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Trolley move and rope
The hanging coil is a pendulum of period 2 pi sqrt(L / g) driven by the trolley acceleration. A smooth or shaped acceleration leaves less swing.
Motion profile
Model checks
The science
What is modelled
A point mass on a rope under a moving support: theta'' = -(g/L) sin theta - (a/L) cos theta - c theta', integrated with RK4 for three acceleration profiles over the same distance.
Equations
- Sway period
T = 2 pi sqrt(L / g). - Constant acceleration response
theta(t) = -(a/g)(1 - cos w t)(linear, undamped). - Zero-vibration shaper: two half-height impulses a half period apart.
Checks run in the page
- The sway period from an integrated pendulum.
- The closed-form step response against RK4.
- Residual swing with and without shaping.
- Travel distance of the profile.
Honest simplifications
Single-mass pendulum in a plane: no hoist motion, rope stretch or mass, coil rotation, wind or sway of the bridge, and the shaper assumes a known rope length. Not a crane control design tool.
What to try
- Press Hard trapezoid then Input-shaped and compare the residual swing.
- Press Short rope, fast: a short rope sways faster but the shaper is harder to time.
- Raise damping and watch the swing die away.