5.4 Connected Pendulums
165
Make and solve the equations of motion of the pendulums taking into account the
force of resistance of the medium.
Modeling and computational experiment
When the pendulums move in a vertical plane, the state of the system is completely
described by two independent parameters—the angles θ 1 and θ 2 of the deviations
of the pendulums from the vertical. Thus, a mechanical system has two degrees of
freedom. We obtain a system of equations, which can be described as a “symmetric”
and an “asymmetric” system.
The equation of free motion for each pendulum is similar to the equation obtained
above for a simple mathematical pendulum:
J
d
2
θ
dt 2 = M
In the absence of viscosity of the medium, the moment of rotation created by
gravity acts on the first pendulum
M 11 = −m 1 gl 1 sin θ 1
created by gravity, and the moment of elasticity
M 12 = kd
2
(sin θ 2 − sin θ 1 ) cos θ 1
Therefore, the equation of motion of the first pendulum will have the form:
m 1 l
2
1
d
2
θ 1
dt 2 = −m 1 gl 1 sin θ 1 + kd
2
(sin θ 2 − sin θ 1 ) cos θ 1
or
d
2
θ 1
dt 2 =
−g
l 1
sin θ 1 +
kd
2
m 1 l
2
1
(sin θ 2 − sin θ 1 ) cos θ 1
When deriving the equation, it is assumed that the deflection angles of the pendulum are limited to 180
◦ , i.e., considered the movement of the pendulum without
a “coup.” A similar formula is obtained for the second pendulum. Thus, for two
pendulums connected by a spring in the absence of external forces, the system of
equations:
dθ 1
dt
= ω 1
dω 1
dt
=
−g
l 1
sin θ 1 +
kd
2
m 1 l
2
1
(sin θ 2 − sin θ 1 ) cos θ 1