Untitled - Aerobib - Universidad Politécnica de Madrid

11.4. SCALING OF ROCKETS FOR NON-STEADY CONDITIONS 271
2
0
Chamber pressure
t
−2
−4
−6
Injection pressure drop
Injection rate
τ i
t
t
−8
−10
Burning rate
τ ch
τ c
t
−12
−14
Effect of burning rate
Half period
t
Figure 11.1: Time delays leading to self-exciting oscillationes.
Figure 11.1, taken from Ref. [13], clearly illustrates the mechanisn of the process
capable of producing self-excited oscillations. In this figure, it is assumed that the
feeding system is of the constant pressure type. Therefore, the pressure drop through
the injector, and with it the rate of fuel, decreases as the pressure of the chamber is
increased. However, the variation in rate of fuel follows the drop in pressure with a
certain delay τ i , determined by the dynamic characteristics of the feeding system. The
fuel injected turns into combustion products with a delay τ ch , which is the time lag,
and such products act in turn on the pressure of the chamber with a delay τ c , which
depends on the dynamic characteristics of the same. If the sum τ i + τ ch + τ c is equal
to half the period of the pressure oscillations, then there is a phase agreement between
cause an effect and the oscillations amplify with the only limitation of damping effects.
The above analysis shows that in order to eliminate oscillations it is necessary
to reduce τ i , τ ch and τ c . The measures taken to this effect confirm theoretical prediction.
Two efficient ways, for example, are to reduce the relation between pressure drop
through the injector and the pressure in the chamber, or to reduce τ ch by eliminating
the propellants having an excessive time lag [15].
Aside from the aforegoin causes, Crocco [15] has pointed out an intrinsic cause
for instability, so called because it depends only on the conditions in the combustion
chamber and not on the interaction between it and the feeding system, indispensable,
as it was established, for the previously analyzed case. The intrinsic instability of
Crocco shows the possible existence of low-frequency oscillations in a rocket with a
constant volume feeding system, which is impossible with the preceding model.