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4 Modeling of Mechanical Oscillatory Systems with One Degree …
thread(shapeType = ”cylinder”, length = l, width =
0.2, height = 0.2, lengthDirection = {l * sin(theta),
-l * cos(theta), 0}, color = {40, 100, 100});
Modelica.Mechanics.MultiBody.Visualizers.Advanced.Shape
bob1(shapeType = ”sphere”, length = 2 * rad, width =
2 * rad, height = 2 * rad, r = {x1, y1, 0}, r_shape =
{-rad, 0, 0}, color = {255, 0, 0});
Modelica.Mechanics.MultiBody.Visualizers.Advanced.Shape
thread1(shapeType = ”cylinder”, length = l, width =
0.2, height = 0.2, lengthDirection = {l * sin(theta1),
-l * cos(theta1), 0}, color = {40, 100, 100});
Modelica.Mechanics.MultiBody.Visualizers.Advanced.Shape
level(shapeType = ”box”, length = l, width = 0.1, height =
0.1, r = {-l / 2, 0, 0}, color = {20, 20, 20});
equation
der(theta) = omega;
der(omega) = -g / l * sin(theta);
x = l * sin(theta);
y = -l * cos(theta);
der(theta1) = omega1;
der(omega1) = -g / l * theta1;
x1 = l * sin(theta1);
y1 = -l * cos(theta1);
annotation(Diagram(coordinateSystem(extent =
{{-148.5, -105}, {148.5, 105}}, preserveAspectRatio =
true, initialScale = 0.1, grid = {5, 5})));
end PendulumViz;
4.2 Galileo Pendulum
Formulation of the problem
The Galileo pendulum is a mathematical pendulum of length L, oscillating near a
vertical wall into which a nail is driven in at a distance l below the suspension point
Fig. 4.7.
Tasks
• Explain why this dynamic model is hybrid;
• Construct differential equations describing this system and find the boundary
conditions at the point of discrete change of parameters;
• Build graphs of the dependence of displacement on the equilibrium position on
time;
• Build a phase portrait of this oscillator.
Modeling and computational experiment
The fundamental difference between this model and the usual mathematical pendulum is the introduction of additional conditions in the equations. This is done using
the means of the Modelica language. Details on the introduction of discrete conditions were described in the introduction on the example of a bouncing ball. First, we