Data and Formulae for Mechanical Engineering Students Department of Mechanical Engineering, Imperial College London September 2009 Contents A General information B Mathematics and computing B.1 Algebra B.1.1 Logarithms B.1.2 Quadratic equations B.1.3 Determinants B.1.4 Vector algebra B.1.5 Series B.1.6 Trigonometry B.1.7 Geometry B.1.8 Analytic geometry B.1.9 Solid geometry B.1.10 Differential calculus B.1.11 Standard Differentials B.1.12 Differential equations B.2 Integral calculus B.3 Laplace transforms B.4 Numerical analysis B.4.1 Approximate solution of an algebraic equation B.4.2 Numerical integration B.4.3 Richardson’s error estimation formula for use with Simpson’s rule B.4.4 Fourier series B.5 Statistics B.6 Probabilities for events B.6.1 Distribution, expectation and variance B.6.2 Probability distributions for a continuous random variable B.6.3 Discrete probability distributions B.6.4 Continuous probability distributions B.6.5 System reliability B.7 Bias, standard error and mean square error B.7.1 Central limit property B.7.2 Confidence intervals 5 5 6 10 10 11 12 12 14 15 15 15 16 16 17 17 17 18 18 19 19 19 19 20 C Mechatronics and control C.1 Charge, current, voltage and power C.2 Networks C.3 Transients 21 21 22 23 i CONTENTS C.4 AC networks C.4.1 Average and root mean square values C.4.2 Phasors and complex impedance C.4.3 Balanced phase a.c supply C.4.4 Electromagnetism C.4.5 DC machines C.4.6 Transformers C.5 Communications C.6 Step function response and frequency response C.6.1 First-order systems C.6.2 Second-order systems C.7 Operational amplifier stages 23 23 24 24 24 25 25 26 26 26 27 27 D Solid Mechanics D.1 Mechanics D.1.1 Square screw threads D.1.2 Flat clutches D.1.3 Kinematics of particle D.1.4 Mass flow problems D.1.5 Kinematics of rigid bodies with sliding contacts D.1.6 Mass moments of inertia D.2 Stress analysis D.2.1 Elastic constants of materials D.2.2 Beam theory D.2.3 Elastic torsion D.2.4 Thin walled pressure vessels D.3 Two-dimensional stress transformation D.4 Yield criteria D.5 Two-dimensional strain transformation D.6 Elastic stress-strain relationships D.7 Thick-walled cylinders 33 33 33 33 33 34 34 34 35 35 35 38 39 39 40 40 40 41 E Thermofluids E.1 Cross-references to table numbers E.2 Dimensionless groups E.3 Heat transfer E.4 Continuity and equation of motion E.4.1 Cylindrical polar coordinates E.4.2 Rectangular Cartesian coordinates E.4.3 Vector form E.5 Equations for compressible flows E.6 Friction factor for flow in circular pipes (Moody diagram) E.7 Perfect gases E.8 Heating (or calorific) values of fuels E.9 Properties of R134a refrigerant E.10 Transport properties of air, water and steam E.11 Approximate physical properties E.12 Thermodynamic property tables for water/steam (IAPWS-IF97 formulation) 43 43 44 45 46 46 46 47 47 48 50 54 55 62 65 67 Page ii Mechanical Engineering Data & Formulae LIST OF TABLES List of Tables A.1 A.2 A.3 A.4 B.1 B.2 B.3 B.4 C.1 C.2 C.3 D.1 D.2 D.3 E.1 E.2 E.3 E.4 E.5 E.6 SI Units and abbreviations Conversion factors from Imperial to SI units Decimal prefixes Physical constants Some indefinite integrals Some definite integrals Standard normal table: values of pdf φ(y) = f (y) and cdf Φ(y) = F (y) Student t table: values tm,p of x for which P (|X | > x) = p, when X is tm Colour codes for resistors etc Standard values for components Operational amplifier signal processing stages Second moments of area for simple cross-sections Beams bent about principal axis Torsion of solid non-circular sections Dimensionless groups for Thermofluids Empirical correlations for forced convection Perfect gases (ideal gases with constant specific heats) Isentropic compressible flow functions for perfect gas with γ = 1.40 −1 ◦ Ideal (semi-perfect) gas specific enthalpy h (kJ kg , 25 C datum) −1 ◦ Molar Enthalpy of Formation hf (kJ kmol at 25 C and atmosphere) as gas or vapour (g), except where indicated as solid (s) or liquid (l) −1 ◦ E.7 Ideal gas molar enthalpy h (kJ kmol , 25 C datum) ◦ E.8 Heating (or calorific) values of gas fuels at 25 C ◦ E.9 Heating (or calorific) values of liquid fuels at 25 C E.10 Saturated Refrigerant 134a — Temperature (−60◦ C to critical point) E.11 Saturated Refrigerant 134a — Pressure (0.2 bar to critical point) E.12 Superheated Refrigerant 134a (0.2 bar to bar) E.13 Superheated Refrigerant 134a (1.5 bar to bar) E.14 Superheated Refrigerant 134a (5 bar to 12 bar) E.15 Superheated Refrigerant 134a (16 bar to 30 bar) E.16 Transport properties of dry air at atmospheric pressure E.17 Transport properties of saturated water and steam ◦ E.18 Approximate physical properties at 20 C, bar ◦ E.19 Saturated water and steam — Temperature (triple point to 100 C) E.20 Saturated water and steam — Pressure (triple point to bar) E.21 Saturated water and steam — Pressure (triple point to bar) E.22 Saturated water and steam — Pressure (triple point to bar) E.23 Subcooled water and Superheated Steam (triple point to 0.1 bar) E.24 Subcooled water and Superheated Steam (0.1 bar to atmosphere) E.25 Subcooled water and Superheated Steam (2 bar to bar) E.26 Subcooled water and Superheated Steam (10 bar to 40 bar) E.27 Subcooled water and Superheated Steam (50 bar to 80 bar) E.28 Subcooled water and Superheated Steam (90 bar to 140 bar) E.29 Subcooled water and Superheated Steam (160 bar to 220 bar) E.30 Supercritical steam (250 bar to 500 bar) E.31 Supercritical steam (600 bar to 1000 bar) Mechanical Engineering Data & Formulae 3 13 13 20 20 21 22 31 36 37 38 44 45 50 51 52 53 53 54 54 56 57 58 59 60 61 63 64 65 68 69 70 71 72 73 74 75 76 77 78 79 80 Page iii LIST OF FIGURES List of Figures C.1 C.2 C.3 C.4 E.1 E.2 Page iv Step response of a first-order low pass filter Bode plot for first-order low and high pass filters Step response of a second-order low pass filter Bode plot for a second-order low pass filter Moody Diagram Psychrometric Chart 27 28 29 30 49 66 Mechanical Engineering Data & Formulae A General information A General information Table A.1: SI Units and abbreviations Quantity Unit Unit symbol Basic units Length Mass Time Electric current Thermodynamic temperature Luminous intensity metre kilogram second ampere kelvin candela m kg s A K cd Derived units Acceleration, linear Acceleration, angular Area Density Force Frequency Impulse, linear Impulse, angular Moment of force Second moment of area Moment of inertia Momentum, linear Momentum, angular Power Pressure, stress Stiffness (linear), spring constant Velocity, linear Velocity, angular Volume Work, energy metre/second radian/second metre kilogram/metre3 newton hertz newton-second newton-metre-second newton-metre metre4 kilogram-metre2 kilogram-metre/second kilogram-metre2 /second watt pascal newton/metre metre/second radian/second metre3 joule ms −2 rad s m kg m−3 N (= kg m s−2 ) (Hz = s−1 ) Ns Nms Nm m4 kg m2 kg m s−1 kg m2 s−1 W (= J s−1 = N m s−1 Pa (= N m−2 ) N m−1 m s−1 rad s−1 m3 J (= N m) Electrical units Potential Resistance Charge Capacitance Electric field strength Electric flux density volt ohm coulomb farad volt/metre coulomb/metre2 V (= W A−1 ) Ω (= V A−1 ) C (= A s) F (= A s V−1 ) V m−1 C m−2 Magnetic units Magnetic flux Inductance Magnetic field strength Magnetic flux density weber henry — — Wb (= V s) H (= V s A−1 ) A m−1 Wb m−2 Mechanical Engineering Data & Formulae −2 Page A General information Table A.2: Conversion factors from Imperial to SI units To convert from to multiply by Acceleration foot/second2 (ft/sec2 ) inch/second2 (in/sec2 ) metre/second2 (m s−2 ) metre/second2 (m s−2 ) 0.3048 0.0254 Area foot2 (ft2 ) inch2 (in.2 ) metre2 (m2 ) metre2 (m2 ) 0.092903 6.4516 × 10−4 Density pound mass/inch3 lbm/in3 kilogram/metre3 (kg m−3 ) 2.7680 × 104 pound mass/foot3 lbm/ft3 kilogram/metre3 (kg m−3 ) 16.018 Force kip (1000 lb) pound force (lb) newton (N) newton (N) 4.4482 × 103 4.4482 Length foot (ft) inch (in) mile (mi), U.S statute mile (mi), international nautical metre (m) metre (m) metre (m) metre (m) 0.3048 0.0254 1.6093 × 103 1.852 × 103 Mass pound mass (lbm) slug (lb-sec2 /ft) ton (2000 lbm) kilogram (kg) kilogram (kg) kilogram (kg) 0.45359 14.594 907.18 Moment of force pound-foot (lb-ft) pound-inch (lb-in.) newton-metre (N m) newton-metre (N m) 1.3558 0.11298 Moment of inertia pound-foot-second (lb-ft-sec ) kilogram-metre (kg m ) Momentum, linear pound-second (lb-sec) kilogram-metre/second (kg m s−1 ) 2 2 1.3558 −1 4.4482 Momentum, angular pound-foot-second (lb-ft-sec) newton-metre-second (kg m s ) 1.3558 Power foot-pound/minute (ft-lb/min) horsepower (550 ft-lb/sec) watt (W) watt (W) 0.022597 745.70 Pressure, stress atmosphere (std) (14.7 lb/in2 ) pound/foot2 (lb/ft2 ) pound/inch2 (lb/in.2 or psi) newton/metre2 (N m−2 or Pa) newton/metre2 (N m−2 or Pa) newton/metre2 (N m−2 or Pa) 1.0133 × 105 47.880 6.8948 × 103 Second moment of area inch4 metre4 (m4 ) 41.623 × 10−8 Stiffness (linear) pound/inch (lb/in.) newton/metre (N m ) 175.13 Velocity foot/second (ft/sec) knot (nautical mi/hr) mile/hour (mi/hr) mile/hour (mi/hr) metre/second (m s−1 ) metre/second (m s−1 ) metre/second (m s−1 ) kilometre/hour (km h−1 ) 0.3048 0.51444 0.44704 1.6093 Volume foot3 (ft3 ) inch3 (in.3 ) UK gallon metre3 (m3 ) metre3 (m3 ) metre3 (m3 ) 0.028317 1.6387 × 10−5 4.546 × 10−3 Work, Energy British thermal unit (BTU) foot-pound force (ft-lb) kilowatt-hour (kw-h) joule (J) joule (J) joule (J) 1.0551 × 103 1.3558 3.60 × 106 Page −1 Mechanical Engineering Data & Formulae A General information Table A.3: Decimal prefixes Multiplication factora 12 Prefix Symbol 000 000 000 000 000 000 000 000 000 000 100 10 = = = = = = 10 109 106 103 10 10 tera giga mega kilo a hecto a deka T G M k h da 0.1 0.01 0.001 0.000 001 0.000 000 001 0.000 000 000 001 = = = = = = 10−1 10−2 10−3 −6 10 −9 10 10−12 deci b centi milli micro nano pico d c m µ n p a Use prefixes to keep numerical values generally between 0.1 and 1000 The use of prefixes hecto, deka, deci and centi should be avoided except for certain areas or volumes where the numbers would otherwise become awkward b Table A.4: Physical constants Avogadro’s numbera N 6.022 × 1023 mol−1 Absolute zero of temperature — K = −273.2 ◦C Boltzmann’s constant k Characteristic impedance of vacuum Z0 1.380 ì 1023 J K1 à0 1/2 = = 120π Ω Electron volt eV 1.602 × 10 −19 1.602 × 10 −19 me e me F 9.109 × 10 −31 1.759 × 10 11 Gas constant R 8.314 J mol Permeability of free space µ0 Permittivity of free space ε0 Planck’s constant h 4π × 10 H m −9 −1 × 10 F m 36π −34 6.626 × 10 Js Standard gravitational acceleration g 9.807 m s Stefan-Boltzmann constant σ 5.67 × 10 Velocity of light in vacuum c 2.9979 × 10 m s — 22.42 × 10 Electronic charge e Electronic rest mass Electronic charge to mass ratio a Faraday’s constant a b Volume of perfect gas at S.T.P J C kg C kg−1 −1 9.65 × 10 C mol −1 −7 K −1 −1 −2 −8 −2 −1 Jm −4 K −1 −3 s m a These are conventional definitions in gram mol units For SI calculations in kg mol units multiply the values given by 103 b At Standard Temperature (0 ◦C) and Pressure (one atmosphere pressure or 1.013 × 105 N m−2 ) Mechanical Engineering Data & Formulae Page A General information Page Mechanical Engineering Data & Formulae B Mathematics and computing B Mathematics and computing Data and formulae for core course examinations in: • Mathematics • Computing and in other, related, optional courses B.1 Algebra B.1.1 Logarithms If by = x, y = logb (x) and: log (x1 x2 ) = log x1 + log x2 log x1 x2 = log x1 − log x2 x log = − log x log x n = n log x log = For natural logarithms b = e = 2.718282 and if ey = x, y = loge (x) = ln (y) Hence log10 x = 0.4343 ln x B.1.2 Quadratic equations If ax + bx + c = 0, then x= −b ± b2 − 4ac 2a and (b > 4ac) for real roots B.1.3 Determinants 2nd order: a1 b1 a2 b2 = a1 b2 − a2 b1 3rd order: a1 a2 a3 b1 b2 b3 c1 c2 c3 = +a1 b2 c3 + a2 b3 c1 + a3 b1 c2 − a3 b2 c1 − a2 b1 c3 − a1 b3 c2 Mechanical Engineering Data & Formulae Page B.1 Algebra B.1.4 Vector algebra a = (a1 i + a2 j + a3 k) = (a1 , a2 , a3 ) etc Scalar (dot) product: a.b = a1 b1 + a2 b2 + a3 b3 Vector (cross) product: a×b= i j k a1 a2 a3 b1 b2 b3 Scalar triple product: a1 a2 a3 b1 b2 b3 c1 c2 c3 [a, b, c] = a.b × c = b.c × a = c.a × b = Vector triple product: a × (b × c) = b (a.c) − c (a.b) B.1.5 Series Binomial series: (1 + x)α = + αx + α(α − 1) α (α − 1) (α − 2) x + x + 2! 3! n ex = + x + (α arbitrary, |x| < 1) x x + ··· + + 2! n! (|x| < ∞) cos x = − sin x = x − x x4 x 2n + − · · · + (−1)n + 2! 4! (2n)! (|x| < ∞) x3 x5 x 2n+1 + − · · · + (−1)n + 3! 5! (2n + 1)! tan x = x + x 2x 17x + + + 15 315 (− sinh x = ex − e−x x3 x5 x7 =x+ + + + 3! 5! 7! cosh x = e +e x −x =1+ π π