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3 Computer Simulation of Dynamic Systems
The equation of motion of the load takes the form
m
d
2 r
dt 2 = −kr + F(r, t)
where
F(r, t) = mω
2
(t)(R 0 + r ),
or
m
d
2 r
dt 2 = −
k − mω
2
(t)
r + mω
2
(t)R 0 ,
and, obviously, when r R 0 , the written linear equation goes over into the equation
with an external exciting force depending on the time F(t) = −mω
2
(t)R 0 . However,
in this case, resonance is impossible, since the external force is always directed in
one direction and is not able to swing the system.
As an example, illustrating motion in non-inertial reference frames, we consider
the so-called elliptical pendulum.
A simple mathematical pendulum is mounted on a trolley of mass m 1 moving
without friction along a smooth rail. The trolley moves with constant transport acceleration a 0 , and a pendulum with a load of mass m 2 at the initial moment of time hangs
vertically down on an inextensible string of length l. After the start of the movement
of the trolley, the pendulum enters into an oscillatory process. It is necessary to
build a mathematical model of the “trolley–pendulum” system and experimentally
investigate the dependence of the kinematic characteristics of the pendulum on the
acceleration of the trolley.
When the mechanical system moves, the tension of the thread changes, which
leads to a change in the restoring force, and, consequently, in the frequency and
period of oscillations of the pendulum.
The equation of motion of a material point in a non-inertial reference frame
without rotation can be represented as:
F = m(a 0 + a rel ),
where a rel is the acceleration of the body relative to the non-inertial reference frame
and a 0 is the portable acceleration of the body. Thus, the acceleration of the load a
consists of a rel —acceleration of the load relative to the trolley and a 0 —acceleration
of the trolley itself:
a = a 0 + a rel
Load speed can also be represented as a decomposition into relative and portable
speeds.
3 Computer Simulation of Dynamic Systems
The equation of motion of the load takes the form
m
d
2 r
dt 2 = −kr + F(r, t)
where
F(r, t) = mω
2
(t)(R 0 + r ),
or
m
d
2 r
dt 2 = −
k − mω
2
(t)
r + mω
2
(t)R 0 ,
and, obviously, when r R 0 , the written linear equation goes over into the equation
with an external exciting force depending on the time F(t) = −mω
2
(t)R 0 . However,
in this case, resonance is impossible, since the external force is always directed in
one direction and is not able to swing the system.
As an example, illustrating motion in non-inertial reference frames, we consider
the so-called elliptical pendulum.
A simple mathematical pendulum is mounted on a trolley of mass m 1 moving
without friction along a smooth rail. The trolley moves with constant transport acceleration a 0 , and a pendulum with a load of mass m 2 at the initial moment of time hangs
vertically down on an inextensible string of length l. After the start of the movement
of the trolley, the pendulum enters into an oscillatory process. It is necessary to
build a mathematical model of the “trolley–pendulum” system and experimentally
investigate the dependence of the kinematic characteristics of the pendulum on the
acceleration of the trolley.
When the mechanical system moves, the tension of the thread changes, which
leads to a change in the restoring force, and, consequently, in the frequency and
period of oscillations of the pendulum.
The equation of motion of a material point in a non-inertial reference frame
without rotation can be represented as:
F = m(a 0 + a rel ),
where a rel is the acceleration of the body relative to the non-inertial reference frame
and a 0 is the portable acceleration of the body. Thus, the acceleration of the load a
consists of a rel —acceleration of the load relative to the trolley and a 0 —acceleration
of the trolley itself:
a = a 0 + a rel
Load speed can also be represented as a decomposition into relative and portable
speeds.