Here's the IMAGE!
Help me solve this problem.
We know:
Length: OA1, A1B1, OA2, A2B2 (and coordinates (x,y) each point)
Angles: α1, α2, β1, β2
Rotation velocities, if you want. Elements can rotate simultaneously (with different v1 and v2) or with equal velocities v1 and v2.
How to find length of curve B1B2? I need an equation.
Thanks!
The main work is to set up a parametric representation of the resulting curve. I shall use a description in terms of complex numbers.
We have the inner arm of length $a>0$ whose endpoint moves with angular velocity $\omega_1$ in a circle described by $$t\mapsto z_1(t)=ae^{i(\phi_1+\omega_1 t)}\ .$$ Add to this the vector formed by the outer arm of length $b>0$. I'm referring the direction of the outer arm in relation to the direction of the inner arm. If the relative angular velocity of the outer arm with respect to the inner arm is $\omega_2$ we obtain the vector $$t\mapsto z_2(t)= b e^{i(\phi_2+\omega_2t)}\ .$$ The total resulting curve then is given by $$t\mapsto z(t)=z_1(t)+e^{i(\phi_1+\omega_1 t)} z_2(t)=e^{i(\phi_1+\omega_1 t)}\bigl(a+be^{i(\phi_2+\omega_2t)}\bigr)\ .$$ In order to obtain the arc length along this curve you have to compute $$|z'(t)|=\sqrt{x'^2(t)+y'^2(t)}\ .$$ Unless the values $a$ and $b$ are related in a special way the resulting integrals $$\int_0^T\bigl|z'(t)\bigr|\>dt$$ will not be elementary.