specific questions. One obtains the response of a system only for particular sets of
parameters, and for particular intervals of time. For nonlinear systems one cannot
just interpolate or extrapolate from discrete data. So, in particular for nonlinear
systems, some theoretical insight is required for turning the huge amount of
computer-generated information into useful understanding.
This chapter provides a set of classical tools for the local analysis of nonlinear
vibrations. Local here means ‘close-to-equilibrium’. A nonlinear system typically
has several states of static and dynamic equilibrium. Local theories concern solutions in the immediate vicinity of such states. Global solutions, that may involve
multiple equilibriums and far-from-equilibrium behavior, are usually inaccessible to
the classical methods presented in this chapter. The term ‘classical’ serves to distinguish the methods described in this chapter from those described in Chap. 6 on
chaotic and global dynamics. Classical methods, though constantly being developed and refined, are based on classical mathematical paradigms, whereas the study
of global dynamics and chaos requires new mathematical concepts and tools.
Classical local analysis is perfectly adequate for the study of numerous nonlinear
problems, and is an important prerequisite for studying global behavior.
The diversity of nonlinear models and tools precludes a general presentation of
the subject. Instead, after describing potential sources of nonlinearity, we go right
into a nonlinear analysis of a simple and illustrative example. This will allow
essential concepts, methods and phenomena to be presented where appropriate.
Subsequent chapters will treat more complicated systems and phenomena.
The main references of this chapter are Blaquiére (1966), Bogoliubov and
Mitropolskii (1961), Bolotin (1964), Cartmell (1990), Guckenheimer and Holmes
(1983), Jackson (1991), Mitropolskii and Nguyen 1997, Nayfeh (1973), Nayfeh and
Balachandran (1995), Nayfeh and Mook (1979), Sanders and Verhulst (1985),
Schmidt and Tondl (2009), Stoker (1950), and Verhulst (1996a).
3.2 Sources of Nonlinearity
Nonlinearities may enter a model in many ways. Their origin may be geometrical or
material, or associated with nonlinear forces or physical configuration. Whatever
their origin, nonlinearities may enter the model equations in similar ways. So, it is
rarely possible to deduce the physical origin of a nonlinearity from its mathematical
representation. Any component of the equations of motion may be nonlinearly
affected: The inertial terms, the terms describing elastic or inelastic restoring forces, the dissipative terms, terms describing external excitation, and the boundary
conditions. Nonlinear terms are recognized by being nonlinear functions of the
dependent variables of the equations of motion. For example, if u(t) describes the
motion of a system, then the terms u
3 , ü, sinu and |u| are all nonlinear, whereas t
2 u,
usint and e
–t
ü are linear terms.
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3 Nonlinear Vibrations: Classical Local Theory
parameters, and for particular intervals of time. For nonlinear systems one cannot
just interpolate or extrapolate from discrete data. So, in particular for nonlinear
systems, some theoretical insight is required for turning the huge amount of
computer-generated information into useful understanding.
This chapter provides a set of classical tools for the local analysis of nonlinear
vibrations. Local here means ‘close-to-equilibrium’. A nonlinear system typically
has several states of static and dynamic equilibrium. Local theories concern solutions in the immediate vicinity of such states. Global solutions, that may involve
multiple equilibriums and far-from-equilibrium behavior, are usually inaccessible to
the classical methods presented in this chapter. The term ‘classical’ serves to distinguish the methods described in this chapter from those described in Chap. 6 on
chaotic and global dynamics. Classical methods, though constantly being developed and refined, are based on classical mathematical paradigms, whereas the study
of global dynamics and chaos requires new mathematical concepts and tools.
Classical local analysis is perfectly adequate for the study of numerous nonlinear
problems, and is an important prerequisite for studying global behavior.
The diversity of nonlinear models and tools precludes a general presentation of
the subject. Instead, after describing potential sources of nonlinearity, we go right
into a nonlinear analysis of a simple and illustrative example. This will allow
essential concepts, methods and phenomena to be presented where appropriate.
Subsequent chapters will treat more complicated systems and phenomena.
The main references of this chapter are Blaquiére (1966), Bogoliubov and
Mitropolskii (1961), Bolotin (1964), Cartmell (1990), Guckenheimer and Holmes
(1983), Jackson (1991), Mitropolskii and Nguyen 1997, Nayfeh (1973), Nayfeh and
Balachandran (1995), Nayfeh and Mook (1979), Sanders and Verhulst (1985),
Schmidt and Tondl (2009), Stoker (1950), and Verhulst (1996a).
3.2 Sources of Nonlinearity
Nonlinearities may enter a model in many ways. Their origin may be geometrical or
material, or associated with nonlinear forces or physical configuration. Whatever
their origin, nonlinearities may enter the model equations in similar ways. So, it is
rarely possible to deduce the physical origin of a nonlinearity from its mathematical
representation. Any component of the equations of motion may be nonlinearly
affected: The inertial terms, the terms describing elastic or inelastic restoring forces, the dissipative terms, terms describing external excitation, and the boundary
conditions. Nonlinear terms are recognized by being nonlinear functions of the
dependent variables of the equations of motion. For example, if u(t) describes the
motion of a system, then the terms u
3 , ü, sinu and |u| are all nonlinear, whereas t
2 u,
usint and e
–t
ü are linear terms.
96
3 Nonlinear Vibrations: Classical Local Theory