Damped oscillations

Classical Mechanics Level 4

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

F = − k x , F=-kx,

where x x is the displacement and k k 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

− b x ˙ = F d r a g . -b\dot { x } ={ F }_{ drag }.

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


The answer is 2.

Mvs Saketh by Mvs Saketh , 24, India

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1 solution

Pranjal Jain
Dec 10, 2014

F = m x ′ ′ = − k x − b x ′ F=mx''=-kx-bx'

Solving this second order linear ODE, we get x ( t ) = c 1 e − t ( b 2 − 4 m k + b ) 2 m + c 2 e t ( b 2 − 4 m k − b ) 2 m x(t)=c_{1}e^{-\dfrac{t(\sqrt{b^{2}-4mk}+b)}{2m}}+c_{2}e^{\dfrac{t(\sqrt{b^{2}-4mk}-b)}{2m}}

For the limiting case of not crossing origin, b 2 − 4 m k = 0 b^{2}-4mk=0 or b = 2 b=2

8 Helpful 0 Interesting 0 Brilliant 0 Confused

Exactly , Because when b falls below 2, we get sinusoidal damped solutions (considering the real class of solutions) which necessarily cross the origin. It is interesting how it oscillates infinite times if b slightly less than 2 and just once when more or equal. Such a harsh discontinuity

Mvs Saketh - 6 years, 6 months ago

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Yeah! Thats somewhat cool!

Pranjal Jain - 6 years, 6 months ago

Can you tell how you solved it I am stuck

Ashwin Gopal - 6 years, 6 months ago

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You don't solve . You guess the solution.If you cannot access the videos reply here.

https://www.youtube.com/watch?v=f8HJ6g1S6OA

https://www.youtube.com/watch?v=XCdPk1pIT1Y

https://www.youtube.com/watch?v=NU5nkbOa1zk

https://www.youtube.com/watch?v=17D4U_r6Gwk

https://www.youtube.com/watch?v=LmFGrx-MSzU

If you want to learn more about Waves and Optics enroll in PHYS201x on EdX.

Arif Ahmed - 6 years, 5 months ago

what do you mean by ODE ? Does This equation has Many Other Solution also ? If So then what is Best Solution for This . ?

I just Use The Standard Solution of this Damping which is given in NCERT, x ( t ) = A 0 e − b t m sin ⁡ ( ω d t + ϕ ) ω d = ( k m ) − ( b 2 m ) 2 \displaystyle{x(t)={ A }_{ 0 }{ e }^{ \cfrac { -bt }{ m } }\sin { { (\omega }_{ d }t+\phi ) } \\ { \omega }_{ d }=\sqrt { { (\cfrac { k }{ m } ) }-{ (\cfrac { b }{ 2m } })^{ 2 } } }

Now For Never crossing X-axis It's Sinusoidal Time Phase Should be constant , Which is only Possible when It's damping frequency should be zero . (or Time period is Infinite) i.e ω d = 0 ⇒ b = 4 m k \displaystyle{{ \omega }_{ d }=0\quad \Rightarrow \boxed { b=\sqrt { 4mk } } }

Deepanshu Gupta - 6 years, 4 months ago

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ODE is ordinary differential equation,

yes, inface ur eqn holds when b is less than 2, when equal, another equation holds whose graph i have plotted. when more than 2, still another set of equations hold,, all solutions are however expressible as

C1e^(tr1)+C2e^(tr2) ,

ur way holds when r1 and r2 are imaginary and C1 = C2 (the good form) , then use e^(ix)=cos(x)+isin(x),

Mvs Saketh - 6 years, 4 months ago

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