Numbers

  NUMBERS

Consider a complex number. We can subdivided from complex number to whole numbers as

1.      Complex number: Real number + imaginary number

2.      Real number:  Rational number + irrational number

3.      Rational number: Integers + Decimal

v  Complex number consists of real part and imaginary part. Ex: 4+3i.The graph used to represent complex number is called argand diagram. It has Real part along abscissa and imaginary part along ordinate.

v  Real numbers consists of rational and irrational number thus the whole number line i.e. from (-∞, + ∞).

·         Rational numbers are those which are terminating or repeating. Ex 14.5869, 48, 27.333333333…….

·         Irrational numbers are those which are non-terminating and non-repeating.

Ex π (3.141592653589793….), e (2.71828182846), 2^1/2 (1.414213562373)

v  Rational numbers are subdivided into integers and decimals.

·         Decimals are those with a decimal point. Ex 0.1424, 1.414, 1.732, 20.689 etc.

§  Proper fractions are those which has numerator lesser than denominator i.e. they are lesser than 1. Ex 2/3, 7/8 etc.

§  Improper fractions are those which has numerator greater than denominator i.e. they are greater than 1. Ex 2/3, 7/8 etc.

§  Mixed fractions are those which are of form 10 1/6. The result is 61/6 (i.e. 6*10 + 1 = 61 and /6 is to be written. Thus 61/6).

The concept of mixed fraction is that it converts improper fraction to proper fraction.

·         Integers are the round figures like 5, 25, 968, -30415 etc. The positive integers are from (0, ∞) and negative integers are from (-∞, 0).

§  The positive integers if started from 0 to ∞ are said to be whole numbers.

§  The positive integers that start from 1 to ∞ are said to be natural numbers.

Understanding Basics of Vibration

What is a Vibration?
Vibration is a motion about a point (equilibrium point). Hence it comes under dynamics.

Example of vibration: 
Newton Mechanics: Vibration of Machines drilling machines, cell phones
Quantum Mechanics: Vibration of electrons during heat transfer (conduction)

Types of Vibration?
1.Free
2.Damped
3.Forced
4.Torsional
5.Longitudinal
6.Transverse
7. Others (Refer Text Book)

Have to understand all types of vibration?
No. Only understand the concept of vibration.

IMAGINE THIS: You are in a vacuum. You have a Spring which has one end fixed and other end with mass. You are pulling it and leaving it (within elastic limit). IT WILL OSCILLATE UNTIL YOU STOP. Because it has no resistance (Vacuum). And no external force acting on it until you stop

Starting with NATURAL FREQUENCY. What is natural frequency?

• What is the point of imagining this?
• Point is that is NATURAL FREQUENCY.
• EVERYBODY HAS MASS AND STIFFNESS. WHEN IT IS OSCILLATED IN A VACUUM IT VIBRATES CONTINUOUSLY AND INDEPENDENT OF TIME. IT GOES ON OSCILLATE. 
• Means there is no energy loss, potential energy (strain energy) is converted to kinetic energy and vice versa
• It follows Simple Harmonic Motion
• From Dynamics, ma = -kx on solving gives  angular natural frequency ωn = √(k/m)
• fn = ωn / 2π
• This is FREE VIBRATION or UNDAMPED VIBRATION

If AIR or OTHER medium Present? not in vacuum

• You have a damping force hence DAMPED VIBRATION
• Damping ratio ζ = damping coefficient (C) / (2*m * ωn ) in other words, Damping ratio is directly proportional to damping coefficient and inversely proportional to natural frequency

• 3 Types of damped vibration are Critical, Underdamped, Overdamped based on damping ratio ζ
• ζ = C / Cc 
• Critical Damping :  ζ = 1
• Underdamped: ζ < 1 {ωd = ωn * √(1- ζ^2)}
• Overdamped: ζ > 1
• Cc = 2√(km)
• 
  

• Logarithmic decrement of underdamped system is the log of ratio of successive amplitudes        δ = log (X1/X2) = 2πζ (ωn/ωd) 

If AIR or OTHER medium Present (or not present) and External Force Present?

• FORCED VIBRATION
• Force may be periodic or aperiodic
• Periodic forces may be harmonic or irregular
• For a general, harmonic periodic force X = F0/[(k - mω^2)^2 + c^2*ω^2]^1/2 when you consider δst = F0/k (for undamped vibration,  X = F0/[(k - mω^2)^2 )

    and for undamped 

• X / δst is the amplification factor or magnification factor M
• Resonance occurs when forcing frequency  to the body ω is equal to natural frequency ωn of the body or in otherwords Magnification factor M = ∞

• For periodic irregular force we can use forier series or trapezoidal rule to form a harmonic 
• For non periodic force forier integral or convolution integral or Laplace transform or numerical methods can be used to solve

For torsional vibration

• linear deflection x becomes torsional deflection θ
• longitudinal stiffness K becomes torsional stiffness Kt
• Mass m replaced by mass moment of inertia J
• Hence change occurs in angular natural frequency formula and other formula

For transverse vibration

• Use Rayliegh method or Dunkerley's Method 

• Rayliegh Method 
  

• Dunkerley's Method 
  
  
  
• Generally these transverse vibration is used in shafts carrying mass 

Mechanical Engineering

Mechanical Engineering can be classified with respect to

1.      Size of object              (geometry)

2.      Motion of object         (position)

3.      Change wrt to Time                           

                                 

Ø  Size of object is whether macromolecular level or micromolecular level. The objects which we see around us are macromolecular level (Newtonian Mechanics to be followed). The objects like e^- , protons, neutrons etc are micromolecular level. Quarks and leptons (particle inside neutrons) are also considered in this level (Quantum Mechanics to be followed).

Ø  Motion of object is whether the body is at rest or moves {in uniform motion (v=c) or moves in non-uniform motion (v != c)}. Body at rest are called Static condition. Body in motion are called Dynamic condition.

Ø  With respect to Time the object can be considered to be steady (change wrt time is constant), unsteady (change wrt time is not constant), independent of time.

 STATIC BRIDGE AND DYNAMIC BOATS
STEADY BRIDGE AND UNSTEADY CLOUDS
NEWTONIAN MECHANICS