Question 1.20: Use Rayleigh’s dissipation function to describe the damped o......
Use Rayleigh’s dissipation function to describe the damped oscillations of a simple pendulum with air resistance proportional to velocity.
Using polar coordinates in the xy-plane of Fig. 1.9, we have r = l and, according to Example 1.16, T = ml² \dot{\theta}^{2}/2. Since V = −mgl cos θ, it follows that
L=\frac{m l^{2}}{2}\dot{\theta}^{2}+mgl \cos \theta . (1.152)
On the assumption that k_{x} = k_{y} = k, Rayleigh’s dissipation function (1.147) is given by
\mathcal{F} =\frac{1}{2} \sum\limits_{i=1}^{N}{\left(k_{ix}v^{2}_{ix}+k_{iy}v^{2}_{iy}+k_{iz}v^{2}_{iz}\right) }, (1.147)
\mathcal{F} = \frac{1}{2}kv^{2} = \frac{1}{2}kl^{2}\dot{\theta}^{2}. (1.153)
Equation (1.151) for θ takes the form
\frac{d}{dt} \left(\frac{\partial L }{\partial \dot{q}_{k} } \right) -\frac{\partial L }{\partial q_{k} }+\frac{\partial \mathcal{F} }{\partial \dot{q}_{k}} =0. (1.151)
\ddot{\theta} + \frac{k}{m}\dot{\theta}+ \frac{g}{l} \sin \theta =0. (1.154)
In the case of small oscillations, this equation of motion reduces to that of a damped harmonic oscillator.
