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