Question

Consider the damped case of the simple pendulum, if damping force affects the motion of the mass m.

1. Find the equation of motion of the damped pendulum.

Write down the solution and discuss its characteristics

Answer 1

There are two forces acting on a damped harmonic oscillator

1. The restoring force due to the spring that causes the oscillation.(gravity in the case of a simple pendulum)

This force is given by

F = -kx

2. The damping force that causes the oscillation to stop th motion

This forcee is proportional to the velocity of motion and is given by

F = -cv.

The total force is then

According to newton's second law,

F = ma

So,

We know that v = dx/dt and a = d^2x/dt^2

So,

Thus we got a differential equation to be solved.

Taking a most general solution of

and substituting in the differential equation,

This is a second degree equation and the solution is given by

Now, the solution of this depends upon the term in the square root.

Case 1.

Then, lambda will have real and imaginary will have two solutions.

This is called a under damped case, where the oscillator will oscillate for a long time.

Case 2

Then, lambda will become real and will have two solutions.

This is called a over damped case, where the oscillator will stop very fast.

Case 3

Then, lambda will become real and will have only one negative solution

This is called a critically damped case, where the oscillator will decay exponentially.