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1 References (1) Watton, J., Fluid Power Systems (1989), Prentice Hall, New Jersey. (2) Esposito, A., Fluid Power with Applications (1994), Prentice Hall, New Jersey. (3) Green, J. W., Aircraft Hydraulic Systems (1985), John Wiley and Sons, New Jersey. (4) Ogata, K., Modern Control Engineering (1997), Prentice Hall, New Jersey. (5) Merritt, H. E., Hydraulic Control System (1967), John Wiley and Sons, New Jersey. (6) Shames, I. H., Mechanics of Fluid (1962), McGraw-Hill, New York. (7) Sullivan, J. A., Fluid Power-Theory and Applications (1989), Prentice Hall, New Jersey. (8) Blackburn, J. F., Fluid Power Control (1960), MIT Press, Massachusetts. (9) Walters, R. B., Hydraulic and Electro-hydraulic Control Systems (1991), Elsevier Applied Science, Amsterdam. (10) Nishiumi, T., Konami, S. and Uchino, T., An Application of the Identification Method Using Self-excited Oscillation to a Hydraulic Motor/load System, Proc. of the 10th Bath International Fluid Power Workshop, Bath (1997), pp (11) Nishiumi, T. and Konami, S., A Design Technique of Hydraulic Servo Actuator Systems for Effective Power Transmission, Proc.ofthe4thJFPSInternational Symposium on Fluid Power, Tokyo (1999), pp (12) Konami, S., Nishiumi, T. and Hata, K., Identification of Linearized Electro- Hydraulic Servo-valve Dynamics by Analyzing Self-excited Oscillations, Journal of the Japan Hydraulic Pneumatic Society, Vol. 27, No. 4 (1997), pp (in Japanese). (13) Nishiumi, T., Ichiyanagi, T., Katoh, H. and Konami, S., Real-time Parameter Estimation of Hydraulic Servo Actuator Systems Using Self-excited Oscillation Method, Transactions of the Japan Fluid Power System Society, Vol. 36, No. 1 (2005), pp (in Japanese). 301

2 302 References (14) Ikebe, Y., Ikebe, J., Nakano, K. and Matushima, K., Servo mechanism and Elements (1965), Ohmsha Ltd., Tokyo (in Japanese). (15) Ishihara, T., Hydraulic Engineering (1968), Asakura Publishing Co., Ltd., Tokyo (in Japanese). (16) Ichikawa, T., Hydraulics and Hydrodynamics (1968), Asakura Publishing Co., Ltd., Tokyo (in Japanese). (17) Ichikawa, T. and Hibi, A., Hydraulic Engineering (1979), Asakura Publishing Co., Ltd., Tokyo (in Japanese). (18) Ichiryuu, K., Electro Hydraulic Control (1993), Nikkan Kogyo Shimbun, Ltd., Tokyo (in Japanese). (19) Imaki, K., Introduction to Hydraulic Engineering (1991), Rikogakusha Publishing Co., Ltd., Tokyo (in Japanese). (20) Koura, M. and Nakamura, T., Expounder of Hydraulic Equipment (1986), Seizando Shoten Publishing Co., Ltd., Tokyo (in Japanese). (21) Konami, S. and Degawa, T., Introduction to Control Engineering (1998), Gakukensha Publishing Co., Ltd., Tokyo (in Japanese). (22) Satoh, T., Design of Hydraulic Servo Systems (1980), Taiga Publishing Co., Ltd., Tokyo (in Japanese). (23) Tsuji, S., Hydraulic Engineering (1982), Nikkan Kogyo Shimbun, Ltd., Tokyo (in Japanese). (24) Takenaka, T. and Urata, E., Hydraulic Control (1975), Maruzen Co., Ltd., Tokyo (in Japanese). (25) Takenaka, T. and Urata, E., Hydraulic Engineering (1970), Yokendo Co., Ltd., Tokyo (in Japanese). (26) Yamaguchi, T. and Tanaka, H., Oil Hydraulics and Pneumatics (1986), Corona Publishing Co., Ltd., Tokyo (in Japanese). (27) Research Group in Dakin Ind. Ltd., Hydraulic Machinery Vol. 1 and Vol. 2 (1974), Japan Machinist-Sha Co., Ltd., Tokyo (in Japanese). (28) Research Group in Nachi Fujikoshi Corp., Elucidation of Oil Hydraulics Application Edition (1984), Japan Machinist-Sha Co., Ltd. (in Japanese). (29) Research Group in Nachi Fujikoshi Corp., Elucidation of Oil Hydraulics- Circuit and Data Edition (1989) Japan Machinist-Sha Co., Ltd. (in Japanese). (30) Japan Fluid Power System Society, Handbook of Oil Hydraulics and Pneumatics (1989), Ohmsha Ltd., Tokyo (in Japanese).

3 Appendix Table A.1 Graphic symbols for hydraulic diagrams. (1) Symbol elements and graphic symbols of mechanical elements. (Continued) 303

4 304 Appendix Table A.1 (Continued) (2) Symbols of hydraulic instruments. (Continued)

5 Appendix 305 Table A.1 (Continued) Table A.2 Prefixes in SI unit system and factor. Symbol Prefix Factor Symbol Prefix Factor G giga m milli M mega µ micro k kilo n nano c centi p pico

6 Table A.3 SI unit systems and other unit systems. Quantity SI unit systems Other unit systems Conversion factors Force newton [N] 1[N]=1[kg m/s 2 ] weight kilogram [kgf] 1[kgf]=1[kg] 9.81 [m/s 2 ] 1[kgf]=9.81[N] Mass kilogram [kg] [kgf/(m/s 2 )] 1 [kg] = [kgf/(m/s 2 )] Work (Energy) joule [J] calorie [cal] 1[kgf m] = 9.81 [J] 1[J]=1[Nm] 1 [kcal] = 427 [kgf m] 1 [cal] = 4.19 [J] Power Pressure watt [W] 1[W]=1[J/s] pascal [Pa] 1[Pa]=1[N/m 2 ] French Horse Power [PS] 1 [PS] = 75 [kgf m/s] kgf/cm 2,atm,bar 1 [atm] = [kgf/cm 2 ] 1[kgf m/s] = 9.81 [W] 1 [PS] = [W] 1[kgf/cm 2 ]= [Pa] 1[atm]= [Pa] 1[bar]=10 5 [Pa] Flow rate [m 3 /s] liter per minute [L/min] 1 [L/min] = [m 3 /s] Angular velocity [rad/s] revolutions per minute [rpm] 1 [rpm] = 1 [min 1 ] = [rad/s] Viscosity [Pa s] poise [P], centi-poise [cp] 1 [P] = 0.1 [Pa s] 1[cP]= [P] Kinematic viscosity [m 2 /s] stokes [St], centi-stokes [cst] 1[St]=1[cm 2 /s] 1[cSt]= [St] 1[St]= [m 2 /s] 306 Appendix

7 Answers to Problems Chapter N 2.2 Volume change: m κ = MPa C, W =3.33 kw N 2.6 f x = ρq u sin θ, Q 1 = 2.7 (1) p 1 = n 1.75 (2) p 1 = 1 p 2 p 2 n 2.8 Q ( ) 19/7 ( ) 4/7 2 d2 l1 = Q 1 d p 2 =0.663 p Q =4.19 cm 3 /s, Q/Q 0 =1+1.5(e/h) 2 l 2 Q(1 cos θ) Q(1 + cos θ), Q 2 = 2 2 Fig. 1 Leakage flow rate through bore-spool clearance. 307

8 308 Answers to Problems 2.11 p s =13.1MPa, h =68.4 µm [ ( ) ] (1) p = 3µvr2 0 r h 3 1 r 0 (2) F = 3πµvr4 0 2h 3 Chapter (1) 2πD p =29.6cm 3 /rev (2) W i =28.6kW (3) T a = 152 N m 3.2 T a =35.2N m 3.3 (1) T m = 989 N m (2) 2πD m = m 3 /rev, α =3.5 (3) ω m =26.4rad/s 3.4 η pm =0.9, p = µω 3.5 ω =5rad/s 3.6 Load torque T = Jθ a ω a sin ω a t Load power W = Jθ2 a ω3 a sin 2ω at 2 r 0 =64.9mm, W max /η m =4.41 kw 3.7 (1) V m1 = V 0(n 1) n ln n, (2) V ln n m2 = V 0 n 1 (3) Fig. 2 Mean non-dimensional velocity for the damping rate. 3.8 p =15.4MPa, F = N (Force opening the port is defined as positive) 3.9 x = πd 2 p 4δk =1.11 mm 4(πC d d p sin 2α + k)

9 3.10 p p s =1 1 [(2C da x cos α)/s] 1+(x/x 0 ) 3.11 (1) Static balancing force equation: Answers to Problems 309 =0.419 A(p s p c ) (ρq 2 /a 0 )=(10+n)kL Flow rate passing through the fixed orifice: 2(p s p c ) Q = C d a 0 ρ Flow rate passing through the variable orifice: 2(p c p L ) Q = C dv w(l nl) ρ (2) Q n2 = =1.024 Q n 1 p 2 = 10 + n 2 1+ p n δ = ρq2 0 A ( 2k(C d a) 2 1 2C da A 3.13 ( a0 ) 2 1 wl (1 n 2) 2 1+ ( a 0 wl ) ) 2 1 (1 n 1) 2 =6.42 Fig. 3 Hydraulic logic circuits Recalling Fig , Polytropic exponent of nitrogen gas is n =1.9for the mean pressure p s = ( )/2 = 16 MPa. Recalling Eq. (3.127) and η =0.95, [ ( ) ] V = 15 ( 15 )( ) 1/1.9 =1.5L.

310 Answers to Problems Chapter 4 4.1 Y (s) X(s) = s +3 s 2 +4s +5, For x(t) =u(t) :y(t) = 2e 2t sin[t +(π/4)] 4.2 t 0:y(t) = 4e t (t +1)+5, t<0:y(t) =0 4.

10 310 Answers to Problems Chapter Y (s) X(s) = s +3 s 2 +4s +5, For x(t) =u(t) :y(t) = 2e 2t sin[t +(π/4)] 4.2 t 0:y(t) = 4e t (t +1)+5, t<0:y(t) =0 4.3 G x (s) =G 1 (s) G 2 (s), G y (s) =G 2 (s)/g 1 (s) 4.4 G x (s) =K a K t, 4.5 G(s) = δ(s) δ R = 1280 s 2 +6s Fig. 4 K a K t K v K m s(t v s +1)(Ts+1)+K a K t K v K m Bode diagrams of the transfer functions. Fig. 5 Bode diagram compared with the straight approximations.

11 Answers to Problems 311 ( x ) ( (1) q =1.7C d 1 0.3x ) πd 2 ps p l d d 4 ρ (2) q = k 1 x + k 2 p l k 1 = 2.55πC ( dd x x ) 0 ps p l0 4 d d ρ k 2 = 1.7πC dd 2 ( x0 ) ( x ) d d ρ(ps p l0 ) 4.9 (1) A =5.6 cm 2 Servo valve: Type15L in Table 4.10 (2) V = F 4.10 (1) Y (s) U(s) = 4 s(s +4) (2) E(s) = R(s) 1+KG(s) (3) K 10 (4) (a) Y (s) R(s) = 4K(1 + T d s) s 2 +4(1+KT d )s +4K (b) T d > x c : q = C d(c + x) 2 ps p l C d(c x) 2 ps + p l 3ρ 3ρ q = k 1 x + k 2 p l k 1 = 2C d 3ρ [ (x0 + c) p s p l0 (x 0 c) p s + p l0 ] k 2 = C [ d 2 (x0 + c) 2 ] (x c)2 + 3ρ ps p l0 ps + p l V m =0.552 m/s, P s =9.96 MPa, A =9.04 cm 2, W p =1.91 kw 4.13 (1) K c =3, (2)G c = s s

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13 Index A absolute pressure, 19 accumulator, 5, 165 air-cooled heat exchanger, 171 all ports close, 144 all ports open, 144 axial piston pump, 78 B balance piston type relief valve, 5, 129 balanced vane pump, 80 bandwidth, 214 bent axis type axial piston pump, 78 Bernoulli s equation, 31 beta ratio, 175 bladder accumulator, 165 Blasius formula, 47 block diagram, 189 Bode diagram, 209 body force, 30 branch point, 189 break point angular frequency, 212 bulk modulus, 22 C cavitation, 70 characteristic equation, 199 characteristic root, 199 chattering, 129 check valve, 2, 142 choke, 40 closed loop control system, 190 closed loop transfer function, 191 command variable, 190 compressibility, 22 contaminant, 174 continuity equation, 35 control surface, 33 control volume, 33 controlled variable, 190 Couette flow, 52 counter balance valve, 136 cracking pressure, 6, 128 cushion device, 110 cut-off characteristics, 9 D damping coefficient, 202 damping ratio, 202 deceleration valve, 141 density, 19 derivative control action, 279 diaphragm accumulator, 6, 166 direct operated servo valve, 160 direct relief valve, 127 directional control valve, 2, 143 directional flow control logic valve element,

14 314 Index directional flow control valve, 142 displacement, 81 dominant roots, 219 driving characteristic curve, 260 dynamic characteristics, 198 E effective bulk modulus, 22 equation of continuity, 36 equation of state, 24, 168 equivalent time constant, 204 Euler s equation of motion, 32 external gear pump, 79 F feedback connection, 191 feedback control system, 190 feedback signal, 190 filter, 174 first order lag system, 201 fixed displacement pump, 4 flow coefficient, 42 flow control valve, 136 flow dividing valve, 139 flow rate, 3 flow rate control logic valve element, 155 Fly-By-Wire, 12 frequency characteristics, 209 frequency response, 209 frequency response method, 236 frequency transfer function, 208 full cutoff, 94 G gage pressure, 18 gain, 208 gain condition, 222 gain constant, 201 gain crossover frequency, 216 gain margin, 216 gas-loaded accumulator, 165 gear motor, 98 gear pump, 77 H Hagen Poiseuille equation, 46 heat exchanger, 171 heat transfer coefficient, 172 high-speed switching valve, 161 Hurwitz stability criterion, 223 hydraulic actuator, 1, 97 hydraulic control system, 187 hydraulic control valve, 117 hydraulic cylinder, 107 Hydraulic fluid, 25 hydraulic lock, 58 hydraulic motor, 77, 87 hydraulic pump, 1, 77 hydraulic source device, 5 hydraulic system, 1 hydraulic system component, 2, 164 hydrostatic bearing, 63 I in-line accumulator, 166 indicial response, 197 integral control action, 278 internal gear pump, 79 inverse Laplace transformation, 193 J journal bearing, 60 K kinematic viscosity, 20 kinetic energy, 31 L laminar flow, 45 Laplace operator, 193 Laplace transformation, 193 linear system, 199 load chart, 256 logic valve, 151 M mass flow rate, 29 mixed valve, 149

15 Index 315 modular stack valves, 150 momentum theory, 38 multi-control valve, 150 N natural angular frequency, 203 Navier Stokes equation, 32 Newton s law of viscosity, 20 Newtonian fluid, 20 Nikuradse s formula, 47 nominal filtration rating, 175 Nyquist stability criterion, 216 O oil hammer, 66 one-dimensional flow, 31 open loop transfer function, 191 orifice, 40 overall pump efficiency, 87 override pressure, 128 P Pascal s law, 18 phase angle, 208 phase angle condition, 222 phase crossover frequency, 216 phase lag compensator, 276 phase lead compensator, 273 phase lead-lag compensator, 278 phase margin, 216 PID controller, 278 pipe fitting, 180 pipe friction factor, 47 piston pump, 77 pole, 200, 221 poppet valve, 118 positive displacement pump, 2, 77 potential energy, 31 power, 3, 84 power density, 3 pressure, 17 pressure compensated flow control valve, 137 pressure control logic valve element, 154 pressure control valve, 127 pressure energy, 31 pressure reducing valve, 132 priority control valve, 140 proportional control action, 278 proportional solenoid control valve, 160 pulse width modulation, 162 R radial piston motor, 100 radial piston pump, 79 redundant system, 12 relief valve, 5, 127 reservoir, 176 resonance angular frequency, 213 resonance peak, 213 response speed, 198 Reynolds number, 45 root locus diagram, 220 rotary motor, 103 S sandwich valve, 150 second order lag system, 202 self-excited oscillation method, 227 sequence valve, 134 series compensation, 272 servo system, 187 servo valve, 159, 232 servo valve capacity, 249 servo valve capacity factor, 242 shut off valve, 144 shuttle valve, 142 solenoid, 147 spool valve, 118 spring-loaded accumulator, 165 squeeze effect, 75 stability, 198 stability limit, 216 standard cylinder, 110 standard second order lag system, 202 static system, 198 steady flow, 29

16 316 Index steady-state error, 206 steady-state position error, 207 steady-state velocity error, 207 stick slip, 109 straight-line approximation, 212 stream tube, 29 streamline, 28 summing point, 189 swash plate type axial piston pump, 4, 78 swept volume, 81 system, 33 T tank unit, 5 theoretical flow rate, 84 theoretical torque, 84 throttle valve, 137 throttle, 40 thrust efficiency, 105, 110 time constant, 201 torque, 83 torque efficiency, 85, 88 transfer function, 189, 199 turbulent flow, 45 two stage servo valve, 159 type-one system, 207 type-two system, 207 type-zero system, 207 U unbalanced vane pump, 80 undamped natural angular frequency, 202 under lapped spool, 240 unloading valve, 133 unsteady flow, 29 V vane motor, 99 vane pump, 77 variable displacement pump, 4, 93 viscosity, 20 volumetric efficiency, 85, 88 W water-cooled heat exchanger, 171 Z zero lapped spool, 240 zero, 221 Ziegler Nichols rules, 281

HYDRAULIC CONTROL SYSTEMS

HYDRAULIC CONTROL SYSTEMS Noah D. Manring Mechanical and Aerospace Engineering Department University of Missouri-Columbia WILEY John Wiley & Sons, Inc. vii Preface Introduction xiii XV FUNDAMENTALS 1 Fluid