3 Nonlinear Vibrations: Classical
Local Theory
3.1 Introduction
A linear system represents a mathematical abstraction – a useful and productive
invention of the human mind. However, the physical systems found in nature and
manmade devices are inherently nonlinear. If in doubt, try imagining any physical
system subjected to a relevant excitation of increasing magnitude: pump energy into
the system and it will respond in some way. Initially, the state of the system may
change in proportion to the applied load, but at some point the change will
inevitably be disproportional. Ultimately, given unlimited power supplies, the
system may break apart into two or more subsystems, or reorganize itself into a
qualitatively different type of system. Rarely does nature recognize linearity and
straight lines.
So, linear models of physical systems are mathematical abstractions, and you
may start wondering why engineering and other scientific educations are then so
heavily dominated by linear systems and theories. One reason is that many real-life
systems indeed operate within a limited range, for which a linear approximation is
sufficient to understand, predict or control their behavior. Another reason is that all
linear models mathematically are more or less alike, and are well understood within
a framework of almost fully developed theories. Nonlinear systems, on the other
hand, are highly diversified – like ‘non-Europeans’, ‘non-metallics’ and other
non-specific descriptors. Few are well understood, and nonlinear theory is still,
more than a hundred years after the pioneering work of Poincaré, by many aspects
in its infancy. For nonlinear systems one has to cope with an impressive number of
competing analytical and numerical tools, some of which are only partly developed
and still subjected to controversies.
Sometimes nonlinear effects cannot be neglected, their inclusion in a model aids
in understanding a problem, or a nonlinear phenomenon is to be suppressed or
utilized in a specific device. We may then turn to the computer, and simulate the
system numerically. Computers, however, provide highly specific answers to highly
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. J. Thomsen, Vibrations and Stability,
https://doi.org/10.1007/978-3-030-68045-9_3
95
Local Theory
3.1 Introduction
A linear system represents a mathematical abstraction – a useful and productive
invention of the human mind. However, the physical systems found in nature and
manmade devices are inherently nonlinear. If in doubt, try imagining any physical
system subjected to a relevant excitation of increasing magnitude: pump energy into
the system and it will respond in some way. Initially, the state of the system may
change in proportion to the applied load, but at some point the change will
inevitably be disproportional. Ultimately, given unlimited power supplies, the
system may break apart into two or more subsystems, or reorganize itself into a
qualitatively different type of system. Rarely does nature recognize linearity and
straight lines.
So, linear models of physical systems are mathematical abstractions, and you
may start wondering why engineering and other scientific educations are then so
heavily dominated by linear systems and theories. One reason is that many real-life
systems indeed operate within a limited range, for which a linear approximation is
sufficient to understand, predict or control their behavior. Another reason is that all
linear models mathematically are more or less alike, and are well understood within
a framework of almost fully developed theories. Nonlinear systems, on the other
hand, are highly diversified – like ‘non-Europeans’, ‘non-metallics’ and other
non-specific descriptors. Few are well understood, and nonlinear theory is still,
more than a hundred years after the pioneering work of Poincaré, by many aspects
in its infancy. For nonlinear systems one has to cope with an impressive number of
competing analytical and numerical tools, some of which are only partly developed
and still subjected to controversies.
Sometimes nonlinear effects cannot be neglected, their inclusion in a model aids
in understanding a problem, or a nonlinear phenomenon is to be suppressed or
utilized in a specific device. We may then turn to the computer, and simulate the
system numerically. Computers, however, provide highly specific answers to highly
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. J. Thomsen, Vibrations and Stability,
https://doi.org/10.1007/978-3-030-68045-9_3
95