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Data and formulae for mechanuical enginerring students

Data and Formulae for Mechanical Engineering Students
Department of Mechanical Engineering, Imperial College London
September 2009

Contents
A General information

1

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 . . . . . . . . . . . . . . . . . . . . . . . . .

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C Mechatronics and control
C.1 Charge, current, voltage and power . . . . . . . . . . . . . . . . . . . . . . . . . .
C.2 Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C.3 Transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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 3 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 . . . . . . . . . . . .

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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 . . . . . . . . . . . . . . . . . .

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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)

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Page ii

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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) . . . . . . . .
0
−1

Molar Enthalpy of Formation hf (kJ kmol at 25 C and 1 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 1 bar) . . . . . . . . . . . . . . . . . .
E.13 Superheated Refrigerant 134a (1.5 bar to 4 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, 1 bar. . . . . . . . . . . . . . . . . . . .

E.19 Saturated water and steam — Temperature (triple point to 100 C) . . . . . . . .
E.20 Saturated water and steam — Pressure (triple point to 2 bar) . . . . . . . . . . .
E.21 Saturated water and steam — Pressure (triple point to 2 bar) . . . . . . . . . . .
E.22 Saturated water and steam — Pressure (triple point to 2 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 1 atmosphere) . . . . . . .
E.25 Subcooled water and Superheated Steam (2 bar to 8 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

1
2
3
3
13
13
20
20
21
22
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36
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59
60
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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 . . . . . . . . . . . . . . . .

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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
2
radian/second
2
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
2
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

−2

Page 1


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

2

2

1.3558

2

−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 2

−1

Mechanical Engineering Data & Formulae


A General information

Table A.3: Decimal prefixes
Multiplication factora
12

Prefix

Symbol

1 000 000 000 000
1 000 000 000
1 000 000
1 000
100
10

=
=
=
=
=
=

10
109
106
103
2
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



0 K = −273.2 ◦C

Boltzmann’s constant

k

Characteristic impedance of vacuum

Z0

1.380 × 10−23 J K−1
µ0 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
1
−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

0

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

4

9.65 × 10 C mol
−1

−7

K

−1

−1

−2
−8

−2 −1

Jm

−4

K

−1

8

−3

s

m

3

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 3


A General information

Page 4

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
1
x

log

= − log x

log x n = n log x
log 1 = 0
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 2 + bx + c = 0, then
x=

−b ±

b2 − 4ac
2a

2

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 5


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 = 1 + αx +

α(α − 1) 2 α (α − 1) (α − 2) 3
x +
x + ...
2!
3!
n

2

ex = 1 + x +

(α arbitrary, |x| < 1).

x
x
+ ··· +
+ ...
2!
n!

(|x| < ∞)

2

cos x = 1 −

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 3 2x 5 17x 7
+
+
+ ...
3
15
315

(−

sinh x =

ex − e−x
x3 x5 x7
=x+
+
+
+ ...
2
3!
5!
7!

cosh x =

e +e
2

x

2

−x

2

=1+
3

4

π
π
2
2
(|x| < ∞)

6

x
x
x
+
+
+ ...
2!
4!
6!

(|x| < ∞)

n+1

x
x
x
ln (1 + x) = x −
+
− · · · + (−1)n
+ ...
2
3
(n + 1)
Page 6

(|x| < ∞)

(−1 < x < 1)

Mechanical Engineering Data & Formulae


B.1

Algebra

Stirling’s formula for n! when n is large:
n! ∼
=
or

n

2πn

1
1
ln n − n + ln (2π)
ln (n!) ∼
= n+
2
2

or

B.1.6

n
e

1
log10 (n!) ∼
log10 n − 0.43429n
= 0.39909 + n +
2

Trigonometry

Definitions:

C
A
B

θ

sin θ =

A
C

cos θ =

B
C

tan θ =

A
B

csc θ =

C
A

sec θ =

C
B

cot θ =

B
A

Signs of trigonometric functions in the four quadrants:

(+)

(+)

II (+)
θ

I
θ
(+)

Quadrant:
sin θ
cos θ
tan θ
csc θ
sec θ
cot θ

(+)

I
+
+
+
+
+
+

II
+


+



(+)

θ
III

(+)
III


+


+

(+)
IV

θ
IV

+


+


Trigonometrical identities
cos2 θ + sin2 θ = 1
1 + tan2 θ = sec2 θ
sin 2θ = 2 sin θ cos θ
cos 2θ = cos θ − sin2 θ = 2 cos2 −1 = 1 − 2 sin2 θ
2

sin

θ
=
2

1
(1 − cos θ)
2

cos

θ
=
2

1
(1 + cos θ)
2

Mechanical Engineering Data & Formulae

Page 7


B.1

Algebra

b

C
a

B

A
c d

A
B
C
=
=
sin a sin b sin c

Sine rule:

C2 = A2 + B2 − 2AB cos c
C2 = A2 + B2 + 2AB cos d

Cosine rule:

sin (a + b) = sin a cos b + cos a sin b
cos (a + b) = cos a cos b − sin a sin b
a+b
a−b
cos
2
2
a+b
a−b
cos
cos a + cos b = 2 cos
2
2

sin (a − b) = sin a cos b − cos a sin b
cos (a + b) = cos a cos b + sin a sin b
a+b
a−b
sin
2
2
a+b
a−b
cos a − cos b = −2 sin
sin
2
2

sin a + sin b = 2 sin

sin a − sin b = 2 cos

sin iz = i sinh z
cos iz = cosh z

sinh iz = i sin z
cosh iz = cos z

B.1.7

Geometry
When the two intersecting lines are, respectively, perpendicular to two other lines,
the angles formed by
each pair are equal.

θ1
θ2
θ 1 = θ2

Similar triangles:
h−y
x
=
b
h

h

x
y

b

h
b

a

x
x

Page 8

b

D

Any triangle:
Area = 21 bh

r

Circle:
circumference = 2πr
Arc length s = rθ
2
Sector area = 12 r θ

s

θ

θ1

θ2

Every triangle inscribed
within
a
semicircle is a right
triangle.

θ1 + θ2 = ½�
θ2
θ1

θ3 θ4

Angles of a triangle:

θ1 + θ2 + θ3 = 180
θ4 + θ1 + θ2

Intersecting chords:
x 2 = ab
x 2 ≈ Db when b
D

Mechanical Engineering Data & Formulae


B.1

B.1.8

Algebra

Analytic geometry

y

y
b

Straight line:
y = b + mx

m

x y
+ =1
a b

b
a x

x
y

y

Circle:
x2 + y 2 = r 2

r

2

2

(x − a) + (y − b) = r

r
b

x

a
b

x

y

x

Ellipse:
y
x 2
+
a
b

2

=1

a
y
b

y

Parabola:
x
y =b
a

b

x=a

2

a

y
b

2

x

a x
y

y

Hyperbola:
xy = a2
a
a

x
a

b

2



y
b

2

=1

x

x
a

Mechanical Engineering Data & Formulae

Page 9

2


B.1

Algebra

B.1.9

Solid geometry

r

Sphere:
3
volume = 43 πr
2
surface area = 4πr

L

Right-circular cone:
2
volume = 13 πr h
lateral area = πrL

L = r 2 + h2

h
r

θ

Spherical wedge:
volume = 32 r 3 θ

B

h

r

B.1.10

Any pyramid or cone:
volume = 13 Bh
where B = area of base.

Differential calculus

Leibnitz’s rule:
Dn (f g) = f Dn g + n (Df ) Dn−1 g +

n(n − 1)
D2 f
2!

Dn−2 g + · · · + Dn f g

d
, f = f (x) and g = g(x)
dx
Taylor’s expansion of f(x) about x = a:

where D =

2

f (x) = f (a) + (x − a)f (a) +

n

n+1

(x − a)
(x − a) (n)
(x − a)
f (n+1) (x)
f (a) + · · · +
f (a) +
2!
n!
(n + 1)!

where a < x < x. Substituting h = x − a gives the following form:
f (a + h) = f (a) + hf (a) +

h2
hn (n)
f (a) + · · · +
f (a) + Rn (h)
2!
n!

n+1

where Rn (h) =

h
f (n+1) (a + θh) , (0 < θ < 1).
(n + 1)!

Taylor’s expansion of f(x, y) about the point (a, b):
f (x, y) = f (a, b) + (x − a)fx + (y − b)fy

a,b

1
+
(x − a)2 fxx + 2 (x − a) (y − b) fxy + (y − b)2 fyy
2!

a,b

+ ...

Substituting h = x − a and k = y − b gives the following form:
f (a + h, b + k) = f (a, b) + hfx + kfy

a,b

+

1
h2 fxx + 2hkfxy + k 2 fyy
2!

a,b

+ ...

Partial differentiation:
If y = Y (x), then f (x, y) = f [x, Y (x)] ≡ F (x) and
dF
∂f
∂f dY
=
+
∂x ∂y dx
dx
Page 10

Mechanical Engineering Data & Formulae


B.1

Algebra

If x = X (t) and y = Y (t), then f (x, y) = F (t) and
dF
∂f dX
∂f dY
=
+
∂x dt
∂y dt
dt
If x = X (u, v) and y = Y (u, v) then f (x, y) = F (u, v) and
∂F
∂f ∂x ∂f ∂y
=
+
∂u
∂x ∂u ∂y ∂u
∂f ∂x ∂f ∂y
∂F
=
+
∂v
∂x ∂v ∂y ∂v
Stationary points of f(x, y):
These occur where fx = 0, fy = 0 simultaneously. Let (a, b) be a stationary point: examine
K = fxx fyy − (fxy )2
a,b

If:
• K < 0, then (a, b) is a saddle point;
• K > 0 and fxx (a, b) < 0, then (a, b) is a maximum;
• K > 0 and fxx (a, b) > 0, then (a, b) is a minimum.
Radius of curvature in Cartesian coordinates:
dy
1+
dx

ρxy =

2 3/2

2

d y
dx 2

B.1.11

Standard Differentials

u
v
sin x

df (x)
dx
nx n−1
dv
du
u
+v
dx
dx
du
dv
v
−u
dx
dx
v2
cos x

cos x

− sin x

tan x

sec2 x

sinh x

cosh x

cosh x

sinh x

tanh x

sech2 x
1
x
aeax

f (x)
xn
uv

loge x = ln x
eax
Mechanical Engineering Data & Formulae

Page 11


B.2

Integral calculus

B.1.12

Differential equations

The first-order linear equation
dy
+ R (x) y = S (x)
dx
has an integrating factor
λ (x) = exp
so that

R (x) dx ,

d
(yλ) = Sλ.
dx
P (x, y) dx + Q (x, y) dy = 0

is an exact equation if
dQ
dP
=
.
dy
dx

B.2

Integral calculus

An important substitution:
tan

θ
= t.
2

Then
sin θ =

2t
(1 + t2 )
2

cos θ =
and
dθ =

Page 12

(1 − t )
(1 + t2 )

2
dt.
(1 + t2 )

Mechanical Engineering Data & Formulae


B.2

Integral calculus

Table B.1: Some indefinite integrals

f (x)

f (x) dx
x π
+
2 4
x
ln (cosec x − cot x) = ln tan
2
x
sin−1
, (|x| < a)
a
ln (sec x + tan x) = ln tan

sec x
cosec x
a2 − x 2
a2 + x 2
x 2 − a2

−1/2

−1/2

−1/2

a2 + x 2
a2 − x 2
x 2 − a2

−1

sinh−1

x
= ln x + a2 + x 2
a

cosh−1

x
= ln x + x 2 − a2
a

1/2

1/2

− ln a = ln

x
x
+ 1+
a
a

− ln a = ln

x
+
a

x
a

2

2 1/2

1/2

−1

, (x ≥ a)

x
1
tan−1
a
a
a+x
1
1
−1 x
ln
tanh
=
, (|x| < a)
a
a
a−x
2a
x−a
1
ln
, (|x| > a)
x+a
2a

−1
−1

Table B.2: Some definite integrals
π/2

π/2

sinn x dx =

In ≡
0

cosn x dx =
0

n−1
π
I , where I0 = and I1 = 1
n n−2
2

π/2

m−1
n−1
Im−2,n =
I
, (m > 1, n > 1)
m+n
m + n m,n−2

b
e−ax sin bx dx =
, (a > 0)
2
a + b2
0
sinm x cosn x dx =

Im,n ≡
0



e−ax cos bx dx =

0


−x

e
0

2

a
a2


π
dx =
2

+ b2

Mechanical Engineering Data & Formulae

, (a > 0)

Page 13


B.3

B.3

Laplace transforms

Laplace transforms

Function

Transform


Definition: f (t)

f (s) =

e−st f (t) dt

0

af (t) + bg(t)
df
dt
d2 f
dt2
−at
e f (t)

af (s) + bg(s)
sf (s) − f (0)
s2 f (s) − sf (0) − f (0)
f (s − a)

0

df (s)
ds
∂f (s, a)
∂a
f (s)
s

f (u)g(t − u) du

f (s)g(s)

tf (t)
∂f (t, a)
∂a
t

f (t) dt



t
0

δ(t0 ), unit impulse at t = t0
1, unit step
tn , n = 1, 2 . . .
eat
e−at
1
tn−1 e−at
(n − 1)!
1 − e−at
1
e−at − e−bt
(b − a)
1
(c − a)e−at − (c − b)e−bt
(b − a)
b
a
1−
e−at +
e−bt
(b − a)
(b − a)
e−at
e−bt
e−ct
+
+
(b − a)(c − a) (c − a)(a − b) (a − c)(b − c)
b(c − a) −at a(c − b) −bt
c−
e +
e
(b − a)
(b − a)

Page 14

1
1
(s > 0)
s
n!
(s > 0)
sn+1
1
(s > a)
s−a
1
s+a
1
(s + a)n
a
s(s + a)
1
(s + a)(s + b)
s+c
(s + a)(s + b)
ab
s(s + a)(s + b)
1
(s + a)(s + a)(s + b)
ab(s + c)
s(s + a)(s + b)

Mechanical Engineering Data & Formulae




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