IM=I-1 T=X IM*FELTIM PHIA=PC1*COSf W*T)+(2.*PC2/PC5+PC3*X(IL-2)/DELTIM-PC4*X(L-1))*PHI(L1 1)-PC2*PHI(L-2)/PC5 PHI = PC2/PC5+PC3*X(L- 1 )/ELTIM PHI (L)=PHIA/PHIB EMF=EMFC N*PHI (L)* 2 PHI( 1)=PHI(L-2) PHI(2)=PHI(L-1) PHI (3)=PHI( L) X(L)=(XKI (XL-X(L-1)) +ABPSBA/IX( L-1)-A)**V+APPSSO/X(L-l)**V-ABPS-AP 1 PS-EMF )*DM+2.*X(L- )-X(L-2) CURT=.468*X(L )*PHI IL) /WINDS WRITE OUTPUT TAPE 6,5002,X(L),T,PHI(L),EMF,CURT X(l)=X(L-2) X(2)=X(L-1) X(3)=X(L) L=4

-36 - I KEEP =I IF (X(3)-S3) 904,904,906 904 RATE=X(3)-X(1 ) IF (RATE) 908,908,902 902 CONTINUE GO TO 60 908 IAB=IKEEP XIM=IKEFP TOTAL=XIM*DELTIM CURT=.468*X(3)*PH I( 3)/WINDS WRITE OUT PUT TAPE 6,5001,X( 3),TOTAL,PHI( 3),EMF,CURT GO TO 700 C -—.PHASE 5 OF CYCLE ---906 IAP=IKEEP I APP= IAP+1 DO 910 I=IAPP,LIMIT XIM=I-1 T=XIM*DELTIM PHIA=PC1*COS(W*T)+(2.*PC2/PC5+PC3*X(L-2)/DELTIM-PC4*X(L-1))*PHI (L1 1 )-PC2*PHI( L-2)/PC5 PHI B=PC2/PC5+PC 3*X( L-1) /DELTIM PHI (L)=PHIA/PHIB EMF=EMFCON*PH I (L)*2 PHI (1)=PHI(L-2) PHI (2)=PHI (L-1) PHI ( 3)=PHI (L) X(L )=(XK*(XL-X( L-1) )+ABPSBA/(X(L-1)-A)**V-ABPS-EMF*DM+2.*X(L-1 )-X 1 (L-2) CURT=.468 *X (L)*PH I ( L)/WINDS WRITE OUTPUT TAPE 6,5002,X(L),T,PHI (L),EMF,CURT X(1)=X(L-2) X(2)=X(L-1) X(3)=X(L) L=4 IKEEP=I IF (X(3)-B) 912,912,812 912 RATE=X(3)-X(1) IF (RATE) 914,914,910 910 CONTINUEF GO TO 60 914 XIM=IKEEP TOTAL=X IM* DELT IM CURT=.468X (3 ) *PH I( 3 )/WI NDS WRITE OUITPUT TAPE 6,50301X(3),TOTAL,PHI(3),EMF,CURT S01=X(3) APPSSO=AP*PS*S 1**V IAB= IKEEP+1 GO TO 700 2003 FORMAT (10F6.0,F10.0,F 1.1) 2005 FORMAT (3F6.1,F8.6, 4) 3002 FORMAT (52H1UNSTEADY-STATE DYNAMIC ANALYSIS OF AXIAL COMPRESSOR) 3003 FORMAT (3H A=,F10.5, 3H B=,F10.5,4H SO=,F0..5,4H XL=,F10.5, 14H XK=,F10.5,4H AP=,F10.5,4H AB=,F10.5,4H PD=,F10.5,4H PS=,F10.5, 23H V=,F5.1) 3004 FORMAT (7H CAPAC=,F6.1,8H RESIST=,F6.1,7H WINDS=,F6.1,8H DELTIM=, 1F 8.6,7H LIMIT=, I4,7H XMASS=,F10.5,3H W=,F10.6) 5001 FORMAT (6H XEND=,F9.5,6H TIME=,F7.5,5H PHI=,F15.5.7H FORCE=,

REFERENCES 1. W. R. Woolrich, Handbook of Refrigerating Engineering, Volo 1 Fundamentals, 4th ed., Avi Publishing Co., Westport, Conn. (1965) Sect. III. 2. Compressed Air and Gas Handbook, 3rd Ed. (Revised) published by Compressed Air and Gas Institute, New York (1966) Chap. 2, 3. H. C. Roters, Electromagnetic Devices, John Wiley and Sons, New York (1941. 4. F. B. Canfield, J. K. Watson and A. L. Blancett, "Electromagnetic Gas Pump for Low Temperature Service", Physics Review of Scientific Instruments, Vol. 34, 1431-3 (1963). 5. G. J. VanWylen, Thermodynamics, John Wiley and Sons, New York (1962). -37 -