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4 Modeling of Mechanical Oscillatory Systems with One Degree …
x = l * sin(theta);
y = -l * cos(theta);
The second stage is the movement of the pendulum to the right of the equilibrium
position θ > 0. The suspension length instantly changed and became equal to L–l.
An event has occurred. New equation of motion:
d
2
θ
dt 2 = −
g
L − l
sin θ
or
dθ
dt
= ω
dω
dt
= −
g
L − l
sin θ
In software implementation:
else
der(omega) = -g / (l - r) * sin(theta);
x = (l - r) * sin(theta);
y = (-r) - (l - r) * cos(theta);
end if;
At the time of the event, the linear velocity does not change, i.e., when moving to
the right (θ < 0), the discrete condition must be satisfied
Lω = (L − l)ω 1
What gives in software implementation redefinition of angular velocity:
when theta < 0 then
reinit(omega, omega * (l - r) / l);
end when;
When moving to the left (θ > 0), the discrete condition must be satisfied:
(L − l)ω = Lω 1
What gives in software implementation redefinition of angular velocity:
when theta > 0 then
reinit(omega, omega * l / (l - r));
end when;
Display the graphs of the main dependencies. On the graphs of displacement and
speed, the moments of the onset of discrete events are clearly visible—they represent
a “stitching” of sinusoids with different periods (Fig 4.9).
We will also construct a series of phase diagrams for the Galileo pendulum at
various initial angles of deviation from the equilibrium position. They are “stitched”
ellipses with different meanings (Fig. 4.10).
Addition: full Galileo pendulum model code (without visualization lines)