154
5 Modeling of Mechanical Oscillatory Systems …
x 1 + x 2 + x 3 = 0
Add an external force to each equation and move on to first-order equations.
Model of coupled spring pendulums:
dx 1
dt
= v 1
m 1
dv 1
dt
= −k 1 (x 1 − x 01 ) − k((x 2 − x 1 ) − (x 02 − x 01 )) − b 1 v 1 + F 1 (t)
dx 2
dt
= v 2
m 2
dv 2
dt
= −k((x 1 − x 2 ) − (x 02 − x 01 )) − k 2 (x 2 − x 02 ) − b 2 v 2 + F 2 (t)
When conducting computational experiments, the external force can be selected
as follows:
– For free oscillations:
F 1 (t) = F 2 (t) = 0
– For forced oscillations with constant exposure:
F 1 (t) = F 1
F 2 (t) = F 2
– To study the phenomenon of resonance with external periodic exposure:
F 1 (t) = F 1 cos ω 1 t
F 2 (t) = F 2 cos ω 2 t
This mechanical system consists of components that are well known to us. It is on
the example of these components in the second chapter that the features of software
and component modeling were compared. Therefore, now we will no longer dwell
on the technical subtleties, but give only the basic steps of analyzing the component
model.
A diagram of a component model of a pendulum with two masses is shown in
Fig. 5.6.
Animation of a computer model of a mechanical system consisting of two bodies,
springs and dampers, can be seen during simulation of the model. An example of a
screenshot of a system animation is shown in Fig. 5.7.
5 Modeling of Mechanical Oscillatory Systems …
x 1 + x 2 + x 3 = 0
Add an external force to each equation and move on to first-order equations.
Model of coupled spring pendulums:
dx 1
dt
= v 1
m 1
dv 1
dt
= −k 1 (x 1 − x 01 ) − k((x 2 − x 1 ) − (x 02 − x 01 )) − b 1 v 1 + F 1 (t)
dx 2
dt
= v 2
m 2
dv 2
dt
= −k((x 1 − x 2 ) − (x 02 − x 01 )) − k 2 (x 2 − x 02 ) − b 2 v 2 + F 2 (t)
When conducting computational experiments, the external force can be selected
as follows:
– For free oscillations:
F 1 (t) = F 2 (t) = 0
– For forced oscillations with constant exposure:
F 1 (t) = F 1
F 2 (t) = F 2
– To study the phenomenon of resonance with external periodic exposure:
F 1 (t) = F 1 cos ω 1 t
F 2 (t) = F 2 cos ω 2 t
This mechanical system consists of components that are well known to us. It is on
the example of these components in the second chapter that the features of software
and component modeling were compared. Therefore, now we will no longer dwell
on the technical subtleties, but give only the basic steps of analyzing the component
model.
A diagram of a component model of a pendulum with two masses is shown in
Fig. 5.6.
Animation of a computer model of a mechanical system consisting of two bodies,
springs and dampers, can be seen during simulation of the model. An example of a
screenshot of a system animation is shown in Fig. 5.7.