equations of motion
simultaneous rotation of a particle about 
two axes

fig 9

rotation about primary (x) axis

x1 = 0
y1 = r cos a

z1 = -r sin a

rotate above system about secondary (z) axis

x2 = y1 sin b
y2 = y1 cos b

z2 = z1

add translation along x axis

x = a(t) + x2
y = y2
z = z2
where  a = some constant

total equation of motion for particle as function of time (t):
for the case under consideration: 

a = 1
r  = .5
b = 22.5(t) degrees
a = 2b


x = t + .5 cos(45t)sin(22.5t)
y = .5 cos(45t)cos(22.5t)
z = -.5 sin(45t)



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