Damped oscillations

Suppose I have an oscillating system, that oscillates under a standard restoring force given by :


where \(x\) is the displacement and \(k\) is the constant of proportionality with \( k > 0 \).

Now suppose it is being damped by a viscous force given by the standard equation of

\[-b\dot { x } ={ F }_{ drag }. \]

Suppose that the particle under the forces have mass \(m\), and at \( t = 0 \) we have \(x={ x }_{ 0 }\).
Then what is the limiting value of \(b\) for which the particle will not cross the origin even once?

Details and Assumptions

  • The graph of the particle will appear as shown.
  • Only consider the positive side, x-axis is time and y-axis displacement
  • m=1 kg
  • K=1 N/m
  • No driving force acts on the object.

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