Hi all
please ignore if post inappropriate.
I have been modeling real time mechanical movements and found myself
retreating to more and more simple setups to cope with the complexity as I
by no means am a specialist. My present concern is about describing
movements in physical pendulums:
A L1 m long bar, p1, with mass m1 hangs from the ceiling in point O. In the
end of this
pendulum hangs another bar, p2, L2 m long and mass m2. p1 is displaced an
angle ø1 from eqvilibrium and p2 an angle ø1 + ø2. How do I describe the
movement of the composite pendulum?
The movement of the composite without individual swings of the 'independent'
pendulums can be expressed as
F(1+2,restore at mc) = - x*k ,k= constant of torsion of composit
x = A*cos(w*t), A=dist(O_cm)*sin(ø_mc), ø_mc the imaginary starting
angle of swing of straight composit
w(1+2) = sqr(k/I)
k = (m1+m2)*g*d, (d=dist(O_cm), cm=center of mass of composit
I = I1 + I2 (with respect to O)
F(1+2, restore at mc) creats a torsion at O from cm. If the same torsion is
produced by the joint, it's magnitude would be
F(1+2, restore at joint) = dist(cm_O)/L1*F(restore at mc)
F(1+2, restore at joint) = - dist(cm_O)/L1*(-
dist(O_cm)*sin(ø_mc)*cos(sqr(k/I)*t)*(m1+m2)*g*dist(O_cm))
The individual swing of the lower pendulum imposes a reactive torque at the
joint as it swings with its own w2 - and thus it modifies the F(1+2, restore
at joint) in a periodic manner
F(2, restore at joint) = - x*k2 ,k2 = constant of torsion of p2 at knee
x = A2*cos(w2*t), A2 = L2*sin(ø2), ø2 the starting angle - ø1 (ø2=its
individual tilt rel to p1).
w2 = sqr(k2/I2)
k2 = m2*g*d, (d = L2/2)
I2 (with respect to joint)
F(2, restore at joint) = - L2*sin(ø2)*cos(sqr(k2/I2)*t)*m2*g*L2/2
F(restore joint) = F(1+2, restore at joint) - F(2, restore at joint)
F(restore joint) = -
dist(cm_O)/L1*dist(O_cm)*sin(ø_mc)*cos(sqr(k/I)*t)*(m1+m2)*g*dist(O_cm) +
L2*sin(ø2)*cos(sqr(k2/I2)*t)*m2*g*L2/2
Finally
torsion at O = F(restore joint)*L1
and
angular accelleration, alfa = (torsion at O)/I
.... hm, I has become timedependant, but it can be expressed if not = I(1+2)
should be used. My problem is, that my intuition tells me, that it all can
be done in a simpler way.
Any proposals?
cheers
Carsten
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