150
5 Modeling of Mechanical Oscillatory Systems …
Fig. 5.1 Movement of two
bodies with friction
given initial velocities under the action of the friction forces that exist between all
the contacting bodies. The areas of application of the friction force are highlighted
in gray, as shown in Fig. 5.1.
Both movements are one-dimensional and have the same direction. It is assumed
that the friction force is proportional to the relative velocity of the two contacting
bodies.
Tasks
Build a model that will describe the change in the velocities of bodies, as well as
their position in space.
Conduct numerical experiments.
The initial velocity of the first body is v 1 (0) = 0.1 m/s; the mass of the first body
is m 1 = 100 kg. The second body at the initial moment is at rest v 2 (0) = 0; the mass
of the second body is m 2 = 10 kg.
For the same masses of bodies, the initial velocity of the second body is v 2 (0) =
0.2 m/s. The first body at the initial moment is at rest v 1 (0) = 0.
Modeling and computational experiment
We will consider the motion of bodies, as well as the change in velocities relative to
the horizontal surface; they will be equal to v 1 and v 2 , respectively. We choose the
following conditional direction for speeds: We assume that speed is positive if the
body moves to the right, and negative if the body moves to the left.
The change in body speeds can be calculated using Newton’s second law.
m
dv
dt
= F
We write these equations for each of the two bodies. We take into account that
friction forces with a horizontal surface and with a second body act on the first body.
The force acting on the second body is due only to friction with the first body.
m 1
dv 1
dt
= −b s1 · v 1 − b 12 · (v 1 − v 2 )
m 2
dv 2
dt
= −b 12 · (v 2 − v 1 )
(v 1 − v 2 ) this is the speed of the first body relative to the second,
(v 2 − v 1 ) this is the speed of the second body relative to the first.
The friction coefficients b s1 and b 12 are the coefficients of friction with the surface
and the first body with the second, respectively.
5 Modeling of Mechanical Oscillatory Systems …
Fig. 5.1 Movement of two
bodies with friction
given initial velocities under the action of the friction forces that exist between all
the contacting bodies. The areas of application of the friction force are highlighted
in gray, as shown in Fig. 5.1.
Both movements are one-dimensional and have the same direction. It is assumed
that the friction force is proportional to the relative velocity of the two contacting
bodies.
Tasks
Build a model that will describe the change in the velocities of bodies, as well as
their position in space.
Conduct numerical experiments.
The initial velocity of the first body is v 1 (0) = 0.1 m/s; the mass of the first body
is m 1 = 100 kg. The second body at the initial moment is at rest v 2 (0) = 0; the mass
of the second body is m 2 = 10 kg.
For the same masses of bodies, the initial velocity of the second body is v 2 (0) =
0.2 m/s. The first body at the initial moment is at rest v 1 (0) = 0.
Modeling and computational experiment
We will consider the motion of bodies, as well as the change in velocities relative to
the horizontal surface; they will be equal to v 1 and v 2 , respectively. We choose the
following conditional direction for speeds: We assume that speed is positive if the
body moves to the right, and negative if the body moves to the left.
The change in body speeds can be calculated using Newton’s second law.
m
dv
dt
= F
We write these equations for each of the two bodies. We take into account that
friction forces with a horizontal surface and with a second body act on the first body.
The force acting on the second body is due only to friction with the first body.
m 1
dv 1
dt
= −b s1 · v 1 − b 12 · (v 1 − v 2 )
m 2
dv 2
dt
= −b 12 · (v 2 − v 1 )
(v 1 − v 2 ) this is the speed of the first body relative to the second,
(v 2 − v 1 ) this is the speed of the second body relative to the first.
The friction coefficients b s1 and b 12 are the coefficients of friction with the surface
and the first body with the second, respectively.