The dynamic index of the system is often given in the form of frequency, so the
displacement of the piston is assumed to be
Y ¼ R sin xt
ð9:2Þ
Then, there is
_
Y ¼ Rx cos xt ¼ _
Y max cos xt
ð9:3Þ
€
Y ¼ ÀRx
2 sin xt
ð9:4Þ
where
R Piston motion amplitude, i.e., maximum displacement (m);
x System bandwidth (rad=s).
Substituting Eqs. (9.2) and (9.4) into Eq. (9.1), it is obtained,
F ¼ ÀmRx
2
þ KR
À
Á
sin xt ¼ F max sin xt
ð9:5Þ
Figure 9.35 shows the load speed and load force–time history curves obtained
by Eqs. (9.3) and (9.5).
Simultaneous Eqs. (9.3) and (9.5), it can be written as
F
ÀmRx 2 þ KR
2
þ
_
Y
Rx
2
¼ 1
ð9:6Þ
As shown in Fig. 9.36, the load trajectory corresponding to the above equation is
a positive ellipse. When the friction moment of rudder and transmission mechanism
is considered, the load trajectory is shown in Fig. 9.37. When the viscous damping
moment is considered, the load trajectory is distorted, as shown in Fig. 9.38.
Reference can be made to the relevant literature.
Fig. 9.35 Time-dependence
curve of load force and load
velocity
116
9 High-Temperature and High-Speed Gas Turbine Pump …