Page 474 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

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Nonlinear Vibrations

Linear system analysis serves to explain much of the behavior of oscillatory
systems. However, there are a number of oscillatory phenomena that cannot be
predicted or explained by the linear theor>\
In the linear systems that we have studied, cause and effect are related
linearly; i.e., if we double the load, the response is doubled. In a nonlinear system,
this relationship between cause and effect is no longer proportional. For example,
the center of an oil can may move proportionally to the force for small loads, but
at a certain critical load, it will snap over to a large displacement. The same
phenomenon is also encountered in the buckling of columns, electrical oscillations
of circuits containing inductance with an iron core, and vibration of mechanical
systems with nonlinear restoring forces.
The differential equation describing a nonlinear oscillatory system can have
the general form
X - h / ( i , x , i ) = 0

Such equations are distinguished from linear equations in that the principle of
superposition does not hold for their solution.
Analytical procedures for the treatment of nonlinear differential equations
are difficult and require extensive mathematical study. Exact solutions that are
known are relatively few, and a large part of the progress in the knowledge of
nonlinear systems comes from approximate and graphical solutions and from
studies made on computing machines. Much can be learned about a nonlinear
system, however, by using the state space approach and studying the motion
presented in the phase plane.

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