Page 143 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Systems with Two

or More Degrees

of Freedom

When a system requires more than one eoordinate to describe its motion, it is
called a multi-DOF system, or an N-DOF system, where N is the number of
coordinates required. Thus, a 2-DOF system requires two independent coordinates
to describe its motion, and it is the simplest of the A^-DOF systems.
The N-DOF system differs from that of the single-DOF system in that it has
N natural frequencies, and for each of the natural frequencies, there corresponds
a natural state of vibration with a displacement configuration known as the normal
mode. Mathematical terms related to these quantities are known as eigenvalues
and eigenvectors. They are established from the N simultaneous equations of
motion of the system and possess certain dynamic properties associated with the
system.
Normal mode vibrations are free undamped vibrations that depend only on
the mass and stiffness of the system and how they are distributed. When vibrating
at one of these normal modes, all points in the system undergo simple harmonic
motion that passes through their equilibrium positions simultaneously. To initiate
a normal mode vibration, the system must be given specific initial conditions
corresponding to its normal mode, For the more general initial conditions, such as
an impulsive blow, the resulting free vibration may contain all the normal modes
simultaneously.
As in the single-DOF system, forced harmonic vibration of the N-DOF
system takes place at the frequency of the excitation. When the excitation fre­
quency coincides with one of the natural frequencies of the system, a condition of
resonance is encountered, with large amplitudes limited only by the damping.
Again, damping is generally omitted except when its concern is of importance in
limiting the amplitude of vibration or in examining the rate of decay of the free
oscillation.
In this chapter, we begin with the determination of the natural frequencies
and normal modes of the 2-DOF system. All of the fundamental concepts of the

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