where
m Mass of sporting water body;
C v Damping coefficient of wave;
K w Elastic coefficient of water;
K a Elastic coefficient of air;
F 0 Amplitude of wave dynamics.
The solution of Eq. (14.13) can be written as
x ¼ Asin xt À u
ð
Þ
ð14:14Þ
Among them, the amplitude A and phase angle u are as follows:
A ¼
F 0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K W þ K a
ð
ÞÀmx 2
½
Š
2
q
þ C q x
À
Á 2
ð14:15Þ
tan u ¼
C V x
K W þ K a
ð
ÞÀmx 2
ð14:16Þ
The natural frequency of the system is
x 0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
K W þ K a
m
r
ð14:17Þ
From Eqs. (14.11) and (14.12), airflow rate Q m can be obtained as follows.
Q m ¼ C p a
p 0 S a
V c À 2DV
Dx
ð14:18Þ
Here, the airflow Q m varies with the volume of air in the chamber, and also
affects the pressure change of the chamber. The pressure change value Dp 1 is
Dp 1 ¼
Àp 0
V c À 2DV 1
Z
Q m dt
ð14:19Þ
where DV 1 ¼
R
Q m dt.
After Laplace transformation, the above mathematical model constitutes the
system block diagram shown in Fig. 14.7. The block diagram of the system consists
of two parts: the transfer function G 1 of moving water body and the transfer
function G 2 of air column.
The air column transfer function G 2 in Fig. 14.7 can be simplified to the form of
Fig. 14.8. Write the following equation according to Fig. 14.8:
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14 Pneumatic Principle and Device of Oscillating Water …