Page 107 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

104 Chapter 1 Fundamentals oF Vibration
TABlE 1.1

2
Time Station, i Time (s), t i Pressure 1kN>m 2, p i
0 0 0
1 0.01 20
2 0.02 34
3 0.03 42
4 0.04 49
5 0.05 53
6 0.06 70
7 0.07 60
8 0.08 36
9 0.09 22
10 0.10 16
11 0.11 7
12 0.12 0

Solution: Since the given pressure fluctuations repeat every 0.12 s, the period is t = 0.12 s and
the circular frequency of the first harmonic is 2p radians per 0.12 s or v = 2p>0.12 = 52.36 rad/s.
As the number of observed values in each wave (N) is 12, we obtain from Eq. (1.97)

2 N 1 12
a 0 = a p i = a p i = 68166.7 (E.1)
N i = 1 6 i = 1
The coefficients a n and b n can be determined from Eqs. (1.98) and (1.99):

2 N 2npt i 1 12 2npt i
a n = a p i cos = a p i cos (E.2)
N i = 1 t 6 i = 1 0.12
2 N 2npt i 1 12 2npt i
b n = a p i sin = a p i sin (E.3)
N i = 1 t 6 i = 1 0.12

The computations involved in Eqs. (E.2) and (E.3) are shown in Table 1.2. From these calculations,
the Fourier series expansion of the pressure fluctuations p(t) can be obtained (see Eq. 1.70):

p1t2 = 34083.3 - 26996.0 cos 52.36t + 8307.7 sin 52.36t
+ 1416.7 cos 104.72t + 3608.3 sin 104.72t - 5833.3 cos 157.08t
- 2333.3 sin 157.08t + g N>m 2 (E.4)