110
3 Computer Simulation of Dynamic Systems
Fig. 3.16 Phase portrait of a strongly damped oscillator (ω 0 = 2 rad/s) with a singular point of the
“node” type at the origin
describing the behavior of a dynamical system. A special point is reached only after
an infinitely long time.
Comment
The same complication, taking into account the force of friction, can be considered
for the model of a mathematical pendulum. We complicate the task and take into
account that when moving in a viscous medium, the pendulum experiences the action
of a resistance force. We write down the rotational moments created by gravity
(M 1 = −mgl sin θ ) and resistance force (M 1 = −bl
2 dθ
dt
), under the assumption that
the resistance force is directly proportional to the speed:
J
d
2
θ
dt 2 = −mgl sin θ − bl
2 dθ
dt
As a result, the equation of free damped oscillations of a mathematical pendulum
is obtained:
d
2
θ
dt 2 = −
g
l
sin θ −
b
m
dθ
dt
The friction in the system causes damping of the oscillations and the value of
b/m characterizes the speed of this damping. If friction is negligible, then in the