Page 96 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

1.11 harmoniC analysis 93
Octave. When the maximum value of a range of frequency is twice its minimum value,
it is known as an octave band. For example, each of the ranges 75–150 Hz, 150–300 Hz,
and 300–600 Hz can be called an octave band. In each case, the maximum and minimum
values of frequency, which have a ratio of 2:1, are said to differ by an octave.

Decibel. The various quantities encountered in the field of vibration and sound (such as
displacement, velocity, acceleration, pressure, and power) are often represented using the
notation of decibel. A decibel (dB) is originally defined as a ratio of electric powers:

P
dB = 10 log¢ ≤ (1.68)
P 0
where P is some reference value of power. Since electric power is proportional to the
0
square of the voltage (X), the decibel can also be expressed as
X 2 X
dB = 10 log¢ ≤ = 20 log¢ ≤ (1.69)
X 0 X 0
where X is a specified reference voltage. In practice, Eq. (1.69) is also used for expressing
0
the ratios of other quantities such as displacements, velocities, accelerations, and pres-
2
-5
sures. The reference values of X in Eq. (1.69) are usually taken as 2 * 10 N>m for
0
2
-6
pressure and 1 mg = 9.81 * 10 m>s for acceleration.
1.11 harmonic analysis 5
Although harmonic motion is simplest to handle, the motion of many vibratory systems is
not harmonic. However, in many cases the vibrations are periodic—for example, the type
shown in Fig. 1.54(a). Fortunately, any periodic function of time can be represented by
Fourier series as an infinite sum of sine and cosine terms [1.36].

1.11.1 If x(t) is a periodic function with period t, its Fourier series representation is given by
Fourier series a 0
expansion x1t2 = 2 + a cos vt + a cos 2 vt + g
1
2
+ b sin vt + b sin 2 vt + g
2
1
a
= 0 + a 1a cos nvt + b sin nvt2 (1.70)
2 n = 1 n n
where v = 2p>t is the fundamental frequency and a , a , a , c, b , b , care constant
1
0
1
2
2
coefficients. To determine the coefficients a and b , we multiply Eq. (1.70) by cos nvt and
n
n
sin nvt, respectively, and integrate over one period t = 2p>v —for example, from 0 to
5 The harmonic analysis forms a basis for Section 4.2.