Question

derive the differential equation for damped harmonic oscillator?

Answer 1

First go for simple harmonic oscillation.

force eqn is,

F = -Kx ........ (1)

(-ve sign due to opposite direction of Force and displacement)

now,

there will be a draging force to oppose the motion and it depends on Velocity....

Fd = -bV ...... (2)

again -ve sign due to opposite direction of Dragging force (Fd) and velocity (V) , and "b" is a constant.

So, net Force in damped Motion,. .

F = -Kx - bV ..... (3)

but we know,
$F = m \times a = m \times \frac{ {d}^{2} x}{ d{t}^{2} }$
$v = \frac{dx}{dt}$
so, putting these values in eqn (3)...

$m \frac{ {d}^{2}x }{ d{t}^{2} } = - kx - b \frac{dx}{dt}$
or,
$m \frac{ {d}^{2}x }{ d{t}^{2} } + kx + b \frac{dx}{dt} = 0$
this is the eqn for damped oscillation...

Answer 2

Answer:

Differential equation for Damped oscillator is  $m\frac{d^{2}x}{d^{2}t} + kx +b\frac{dx}{dt} =0$

Explanation:

• As we know the force equation for simple harmonic oscillator is  

             $F=-kx$                                                             ...........(1)

           where F = restoring force

                       k = force constant

                        x = displacement

         and -ve sign shows that both displacement and restoring force                                 acting in the opposite direction to each other.

• There must be a dragging force to oppose the motion, which depends upon the velocity.

           $Fd=-bV$                                                            ............(2)

           Where Fd = dragging force

                        b = constant

                         V = velocity

•  From eq. (1) and (2), the net force in the damped motion,

             $F_{net} =kx-bV$                                                       .............(3)

•  We know that, $F=ma$

                where  a = acceleration =$\frac{d^{2}x}{d^{2}t}$

                             V =  $\frac{dx}{dy}$

• Put the value of F and V in eq.(3),

         we will get ,

                $m\frac{d^{2}x}{d^{2}y} = -kx-b\frac{dx}{dy}$

        ∴      $m\frac{d^{2}x}{d^{2}y} +kx+b\frac{dx}{dy}=0$

    This is the differential equation for the damped oscillator.