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3 Computer Simulation of Dynamic Systems
N = −kθ,
The proportionality coefficient k is called the spring stiffness. Applying the basic
equation of the dynamics of rotation of a rigid body around a fixed axis to the motion
of a disk with an inertia moment I,
N = I
dω
dt
= I
d
2
θ
dt 2
we obtain the following differential equation of the natural oscillations of the torsion
spring oscillator:
I
d
2
θ
dt 2 = −kθ
or
d
2
θ
dt 2 +
k
I
θ = 0
The general solution of this equation is a simple harmonic oscillation. The oscillations occur with an angular frequency ω 0 , the square of which is proportional to
the spring stiffness k and inversely proportional to the moment of inertia I of the
disk, i.e. this mathematical model is equivalent to the spring pendulum model, but
with other physical parameters.
In the presence of a viscous friction force, the braking moment of this force
proportional to the angular velocity of the disk will be present in the differential
equation ˙
θ :
I ¨
θ + b ˙
θ + kθ = 0
And again, they got a model similar to the spring pendulum model with damping.
The graphs characterizing the kinematic and dynamic behavior of such a model are
already known to us—they will coincide with the graphs for a spring pendulum with
the difference that instead of linear displacement and velocity, angular displacement
and angular velocity will act as kinematic characteristics.
Apply external torque to the disk:
I ¨
θ + b ˙
θ + kθ = T
Thus, we will be able to additionally investigate the constant effect on the system,
which we have not studied above.
Let the system consists of a disk with an inertia moment of I = 0.5 kg m
2 and a
spring of rigidity k = 0.1 kg/s
2 with a fixed end. Let the system be at rest at the initial
3 Computer Simulation of Dynamic Systems
N = −kθ,
The proportionality coefficient k is called the spring stiffness. Applying the basic
equation of the dynamics of rotation of a rigid body around a fixed axis to the motion
of a disk with an inertia moment I,
N = I
dω
dt
= I
d
2
θ
dt 2
we obtain the following differential equation of the natural oscillations of the torsion
spring oscillator:
I
d
2
θ
dt 2 = −kθ
or
d
2
θ
dt 2 +
k
I
θ = 0
The general solution of this equation is a simple harmonic oscillation. The oscillations occur with an angular frequency ω 0 , the square of which is proportional to
the spring stiffness k and inversely proportional to the moment of inertia I of the
disk, i.e. this mathematical model is equivalent to the spring pendulum model, but
with other physical parameters.
In the presence of a viscous friction force, the braking moment of this force
proportional to the angular velocity of the disk will be present in the differential
equation ˙
θ :
I ¨
θ + b ˙
θ + kθ = 0
And again, they got a model similar to the spring pendulum model with damping.
The graphs characterizing the kinematic and dynamic behavior of such a model are
already known to us—they will coincide with the graphs for a spring pendulum with
the difference that instead of linear displacement and velocity, angular displacement
and angular velocity will act as kinematic characteristics.
Apply external torque to the disk:
I ¨
θ + b ˙
θ + kθ = T
Thus, we will be able to additionally investigate the constant effect on the system,
which we have not studied above.
Let the system consists of a disk with an inertia moment of I = 0.5 kg m
2 and a
spring of rigidity k = 0.1 kg/s
2 with a fixed end. Let the system be at rest at the initial