Aeronautical Engineer’s Data Book - part 3 pdf

46 Aeronautical Engineer’s Data Book The parametric form of the equation is x = a sec ␪ , y = b tan ␪ where ␪ s the eccenteric angle. The equation of the tangent at (x 1 , y 1 ) is xx 1 yy 1 ᎏᎏ – ᎏᎏ = 1. a 2 b 2 Sine Wave (see Figure 2.13) y = a sin(bx + c) y = a cos(bx + c') = a sin(bx + c) (where c = c'+π/2) y = m sin bx + n cos bx = a sin(bx + c)  2 where a =  m 2 + n  , c = tan –1 (n/m). y axis x axis c/b a 2π/ b 0 Fig. 2.13 Sine wave Helix (see Figure 2.14) A helix is a curve generated by a point moving on a cylinder with the distance it transverses parallel to the axis of the cylinder being proportional to the angle of rotation about the axis: x = a cos ␪ y = a sin ␪ z = k ␪ where a = radius of cylinder, 2πk = pitch. 47 Fundamental dimensions and units a z y x 2π k Fig. 2.14 Helix 2.9 Useful references and standards For links to ‘The Reference Desk’ – a website containing over 6000 on-line units conversions ‘calculators’ – go to: www.flinthills.com/ ~ramsdale/EngZone/refer.htm United States Metric Association, go to: http://lamar.colostate.edu/~hillger/ This site contains links to over 20 units-related sites. For guidance on correct units usage go to: http://lamar.colostate.edu/~hillger/correct.htm Standards 1. ASTM/IEEE SI 10: 1997: Use of the SI system of units (replaces ASTM E380 and IEEE 268). 2. Taylor, B.N. Guide for the use of the Inter- national System of units (SI): 1995. NIST special publication No 8111. 48 Aeronautical Engineer’s Data Book 3. Federal Standard 376B: 1993: Preferred Metric Units for general use by the Federal Government. General Services Administra- tion, Washington DC, 20406. Section 3 Symbols and notations 3.1 Parameters and constants See Table 3.1. Table 3.1 Important parameters and constants Planck’s constant (h) Universal gas constant (R) Stefan–Boltzmann constant (␴) Acceleration due to gravity (g) Absolute zero Volume of 1 kg mol of ideal gas at 1 atm, 0°C Avagadro’s number (N) Speed of sound at sea level (a 0 ) Air pressure at sea level (p 0 ) 6.6260755 ϫ 10 –34 J s 8.314510 J/mol/K 5.67051 ϫ 10 –8 W/m 2 K 4 9.80665 m/s 2 (32.17405 ft/s 2 ) –273.16°C (–459.688°F) 22.41 m 3 6.023 ϫ 10 26 /kg mol 340.29 m/s (1116.44 ft/sec) 760 mmHg = 1.01325 ϫ 10 5 N/m 2 = 2116.22 lb/ft 2 Air temperature at sea level (T 0 ) 15.0°C (59°F) Air density at sea level (␳ 0 ) 1.22492 kg/m 3 (0.002378 slug/ft 3 ) Air dynamic viscosity at sea 1.4607 ϫ 10 –5 m 2 /s level (µ o ) (1.5723 ϫ 10 –4 ft 2 /s) 3.2 Weights of gases See Table 3.2. Table 3.2 Weights of gases Gas kg/m 3 lb/ft 3 Air 1.22569 0.07651 (at 59.0°C) Carbon dioxide 1.97702 0.12341 Carbon monoxide 1.25052 0.07806 Helium 0.17846 0.01114 Hydrogen 0.08988 0.005611 Nitrogen 1.25068 0.07807 Oxygen 1.42917 0.089212 All values at atmospheric pressure and 0°C. 50 Aeronautical Engineer’s Data Book 3.3 Densities of liquids at 0°C See Table 3.3. Table 3.3 Densities of liquids at 0°C Liquid kg/m 3 lb/ft 3 Specific gravity Water 1000 62.43 1 Sea water 1025 63.99 1.025 Jet fuel JP 1 800 49.9 0.8 JP 3 775 48.4 0.775 JP 4 785 49 0.785 JP 5 817 51 0.817 Kerosine 820 51.2 0.82 Alcohol 801 50 0.801 Gasoline (petrol) Benzine 720 899 44.9 56.12 0.72 0.899 Oil 890 55.56 0.89 3.4 Notation: aerodynamics and fluid mechanics See Table 3.4. Table 3.4 Notation: aerodynamics and fluid mechanics The complexity of aeronautics means that symbols may have several meanings, depending on the context in which they are used. a Lift curve slope. Acceleration or deceleration. a Local speed of sound. Radius of vortex core. a' Inertial or absolute acceleration. 0 Speed of sound at sea level. Tailplane zero a a a a a incidence lift coefficient. 1 Tailplane lift curve slope. 2 Elevator lift curve slope. 3 Elevator tab lift curve slope. ∞ Lift curve slope of an infinite span wing. h Local lift curve slope at spanwise co-ordinate h. a Local lift curve slope at spanwise co-ordinate y. y ac Aerodynamic centre. A Aspect ratio. Moment of inertia. Area. A State matrix. AF Activity factor of propeller. b Total wing-span (= 2s). Hinge moment b b coefficient slope. Rotational factor in propeller theory. General width. 1 Elevator hinge moment derivative with respect to ␣ T . 2 Elevator hinge moment derivative with respect to ␩ . 51 Symbols and notations Table 3.4 Continued b 3 Elevator hinge moment derivative with respect to ␤ ␩ . B Input matrix. Number of blades on a propeller. c Wing chord. Viscous damping coefficient. Pitot tube coefficient. c 0 Root chord. c t Tip chord. c Local chord at spanwise co-ordinate y. y cg Centre of gravity. cp Centre of pressure. C Output matrix. C C Coefficient of contraction. C D Total drag coefficient. C DO Zero lift drag coefficient. C f Frictional drag coefficient. C L Lift coefficient. C LW Wing lift coefficient. C LT Tailplane lift coefficient. C H Elevator hinge moment coefficient. C m Pitching moment coefficient. C MO Pitching moment coefficient about aerodynamic centre of wing. C n Yawing moment coefficient. C Pressure coefficient. Power coefficient for p propellers. C R Resultant force coefficient. C v Coefficient of velocity. CP Centre of pressure. D Drag. Propeller diameter. D' Drag in a lateral-directional perturbation. D Direction cosine matrix. Direct matrix. D c Camber drag. D f Friction drag. D Pressure drag. p D ␣ Incidence drag. f Coefficient of friction. F Aerodynamic force. Feed-forward path transfer function. Fractional flap chord. F c Aerodynamic force due to camber. F r Froude number. F ␣ Aerodynamic force due to incidence. F ␩ Elevator control force g Acceleration due to gravity. G Controlled system transfer function. h Height. Centre of gravity position on reference chord. Enthalpy (specific). h 0 Aerodynamic centre position. h F Fin height co-ordinate above roll axis. h m Controls-fixed manoeuvre point position on reference chord. h' m Controls-free manoeuvre point position on reference chord. 52 Aeronautical Engineer’s Data Book Table 3.4 Continued h n Controls-fixed neutral point position on reference chord. h' n Control-free neutral point position on reference chord. H Hinge moment. Feedback path transfer function. Total pressure. Shape factor. H F Fin span measured perpendicular to the roll axis. H m Controls fixed manoeuvre margin. H" m Controls free manoeuvre margin. i x Moment of inertia in roll (dimensionless). i Moment of inertia in pitch (dimensionless). y i z Moment of inertia in yaw (dimensionless). I" Normalized inertia. I x Moment of inertia in roll. I Moment of inertia in pitch. y I z Moment of inertia in yaw. J Propeller ratio of advance. Moment of inertia. j (or i) The imaginary operator ( ͙ –1 ෆ ). k Spring stiffness coefficient. Lift-dependent drag factor. Interference factor. k Centre of pressure coefficient. cp k d Cavitation number. k Pitch rate transfer function gain constant. M M M M L L L L L l l l K K k k k k q u Axial velocity transfer function gain constant. w Normal velocity transfer function gain constant. ␪ Pitch attitude transfer function gain constant. ␶ Turbo-jet engine gain constant. K Feedback gain. Circulation. Bulk modulus. K Feedback gain matrix. 0 Circulation at wing mid-section. n Controls-fixed static stability margin. K' n Controls-free static stability margin. l Lift per unit span. d Disc loading (helicopter). f Fin arm. t Tail arm. L Lift. Rolling moment. Temperature lapse rate. c Lift due to camber. w Wing lift. F Fin lift. T Tailplane lift. ␣ Lift due to incidence. m Mass. Strength of a source or sink (fluid mechanics). Hydraulic depth. m' Rate of mass flow. M Mach number. 0 Free stream Mach number. crit Critical Mach number. M Pitching moment. 0 Wing–body pitching moment. T Tailplane pitching moment 53 Symbols and notations Table 3.4 Continued n Frequency. Number of revs per second. Polytropic exponent. N Yawing moment. P P P o Origin of co-ordinates. p Roll rate perturbation. Static pressure in a fluid. P Power. Total pressure. 0 Stagnation pressure. s Static pressure. t Total pressure. q Pitch rate perturbation. A propeller coefficient. Discharge quantity. Q Dynamic pressure. r Yaw rate perturbation. General response variable. Radius vector. R Radius of turn. Resultant force. Characteristic gas constant. Re Reynolds number. s Wing semi-span. Laplace operator. Specific U U T T S S S entropy. Distance or displacement. S Wing area. B Projected body side reference area. F Fin reference area. T Tailplane reference area. t Time. Maximum airfoil section thickness. T Time constant. Thrust. Temperature. r Roll time constant. s Spiral time constant. u Velocity component. Internal energy. u Input vector. U Total axial velocity. e Axial component of steady equilibrium velocity. E Axial velocity component referred to datum-path V V earth axes. v Lateral velocity perturbation. v Eigenvector. V Total lateral velocity. e Lateral component of steady equilibrium velocity. E Lateral velocity component referred to datum- V V V V V path earth axes. 0 Steady equilibrium velocity. F Fin volume ratio. R Resultant speed. S Stalling speed. T Tailplane volume ratio. V Eigenvector matrix. w Normal velocity perturbation. Wing loading. W W Downwash velocity. W Total nomal velocity. Weight. e Normal component of steady equilibrium velocity. E Normal velocity component referred to datum- path earth axes. 54 Aeronautical Engineer’s Data Book Table 3.4 Continued y x Longitudinal co-ordinate in axis system. x State vector. X Axial force component. y Lateral co-ordinate. B Lateral body ‘drag’ coefficient. y Output vector. Y Lateral force component. z Normal co-ordinate in axis system. Spanwise co- ordinate. z Transformed state vector. Z Normal force component. Greek symbols ␣ Angle of incidence or attack. Acceleration ␥ ␤ ␤ ␣ ␣ (angular). ␣ ' Incidence perturbation. e Equilibrium incidence. T Local tailplane incidence. ␤ Sideslip angle perturbation. Compressibility. e Equilibrium sideslip angle. ␩ Elevator trim tab angle. ␥ Flight path angle perturbation. e Equilibrium flight path angle. ⌫ Wing dihedral angle (half). Circulation. Strength of vortex. ␦ Airfoil section camber. Boundary layer thickness. ␦ m Mass increment. ⑀ Throttle lever angle. Downwash angle. ␨ Rudder angle perturbation. Damping ratio. ␪ Vorticity. ␩ Efficiency. ␪ Pitch angle perturbation. Angle. e Equilibrium pitch angle. Angular co-ordinate (polar). Propeller helix angle. ␭ Eigenvalue. Wavelength. Friction coefficient in a ␻ ␻ µ µ pipe. ⌳ Wing sweep angle. µ Viscosity (dynamic). 1 Longitudinal relative density factor. 2 Lateral relative density factor. ␯ Viscosity (kinematic). ␰ Aileron angle perturbation. ␳ Density. ␴ Aerodynamic time parameter. Tensile stress. ␶ Engine thrust perturbation. Shear stress. ␾ Phase angle. A general angle. ⌽ State transition matrix. ⌿ Yaw angle perturbation. Stream function. ␻ Natural frequency. Angular velocity. b Bandwidth frequency. n Damped natural frequency. 55 Symbols and notations Table 3.4 Continued Subscripts 0 Datum axes. Normal earth-fixed axes. Straight/level flight. Free stream flow conditions. Sea level. 1/4 Quarter chord. 2 Double or twice. ∞ Infinity condition. a Aerodynamic. Available. b Aeroplane body axes. Bandwidth. c Chord. Compressible flow. Camber line. D Drag. e Equilibrium. E Earth axes. F Fin. g Gravitational. Ground. h Horizontal. H Elevator hinge moment. i Incompressible. Ideal. l Rolling moment. LE Leading edge. L Lift. m Pitching moment. Manoeuvre. n Damped natural frequency. n Neutral point. Yawing moment. p Power. Phugoid. p Roll rate. q Pitch rate. r Roll mode. r Yaw rate. s Short period pitching oscillation. Spiral. Stagnation. Surface. t Tangential. TE Trailing edge. T Tailplane. u Axial velocity. U Upper. v Lateral velocity. V Vertical. w Wing. w Normal velocity. x ox axis. y oy axis. z oz axis. ␣ Angle of attack or incidence. ⑀ Throttle lever. ␨ Rudder. ␩ Elevator. ␪ Pitch. ␰ Ailerons. ␶ Thrust. 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3 2. 833 7E 3 2.1708E 3 1.6727E 3 1.2961E 3 1.0095E 3 7.8728E–4 6. 139 5E–4 4.7700E–4 3. 6869E–4 2. 834 4E–4 2.1668E–4... 130 00 135 00 14000 14500 15000 –5.6 –6.6 –7.6 –8.6 –9.6 –10.6 –11.5 –12.5 – 13. 5 –14.5 0.6745 0.6614 0.6486 0. 636 0 0.6 236 0.61 13 0.59 93 0.5875 0.5758 0.56 43 0.7269 0.7155 0.70 43 0.6 932 0.6822 0.67 13 0.6606 0.6500 0. 639 6 0.6292 0.9 430 0.9402 0. 937 4 0. 934 7 0. 931 9 0.9291 0.92 63 0.9 235 0.9207 0.9179 1.2972 1 .31 40 1 .33 10 1 .34 84 1 .36 60 1 .38 40 1.4022 1.4207 1. 439 6 1.4588 32 7.8 32 7.2 32 6.6 32 6.0 32 5.4 32 4.7 32 4.1... 0.4406 0. 431 4 0.42 23 0.4 134 0.4046 0 .39 60 0 .38 76 0 .37 93 0 .37 11 0.5 238 0.5150 0.5062 0.4976 0.4891 0.4806 0.47 23 0.4642 0.4561 0.4481 0.8867 0.8 838 0.8809 0.8781 0.8752 0.87 23 0.8694 0.8665 0.8 636 0.8607 1.6927 1.71 63 1.74 03 1.7647 1.7895 1.8148 1.8406 1.8668 1.8 935 1.9207 31 5.4 31 4.8 31 4.1 31 3.5 31 2.9 31 2.2 31 1.6 31 1.0 31 0 .3 309.7 7772 25500 35 .3 0 .36 31 0.4402 0.8578 1.9484 30 9.0 26000 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0 .38 81 0. 837 2 2.1571 30 4.5 8991 29500 – 43. 2 0 .30 38 0 .38 11 0. 834 2 2.1892 30 3.8