Page 90 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 3.10 Sharpness of Resonance 77

Figure 3.9-1. Frequency response
with structural damping.
The following algebra leads to

_ _ (1 - r ^ )
2y[il - r^Ÿ + y^]

1 Ÿ 4 y ^ ( l — r^) + (1 — r^) —2 y ^ ( l — r^) +
+ y +
4 y ^ [ ( l - r ^ f + r ’

2yj

-\- \ y + l Ÿ ^ Î ' ]
2yJ [2yJ

This is a eircle of radius l/2 y with center —1 /2y, as shown in Fig. 3.9-1.
Every point on the circle represents a different frequency ratio r. At

resonance, r = 1, x = 0, y = —1/y, and H(r) = —i/y.

3.10 SHARPNESS OF RESONANCE
In forced vibration, there is a quantity Q related to damping that is a measure of
the sharpness of resonance. To determine this quantity, we assume viscous damp­
ing and start with Eq. (3.1-7).
When co/co^ = 1, the resonant amplitude is (To//c)/2^'. We now seek
the two frequencies on either side of resonance (often referred to as sidebands),
where X is 0.707Aj.g^. These points are also referred to as the half-power points
and are shown in Fig. 3.10-1.
Letting X = O.IOIX^^^ and squaring Eq. (3.1-7), we obtain
1
i2
1 - — 2^