Calculating Time for Window Shade Roll-Up: M, W, L, T, R, m, g Dynamics

[Loren Booda]

A window shade of mass M, thickness W and length L is driven by a linear spring having tension T (with shade unfurled) and tension 0 (with shade rolled up). The solid cylinder upon which the shade winds has radius R and mass m. What is the time elapsed for the shade to roll up completely when released from its unfurled position? Assume zero friction and gravitational acceleration g.

 

[mu]

To the best of my knowledge, this gets real ugly. The equations are easy to obtain, but because of their nonlinearity you'll have a hard time solving them. I think a numerical solution would more appropriate for this problem. I can however point you in the right direction.

First, you need to determine how the shade rolls around the cylinder: radius of the (cylinder + shade) system while it's rolling and length around the cylinder for any angle. The length is obtained by integrating the radius function.

Next, calculate the (angle dependent) moment of inertia. Fairly easy once you know the thickness of the shade on the cylinder.

Finally, find the expressions for the forces acting on the cylinder (weight of remaining shade and spring). Don't forget they depend on the angle. Plug them in Newton's 2nd law:
$$I\left( \theta \right)\ddot \theta = \sum {\tau \left( \theta \right)}$$
You now need to solve the resulting equation. Not easy.

That's about it. If you need some more details (like equations), let me know

 

[Loren Booda]

Thanks, mu. I got my inspiration while tugging the shade of my bathroom window. Such a mundane process, fairly easily stated, but hard as heck to solve - one needs a computer. I am aware of the basic equations, just stumped on how they are ultimately coupled (nonlinearly).