Page 64 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Harmonically

Excited Vibration

When a system is subjected to harmonic excitation, it is forced to vibrate at the
same frequency as that of the excitation. Common sources of harmonic excitation
are unbalance in rotating machines, forces produced by reciprocating machines,
and the motion of the machine itself. These excitations may be undesirable for
equipment whose operation may be disturbed or for the safety of the structure if
large vibration amplitudes develop. Resonance is to be avoided in most cases, and
to prevent large amplitudes from developing, dampers and absorbers are often
used. Discussion of their behavior is of importance for their intelligent use. Finally,
the theory of vibration-measuring instruments is presented as a tool for vibration
analysis.

3.1 FORCED HARMONIC VIBRATION

Harmonic excitation is often encountered in engineering systems. It is commonly
produced by the unbalance in rotating machinery. Although pure harmonic excita­
tion is less likely to occur than periodic or other types of excitation, understanding
the behavior of a system undergoing harmonic excitation is essential in order to
comprehend how the system will respond to more general types of excitation.
Harmonic excitation may be in the form of a force or displacement of some point
in the system.
We will first consider a single-DOF system with viscous damping, excited by a
harmonic force sin cot, as shown in Fig. 3.1-1. Its differential equation of motion
is found from the free-body diagram to be
mx cx -\- kx = Ff. sin cot (3.1-1)

The solution to this equation consists of two parts, the complementary
function, which is the solution of the homogeneous equation, and the particular

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