**
9.2 RESONANCE
**

The
natural frequency (and its overtones) are of great interest to the designer as
they define the frequencies at which the system will
**
resonate
**
. The single-
*
DOF
*
lumped parameter systems shown in Figures 9-1
and 9-2 (pp. 214-215) are the simplest possible to describe a dynamic system,
yet they contain all the basic dynamic elements. Masses and springs are energy
storage elements. A mass stores kinetic energy, and a spring stores potential
energy. The damper is a dissipative element. It uses energy and converts it to
heat. Thus all the losses in the model of Figure 9-1 occur through the damper.

These
are “pure” idealized elements which possess only their own special
characteristics. That is, the spring has no damping and the damper no
springiness, etc. Any system that contains more than one energy storage device,
such as a mass and a spring, will possess at least one natural frequency. If we
excite the system at its natural frequency, we will set up the condition called
resonance in which the energy stored in the system’s elements will oscillate
from one element to the other at that frequency. The result can be violent
oscillations in the displacements of the movable elements in the system as the
energy moves from potential to kinetic form and vice versa.

Figure
9-6a shows a plot of the amplitude and phase angle of the displacement response
*
Y
*
of the system to a sinusoidal input forcing
function at various frequencies
*
f
*
. In our case,
the forcing frequency is the angular velocity at which the cam is rotating. The
plot normalizes the forcing frequency as the frequency ratio
*
f
*
/
*
n
*
. The forced response amplitude
*
Y
*
is
normalized by dividing the dynamic deflection
*
y
*
by the static
deflection
*
F
*
0
/
*
k
*
that the same force amplitude
would create on the system. Thus at a frequency of zero, the output is one,
equal to the static deflection of the spring at the amplitude of the input
force. As the forcing frequency increases toward the natural frequency
*
n
*
,
the amplitude of the output motion, for zero damping, increases rapidly and
becomes theoretically infinite when
*
f
*
=
*
n
*
.
Beyond this point the amplitude decreases rapidly and asymptotically toward
zero at high frequency ratios. Figure 9-6c shows that the phase angle between
input and output of a forced system switches abruptly at resonance.

Figure
9-6b shows the amplitude response of a “self-excited” system for which there is
no externally applied force. An example might be a shaft coupling a motor and a
generator. The loading is theoretically pure torsion. However, if there is any
unbalance in the rotors on the shaft, the centrifugal force will provide a
forcing function proportional to angular velocity. Thus, when stopped there is
no dynamic deflection, so the amplitude is zero. As the system passes through
resonance, the same large response as the forced case is seen. At large
frequency ratios, well above critical, the deflection becomes static at an
amplitude ratio of 1. Cam-follower systems are subject to both of these types
of vibratory behavior. An unbalanced camshaft will self-excite, and the
follower force creates a forced response.