Page 281 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Vibration of

Continuous Systems

The systems to be studied in this chapter have continuously distributed mass and
elasticity. These bodies are assumed to be homogeneous and isotropic, obeying
Hooke’s law within the elastic limit. To specify the position of every point in the
elastic body, an infinite number of coordinates is necessary, and such bodies,
therefore, possess an infinite number of degrees of freedom.
In general, the free vibration of these bodies is the sum of the principal or
normal modes, as previously stated. For the normal mode vibration, every particle
of the body performs simple harmonic motion at the frequency corresponding to
the particular root of the frequency equation, each particle passing simultaneously
through its respective equilibrium position. If the elastic curve of the body under
which the motion is started coincides exactly with one of the normal modes, only
that normal mode will be produced. However, the elastic curve resulting from a
blow or a sudden removal of forces seldom corresponds to that of a normal mode,
and thus all modes are excited. In many cases, however, a particular normal mode
can be excited by proper initial conditions.
For the forced vibration of the continuously distributed system, the mode
summation method, previously touched upon in Chapter 6, makes possible its
analysis as a system with a finite number of degrees of freedom. Constraints are
often treated as additional supports of the structure, and they alter the normal
modes of the system. The modes used in representing the deflection of the system
need not always be orthogonal, and a synthesis of the system using nonorthogonal
functions is possible.

9.1 VIBRATING STRING
A flexible string of mass p per unit length is stretched under tension T. By
assuming the lateral deflection y of the string to be small, the change in tension
with deflection is negligible and can be ignored.

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