178
5 Modeling of Mechanical Oscillatory Systems …
Modeling and computational experiment
Compilation of the differential equation of motion of a double pendulum in Cartesian coordinates is difficult due to the presence of reactions that occur in the hinge
joints. Similar problems are solved by compiling equations of motion for generalized
coordinates (using the Lagrange equations). The fact is that setting the position of
a system of points fastened by bonds in Cartesian coordinates is not always convenient. The choice of parameters necessary to describe the position of all points of the
mechanical system (i.e., generalized coordinates) should be determined primarily by
expediency.
In our case, it is convenient to take the angles of deviation of each of the pendulums
from the vertical (θ 1 and θ 2 ) as generalized coordinates.
First, it is necessary to introduce a reference frame and express the Cartesian
coordinates in terms of generalized:
x 1 = l 1 sin θ 1
y 1 = −l 1 cos θ 1
and
x 2 = l 1 sin θ 1 + l 2 sin θ 2
y 2 = −(l 1 cos θ 1 + l 2 cos θ 2 )
Then, by differentiating these equalities, we obtain the Cartesian velocity components expressed in terms of the generalized coordinates (θ 1 and θ 2 ) and the
generalized velocities ( ˙
θ 1 and ˙
θ 2 ):
˙
x 1 = l 1 ˙
θ 1 cos θ 1
˙
y 1 = l 1 ˙
θ 1 sin θ 1
and
˙
x 2 = l 1 ˙
θ 1 cos θ 1 + l 2 ˙
θ 2 cos θ 2
˙
y 2 = l 1 ˙
θ 1 sin θ 1 + l 2 ˙
θ 2 sin θ 2
Then, you can calculate the squares of the speeds of each pendulum (included in
the kinetic energy):
v
2
1 = ˙
x
2
1 + ˙
y
2
1
v
2
2 = ˙
x
2
2 + ˙
y
2
2
Cartesian coordinates included in the potential energy formula,
W = m 1 gy 1 + m 2 gy 2
They are replaced with generalized coordinates. The velocities included in the
kinetic energy formula should be replaced by generalized velocities, so that kinetic
energy in the general case will depend on the generalized coordinates
T =
m 1 v
2
1
2
+
m 2 v
2
2
2
Write down the Lagrangian, and compose the Lagrange equations for the first and
second pendulums in the absence of non-potential forces:
5 Modeling of Mechanical Oscillatory Systems …
Modeling and computational experiment
Compilation of the differential equation of motion of a double pendulum in Cartesian coordinates is difficult due to the presence of reactions that occur in the hinge
joints. Similar problems are solved by compiling equations of motion for generalized
coordinates (using the Lagrange equations). The fact is that setting the position of
a system of points fastened by bonds in Cartesian coordinates is not always convenient. The choice of parameters necessary to describe the position of all points of the
mechanical system (i.e., generalized coordinates) should be determined primarily by
expediency.
In our case, it is convenient to take the angles of deviation of each of the pendulums
from the vertical (θ 1 and θ 2 ) as generalized coordinates.
First, it is necessary to introduce a reference frame and express the Cartesian
coordinates in terms of generalized:
x 1 = l 1 sin θ 1
y 1 = −l 1 cos θ 1
and
x 2 = l 1 sin θ 1 + l 2 sin θ 2
y 2 = −(l 1 cos θ 1 + l 2 cos θ 2 )
Then, by differentiating these equalities, we obtain the Cartesian velocity components expressed in terms of the generalized coordinates (θ 1 and θ 2 ) and the
generalized velocities ( ˙
θ 1 and ˙
θ 2 ):
˙
x 1 = l 1 ˙
θ 1 cos θ 1
˙
y 1 = l 1 ˙
θ 1 sin θ 1
and
˙
x 2 = l 1 ˙
θ 1 cos θ 1 + l 2 ˙
θ 2 cos θ 2
˙
y 2 = l 1 ˙
θ 1 sin θ 1 + l 2 ˙
θ 2 sin θ 2
Then, you can calculate the squares of the speeds of each pendulum (included in
the kinetic energy):
v
2
1 = ˙
x
2
1 + ˙
y
2
1
v
2
2 = ˙
x
2
2 + ˙
y
2
2
Cartesian coordinates included in the potential energy formula,
W = m 1 gy 1 + m 2 gy 2
They are replaced with generalized coordinates. The velocities included in the
kinetic energy formula should be replaced by generalized velocities, so that kinetic
energy in the general case will depend on the generalized coordinates
T =
m 1 v
2
1
2
+
m 2 v
2
2
2
Write down the Lagrangian, and compose the Lagrange equations for the first and
second pendulums in the absence of non-potential forces: