Question and Numerical - Ask Your Doubt - B-Adv-CM-Q2

For a system shown below a) Write down the equation of motion of the system b) Find out if the system underdamped, critically damped or overdamped

B-Adv-CM-Q2

For a system shown below

a) Write down the equation of motion of the system
b) Find out if the system underdamped, critically damped or overdamped
c) Determine x(t) for given initial condition x(0)=3cm ; x'(0)=0

Damping Definitions in Harmonic Motion

In oscillatory systems (like a mass-spring system or an RLC circuit), damping refers to the effect of a resistive force (such as friction or resistance) that reduces the amplitude of oscillations over time. The damped harmonic oscillator is a key example of such a system.

Based on the amount of damping present, there are three main types of damping:

1. Under damping:

Definition: In underdamped systems, the damping is weak, and the system oscillates with a gradually decreasing amplitude. The system still oscillates back and forth, but the amplitude of these oscillations decreases exponentially over time due to the damping force.
Behavior: The system oscillates with decreasing amplitude and eventually comes to rest. The oscillation frequency is slightly lower than the natural frequency of the undamped system.

Example:

A pendulum swinging in the air (with air resistance) is an example of an underdamped system. It oscillates back and forth before eventually coming to rest.

2. Critical Damping:

Definition: In critically damped systems, the damping is exactly strong enough to return the system to equilibrium as quickly as possible without oscillating. This is the threshold between oscillatory (underdamped) and non-oscillatory (overdamped) motion.
Behavior: The system returns to equilibrium in the shortest time possible without oscillating. Any increase in damping would cause the system to slow down further (overdamped).

Example:

The suspension in car shock absorbers is often designed to be critically damped so that the car returns to a stable position as quickly as possible without oscillating when going over bumps.

3. Overdamping :

Definition: In overdamped systems, the damping is so strong that the system returns to equilibrium without oscillating, but it does so more slowly than in the critically damped case. The system is sluggish and takes a long time to return to its resting state.
Behavior: The system slowly returns to equilibrium without oscillating. As damping increases, the system becomes more sluggish.

Example:

A door closer mechanism that prevents a door from slamming shut is an example of an overdamped system. The door closes slowly and smoothly, without oscillations, to prevent it from banging.