Chapter 5
Modeling of Mechanical Oscillatory
Systems with Several Degrees of Freedom
Until now, we have considered systems with one degree of freedom, having one natural frequency. If the design of the system is complicated, it can make a more complex
movement. In many cases (in the absence of external forces), it can be reduced to the
sum of two oscillations with different frequencies, depending on the properties of
the system. Such a system has two degrees of freedom. Even a simple mathematical
pendulum can oscillate in two mutually perpendicular directions, i.e., in the general
case, it is a system with two degrees of freedom. Most often, several degrees of freedom are possessed by the so-called coupled systems—systems with many degrees
of freedom—between which there are bonds that provide the possibility of energy
exchange between different degrees of freedom. The main feature characteristic of
any connected system is that its own oscillations are generally inharmonious and,
depending on the method of observation, can be perceived either as beats that occur
in such a way that the oscillation energy is periodically pumped (completely or partially) from one parts of the system to another and vice versa, or as the sum of two
harmonic oscillations with frequencies ω
+ and ω
− determined by the structure of
the system as a whole.
We begin the study of mechanical models with several degrees of freedom not
from oscillation processes, but from the relative motion of two bodies. This is a
comparative simple task that does not require any additional information, except
for the basics of modeling studied in the previous chapter. The models given is this
chapter are thoroughly considered in [1, 2].
5.1 The Movement of Two Bodies with Friction
The dynamic system consists of two bodies. The first body of mass m 1 moves along
the horizontal plane, and the second body of mass m 2 moves along the first body with
© Springer Nature Singapore Pte Ltd. 2020
K. Rozhdestvensky et al., Computer Modeling and Simulation
of Dynamic Systems Using Wolfram SystemModeler,
https://doi.org/10.1007/978-981-15-2803-3_5
149
Modeling of Mechanical Oscillatory
Systems with Several Degrees of Freedom
Until now, we have considered systems with one degree of freedom, having one natural frequency. If the design of the system is complicated, it can make a more complex
movement. In many cases (in the absence of external forces), it can be reduced to the
sum of two oscillations with different frequencies, depending on the properties of
the system. Such a system has two degrees of freedom. Even a simple mathematical
pendulum can oscillate in two mutually perpendicular directions, i.e., in the general
case, it is a system with two degrees of freedom. Most often, several degrees of freedom are possessed by the so-called coupled systems—systems with many degrees
of freedom—between which there are bonds that provide the possibility of energy
exchange between different degrees of freedom. The main feature characteristic of
any connected system is that its own oscillations are generally inharmonious and,
depending on the method of observation, can be perceived either as beats that occur
in such a way that the oscillation energy is periodically pumped (completely or partially) from one parts of the system to another and vice versa, or as the sum of two
harmonic oscillations with frequencies ω
+ and ω
− determined by the structure of
the system as a whole.
We begin the study of mechanical models with several degrees of freedom not
from oscillation processes, but from the relative motion of two bodies. This is a
comparative simple task that does not require any additional information, except
for the basics of modeling studied in the previous chapter. The models given is this
chapter are thoroughly considered in [1, 2].
5.1 The Movement of Two Bodies with Friction
The dynamic system consists of two bodies. The first body of mass m 1 moves along
the horizontal plane, and the second body of mass m 2 moves along the first body with
© Springer Nature Singapore Pte Ltd. 2020
K. Rozhdestvensky et al., Computer Modeling and Simulation
of Dynamic Systems Using Wolfram SystemModeler,
https://doi.org/10.1007/978-981-15-2803-3_5
149