Nonuniform circular motion of an unbalanced wheel with position-dependent acceleration

[Jonathan]

How does one analyse nonuniform circular motion of an unbalanced wheel where the instantaneous acceleration of the anomalous point mass (the one that makes it unbalanced) depends on the position in rotation? In this case, the acceleration depends on the cosine of the angle relative to right-hand side of a horizontal line (0 = 3 o'clock, π/2 = 12 o'clock, etc. as usual).

 

[HallsofIvy]

I wanted to think about this for a while but as soon as I started actually working on it I noticed something: Since the weight is concentrated at a single point, we can ignore the disk and think of the weight as attached to the center of the wheel by a rod: this is the classic "pendulum problem"!

Drawing a force diagram and, of course, using "F= ma", we get
m r d²θ/dt²= -mg sinθ where θ is 0 when the weight is directly below the center of the wheel and r is the distance from the center of the wheel to the weght.

This is a (very) non-linear equation so there is no general method of solution. If θ is small, we can approximate sinθ by θ and get r d²θ/dt² = - g θ or d²θ/dt²+ g/rθ= 0.


That's a linear homogeneous equation with constant coefficients and its general solution is θ(t)= C1 cos([squ](g/r)θ)+ C2 sin([squ](g/r)θ). In particular, if we hold the wheel so that the weight makes initial angle Θ with the vertical and release it, θ(t)= Θcos([squ](g/r)θ). The weight moves through the vertical and to an equal height on the other side then repeats periodically.

More generally, we can use "quadrature". If we let ω= dθ/dt, we have d²θ/dt²= dω/dt and then, using the chain rule, dθ/dt dω/dθ= ω dω/dθ.

The equation becomes ωdω/dθ= -g/r sinθ so ωdω= (-g/r) sinθdθ and
(1/2)ω²= (g/r)cosθ+ C.

Theoretically, one could solve for ω= dθ/dt and then integrate that but it gives an "elliptic integral" which cannot be done in closed form. What we can do is draw the "phase plane diagram". For a number of different values of C, graph ω against θ. For some values of C you get "circular" graphs (periodic solutions- the wheel swings back and forth). For other values it's not: the wheel just keeps going around in the same direction.


edit: fixed θs and ωs

 

[Jonathan]

Finally a post!

Well, just as I thought, no (simple) solution. I hate it when that happens, and it always does because I don't work with simple problems. Thanks!

 

[schwarzchildradius]

It's an undriven, damped harmonic oscillator, as said- there is an excellent solution. You're trying to solve this differential equation:

m d²x/dt² + b dx/dt + kx =0

(the right-hand side would be a force function if the oscillator were driven)
solve it for x and you can find the instantaneous position of any particle on the rim of the wheel from time just after initial acceleration.