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Wize University Physics Textbook (Master) > Periodic Motion: Oscillations

Critical Damping and Over-Damping Conditions

Underdamped OscillationsPrevious Section
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Critical Damping and Over-Damping Conditions


If the damping constant is relatively large, the system does not do even a single oscillation. In this case, the mass just decays to its equilibrium exponentially.

Critical Damping Condition

In underdamped conditions, the oscillation decays faster by increasing bb. At a very special point when ωo=b∗2m\omega_o=\frac{b^*}{2m}, there would be no oscillation anymore. This condition is called critical damping condition.



  • In this condition, ω′=0\omega'=0. So: x(t)=Aoe−b∗t2mcos⁡φx\left(t\right)=A_oe^{-\frac{b^*t}{2m}}\cos\varphi
  • There is no oscillation and the displacement decays to zero exponentially



Wize Concept
The system has critical damping condition if:
b∗=2mωo=2kmb^*=2m\omega_o=2\sqrt{km}




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Overdamping Condition

If b>b∗b>b^*, the system would be in overdamping condition

b2m>ωo\boxed{\dfrac{b}{2m}>\omega_o}
  • There is again no oscillation in overdamped condition.
  • x(t)x\left(t\right)decays to zero at a slower rate compared to a critically damped condition.
  • The bigger the value of the damping constant is, the slower x(t)x\left(t\right)goes to zero

Wize Tip
The bigger the value of the damping constant is, the slower x(t)x\left(t\right) goes to zero.

Watch Out!
Usually you only need to describe these two conditions qualitatively.














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An oscillator is currently undergoing critically damping motion. What kind of motion would it undergo if mass is increased?




UNDERDAMPED


An oscillator is currently undergoing critically damping motion. What kind of motion would it undergo if friction is increased?
Extra Practice
Critical Damping and Over-Damping Conditions
A mass of m=500 gm = 500\ \mathrm{g} is attached to a spring with spring constant k=10 N/mk = 10\ \mathrm{N}/\mathrm{m}. This system has a damping coefficient of b=3.2 kg/sb=3.2\ \mathrm{kg}/\mathrm{s}. If we want to modify the mass to obtain critically damped motion, what shall we do?
An oscillator is currently undergoing critically damping motion. What kind of motion would it undergo if the amplitude of the oscillation is increased?
Damping motion
An oscillator is currently undergoing critically damping motion. What kind of motion would it undergo if mass is increased?
An oscillator is currently undergoing critically damping motion. What kind of motion would it undergo if friction is increased?
50 kgkg person taking a bungee jump:
a. If the initial length of the bungee cord is 20 mm and at the lowest point, the length of the cord is 25mm, what is the spring constant of the cord?
b. After the jump, the person is experiencing a damping oscillation. What is the damping constant if the amplitude of its oscillation reduces to 15\frac{1}{5} of its original amplitude after 20 s?
Practice: Damped Oscillation
A 0.3 kg0.3\ kgmass undergoes a damped oscillation with the damping constant 5 kgs5\ \frac{kg}{s}. How long does it take for the amplitude and total energy of the oscillation to be 14\frac{1}{4}of their initial values respectively?