Differential equation for motion of a pendulum

[fiso]

Hello,

I'm trying to create a model of pendulum motion in Matlab to fit the curve we measured in class, and it has some non-linear components (friction). It looks like the best fit is a sinusoid with an envelope given by a parabola (see attached file).

The equations of motion are given by:
$\ddot{\phi}(t) = -k_2\dot{\phi}(t)-k_3\sin(\phi(t))$

How this equation can be changed so that the solution of it is a searched sinusoid?

Thank you.

// edit: please move this to Differential Equations

 

[mathman]

The usual approach is to linearize, assuming the angle of the swing is small, so replace sin(φ(t)) by φ(t).