# Pendulum Damping force effect on amplitude over time

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1. May 1, 2015

### Dreshawn

I am struggling with setting up a problem to solve for the change in amplitude of a pendulum affected by a damping force (presumable air friction) over a time period.
The original amplitude of the pendulum is 1.4 m from the equilibrium on a 15 m massless wire with a 110 kg brass bob at the end. The damping force is 0.010 kg/s. This force is dependent on velocity and I have set up an equation for velocity dependent on time -
dv/dt = -ωAsin(ωt+Φ0). I know that this force will always oppose motion but I am having trouble putting these together to see the affect on the pendulum bob's acceleration and amplitude. I was wondering if I should just use the pendulums max velocity at the bottom to approximate this damping force & then apply this to the # of oscillations to get a rough estimate on the decrease of acceleration at the bottom & then solve for amplitude or if there is another more accurate way to solve this problem.

Thank you

2. May 1, 2015

### Hesch

The damping force is not constant: Fair(t) = -k * v(t), (assuming laminar airflow). k is a constant.

If you are able to, then setup a model using Laplace transform. You will get a position output from the model in the form:

Y(s) = A / (s2 + Bs + C) that can be rewritten:

Y(s) = A / (s2 + 2ζωns + ωn2), ξ = damping ratio, ωn = resonance frequency [rad/s].

The inverse transform of Y(s) will give you position(t).

Last edited: May 1, 2015