Chapter 4
Modeling of Mechanical Oscillatory
Systems with One Degree of Freedom
Oscillatory movement, or simply oscillations, is called any movement or change of
state, characterized by a varying degree of repeatability over time of the values of the
physical quantities that determine this movement or state. We encounter oscillations
in the study of various physical phenomena: sound, light, alternating currents, radio
waves, pendulum swings, etc. It turns out that there is a commonality of the laws of
these phenomena and mathematical methods for their study. Examples of oscillatory
motion are oscillations of pendulums, strings, telephone membranes, charge, and
current in an oscillatory circuit, etc.
More details about oscillatory systems can be found, for example, in references
[1–3].
Oscillations are accompanied by the alternate conversion of the energy of one type
into the energy of another type. Oscillatory motion is called periodic if the values of
physical quantities that change during oscillations are repeated at regular intervals.
All types of vibrations can be classified by the following parameters:
• by physical nature (mechanical and electromagnetic);
• by the nature of occurrence and existence (free, forced, parametric, and selfoscillations);
• by the nature of the dependence of the oscillating quantity on time (harmonic and
non-harmonic).
Despite the different nature of the oscillations, the same physical laws are found in
them; they are described by the same equations, investigated by general methods. In
this section, we consider mechanical vibrations, i.e., repeated changes in the positions
and velocities of any bodies or parts of bodies that occur in the presence of elastic
forces, gravity, and other forces.
In the two previous chapters, the principles of constructing basic oscillatory
models—the mathematical and spring pendulum—were considered. Now, we will
move on to a more comprehensive study of their dynamic characteristics and the
construction of more complex oscillatory models on their basis.
Let us start with the study of mathematical pendulum models.
© 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_4
131
Modeling of Mechanical Oscillatory
Systems with One Degree of Freedom
Oscillatory movement, or simply oscillations, is called any movement or change of
state, characterized by a varying degree of repeatability over time of the values of the
physical quantities that determine this movement or state. We encounter oscillations
in the study of various physical phenomena: sound, light, alternating currents, radio
waves, pendulum swings, etc. It turns out that there is a commonality of the laws of
these phenomena and mathematical methods for their study. Examples of oscillatory
motion are oscillations of pendulums, strings, telephone membranes, charge, and
current in an oscillatory circuit, etc.
More details about oscillatory systems can be found, for example, in references
[1–3].
Oscillations are accompanied by the alternate conversion of the energy of one type
into the energy of another type. Oscillatory motion is called periodic if the values of
physical quantities that change during oscillations are repeated at regular intervals.
All types of vibrations can be classified by the following parameters:
• by physical nature (mechanical and electromagnetic);
• by the nature of occurrence and existence (free, forced, parametric, and selfoscillations);
• by the nature of the dependence of the oscillating quantity on time (harmonic and
non-harmonic).
Despite the different nature of the oscillations, the same physical laws are found in
them; they are described by the same equations, investigated by general methods. In
this section, we consider mechanical vibrations, i.e., repeated changes in the positions
and velocities of any bodies or parts of bodies that occur in the presence of elastic
forces, gravity, and other forces.
In the two previous chapters, the principles of constructing basic oscillatory
models—the mathematical and spring pendulum—were considered. Now, we will
move on to a more comprehensive study of their dynamic characteristics and the
construction of more complex oscillatory models on their basis.
Let us start with the study of mathematical pendulum models.
© 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_4
131