Tracking the position of trebuchet payload

[Green Lantern]

Homework Statement

For the parts of the Trebuchet shown, develop expressions for the position, velocity,
and acceleration, x, x', and x'' of the payload as a function of the arm angle θ and its
derivatives θ', θ''. Use the vector tools in Mathematica to help develop the
expressions needed. Note that a closed kinematic loop is present during this phase of
the motion if the rope remains taut. By either using max/min calculus tools or plots of
your own design, find a set of parameters L, R, and H that will provide the maximum
velocity x' at the point the payload lifts off the ground. Provide plots showing the
position, velocity and acceleration of the payload as a function of the arm angle θ.
Assume the arm angle increases as a quadratic function of time. Bonus points are
available if animations of the motion are provided. Provide discussion of what is
observed as your analysis proceeds. Use the Mathematica notebooks as the report
medium. Report on your work immediately near each plot. Label all axes and title
each plot.

Problem diagram is trebuchet.jpg

Homework Equations

v = dr/dt + ω cross r
a = dv/dt + ω cross v

The Attempt at a Solution

My coordinate system is trebwork.jpg

For position of the payload I have:

r = La1 + Rb1

Where

A-Frame:

a1 = sinθn1 - cosθn2
a2 = cosθn1 + sinθn2

and

B-Frame:

b1 = -cosβa1 - sinβa2
b2 = -sinβa1 + cosβa2

and

ω(Frame:N-B) = θ'a3
ω(Frame:N-A) = β'b3

My vector loops is:

r = La1 + Rb1 - xn2 + Hn1 = 0

I derive this to find velocity and again for acceleration.

First, how do I solve for β?

 

[AvroArrow]

How do I use the vector tools in Mathematica to help develop the expressions needed? Also, how do I use max/min calculus tools or plots of my own design to find the set of parameters L, R, and H that will provide the maximum velocity x'?