- 1. Engineering Formula Sheet Statistics Mode Mean Place data in ascending order. Mode = most frequently occurring value ∑ xi n µ= µ = mean value Σxi = sum of all data values (x1, x2, x3, …) n = number of data values Median Place data in ascending order. If n is odd, median = central value If n is even, median = mean of two central values Standard Deviation σ=ඨ If two values occur at the maximum frequency the data set is bimodal. If three or more values occur at the maximum frequency the data set is multi-modal. ∑(xi - µ)2 n n = number of data values σ = standard deviation xi = individual data value ( x1, x2, x3, …) ߤ = mean value n = number of data values Range Range = xmax - xmin xmax = maximum data value xmin = minimum data value Probability Independent Events P (A and B and C) = PAPBPC Frequency fx = nx n Px = fx fa P (A and B and C) = probability of independent events A and B and C occurring in sequence PA = probability of event A Mutually Exclusive Events fx = relative frequency of outcome x nx = number of events with outcome x n = total number of events Px = probability of outcome x fa = frequency of all events Binomial Probability (order doesn’t matter) Pk = n!(pk )(qn-k ) k!(n-k)! P (A or B) = probability of either mutually exclusive event A or B occurring in a trial PA = probability of event A Σxi = sum of all data values (x1, x2, x3, …) n = number of data values Conditional Probability Pk = binomial probability of k successes in n trials p = probability of a success q = 1 – p = probability of failure k = number of successes n = number of trials PLTW, Inc. P (A or B) = PA + PB ܲ(= )ܦ|ܣ ܲ()ܣ|ܦ(ܲ ∙ )ܣ ܲ()ܣ~|ܦ(ܲ ∙ )ܣ~(ܲ + )ܣ|ܦ(ܲ ∙ )ܣ P (A|D) = probability of event A given event D P(A) = probability of event A occurring P(~A) = probability of event A not occurring P(D|~A) = probability of event D given event A did not occur Engineering Formulas IED POE DE CEA AE BE CIM EDD 1
- 2. Plane Geometry Ellipse Rectangle 2b Area = π a b Circle Perimeter = 2a + 2b Area = ab 2a Circumference =2 π r Area = π r2 B Triangle Parallelogram Area = ½ bh a = b + c – 2bc·cos∠A 2 2 2 b = a + c – 2ac·cos∠B 2 2 2 c = a + b – 2ab·cos∠C h Area = bh 2 b 2 cos θ = a c a c c h 2 A C b s s(ଵ f) Area = n ଶ 2 2 c =a +b sin θ = 2 Regular Polygons Right Triangle 2 a b f n = number of sides θ c b a tan θ = b a h Trapezoid Area = ½(a + b)h h h b h Solid Geometry Cube Sphere 3 Volume = s 2 Surface Area = 6s s s ସ r 3 Volume π r ଷ 2 Surface Area = 4 π r s Rectangular Prism Cylinder r h Volume = wdh Surface Area = 2(wd + wh + dh) d w h 2 Volume = π r h 2 Surface Area = 2 π r h+2 π r Right Circular Cone πr2 h Volume = 3 Surface Area = π r ඥr2 +h2 h Irregular Prism r h Volume = Ah A = area of base Pyramid Volume = Ah 3 A = area of base PLTW, Inc. h Constants 2 g = 9.8 m/s = 32.27 ft/s -11 3 2 G = 6.67 x 10 m /kg·s π = 3.14159 Engineering Formulas IED POE DE 2 CEA AE BE CIM EDD 2
- 3. Conversions Mass Force Area 2 1 acre = 4047 m 2 = 43,560 ft 2 = 0.00156 mi 1 kg = 2.205 lbm 1 slug = 32.2 lbm 1 ton = 2000 lbm 1N 1 kip Energy = 0.225 lbf = 1,000 lbf 1J = 0.239 cal -4 = 9.48 x 10 Btu = 0.7376 ft·lbf 1kW h = 3,6000,000 J Pressure Length 1m 1 km 1 in. 1 mi 1 yd 1 atm Volume = 3.28 ft = 0.621 mi = 2.54 cm = 5280 ft = 3 ft 1L 1mL = 0.264 gal 3 = 0.0353 ft = 33.8 fl oz 3 = 1 cm = 1 cc 1psi = 1.01325 bar = 33.9 ft H2O = 29.92 in. Hg = 760 mm Hg = 101,325 Pa = 14.7 psi = 2.31 ft of H2O Defined Units 1J 1N 1 Pa 1V 1W 1W 1 Hz 1F 1H Time Temperature Change 1K = 1 ºC = 1.8 ºF = 1.8 ºR 1d 1h 1 min 1 yr = 24 h = 60 min = 60 s = 365 d Power 1W = 3.412 Btu/h = 0.00134 hp = 14.34 cal/min = 0.7376 ft·lbf/s = 1 N·m = 1 kg·m / s2 = 1 N / m2 =1W/A =1J/s =1V/A = 1 s-1 = 1 A·s / V = 1 V·s / V SI Prefixes Numbers Less Than One Power of 10 Prefix Abbreviation 10-1 10-2 10-3 10-6 10-9 10-12 10-15 10-18 10-21 10-24 decicentimillimicronanopicofemtoattozeptoyocto- Equations d c m µ n p f a z y Numbers Greater Than One Power of 10 Prefix Abbreviation 101 102 103 106 109 1012 1015 1018 1021 1024 Temperature TK = TC + 273 Mass and Weight M = VDm W = mg W = VDw V = volume Dm = mass density m = mass Dw = weight density g = acceleration due to gravity PLTW, Inc. TR = TF + 460 TF - 32 TC = 180 100 TK = temperature in Kelvin TC = temperature in Celsius TR = temperature in Rankin TF = temperature in Fahrenheit Engineering Formulas decahectokiloMegaGigaTeraPetaExaZettaYotta- da h k M G T P E Z Y Force F = ma F = force m = mass a = acceleration Equations of Static Equilibrium ΣFx = 0 ΣFy = 0 ΣMP = 0 Fx = force in the x-direction Fy = force in the y-direction MP = moment about point P IED POE DE CEA AE BE CIM EDD 3
- 4. Equations (Continued) Electricity Fluid Mechanics Energy: Work W = F∙d P= V1 W = work F = force d = distance F A V2 T1 P1 T1 Power T2 P2 = T2 RT (series) = R1 + R2+ ··· + Rn (Charles’ Law) RT (parallel) = (Guy-Lussanc’s Law) 1 1 1 1 + + ∙∙∙ +R R1 R2 n Kirchhoff’s Current Law Q = Av P = power E = energy W = work t = time τ = torque rpm = revolutions per minute Efficiency Pout ∙100% Pin Pout = useful power output Pin = total power input IT = I1 + I2 + ··· + In n or IT = ∑k=1 Ik A1v1 = A2v2 E W = t t τ∙rpm P= 5252 Kirchhoff’s Voltage Law Horsepower = P = absolute pressure F = Force A = Area V = volume T = absolute temperature Q = flow rate v = flow velocity U = potential energy m =mass g = acceleration due to gravity h = height t ܌ t 1 2 Q ∆t (where acceleration = 0) U= 1 k = R L kA∆T L a= vf ି vi t P= X= vi sin(2θ) -g A1v1 = A2v2 Pnet = σAe(T2 4 -T1 4 ) 2 d = d0 + v0t + ½at K = mv2 v = v0 + 2a(d – d0) K = kinetic energy m = mass v = velocity τ = dFsinθ Energy: Thermal Q =mc∆T Q = thermal energy m = mass c = specific heat ∆T = change in temperature Thermodynamics P= v = v0 + at Energy: Kinetic V = voltage VT = total voltage I = current IT = total current R = resistance RT = total resistance P = power (where acceleration = 0) d v= VT = V1 + V2 + ··· + Vn n or VT = ∑k=1 Vk P = Q′ = AU∆T Mechanics Energy: Potential U = mgh QP 1714 absolute pressure = gauge pressure + atmospheric pressure s= PLTW, Inc. V = IR P = IV P1V1 = P2V2 (Boyle’s Law) P= Efficiency (%) = = Ohm’s Law 2 2 s = speed v = velocity a = acceleration X = range t = time d = distance g = acceleration due to gravity d = distance θ = angle τ = torque F = force Engineering Formulas P = rate of heat transfer Q = thermal energy A = Area of thermal conductivity U = coefficient of heat conductivity (U-factor) ∆T = change in temperature ∆t = change in time R = resistance to heat flow ( R-value) k = thermal conductivity v = velocity Pnet = net power radiated σ = 5.6696 x 10 -8 W m2 ∙K 4 e = emissivity constant T , T = temperature at time 1, time 2 POE 4 DE 4
- 5. Section Properties Rectangle Centroid Moment of Inertia h Ixx x x 3 bh = 12 b Ixx = moment of inertia of a rectangular section about x-x axis ഥ x= ∑ Ai ഥ and y = b 2 ഥ= and y h 2 Right Triangle Centroid ഥ= x b 3 ഥ= and y h 3 Semi-circle Centroid Complex Shapes Centroid ∑ xi Ai ഥ= x ഥ x = r and y = ഥ ∑ yi A i ∑ Ai ഥ= x x-distance to the centroid ത y = y-distance to the centroid xi = x distance to centroid of shape i yi = y distance to centroid of shape i Ai = Area of shape i 4r 3π ഥ= x x-distance to the centroid ത y = y-distance to the centroid Structural Analysis Material Properties Beam Formulas Stress (axial) Reaction F σ= A Moment Deflection σ = stress F = axial force A = cross-sectional area Reaction Moment Strain (axial) Deflection ϵ= δ L0 Reaction ϵ = strain L0 = original length δ = change in length Deflection Moment PL Mmax = 4 max = ߪ(F2 -F1 )L0 (ߜଶ − ߜଵ )A E = modulus of elasticity σ = stress ε = strain A = cross-sectional area F = axial force δ = deformation PLTW, Inc. ωL 2 ωL2 Mmax = (at center) 8 5ωL4 384EI (at center) RA = RB = P Mmax = Pa (between loads) Pa = 24EIቀ3L2 -4a2ቁ max Pb Moment E= 3 RA = RB = RA = Mmax = Deflection 2 (at point of load) PL max = 48EI (at point of load) Reaction Modulus of Elasticity σ E= ε P RA = RB = ୫ୟ୶ L and RB = Pab L (at center) Pa L (at Point of Load) = ౌaౘ(aశమౘ)ඥయa(aశమౘ) మళు (at x = ට a(aାଶୠ) Deformation: Axial when a > b ) Truss Analysis FL0 δ = AE ଷ, 2J = M + R δ = deformation F = axial force L0 = original length A = cross-sectional area E = modulus of elasticity Engineering Formulas J = number of joints M =number of members R = number of reaction forces POE 5 AE 4 CEA 4
- 6. Simple Machines Inclined Plane Mechanical Advantage (MA) DE IMA= DR % Efficiency= ൬ FR AMA= FE AMA ൰ 100 IMA IMA= L (slope) H Wedge IMA = Ideal Mechanical Advantage AMA = Actual Mechanical Advantage DE = Effort Distance DR = Resistance Distance FE = Effort Force FR = Resistance Force IMA= L (⊥ to height) H Lever Screw 1st Class IMA = C Pitch Pitch = 2nd Class 1 TPI C = Circumference r = radius Pitch = distance between threads TPI = Threads Per Inch 3rd Class Compound Machines MATOTAL = (MA1) (MA2) (MA3) . . . Wheel and Axle Gears; Sprockets with Chains; and Pulleys with Belts Ratios Nout dout ωin τout GR= = = = Nin din ωout τin Effort at Axle dout ωin τout = = (pulleys) din ωout τin Compound Gears B D GRTOTAL = ቀ ቁ ቀ ቁ A C Effort at Wheel Pulley Systems IMA = Total number of strands of a single string supporting the resistance IMA = DE (string pulled) DR (resistance lifted) PLTW, Inc. GR = Gear Ratio ωin = Angular Velocity - driver ωout = Angular Velocity - driven Nin = Number of Teeth - driver Nout = Number of Teeth - driven din = Diameter - driver dout = Diameter - driven τin = Torque - driver τout = Torque - driven Engineering Formulas POE 6
- 7. Structural Design Steel Beam Design: Shear Va = Steel Beam Design: Moment Vn Ma = v Mn b Vn = 0.6FyAw Mn = FyZx Va = allowable shear strength Vn = nominal shear strength v = 1.5 = factor of safety for shear Fy = yield stress Aw = area of web Ma = allowable bending moment Mn = nominal moment strength b = 1.67 = factor of safety for bending moment Fy = yield stress Zx = plastic section modulus about neutral axis Storm Water Runoff Storm Water Drainage Q = CfCiA Cc = C1 A1 + C2 A2 + ∙∙∙ A1 + A2 + ∙∙∙ 3 Q = peak storm water runoff rate (ft /s) Cf = runoff coefficient adjustment factor C = runoff coefficient i = rainfall intensity (in./h) A = drainage area (acres) Runoff Coefficient Adjustment Factor Return Period Cf 1, 2, 5, 10 1.0 25 1.1 50 1.2 100 1.25 Water Supply Hazen-Williams Formula hf = 10.44LQ C 1.85 1.85 4.8655 d hf = head loss due to friction (ft of H2O) L = length of pipe (ft) Q = water flow rate (gpm) C = Hazen-Williams constant d = diameter of pipe (in.) Dynamic Head Rational Method Runoff Coefficients Categorized by Surface Forested 0.059—0.2 Asphalt 0.7—0.95 Brick 0.7—0.85 Concrete 0.8—0.95 Shingle roof 0.75—0.95 Lawns, well drained (sandy soil) Up to 2% slope 0.05—0.1 2% to 7% slope 0.10—0.15 Over 7% slope 0.15—0.2 Lawns, poor drainage (clay soil) Up to 2% slope 0.13—0.17 2% to 7% slope 0.18—0.22 Over 7% slope 0.25—0.35 Driveways, 0.75—0.85 walkways Categorized by Use Farmland 0.05—0.3 Pasture 0.05—0.3 Unimproved 0.1—0.3 Parks 0.1—0.25 Cemeteries 0.1—0.25 Railroad yard 0.2—0.40 Playgrounds 0.2—0.35 (except asphalt or Districts Business Neighborhood 0.5—0.7 City (downtown) 0.7—0.95 Residential Single-family 0.3—0.5 Multi-plexes, 0.4—0.6 detached Multi-plexes, 0.6—0.75 attached Suburban 0.25—0.4 Apartments, 0.5—0.7 condominiums Industrial Light 0.5—0.8 Heavy 0.6—0.9 Spread Footing Design qnet = qallowable - pfooting pfooting = tfooting ∙150 q= lb 2 ft P A qnet = net allowable soil bearing pressure qallowable = total allowable soil bearing pressure pfooting = soil bearing pressure due to footing weight tfooting = thickness of footing q = soil bearing pressure P = column load applied A = area of footing dynamic head = static head – head loss PLTW, Inc. Engineering Formulas CEA 5
- 8. PLTW, Inc. Engineering Formulas CEA 6 Equivalent Length of (Generic) Fittings Hazen-Williams Constants
- 9. 555 Timer Design Equations T = 0.693 (RA + 2RB)C f = 1 T duty-cycle = (RA + RB ) ∙100% (RA +2RB ) T = period f = frequency RA = resistance A RB = resistance B C = capacitance Boolean Algebra Boolean Theorems Commutative Law Consensus Theorems X• 0 = 0 X•Y = Y•X ഥ X + XY = X + Y X•1 = X X+Y = Y+X ഥഥ ഥ X + XY = X + Y X• X =X Associative Law ഥ X • X=0 X(YZ) = (XY)Z X+0=X ഥ തത X + XY =തത + Y X ഥ ഥ ഥ ഥ X + XY = X + Y X + (Y + Z) = (X + Y) + Z DeMorgan’s Theorems X+1=1 X+X=X Distributive Law തതതതത= X + ഥ XY ഥ Y X+ഥ =1 X X(Y+Z) = XY + XZ തതതതതതത= ഥ • Y X+Y X ഥ ന X=X (X+Y)(W+Z) = XW+XZ+YW+YZ Speeds and Feeds N= CSቀ12in.ቁ ft πd fm = ft·nt·N Plunge Rate = ½·fm N = spindle speed (rpm) CS = cutting speed (in./min) d = diameter (in.) fm = feed rate (in./min) ft = feed (in./tooth) nt = number of teeth PLTW, Inc. Engineering Formulas DE 5 CIM 4
- 10. Aerospace Equations R e= CL = Orbital Mechanics I = Fave ∆t F N = W൫vj - vo ൯ 2D Aρv2 Fnet = Favg - Fg ݁ =ඨ1 - ρvl a = vf ∆t T = 2π Forces of Flight CD = Propulsion 2L Aρv2 M = Fd CL = coefficient of lift CD = coefficient of drag L = lift D = drag A = wing area ρ = density Re = Reynolds number v = velocity l = length of fluid travel = fluid viscosity F = force m = mass g = acceleration due to gravity M = moment d = moment arm (distance from datum perpendicular to F) FN = net thrust W = air mass flow vo = flight velocity vj = jet velocity I = total impulse Fave = average thrust force t = change in time (thrust duration) Fnet = net force Favg = average force Fg = force of gravity vf = final velocity a = acceleration t = change in time (thrust duration) NOTE: Fave and Favg are easily confused. 1 K = 2 mv2 U= − GMm R E=U+K=− √ య = 2π aమ √GM ݁ = eccentricity b = semi-minor axis a =semi-major axis T = orbital period a = semi-major axis = gravitational parameter F = force of gravity between two bodies G = universal gravitation constant M =mass of central body m = mass of orbiting object r = distance between center of two objects Bernoulli’s Law ρv2 ρv2 ቇ = ቆPs + ቇ 2 1 2 2 PS = static pressure v = velocity ρ = density GMm 2R K = kinetic energy m =mass v = velocity U = gravitational potential energy G = universal gravitation constant M =mass of central body m = mass of orbiting object R = Distance center main body to center of orbiting object E = Total Energy of an orbit PLTW, Inc. య aమ GMm F= r2 ቆPs + Energy b2 a2 Engineering Formulas Atmosphere Parameters T = 15.04 - 0.00649h p = 101.29 ቈ ρ= (T + 273.1) 288.08 5.256 p 0.2869(T + 273.1) T = temperature h = height p = pressure ρ = density AE 5