Damped oscillations

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

F=kx,F=-kx,

where xx is the displacement and kk is the constant of proportionality with k>0 k > 0 .

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

bx˙=Fdrag.-b\dot { x } ={ F }_{ drag }.

Suppose that the particle under the forces have mass mm, and at t=0 t = 0 we have x=x0x={ x }_{ 0 }.
Then what is the limiting value of bb 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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