FreitagC

Principles of Engineering

Area:

Rectangle: A = bh
Parallelogram: A = bh
Triangle: A = 1/2 bh
Circle: A = πr2
Cone: A = π x Diameter x Length ÷ 2
Sphere: A = π x Diameter²

Volume:

Cylinder = πr²h
Rectangular Solid = whd
Pyramid = 1/3 x (Area of Base) x (Height)
Cone = 1/3 x (Area of Base) x (Height)
Sphere = 4πr³ / 3
1 Cubic inch = 1.804 Fluid Ounces
Right Tri. Prism = 1/2 Base x Height

Surface Area:

Total Surface Area = Sum of all Exterior Surfaces
Prism = 2(wd + wh + dh)
Cylinder = 2πrh + 2πr²
Cone = πr² + πrs
Sphere = 4πr²

Circumference: C = πD

Weight = Volume x Density

Trigonometry:

Right Triangles
- SOHCAHTOA
θ = angle
sin θ= opposite / hypotenuse
cos θ= adjacent / hypotenuse
tan θ= opposite / adjacent

Pythagorean Theorem:

Right Triangles

a² + b² = c²

Conversion:

Decimal Inches to Millimeters, multiply by 25.4

2 in x 25.4 = 51mm
.75 x 25.4 = 19mm

Millimeters to Decimal Inches, divide by 25.4

75mm ÷ 25.4 = 2.95 in

10mm ÷ 25.4 = .39 in

Physics:

Density = mass/Volume; D = m/V
Total Distance = average velocity/time; d=vt
Momentum = Mass * Velocity; p=mv
Work = Force * Distance; W=Fd
Power = Work / Time; P=W/t
Energy Transfer = Mass * Specific Heat * Change in temp.
q=mc/\T
Torque = Force * Distance
Moment = Force * Distance

Unit 4 - Mechanisms

Simple Machines

MA = Mechanical Advantage
MA = R ÷ E
R = Load
E = effort force
LE = length to effort
LR = length to resistance
L = slope length
H = slope height
C = circumference
Pitch = screw pitch
TPI = threads per inch

Lever

MA = LE ÷ LR
LE x E = LR x R
Load (R) = MA * Effort (E)
Effort (E) = Load (R) ÷ MA

Wedge

MA = Length ÷ Height
Load (R) = MA * Effort (E)
Effort (E) = Load (R) ÷ MA

Inclined Plane

MA = Length ÷ Height (Length = Slope)
Load (R) = MA * Effort (E)
Effort (E) = Load (R) ÷ MA

Wheel & Axle

MA = Wheel Radius / Axle Radius
MA = LE ÷ LR
Torque = Force * Radius
Circumference = π * D
Load (R) = MA * Effort (E)
Effort (E) = load (R) ÷ MA

Screw

Pitch = 1 ÷ (teeth per inch)
Circumference = π * Diameter
MA = Circumference ÷ Pitch
MA = (π * Diameter) ÷ Pitch

Pulley

MA = Total number of strands supporting the load (count last strand if going up only!)
MA = Load (R) ÷ Effort (E)
Load (R) = MA * Effort (E)
Effort (E) = Load (R) ÷ MA

Gear

GR = Gear Ratio
Nin = number of teeth on driver gear
Nout = number of teeth on driven gear
Din = driver gear diameter, in
SR = Win / Wout
Dout = driven gear diameter, in
Win = driver gear speed, rpm
Wout = driven gear speed, rpm
Tin = torque of driver gear, ft lbs.
Tout = torque of driven gear, ft lbs.

GR = Input Rate / Output Rate
SR = speed ratio
Win / Wout = Dout / Din
Tin / Tout = Din / Dout

Belt

D in = Driver pulley diameter, in
SR = Speed Ratio = Input Rate / Output Rate
D out = Driven pulley diameter, in
w in = Driver pulley speed, rpm
w out = Driven pulley speed, rpm
T in = Torque of driver pulley, ft lbs.
T out = Torque of driven pulley, ft lbs.
Win/Wout = Dout/Din
Tin/Tout = Din/Dout
SR = Win/Wout

Thermodynamics

1st Law (closed system):

Q - W = /\U
Q = heat transfer
W = work
/\U = m (u2 - u1)

1st Law (open system):

Q - W = /\H
Q = heat transfer
W = work
/\H = m (h2 - h1)

Fluid Systems

Pascal's Law:

The pressure can be calculated with the formula:

or F = p x A

p = pressure (psi)

F = force (pound)

A = area (square inch)

Charles Law:

$\frac{V_1}{T_1} = \frac{V_2}{T_2} \qquad \mathrm{or} \qquad \frac {V_2}{V_1} = \frac{T_2}{T_1} \qquad \mathrm{or} \qquad V_1T_2 = V_2T_1$

V1 = initial volume

V2 = resulting volume

T1 = initial absolute temperature

T2 = resulting absolute temperature

Boyle's Law:

P₁V₁ = P₂V₂

Were P = Pressure and V =Volume

Bernoulli's Principal:

As the speed of a moving fluid (liquid or gas) increases, the pressure within that fluid decreases.

Electrical Systems

Ohms Law

Unit 5 - Statics & Strength of Material

English System:

1 foot (ft) = 12 inches
1 yard (yd) = 3 feet
1 mile (mi) = 1760 yards
1 sq. foot = 144 sq. inches
1 acre = 4,840 sq. yards; = 43,560 sq ft
1 sq. mile = 640 acres

Units of System:

mm = millimeter = .001 m
cm = centimeter = .01 m
dm = decimeter = .1 m
m = meter = 1 m
dam = decameter = 10 m
hm = hectometer = 100 m
km = kilometer = 1000 m

Conversion:

1 lb = 4.4482 N
1 slug = 14.5939 kg
1 ft = 0.3048 m
1 in = 2.54 cm
1 ft = 0.3048 m
1 mi = 1.609 km
1 cm = 0.3937 in
1 m = 3.2808 ft
1 km = 0.6214 mi
1 oz = 28.35 g
1 lb = 0.4536 kg
1 gal = 3.785 L
1 g = 0.0353 oz
1 kg = 2.2046 lb
1 L = 0.2642 gal

Statics

∑ = sum

F = force

M = moment about a point

CCW = counter-clockwise

CW = Clockwise

M = F×D

∑F_X = 0 = X(right) – X(left)

∑F_Y = 0 = Y(up) – Y(down)

∑M = 0 = CCW + CW

Free Body Diagrams
Centroid

center of gravity or center of mass

Symbol = and Safety
Factors of

Centroid = Center of Gravity

Triangle:

Centroid = h/3; b/3

Irregular Shapes:

Section off shape for easy calculations

Label each section i.e. A₁, A₂, A₃, etc.

Find centroid of each section ( X & Y)

Calculate Area of each section

Use table below:

Section Area X Y A x X A x Y

A₁

A₂

A₃

add above

add above

add above

Total Area

divide by total area divide by total area

Centroid X = Y =

Loads:

Axial Load: s = P / A        A = Cross Sectional Area

Shear Load: t = P / A       A = Cross Sectional Area

Bending Stress:

   or

s = bending stress

M = moment

C = maximum fiber distance

I = moment of inertia of cross section

S= section modulus= I / C

Strain:

Strain = Deformation

              Original Length

or

ε= d / L

Deflection:

d = P * L

      A * E

P is the applied load

L is the length

A is the cross section area

E is the elastic modulus

Cantilever Beam with Concentrated Load:

d max = - P * L3

                 3*E * I

d max is the maximum deflection

P is the applied load

L is the length

I is the cross section moment of inertia

E is the elastic modulus

Simply Supported Beam with Concentrated Load

d max = - P * L3

                 48*E * I

d max is the maximum deflection

P is the applied load

L is the length

I is the cross section moment of inertia

E is the elastic modulus

Moment of Inertia = 1/12 (bh3)

M of I (Plank) = 1/12 ((6)(2)3) = 4 in4
M of I (Joist) = 1/12 ((2)(6)3) = 36 in4

Unit 6 - Strength of Material:

Variables:

∆ = delta = the change in

d = deformation

σ = stress (force per unit area)

Î = strain

E = modulus of elasticity, Young’s Modulus

P = axial force

A = cross section area

A_o = original area

L_o= oringinal test length

L = length

r = radius of a circle

d = diameter of a circle

F = Force

Press = Pressure

T = Shear Strength

Formulas:

Stress = σ = P ÷ A_o

Strain = Î= d ÷ L_o

Deformation = d= PL ÷ A_oE

Modulus of Elasticity = E= ∆σ ÷ ∆Î

Modulus of Elasticity =

Modulus of Elasticity = ((P1-P2) * L_o) ÷ ((d1-d2) * A_o )

A=πr² (area of circle when using the radius)

A= .7854d² (area of circle when using the diameter)

Axial Force = P = F ÷ A

Press = P ÷ A

Statistics:

Mean = Add up all the data, and then divide this total by the number of values in the data.

Mode = Number which occurs most often

Median = Put the numbers in order and find the middle number

Range = Subtract the lowest value from the highest value.

Standard Deviation:

S = standard deviation
Σ = sum of
X = individual score
M = mean of all scores
n = sample size (number of scores)

Unit 8 - Kinematics

Acceleration of Gravity:

The acceleration of gravity on earth is approximately:
9.81 m/s² in SI units (Metric)
32.15 ft/s² in US Customary units.

Force = Mass (m) x Acceleration (a)

Weight (W) = Force (F)

Force (F) = Mass (m) x Gravity (g)

Weight (W) = Mass (m) x Gravity (g)

Variable

Variable Name

s

Displacement

x

Horizontal displacement; Range

y

Vertical displacement

t

Time

v_i

Initial velocity

q

Theta

v_ix

Initial horizontal velocity

v_iy

Initial vertical velocity

a

Acceleration

a_x

Horizontal

acceleration

a_y

Vertical

acceleration

g

Acceleration due to gravity

Potential Energy = mass * gravity * height

Initial Velocity:

Horizontal Component: V_ix = ViCosq

Vertical Component: V_iy = ViSinq

Range: