Engineering Formulas. Statistics. Probability. (x ) Mode. Place data in ascending order. Mode = most frequently occurring value

Engineering Formula Sheet Statistics Mode Mean Place data in ascending order. Mode = most frequently occurring value ∑x µ = mean value Σxi = sum o...
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Engineering Formula Sheet Statistics

Mode

Mean

Place data in ascending order. Mode = most frequently occurring value

∑x µ = mean value Σxi = sum of all data values (x1, x2, x3, … n = number of data values

∑(x

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



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.

)

n = number of data values

σ = standard deviation xi = individual data value ( x1, x2, x3, …

Range

n = number of data values

xmax = maximum data value xmin = minimum data value

Range = xmax - xmin

Probability Independent Events P (A and B and C) = PAPBPC

Frequency

P (A and B and C) = probability of independent events A and B and C occurring in sequence PA = probability of event A

x

x

x

x

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)

P (A or B) = PA + PB 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

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( | )

( )

( ) ( | )

( | ) ( )

( |

)

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

Plane Geometry

Ellipse

Rectangle

2b

Circle

Perimeter = 2a + 2b Area = ab

2a

B

Triangle Parallelogram h

Area = 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

C

2

c

h

2

A b

s

Regular Polygons

Right Triangle 2

a

2

b

2

Area = ½ bh

f

2

c =a +b

c

a

n = number of sides θ

b

a h

Trapezoid Area = ½(a + b)h

h h b h

Solid Geometry Cube

Sphere

s

3

Volume = s 2 Surface Area = 6s

r

3

s

Volume = r Surface Area = 4

s

r

2

Rectangular Prism Cylinder

r

h Volume = wdh Surface Area = 2(wd + wh + dh)

d

w

h

2

Volume = r h Surface Area = 2

r h+2

r

2

Right Circular Cone h

Irregular Prism r



h

Volume = Ah A = area of base

Pyramid

h A = area of base

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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

Conversions Mass

Area

Force 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 = 3,600,000 J

Pressure Length

Volume

1m 1 km 1 in. 1 mi 1 yd

= 3.28 ft = 0.621 mi = 2.54 cm = 5280 ft = 3 ft

1L

1mL

1 atm = 0.264 gal 3 = 0.0353 ft = 33.8 fl oz 3 = 1 cm = 1 cc 1psi

Temperature Unit Equivalents

Time 1d 1h 1 min 1 yr

1K

= 24 h = 60 min = 60 s = 365 d

= 1 ºC = 1.8 ºF = 1.8 ºR

= 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

Power 1W

See below for temperature calculation

= 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 Mass and Weight

Numbers Greater Than One Power of 10 Prefix Abbreviation 101 102 103 106 109 1012 1015 1018 1021 1024

d c m µ n p f a z y Temperature TK = TC + 273

M = VDm

TR = TF + 460

W = mg

TF =

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da h k M G T P E Z Y

Force F = ma F = force m = mass a = acceleration

Tc + 32

W = VDw V = volume Dm = mass density m = mass Dw = weight density g = acceleration due to gravity

decahectokiloMegaGigaTeraPetaExaZettaYotta-

Equations of Static Equilibrium TK = temperature in Kelvin TC = temperature in Celsius TR = temperature in Rankin TF = temperature in Fahrenheit

Engineering Formulas

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

Equations (Continued) Energy: Work

Electricity Ohm’s Law

Fluid Mechanics

V = IR P = IV

W = work F = force parallel to direction of displacement d = displacement Power

Efficiency y Pout = useful power output Pin = total power input

’ L

(Gay-L p1V1 = p2V2

P = power E = energy W = work t = time τ = torque rpm = revolutions per minute

RT (series) = R1 + R2+ ··· + Rn

’L

B y ’ L

Kirchhoff’s Current Law

Q = Av

IT = I1 + I2 + ··· + In ∑ or

A1v1 = A2v2

Kirchhoff’s Voltage Law

VT = V1 + V2 + ··· + Vn ∑ or absolute pressure = gauge pressure + atmospheric pressure

p = absolute pressure F = Force A = Area V = volume T = absolute temperature Q = flow rate v = flow velocity

V = voltage VT = total voltage I = current IT = total current R = resistance RT = total resistance P = power Thermodynamics ′

Mechanics

∆T

∆ ̅

Energy: Potential g

L

̅

U = potential energy m =mass g = acceleration due to gravity h = height Energy: Kinetic

L A1v1 = A2v2 g v = v0 + at d = d0 + v0t + ½at 2

2

2

v = v0 + 2a(d – d0) K = kinetic energy m = mass v = velocity Energy: Thermal

Q = thermal energy m = mass c = specific heat ∆T = change in temperature

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τ = dFsinθ ̅ g ̅ g y v = velocity a = acceleration X = range t = time ∆d = change in displacement d = distance g = acceleration due to gravity 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 g R = resistance to heat flow ( R-value) k = thermal conductivity v = velocity Pnet = net power radiated = 5.6696 x 10

-8

e = emissivity constant L = thickness T1, T2 = temperature at time 1, time 2

v = flow velocity

POE 4 DE 4

Section Properties Moment of Inertia

Rectangle Centroid h

x

x xx

b

Ixx = moment of inertia of a rectangular section about x-x axis

∑x ∑

and y̅

and y̅

Right Triangle Centroid x̅

and y̅

Semi-circle Centroid

Complex Shapes Centroid







∑y





x̅ x y̅ y xi = x distance to centroid of shape i yi = y distance to centroid of shape i Ai = Area of shape i

x̅ x y̅ y

Structural Analysis Material Properties Beam Formulas Reaction

Stress (axial)

B L

Moment Deflection = stress F = axial force A = cross-sectional area

L

x

B L

Moment

L

x

Reaction

= strain L0 = original length δ = change in length

Moment

x

Deflection

x

x

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and

Deformation: Axial δ

L

δ = deformation F = axial force L0 = original length A = cross-sectional area E = modulus of elasticity

Engineering Formulas



) (at center)

B

L

(at Point of Load)

L

Deflection (at

E = modulus of elasticity = stress = strain A = cross-sectional area F = axial force δ = deformation

( L L

Moment

(

(at center)

B

Reaction

Modulus of Elasticity

(at center)

x

Deflection

L

(at point of load) L

Reaction

Strain (axial)

(at point of load)

x

(

)√

(

)

(

)

)

Truss Analysis 2J = M + R J = number of joints M =number of members R = number of reaction forces

POE 5 AE 4 CEA 4

Simple Machines Inclined Plane Mechanical Advantage (MA)

y (

L

) Wedge

IMA = Ideal Mechanical Advantage AMA = Actual Mechanical Advantage DE = Effort Distance DR = Resistance Distance FE = Effort Force FR = Resistance Force

L

g

Lever Screw 1st Class

IMA =

Pitch = 2nd Class

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

Effort at Axle (

)

Compound Gears B GRTOTAL = ( ) (

Effort at Wheel

Pulley Systems IMA = Total number of strands of a single string supporting the resistance IMA =

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g

)

GR = Gear Ratio in = Angular Velocity - driver out = Angular Velocity - driven Nin = Number of Teeth - driver Nout = Number of Teeth - driven din = Diameter - driver dout = Diameter - driven in = Torque - driver out = Torque - driven

Engineering Formulas

POE 6