3.2 Fundamental Principles for Mathematical Models Development
101
r = A sin(ωt) + B cos(ωt)
where
ω =
g
l
the natural frequency of small oscillations, and the quantities A, B are determined
through θ (t = 0),
dα
dt (t = 0).
Let us return to the question of constructing a phase portrait. Let us construct
a phase portrait of a mathematical pendulum. For definiteness, we assume that the
pendulum is a load of mass m, which can freely oscillate on a rigid suspension of
length l. We write the total energy of the pendulum E when it is deflected by an
arbitrary angle θ :
E =
m L
2
ω
2
2
+ mgl(1 − cos θ )
We express ω from this equation:
ω = ±
2E/
ml 2
− 2ω
2
0 (1 − cos θ ),
where ω 0 =
√
g/l, then we can construct a phase portrait of the pendulum for
arbitrary deflection angles.
We choose the initial values of the angular velocity of the pendulum from the range
of −8 < ω 0 < 8 rad/s in increments of 2 rad/s with zero initial deviation ϕ 0 = 0.
It is clear that when the initial value of the angle is shifted by 2π , the type of phase
portrait for oscillations does not change. Therefore, the graph can be immediately
supplemented with phase trajectories for the cases ϕ 0 = 2π and ϕ 0 = −2π .
It can be seen from the figure that when the initial value of the angular velocity
modulo exceeds a certain number, the nature of the movement of the pendulum
changes, namely the oscillations are replaced by rotation around the suspension point.
Thus, there are two types of phase trajectories corresponding to two types of motion:
closed trajectories (oscillations) and open paths (rotation around a suspension point).
We find a trajectory that will be a transition between the two types of motion.
It corresponds to a certain initial value of the angular velocity, which can be found
from the law of conservation of energy.
At the initial moment of time, the potential energy can be considered equal to zero
(since the load is at the lowest point), and the total energy is equal to the kinetic:
E = E k =
mv
2
x0
2
=
mω
2
0 l
2
2
where v x0 = ω 0 l—horizontal speed reported to the load at the initial moment of
time.
101
r = A sin(ωt) + B cos(ωt)
where
ω =
g
l
the natural frequency of small oscillations, and the quantities A, B are determined
through θ (t = 0),
dα
dt (t = 0).
Let us return to the question of constructing a phase portrait. Let us construct
a phase portrait of a mathematical pendulum. For definiteness, we assume that the
pendulum is a load of mass m, which can freely oscillate on a rigid suspension of
length l. We write the total energy of the pendulum E when it is deflected by an
arbitrary angle θ :
E =
m L
2
ω
2
2
+ mgl(1 − cos θ )
We express ω from this equation:
ω = ±
2E/
ml 2
− 2ω
2
0 (1 − cos θ ),
where ω 0 =
√
g/l, then we can construct a phase portrait of the pendulum for
arbitrary deflection angles.
We choose the initial values of the angular velocity of the pendulum from the range
of −8 < ω 0 < 8 rad/s in increments of 2 rad/s with zero initial deviation ϕ 0 = 0.
It is clear that when the initial value of the angle is shifted by 2π , the type of phase
portrait for oscillations does not change. Therefore, the graph can be immediately
supplemented with phase trajectories for the cases ϕ 0 = 2π and ϕ 0 = −2π .
It can be seen from the figure that when the initial value of the angular velocity
modulo exceeds a certain number, the nature of the movement of the pendulum
changes, namely the oscillations are replaced by rotation around the suspension point.
Thus, there are two types of phase trajectories corresponding to two types of motion:
closed trajectories (oscillations) and open paths (rotation around a suspension point).
We find a trajectory that will be a transition between the two types of motion.
It corresponds to a certain initial value of the angular velocity, which can be found
from the law of conservation of energy.
At the initial moment of time, the potential energy can be considered equal to zero
(since the load is at the lowest point), and the total energy is equal to the kinetic:
E = E k =
mv
2
x0
2
=
mω
2
0 l
2
2
where v x0 = ω 0 l—horizontal speed reported to the load at the initial moment of
time.