MATH – ECON – ENGG SCIENCE
PLANE GEOMETRY
Polygons
Number of diagonal: Nd = nC2 – n
⁄ )
(
Interior angle:
Area of n-side polygon:
side b:
(
)
Cylinder
circumscribed in a circle:
√(
)(
(
)
Segment Area:
Circle Theorems
Inscribed angle (a); Tangent & chord (b):
̂
)
)(
)(
)(
)(
Frustum of Cone
(
(
Sphere
)
A
( )( ) ( )( )
Intersecting secants (d):
̂)
(̂
= sum of prod. of opposite sides
3
A
D
A
θ
C
(d)
C
B
(e)
Triangles
)(
)⁄
)(
Inscribed in a circle:
⁄
Circumscribes a circle:
Circle tangent to side a:
(
)
)
H
2
1   y ' dx

2
A
(
(
(
Wedge:
S A
T C
O
)
)
)
x2
y2
x1
x2
y1
(
)
)
Zone:
Cone:
Pyramid:
(
)
(
Torus
)
√
√
( )
( )
Sine Law
Cosine Law
Tangent Law
Ellipsoid
A
Oblate Spheroid minor axis
B
B
θ
)
(
ab
SOLID GEOMETRY
Prism
C
B
TRIGONOMETRY
SOH CAH TOA
CHO SHA CAO
)
Segment:
A    ycurve,top  ycurve ,bottom  dx
x1
1 2
A   R 2 d
2 1
(c)
R
Lune:
A   ycurve dx   xcurve dy
D
θ
B
x2
S
Spherical:
θ x
θ
)
θ
Plane Area
( )( ) ( )( )
Tangent and Secant (e):
̂)
(̂
( )( ) ( )
(b)
2
y1
(
)
(
)
SPHERICAL TRIGONOMETRY
Truncated Prism
A
Prolate Spheroid major axis
Prismatoid
(
Regular Polyhedron
Two bases:
[
Hyperboloid
[
Conoid
Regular
Polyhedron
F
Tetrahedron
4
4
4
6
Hexahedron
6
6
8
12
Octahedron
8
8
6
12
Dodecahedron
12 12
20
30
= 7.66
3
Icosahedron
20 20
12
30
= 2.18
3
A
V
c
B
Paraboloid
)
E
(F+V-2)
Volume
=
2 3
12
=
=
3
2 3
3
b
a
C
Ac
cc
B
b
a
180 < A + B + C < 540
Napier’s Rule I: Sin-Tan-Ad
]
Napier’s Rule II: Sin-Cos-Op
]
Sine Law
Cosine Laws: “SPAN”
Volume
Circular Disk: V  

x2
x1

 r 2 dy
Propositions of Pappus
First Proposition:
A  2 R  S
Second Theorem:
V  2 R  A
Length of an Arc
)
Ptolemy’s theorem:
Ellipse
V 
x1
y2
x1
)
√(
Parabolic segment
Intersecting chords (c):
̂)
(̂
A
)(
)(
√(
A
Frustum of a Pyramid
(
√
Cone
)
Cyclic Quadrilateral
Bramaguptha’s Formula:
Circles
Arc length:
Sector Area:
(a)
Circular Ring:
)
Pyramid
)(
√(
x2
(
Rhombus:
Trapezoid
(
Trapezium
Cylindrical Shell: V  2 xycurve dx

s: semiperimeter
θ: average of opposite angles
inscribed in a circle:
(
)
√ (
(
Parallelogram
ycurve dx
Spherical Defect, d:
(
Spherical Coordinates
z
)
ϕ
ANALYTIC GEOMETRY
Division of Line Segment
√
Distance bet. line and a point
√
Area of n-sided polygon
*
+
Conic Sections
Conics
Eccentricity
e f /d
Discriminant
B 2  4 AC
Hyperbola
>1
>0
Parabola
=1
=0
Ellipse
<1
< 0 (A ≠ C)
Circle
=0
< 0 (A = C)
1   y '2 

R
y ''
Hyperbola:
Limacon
Cycloid
Cylindrical Coordinates
dx
x n 1
 x dx  n  1  x  ln x
  sin x  dx   cos x
 
r 2  a 2 cos 2
r  a sin 2
r  a 1  cos  
r  b  a cos2
x  a   sin  
y  a 1  sin  
z
P(r,θ,z)
x
θ r
z
y
  cos x  dx  sin x
  tan x  dx  ln sec x
 u dv  uv   v du
 f  x  dx    f  x  dx
 f  x  dx   f  x  dx   f  x  dx
b
DIFFERENTIAL EQUATION
Variable Separable
Homogeneous DE
( )
( )
degree of M = degree of N
Sol’n: y = vx or x = vy
Exact DE
(
)
( )
⁄
⁄
Sol’n: Integrate Mdx and Ndy,
Equate to solve g(y) or h(x)
Unexact DE
⁄
⁄
(
)
(
∫ *
∫ ( )
( )
+
)
*
∫
+
2
+
Solve for u’, integrate, and substitute.
LOGARITHM
(
)
(
) (
)
∫ ( ) ∫ (
Bernoulli’s Diff. Equation
( )
( )
(
c
b
a
a
c
(
(
(
Trigonometric Substitution
m
n
 sin u cos u du :
1 2 3
4 6
M12  4 5 6 
7 9
7 8 9
)∫ ( )
Cofactor
( )
Adjoint Matrix
( )
Pivotal Method
Sol’n: Solve for roots.
(a) Real and distinct
(b) Real and repeated
(c) Complex,
(
Non-homogeneous LDE
(
)
)
Sol’n:
)
MATRIX AND DETERMINANTS
Minor, Mij
)∫ ( )
∫ ( )
2ND order LDE
)
( )
( )
)
b
If m or n is odd,
2
∫ ( )
∫ ( )
( )
∫ ( )
a
a
b
For
+
; sin
cos + sin
]
cos ; [ 0 + 1 + +
cos
]
sin
+[ 0 + 1 + +
sin
Substitute and solve for coefficients.
yp by MVP,
Substitute u to coefficients of yc.
Linear Differential Equation
( )
( )
n
x3  y 3  2axy  0
1
COMPLEX NUMBERS
3/2
INTEGRAL CALCULUS
x
x
  0  dx  C
  e  dx  e
Diameter of the conics:
Differential, y=x/m
Polar Coordinates
;
;
+
u du :
Radius of Curvature
L’hôpital’s Rule
f  x
f ' x
f n  x
Lim
 Lim
 ...  Lim n
x a g  x 
x a g '  x 
x a g  x 
Parabola:
Cardiod
 sec
n
cos
d
d
c  0
 a x    log a e  a x
dx
dx
d
d
1
 ln x  
 x n   nx n1 dx
dx
x
d x
d
1
x
 log a x    log a e 
e   e
dx
dx
x
d
1
1
d
sin
x



 sin x   cos x dx
1  x2
dx
d
d
 cos x    sin x
 cos1 x    1 2
dx
dx
1 x
d
2
d
1

1
 tan x   sec x
 tan x  
dx
dx
1  x2
d
dv
du
 uv   u  v
dx
dx
dx
du
dv
v
u
d
dx
dx
u / v  
dx
v2
Ellipse:
Four-leaved Rose
n
DIFFERENTIAL CALCULUS
Distance bet. 2 parallel lines
𝑝
0

Angle bet. 2 intersecting lines
(
)
Lemniscate of Bernoulli
r
yp by MUC,
𝑓( )
y
For tan u du or
Angle of Inclination
Folium of Descartes
θ
x
√
If m and n are odd,
P(r,θ,ϕ)
( )
4 1 3
1   3 0 
2  3 4   3 2 
x  2 0 1   1
10   5 2  6   3 0 
10 6 5
Inverse Matrix, A-1
Transpose
Form Adjoint Matrix
Divide by determinant
ALGEBRA
Binomial Expansion
term with yr:
Non-mutual exclusive
n r r
y
n Cr x
Arithmetic Progression
(
)
(
)
Deferred Annuity
*
Conditional
)
Pyramidal Number (square base)
(
)(
(
Population:
)]
Pyramidal Number (triangle base)
)(
)
Sample:
(
(
)(
)
Work Problems
Unit work * time = 1
Total man-time = Σ each man-time
(
)
(
)
Mixture Problems
Quantity: A + B = C
Composition: Ax + By = Cz
Permutation: order
(
)
Alike things:
Ring:
(
)
Combination: group
(
)
taken 1 or 2 or n
Benefit to Cost Ratio
Annual Equivalent Cost
C0
CL
C

1  1  i  n   1  i n  1

 

i
i

 

Benefit to Cost Ratio
⁄
)
Sinking Fund
Annual depreciation, d
̅)
(
̅)
)
(
(
ENGINEERING SCIENCE
Vectors
)
A  Ax i  Ay j  Az k
+
Dot Product
Book Value, Cn
A  B  A B cos 
Declining Balance
Depreciation at nth year
(
(
)
Ordinary: 360 days
Exact: 365/366 days
Compound Interest
(
)
(
)
Nominal rate of Interest
√
√
* +
(
*
Friction
Sliding Block
)
Rolling Friction
Belting Friction
Double Declining Balance, DDB
Same with Declining Balance but
Sum of Years Digit, SYD
Depreciation at nth year
(
(
(
)
)
√(
)
Service Output Method
+
+
⁄ )
Catenary: uniformly dist. along length
)
(
Working Hours Method
(
)
Bonds
*
Cable
Parabolic: uniformly dist. horizontally
⁄
)
Total depreciation
Annuity:
(
A  B  a n A B sin 
)
Scrap value
(
)
Total depreciation
Discount
Discount:
Rate of discount:
(
)
Rate of discount vs interest
*
Cross Product
Book value
Effective rate
(
)
Continuous Compounding
Annuity Due
)
*
ECONOMICS
Simple Interest
Joint
)
Total depreciation, Dn
Standard Deviation:
Relative Variability: SD/mean
Z-score
Ordinary:
(
(
)
(
PROBABILITY
Complementary
Breakeven
Book Value, Cn
Pyramidal Number (rectangle base)
(
Total Investment  Salvage Value
Payout

Period
Net Annual Cash Flow
Total depreciation, Dn
STATISTICS
Median: middle of arranged set
Mode: most frequent value
Mean: ̅ , average
Variance
Other Sequences:
M-gonal Numbers
(
Pay out Period
p: success, f: failure
√
)(
)
Depreciation:
Straight Line, SLD
Annual depreciation, d
Repeated Trials
)
(
+(
Independent
Geometric Progression
[
)
Perpetuity
P  n Cr pr qnr
(
Rate of Return, ROR
(
(
)
+
(
)
)
Centroid (1ST Moment)
Centrifugal Force
Ft  mat
at  V t  r
S x   x dS
Ax   x dA
V x   x dV

Polar Moment of Inertia
Jz  Ix  I y
Mass moment of Inertia
I   r 2 dm
I x  I x0  mr
2
Dynamics (Kinematics)
Uniform Accel. Motion (Free fall, a=-g)
V f2  V02  2a x
x  V0 t  1 2 at 2
V f  V0  at
x  1 2 V f  V0 t

Projectile Motion
 y  V0 sin   t  1 2 gt 2
h
V0 sin  
2

x  V0 cos   t
V 2 sin 2
R 0
g
2g
Rotational Kinematics
same with linear but replace
s   , v  , a  
Linear and Angular Relations
s  r , v  r , a  r
Dynamics (Kinetics)
Newton’s Law of Motion
1st Law:
F  0
2nd Law:
F  ma
3rd Law:
FR
Newton’s Law of Universal Gravitation
mm
F  G 12 2
s
G  6.67  1011 N-m2 /kg 2
D’Alembert’s Principle
F  REF  0
REF  ma
Circular Motion
Centripetal Force
Fn  man
an  V 2 r  r 2
Wnet  KE
x
y
16
4a
3
I x0 
F t   P
F  t f  t0   m  v f  v0 

 ab3
bh3
21
3
x b
4
Iy 

2bh3
7
3
x b
8
Iy 
PARABOLA 2
e=1
0<e<1
e=0
e
Ix 
SPHERE:
h2 h1
ROD,
SEMI CIRCLE
4
x
J 
y
cg
4
y
x
HOLLOW
2 2
mr
3

1
m R2  r 2
2
ONE END
I
1 2
mL
3

1
m 3R 2  3r 2  4 L2
12
I
3
mr 2
10
RECT. PLATE thru CENTER
r
2
4
I
x
cg
HOLLOW, at end
CONE:
h
b
h
I
CENTER
I
h
2
y
 ab3
HOLLOW
1
I
mL2
12
OTHER GEOMETRIC PROPERTIES
TRIANGLE
bh3
bh3
h
Ib 
I x0 
y
12
36
3
RECTANGLE
r
cg
θ
θ
I
1
I  mr 2
2
H  P  r  I 0
I x0  I y0 
y
b
SOLID
Angular Momentum
4
x
cg
2
I  mr 2
5
J  F  r  t
bh3
12
cg
a
y
2b3 h
15
3
y h
5
CYLINDER: SOLID
CIRCLE
b
b3 h
5
3
y h
10
Thrown at angle: e  tan  2 cot 1
Angular Impulse
I x0 
16
4b
3
I y0 
4
Ix 
 P before impact   P after impact
e  V2 '  V1 ' V2  V1
bh3
3
x
y
 a3b
PARABOLA 1
Momentum
Ib 
cg
ELLIPSE
Impulse-Momentum Theorem
Perfectly elastic:
Inelastic collision:
Perfectly inelastic:
Special Case:
Bounce:
Iy 
16
4r
3
1 4
sin 2 
r  

4 
2 
1 
sin 2 
I y0  r 4  

4 
2 
2r sin 
x
3
Work-Energy Theorem

Ix 
y
 r4
I x0 
M  I 
I  mk 2
KEr  1 2 I  2
Parallel Axis Theorem
I x  I x 0  Ar 2
8
SECTOR
Centroidal Rotation
m
y
QUARTER CIRCLE
 ab3
V
tan     
gr
I   t  r 2 dA
 r4
Iy 
16
4r
x
3
2
Thin Plate
 r4
Ix 
Banking of Highway
m
Iy 
QUARTER CIRCLE
T  W cos 
F V2
tan   n 
W
gr
t  2 h / g
2
 r4
8
4r
y
3
Total accel  an 2  at 2
Conical Pendulum
Moment Of Inertia (2ND Moment)
I y   x 2 dA  A  x
Ix 

1
m a 2  b2
12



1 lbf
CONVERSIONS
10^X
18
15
12
9
6
3
2
1
PREFIX
Exa
Peta
Tera
Giga
Mega
Kilo
Hecto
Deka
10^X
-1
-2
-3
-6
-9
-12
-15
-18
PREFIX
deci
centi
milli
micro
nano
pico
femto
atto
DISTANCE/SPEED/ACCEL
1 in
= 1000 mil
1 ft
= .3048 m = 3 hands
1 yd
=
3 ft
1 fathom
=
6 ft
1 chain
= 66 ft
1 furlong
= 660 ft
1 mile
= 5280 ft
1 n. mile
= 6080 ft = 1/60 degree
1 knot
=
1 naut. mile/hr
1 m/s
= 3.6 kph
1 lightyear = 9.46 x 1012 m
1 parsec
= 3.084 x 1013 m
1 Angstrom = 10-10 m
9.81 m/s2 = 32.2 ft/s2
AREA
1 acre
1 are
1 hectare
= 1 furlong x 1 chain
= 100 m2
= 10000 m2
VOLUME/FLOW RATE
1 gal
= 3.785 L = 0.1337 ft3
1 bbl
= 42 gal
1 m3
= 1000 L
1 ganta
=
8 chupas = 3 L
MASS
1 kg
1 lbm
1 slug
1 tonne
1 short ton
1 long ton
= 2.2
= 16
= 32.2
=
1
= 2000
= 2240
lbm
oz
lbm
MT = 1000 kg
lbm
lbm
DENSITY/CONCENTRATION
1 kg/L
= 62.4 lbm/ft3
1 ppm
=
1 mg/L or 1 mg/kg
FORCE
1N
1 kgf
= 100 000 dynes
=
9.81 N
PRESSURE
1 atm
1 bar
1 MPa
ENERGY
1 Btu
1 kcal
1J
1 chu
1 eV
=
4.448 N
= 101.325 kPa
=
14.7 psi
=
29.92 inHg = 760 mmHg
=
760 torr
=
100 kPa
=
1 N/mm2
=
1055 J
=
252 cal
=
778 ft-lbf
=
4.187 kJ
=
107 erg
=
1.8 Btu
= 1.602 x 10-19 J
POWER
1 hp
=
0.746
=
550
=
2545
1 metric hp =
736
1 kW
=
3412
1 TOR
=
3.516
= 12 000
1 BoHP
= 35 322
kW
ft.lbf/s
Btu/h
W
Btu/h
kW
Btu/hr
kJ/hr
TEMPERATURE
F
= 1.8C + 32
R
= F + 460
K
= C + 273
R
= 1.8 K
F
= 1.8 C
DYNAMIC VISCOSITY
1 poise
=
0.1 Pa-s
KINEMATIC VISCOSITY
1 stoke
=
1 cm2/s
ANGLE
1 rev
= 360
=
2π rad
= 400 grad
= 400 gons
= 6400 mils
CONSTANTS
GENERAL
̅ =
=
=
8.3143 J/mol . K
1545 lbf-ft/lbm.mol.R
0.0821 L-atm/mol-K
c
=
3 x 108
NA = 6.02 x 1023
ς = 5.67 x 10-8
Solar Constant =
Radius of Earth:
Earth Escape V:
Human Heat:
m/s
/mole
W/m2K4
1353
6.38 x 106
11.2
225
W/m2
m
km/s
Btu/hr
WATER/ICE/LIQUIDS
Cp =
4.186 kJ/kg.K
Lf =
334 kJ/kg
=
144 Btu/lbm
Lv =
2257 kJ/kg
=
97 0 Btu/lbm
E = 2.1 x 106 kPa
Surface tension, ς
@ 0C
ς = 0.076 N/m
@ 100C ς = 0.059 N/m
Cp of ice
= 0.5(Cp water)
Liquids:
SGmercury
= 13.55
SGsea water
= 1.03
AIR/GASES
k = 1.4 or 1.3 (hot)
Cp = 1 kJ/kg-K = 0.24 Btu/lbm.R
Cv =
0.7186 kJ/kg
R =
0.287 kJ/kg.K
=
53.34 lbf-ft/lbm.R
ρ =
1.2 kg/m3
Latent hv = 2442 kJ/kg
Specific heat ratio:
He, noble gases k = 1.667
Carbon dioxide k = 1.287
Nitrogen
k = 1.399
STEEL
E =
G =
α =
ρ =
30 x 106
12 x 106
12 x 10-6
7860
psi
psi
/C
kg/m3
OTHERS
Molecular Weights:
H(1),He(4), C(12),N(14),O(16)
S(32), Air(29)
OTHERS
1 clo
= 0.880 [Btu/h· ft²·°F]-1
1 board ft = 1 ft x 1 ft x 1 in
MACHINE DESIGN & SHOP PRACTICE
With shock factors
)
[(
Stmax =
STRESSES
Axial Stress
St =
Ssmax =
Shear Stress
Ss =
Vertical Shear
Torsion
Ss =
=
=
(
)
=
Sf =
=
Thermal Stress ST = (
δ = (
Design Stress Sd = =
Modulus of Elasticity S =
Modulus of Rigidity
G =
)
(
)
√
[
]
Ssmax =
Stmax =
Ssmax =
*(
√(
)
√(
)
Variable Stresses
Ductile Materials
=
Brittle Materials
=
SHAFTINGS
Power Transmission
P=
Line Shaft
P=
Short Shaft
P=
*units in hp, inches, rpm
Diameter
Power
*kW,N-mm,rpm
*hp,lbf-in,rpm
)
SV =
θ =
SC
=
L
=1.18D
SS
=
Compression
SC
=
Total Torque
Total Capacity
T =
TC =
D=√
P =
P =
)
+
Shearing of Bolt
T =
Compression of Bolt
THREADED MEMBERS
Stresses
Valiance
T =
Power Screw
Collar friction TC = (
Raising & Lowering
Square
Tf =
Brass Tubes
)
(
)
ACME
Tf
=
*
+
Trapezoid
Tf
=
*
+
American
Total Torque
Efficiency
Tf
T
e
=
=
=
*
+
=
Friction angle β
Linear Velocity V
Lead Angle
λ
=
= NL
=
Lead
L
L
L
Do
=P
= 2P
= 3P
=
d
=*
Screw D
()
)
+
SW =
=
Axial
Sa
=
Dbc =
Max. Tensile Stmax =
h
h
h
=
=
=D
Max. Shear
Ssmax=
Eqv. Max.T.
Stmax =
T
=
T
T
Fi
=
=
=
Pcr
=
t/do > 0.03
Pcr
=
⁄
]
)
( )
( )
Lap-welded Steel Tubes
t/do > 0.03
Pcr
=
=
( )
( )
( )
Short Tube
Collapsing/Critical Pressure
Pcr =
(
)
Crushing Stress
Sc =
RIVETS AND WELDED JOINT
Rivet
St
=
=
Critical Pressure Thin Tubes
Stainless Steel Tubes
t/do < 0.025
Pcr
e
[√
(
t/do > 0.03
single
double
triple
Longitudinal SL =
Thin walled Sphere (t >0.1ri)
Tangential St =
Thick walled
t
=
=
Shaft D
D = ( ) or ( )
*diam. (inch); L (ft); rpm; hp
PRESSURE VESSEL
Thin walled Cylinder
Tangential St =
Thickness
Pcr
FS
⁄
(
t/do < 0.025
)
(
Trms Power HP = ( )
Faires
Sd =
Applied Load
Valiance
Fa =
Faires
Fe =
Bolt Constant, C
Bronze
c
= 10 000
Carbon Steel
c
= 5 000
Alloy Steel
c
= 1 500
Working Strength of Bolt
]
Ws = [
Bolt Spacing
Z =
Bolt Circle Diameter
Depth Tap
Brittle
Valiance (Steel)
Faires (S, WI)
Initial Torque
Valiance
Faires
Lubricated
As received
Initial Tension
) ]
Outside D
Handbook
COUPLING
Ssmax = √
Stmax =
(
(
=
Same Material
SPLINES
Shearing
)
)
)
)
SS
Compressive Stress
(
Combined Stresses
S
=
Stmax =
Angular Deformation
KEYS
Shearing Stress
Bearing Stress Sb =
Bending Stress Sf =
√(
√(
Weld
Ave. Shear
Ss
=
FS
FS
=
Max. Shear
Max. Tensile
Ssmax =
Ssmax =
BEARINGS
Bearling Pressure F
=
Max. Contact Stresses
Balls
Ssmax = 0.31 Smax
Cylinders
Ssmax = 0.31 Smax
Life in million revs
)
Balls
L
=(
)
Cylinders
L
=(
Compressive Breaking Load
FC =
Carbon steel
k
= 100,000
Alloy Steel
k
= 125,000
FS
FS = 10
Maximum Load
Fmax =
Diam. Clearance
Cd
=
SPRING
End Type
Actual n Solid L
Free L
Ground
n
(
)
Plain
n
Squared &
)
n+2 (
Ground
)
Squared
n+2 (
Spring Index
c
=
Whal Factor
Stresses
k
=
=
Round Wire
S
=
Square Wire
S
=
Rect. Wire
Deflections
S
=
Round Wire
δ
=
Square Wire
δ
=
Rect. Wire
δ
=
Stress (Torsion)
S
Deflection (Torsion)
=
Helical round
δ
=
Spiral round
δ
=
Spiral rect.
δ
=
(
)
(
S
=
Multiple
S
=
)
δ
=
Multiple
δ
Length of Wire
Free Length
L
FL
Impact Load
Spring Rate
Spring System
Series
Parallel
FLYWHEEL
Total Weight
Rim weight
(
Punch hole Energy
Punching Force
Steel round
Steel square
Brass rect.
k
k
k
=
(
)
E
=
F
=
=
Coef. Of Fluctuation Cf
=
=
=
( )
Fa,cw =
( )
Fa
T=
=
(
(
(
(
)
)
)
Uniform Wear
Fa =
T=
FC =
Belt cross-section
)
Base circle D
Center distance
external
internal
= F1 – F2
A=
(
* (
(
Fs
=
(
)
)
Dynamic Load
Fd
=
)
(
)
T=
Fa =
Fa =
Engagement Force
Max. Pressure
Pmax =
*
+
(
)
Radial spring
S =
Garter spring
S =
Crossed L = (
Belt Speed
)
V=
=
(
)
(
)
(
(
)
(
)
(
)
)
)
)(
Factor of Safety
)
Outside Diam.
FS =
Di =
(
⁄
)
D0i = *
(
(
=
Pd
PC
a
=
=
=
Whole depth
Working depth
Clearance
Tooth thickness
d
d
Do
Drp
Drg
W
Wr
c
t
=
=
=
=
=
=
=
=
=
Backlash
B
=
Face width
b
=
(
)+
)
Fd = [
(2000  Vm  4000 fpm)
+
(Vm > 4000 fpm)
√
]
Normal Circular Pitch
Axial Pitch
Pcn =
Pa =
Lead
single helix
double helix
triple helix
multiple helix
BF Strength
L
L
L
L
Fs
=
= 1.0 to 1.25
= 1.25 to 1.5
= 1.5 to 1.75
= 1.75 to 2.0
 Fd
=
⁄
=
= Pa
= 2Pa
= 3Pa
= nPa
=
Dynamic Load
Fd =
=
)
√
Failure based on fatigue
Nsf
Uniform load w/o shock Nsf
Medium shock
Nsf
Moderately heavy shock Nsf
Heavy shock
Nsf
Failure based on wear
Fw
Wear Load
Fw
GEARS (HELICAL)
Radial Force
Fr =
Tangential Force
Ft =
Axial Force
Fa =
Normal Pressure Angle
(
)
ϕn =
Normal Diametral Pitch Pdn =
=
Chain Length
L
GEAR (SPUR)
Diametral Pitch
)
Circular Pitch
)
Addendum
Dedendum
14.5 and 22.5
20 and 25
Outside D
Root D
(
(
(
)⁄
)⁄
Intermittent Service
Commercial cut (Vm  2000 fpm)
Fd = *
+
Carefully cut
Fd = *
Precision cut
*units in lbf, inches
POWER CHAIN
Pitch Diameter
Expanding ring clutch
T =
Band Clutch (same with band brake)
Centrifugal Clutch
Torque
T=
Radial spring force
)
Ultimate Strength for plow steel
6 x 7; 6 x 19; 6 x 37
Fu =
)
=(
=(
BF Strength
(
WIRE ROPES
Bending Load
Fb =
Weight of rope Wr =
)+
Total Tension Ft = (
C
C
)
)
V-belts
Tension Ratio
Db = D
(
L = (
Open
Fa =
T=
Fr =
=
Fe =
BELTS
Centrifugal Force
Crossed
θ =
Power transmitted
P =(
H =
H =
tr =
T=
*units in lb, inches, rpm
=
Belt length
Uniform Pressure
Block Clutch
Torque
Radial Force
Belt tension ratio
Effective Belt pull
Angle of Contact
Open
θ =
( )
CLUTCH
Plate/Disk Clutch
Cone Clutch
Torque
Axial Force
= WA+WH +WR
( )
=
=
S
BRAKES
Band Brake
Tension Ratio
)
= [ ( ⁄ )]
=
*units in tons, inches
Hoop Stress
(
=
=
=
) =( )
=
=
=
Smax =
)
cast iron
C = 0.13 Btu/lb.F
cast steel
C = 0.116 Btu/lb.F
Spot Brake
Braking torque capacity
T
=
)
= (
Wf
WR
F
F
Max. stress
Brake Shoe
Heat dissipated
in brakes
for lowering brakes
Temperature rise
Deflections (Leaf)
Single
T
=(
Pmax =
Actuating Force
Differential Brake
Actuating Force
Block Brake
Braking Torque
*a-moment arm; L-wire length
Stresses (Leaf)
Single
Torque
Max. unit pressure
(
)
√
Wear Load
Fw =
Formative no. of teeth
Nev =
GEARS (WORM)
Diametral Pitch
Lead
Lead Angle
Pitch line velocity
Pd =
L =
λ =
Vw =
⁄
Vg =
Worm Force
Fw =
Separating Force
FS =
*
+
Tangential Force on worm
FG = *
+
Efficiency of the worm gear
e =
*
+
Face width
Worm OD
b =
Dwo =
Worm Diameter
Dw =
Teeth BFS
Fs =
Dynamic Load
Worm Load
Thermal Capacity
GEARS (BEVEL)
Cutting or Root angle
Face angle
Pitch angle
Fd = *
Fw =
Q =(
Face width
ω
β
γp
γg
b
Length of cone
L =√
Strength
Fs =
Dynamic Load
MACHINE SHOP
Time =
RPM(speed)
+
)
=
=
=
=

(
(
*
⁄
⁄
)
)
+
Fd = *
+
=
Feed (in/min) =
(
)
OTHERS
Petrox Formula
Tf =
, N.m
POWER AND INDUSTRIAL PLANT
ENGINEERING
S =
=
(
→
Water:
θ = 0
Mercury: θ = 140
Variation in pressure
)
(
=
= ̅
=
=
=
Dalton’s Law of Partial Pressure
=
Processes
Nonflow Work:
)
=∫
= (
Steady flow Work:
)
=∫
= (
Heat Transferred:
=
Isometric: V = C
Wn = 0
)
Ws = (
Q = U
S =
( )
)
SVSV
SPSP
SPSP
PTPT
( )
OTTO
BRAYTON
RANKINE
ERICCSON
Isothermal: PV = C
Wn =
( )=
( )
= Wn
= H = 0
= Wn
=
( )
Flow in Pipes
Continuity Eqn
Compressible:
̇ = ̇ →
Incompressible:
=
→
Bernoulli’s Eqn
INTERNAL COMBUSTION ENGINE
Otto Cycle:
Compression ratio
rk = =
Volume Displacement VD =
Percentage clearance
c =
Clearance Volume
Efficiency
VC =
e =
Mean eff. Pressure,
Diesel Cycle:
Compression ratio
Pm =
rk
=
Cut-off Ratio
rc
=
Expansion Ratio
re
=
e =
Dual Cycle
Pressure ratio
( )
Pressure decreases upwards
Pressure increases downwards
Buoyancy
CARNOT
STIRLING
DIESEL
DUAL
Carnot Cycle
Efficiency
Isentropic: PVk = C
Wn =
Ws = k Wn
U = -Wn
H =
Q = S = 0
STST
TVTV
SPSV
SVPSV
Baume =
Liquids:
Gases:
Manometer
)
CYCLES
=
Ws
U
Q
S
=( )
Polytropic: PVn = C
Wn =
Ws = n W n
Q =
FUELS AND COMBUSTION
API
=
)⁄
(
=( )
THERMODYNAMICS
Pabs =
QS =

QL =
H =
Ideal Gases
=
=
=
( )
Isobaric: P = C
Wn = (
Ws = 0
Q = H
S =
Capillary Action
=
Efficiency
rp
=
=
Laminar flow:
Turbulent flow:
Friction Losses
Orifices
Weirs
√
⁄
*(
⁄
)
( )
+
⁄
e =
(
)
√
;
⁄
FLUID MECHANICS
SG =
=
Wta
Waa
=( ⁄ ) =
=( ⁄ )
=
(
)
(
)
Lift
)
77% N2
79% N2
=
Electrical efficiency:
egen =
Thermal efficiency:
eti
=
etb =
etc =
Viscosity
Stokes’ Law
Surface Tension
Soap: σ =
Liquid: σ =
Velocity of Sound
⁄
⁄
)
DIESEL POWERPLANT
Piston Disp.: VD =
Piston Speed = 2 L N
Indicated Power
Pind = Pmi VD
Brake Power
Pb
=
=
T
= Fr
Pb
= Pmb VD
Friction Power
Pf
= Pind - Pb
Mechanical efficiency: em =
Drag
⁄
)
)
)
Bulk Modulus of Elasticity
(
( ⁄
Volumetric Analysis: %V =
%G = %V(
Noncircular:
;
=
Wta
Gravimetric Analysis: %G =
Reynold’s Number
)
[
(
)]
=
=
kJ/kg
= 33820C+144212(H- )+9304S kJ/kg
= 13500C + 60890H
Btu/lb
=
(
)
(
=
Composition of air:
By Weight: 23% O2
By Volume: 21% O2
( ⁄ )
 =( ⁄ )
(
(
SGt
Qh
Qh
Qh
Wta
√
(
)
Engine efficiency:
eei =
eeb =
eec =
Volumetric efficiency: ev
=
Va
=
Specific fuel consumption
(
)
mi
=
mb
=
(
)
mc
=
(
)
Heat rate
HRi =
(
)
HRb
=
(
)
HRc
=
(
)
Equivalent Specific Evaporation
ESE =
Boiler Efficiency
(
)
ebo =
*mC is amount of carbon in ash
Engine at High altitudes
=
Pact
=
(
√
)
inHg
T
=
*h in feet
R
GAS TURBINE POWERPLANT
Thermal efficiency
eth
=
Overall efficiency
eth
=
Combustor efficiency
Eh
=
Net heat plant rate
NHR =
STEAM POWERPLANT
Steam Rate
SR
=
Rated Boiler HP
Water tube: RBoHP =
Fire tube:
RBoHP =
*A in m2
Developed Boiler HP
(
)
DBoHP =
*ms in kg/hr
*h in kJ/kg
Percent Rating
Percent Rating =
Turbine
Wt =
Wact = (
Pump
Wp =
Wact = (
)
)
GAS AND FEEDWATER LOOP
Draft Loss
D=
) cm H2O
(
*Units in SI
Friction factor, f
Air-steel:
Air-concrete:
Fluegas-steel:
Fluegas-concrete:
Fan Work
Specific Speed
Use Factor
U sF =
NS = ⁄
Total efficiency
etotal = ehemev
NONCONVENTIONAL POWERPLANT
Solar Power
Qsun = Qw + PE + Qloss
MACHINE FOUNDATION
Clearance, c
Bedplate to edge: 6 in to 12 in
To ground: 6 in min
Upper width
a = w +2c
Weight of foundation
Wf = 3 to 5 times Wm
Vf = W f / ρ
Lower width, b
=
Depth, h
h = 3.2 to 4.2 times stroke
Vf = ( )
Weight of Steel bar reinforcements
WSB =
Anchor bolts
Depth =
Grate Efficiency
egrate =
Generator Speed
N
=
P
ASME Evaporation Units
)
AEU = (
Factor of Evaporation
FE =
Equivalent Evaporation
EE =
Actual Specific Evap. or Boiler Economy
ASE =
f = 0.005
f = 0.007
f = 0.014
f = 0.014
VARIABLE LOAD PROBLEMS
KW LOAD
PLANT CAPACITY
PEAK LOAD
W=
Air Horsepower
AVERAGE LOAD
kW-hrs
Draft per 30m chimney
D30 = (
)
*Brick and steel: k =2.7
HYDROELECTRIC POWERPLANT
Pwater =
Pelton
h=
Reaction (Francis and Kaplan)
h=
Peripheral Coefficient
=
√
TIME (hrs)
√
Operation Factor
OF =
Plant Factor
PF =
CHIMNEY
Densities ρair =
ρgas =
Draft head
hw = (
)
Volume Flow Rate of Flue gas
Qg =
Theoretical Velocity of Flue gas
Vt = √
Reserve over Peak
ROP = Plant Capacity – Peak Load
Average Load
Ave. Load =
Load Factor
LF =
Capacity Factor
CF =
Annual Capacity Factor
ACF =
(
)
Actual Velocity of Flue gas
)
Va = (
Chimney Inside Diameter, D
Qg = (
Reserve
Over Peak
HPt =
Demand Factor
DeF =
Diversity Factor
DiF =
Utilization Factor
UtF =
)
PIPING
GREEN
SILVER-GRAY
VIOLET
LIGHT BLUE
LIGHT ORANGE
WHITE
BROWN
YELLOW OCHRE
BLACK
SAFETY RED
SAFETY YELLOW
Water
Steam
Acid/Alkali
Air
Electricity
Communications
Flammable, Oil
Gases
Other Fluids, Drainage
Fire fighting
Hazardous
Pipe wall thickness
Power Piping Systems:
tmin =
tnominal =
Industrial and Gas Piping Systems:
tmin =
Refrigeration Piping Systems:
tmin =
Composite Pipe
Q= ( ⁄ ) (
⁄
Adiabatic Compressor Efficiency
ec(adiabatic) =
MULTISTAGE COMPRESSOR
( )
Pm = √( )
*( )
]
)
COMPRESSORS
SINGLE STAGE
NPr =
VD =
Capacity:
c = n, k, or 1
V1 ’ =
Clearance: c = VC/VD
Volumetric Efficiency
=
( )
⁄
ev = [
( )
Work: WS
Polytropic and Isentropic (n=k)
*( )
NS =
√
FANS AND BLOWERS
Static Head,
hs
SATURATION LINE
SPECIFIC VOLUME
WET BULB TEMP
DEW POINT TEMP
REL. HUMIDITY
DRY-BULB TEMP
Air mixing
Mass:
Energy:
Moisture:
Temp:
Air conditioner
RC = (
)
Rate of moisture removal =
Volume flow rate: V1’ =
Cooling tower “drawing
Range: TR =
Approach: TA =
–
Cooling tower efficiency
e=
=
Total head:
h
Capacity:
Power output:
Power input:
Q
= AV
Pair =
Pbrake =
+
Compressor Efficiency
ec =
Piston Speed: V = 2LN
Indicated Power: Pind = PmiVD
H
2
2
1
= hs +
=
( )
P
3 N
5 D
1 ρ
REFRIGERATION
Reverse Carnot COP
Isothermal:
( )
Q
1
3
0
( )
Turbine specific speed, ns =
rk = rcre
=
Psychometric chart
⁄
Dryer
Regain
COP =
Refrigeration Load
Q=
Vapor Compression Cycle
P
1-2: compression
2-3: condensation
3-4: expansion
4-1: evaporation
(
)
=
Moisture content =
3
4
2
1
h
,m3/s
, m2
OTHERS
1 yd3 = 6 sacks cement
Enthalpy
H=
Degree of saturation
D=
=
(
)
Similar Fans:
]
=
Specific Volume
υ=
= e pe m
Static efficiency, es
Actual:
⁄
)
Q H P
1 2 3 N
3 2 5 D
NGr =
MACHINERY ROOM
Exhaust air, Q =
Free aperture, F =
*G in kg
Relative Humidity
RH =
Specific Speed
Similar pumps:
NNu =
,
(
evol
= Q/VD
Volume flow rate: Q = VA
Slip:
S = VD - Q
Percent Slip:
%S = S/VD
Re =
(buoyancy/viscous)
)
Power: P =
Efficiencies:
epump =
eoverall =
Prandtl Number
(momentum/heat)
Nusselt Number
(Tgradient/overall T)
Grashof Number
ev =
(
H=
Sensible Heat Ratio: SHR =
Recirculated air: mr =
(
Ventilation load: QV =
)
)
AIRCONDITIONING
Pressure
Pt = Pa + Pv
Humidity ratio
w= =
emotor =
AMTD =
Reynolds Number
(inertial/viscous)
COP =
+
PUMPS
Total dynamic head:
)
Critical radius:
rc =
Radiation
Q/t = σ [
Perfect Black Body
Convection
Q= (
Heat Exchangers
LMTD = (
⁄
)
Volume flow rate:
V1’ = mv1
Heat rejected,
QR = (
Refrigerating Capacity, QA = (
Refrigerating Effect, RE =
Coefficient of Performance
SPECIFIC HUMIDITY
HEAT TRANSFER
Conduction
(
)
Q=
Fluid to Wall to Fluid
(
)
Q=
Aircon calculation
Sensible:
Qs =
Latent:
QL =
Total:
QT =
(
(
)
)
⁄
)