5.9 Sophisticated Mechanical System with Springs and Block
203
Tasks
It is necessary to study this system, making equations describing the change in the
system over time, and build its computer model.
Write equations describing the change in elongation time and spring speeds.
Conduct a simulation and computational experiment in a WSM environment.
Modeling and computational experiment
When modeling the system, we make the following simplifying assumption. We
assume that the force applied to the left end of the lever is instantly balanced by the
forces applied to the right of the fulcrum, i.e., the elasticity of the second spring and
the force of viscous friction. Under this assumption, the lever is in equilibrium at
each instant of time, i.e., the rule of moments is fulfilled for him:
k 1 e 1 · L 2 = k 2 e 2 · L 3 + b 1 v 4 · L 4
Here e 1 and e 2 are the extensions of the springs. In addition, at each point of the
lever, its angular velocity is the same:
ω =
v 2
L 2
=
v 3
L 3
=
v 4
L 4
The module in this formula indicates that linear velocities v 2 , v 3 , and v 4 may have
different signs.
The spring extension speeds are related to the difference in the speeds of the points
of their attachment:
de 1
dt
= v 1 − v 2
de 2
dt
= v 3
The above relations make up the mathematical model of the problem (Fig. 5.71).
Let us conduct a numerical experiment. We plot the velocities in the system. It can
be seen that with external periodic exposure, oscillations are excited in the system,
which cease when the exposure ceases (Fig. 5.72).
According to the schedule, it is possible to analyze the fluctuations of the lever—
the right end of the lever moves periodically with a constant offset relative to external
influences. The left end of the lever is out of phase with the right.
Here are the spring extension graphs (yellow graph—left spring, blue graph—right
spring) (Fig. 5.73).
203
Tasks
It is necessary to study this system, making equations describing the change in the
system over time, and build its computer model.
Write equations describing the change in elongation time and spring speeds.
Conduct a simulation and computational experiment in a WSM environment.
Modeling and computational experiment
When modeling the system, we make the following simplifying assumption. We
assume that the force applied to the left end of the lever is instantly balanced by the
forces applied to the right of the fulcrum, i.e., the elasticity of the second spring and
the force of viscous friction. Under this assumption, the lever is in equilibrium at
each instant of time, i.e., the rule of moments is fulfilled for him:
k 1 e 1 · L 2 = k 2 e 2 · L 3 + b 1 v 4 · L 4
Here e 1 and e 2 are the extensions of the springs. In addition, at each point of the
lever, its angular velocity is the same:
ω =
v 2
L 2
=
v 3
L 3
=
v 4
L 4
The module in this formula indicates that linear velocities v 2 , v 3 , and v 4 may have
different signs.
The spring extension speeds are related to the difference in the speeds of the points
of their attachment:
de 1
dt
= v 1 − v 2
de 2
dt
= v 3
The above relations make up the mathematical model of the problem (Fig. 5.71).
Let us conduct a numerical experiment. We plot the velocities in the system. It can
be seen that with external periodic exposure, oscillations are excited in the system,
which cease when the exposure ceases (Fig. 5.72).
According to the schedule, it is possible to analyze the fluctuations of the lever—
the right end of the lever moves periodically with a constant offset relative to external
influences. The left end of the lever is out of phase with the right.
Here are the spring extension graphs (yellow graph—left spring, blue graph—right
spring) (Fig. 5.73).