Explain the following terms with a neat sketch a) Under damping, over damping and critical damping b) Longitudinal, Transverse and Torsional vibrations

 

a) Under Damping, Over Damping, and Critical Damping

These terms describe how a system responds when disturbed from equilibrium:

1. Under Damping:

   • System oscillates with gradually decreasing amplitude

   • Returns to equilibrium fastest but overshoots multiple times

   • Common in mechanical systems with low friction

   • Sketch shows oscillating curve with decreasing peaks

2. Critical Damping:

   • System returns to equilibrium in shortest time without oscillating

   • Ideal for many engineering applications (e.g., vehicle suspensions)

   • Sketch shows smooth curve approaching equilibrium without crossing it

3. Over Damping:

• System returns to equilibrium very slowly without oscillating

• Excessive resistance to motion

• Sketch shows very gradual curve approaching equilibrium

• Displacement
  ^
  |    Underdamped (oscillating decay)
  |   / \       / \       / 
  |  /   \     /   \     /
  | /     \   /     \   /
  |/       \_/       \_/
  +----------------------> Time
  |
  |    Critically damped
  |    /
  |   /
  |  /
  | /
  |/
  +---------------------->
  |
  |    Overdamped
  |    /
  |   /
  |  /
  | /
  |/
  +---------------------->

        

b) Longitudinal, Transverse, and Torsional Vibrations

These describe different vibration modes based on displacement direction:

1. Longitudinal Vibrations:

   • Particles move parallel to wave propagation direction

   • Compression and rarefaction occur

   • Example: Spring vibrating along its length

   • Sketch shows spring with alternating compressed and stretched sections

2. Transverse Vibrations:

   • Particles move perpendicular to wave direction

   • Creates crests and troughs

   • Example: Guitar string vibration

   • Sketch shows wavy string with up-down motion

3. Torsional Vibrations:

• Twisting/rotational motion about the axis

• Angular displacement occurs

• Example: Crankshaft twisting

• Sketch shows rod with alternating clockwise/counter-clockwise twists

             Longitudinal:

[|||]  [| |]  [|||]  [| |]  (spring compression/extension)

Transverse:
   ^        ^        ^
  / \      / \      / \
 /   \____/   \____/   \___ (string moving up/down)

Torsional:
<--)   (---)   (--->   (---) (rod twisting left/right)

Key Characteristics:
Longitudinal: Requires medium, travels through solids/liquids/gases
Transverse: Only in solids and liquid surfaces, not through fluids
Torsional: Occurs in shafts, involves shear stress
These vibration types are fundamental in analyzing mechanical systems, from vehicle components to building structures. Each has distinct mathematical models and engineering solutions for vibration control.