Nozzle Blocking Fault

We will first model the nozzle blocking fault as an efficient reduction of the nozzle throat area (see Figure 11, below). This fault is described by above equations and cross section of the nozzle throat is a time function. The value of is not controlled during a flight of the rocket. We have to solve an inverse problem: to find the hidden variable using data of sensors of head gas pressure and temperature in combustion chamber. In this case these data were obtained by numerical simulation of SPDE (1)-(4), (6)-(8) [see Figure 11, right]. As an example, for the simulations we use typical parameters for the Titan IV: r0 = 0.74 m, rt = 0.63 m, rex =1.61 m, L0 = 41.25 m, = 1800 kg.m-3, H = 2.9x106 J.kg-1, cp =1.005 J/kgK, , rc = 0.01 m.sec-1, pc = 7x106 Pa, n=0.35, .=1.4.

*Click to enlarge images above*

Figure 11. (left) The geometry of the case and the grain before the fault is shown by the black lines and after the fault is shown by the dashed green lines. (right). Synthetic data for the temperature (red line) and the pressure (blue line) of the SRB operation are shown with a neutral thrust curve. Fault occurs at t=15 sec, T_{0}=273K.

*Estimation of the parameters of the low-dimensional model*

Figure 12. Schematic drawing of low-dimensional model of nozzle blocking fault

Figure 12 is a diagrammatic representation of low-dimensional model of nozzle blocking fault. The model includes state driving noises, which are assumed to have unknown parameters. Noise characterization is important for designing the fault estimation algorithm.

The parameters of the model can be estimated within the framework we have developed. Diagrammatic representation of fault estimation based on low-dimensional dynamic model is shown in Figure 13. An example of estimation of the parameters of the model is shown in the Table 1. In this test, the propellant grain has cylindrical geometry and we consider the progressive thrust curve. It can be seen from the table that the high accuracy of estimation can be achieved in a short time interval. This fact allows us to detect changes in the parameter values almost in real time.

Figure 13. Schematic drawing of fault estimation based on the low-dimensional model

Table 2 The results of the parameter estimation of the model. The total time of the measurements in this test was T=0.5 sec, the sampling time was DELTAt=0.001 sec, and the number of measured points was N=500. The propellant grain has cylindrical geometry.

We applied the method of dynamical inference to estimate the parameters of the fault. The fault is introduced at time t=15sec, when the area of the nozzle throat is reduced by 10%. The results of the estimation of the value of the parameter c0GSt/(pL) in the flight with progressive thrust curve are shown in Figure 14. It can be seen that the shift in the model parameters can be accurately estimated within 0.1 sec after the fault.

*Building a probability density function of hazardous events*

The estimated parameters can be used to predict the probability of hazardous events at any given time. For example, consider the prognostic problem for the nonlinear time evolution of the pressure buildup induced by the nozzle blocking at time t=15sec for the SRM with the same cylindrical geometry as above, but with a neutral trust curve. The time evolution of the nozzle area in this fault and is highly nonlinear, as shown in Figure 15.

Synthetic Data

*Click to enlarge image above*

Figure 14. (left) Estimation of the value of the parameter -c0GSt/(pL) before (left curve) and after (right curve) the fault. The dashed line shows the actual value of the parameter. The solid lines show the PDF of the parameter estimation with T=0.1 sec, .t=0.001 sec, N=500 (see the caption for the Table 1).

Figure 15. (right) Nonlinear time evolution of the nozzle blocking fault is shown in the bottom figure. The corresponding time evolution of the pressure build up is shown in the top figure. The red line shows margins for the pressure.The growth of leads to the overpressure fault and possibly to the case burst with a significant time delay of 7 sec after the fault. We repeat the simulations of the fault and analyze the distribution of the predicted pressures and predicted times of the overpressure faults as a function of the time interval of inference, which is the time elapsed after the fault. Results of the analysis are shown in Figure 16 and Figure 17.

Synthetic Data

Figure 16. Nonlinear time evolution of the pressure build up after the nozzle blocking fault is shown by the back solid line. Predicted dynamics of the pressure is shown by the jagged lines. The results of the predictions build 1 sec, 1.5 sec, and 2.1 sec after the fault are shown by green, cyan, and blue lines respectively. The values of the pressure at t=14 sec, which are used to build the PDF of the pressure are shown by red circles. The time moments of the predicted overpressure faults used to build the PDF of the case burst times as shown by the black circles on the red margin line.Synthetic Data

Figure 17. (left) The PDF of the predicted values of pressure at t=14 sec build 1 sec (green line), 1.5 sec (cyan line), and 2.1 sec (blue lines) after the fault. The dashed vertical line shows the dangerous level of the pressure. (right) The PDF of the predicted times of the overpressure fault build 1sec (green line), 1.5 sec (cyan line), and 2.1 sec (blue lines after the fault). The dashed vertical line shows the actual time when the overpressure fault will occur.

It can be seen from Figure 16 and Figure 17 that the accuracy of the prediction is a strong function of the time elapsed after the fault used to infer model parameters and to predict the pressure dynamics. In particular, it can be seen that the predicted values converge to the correct ones in 2.1 sec.

We note that the derived LDPM can be used to analyze effectively the results of the ground and flight firing tests, to provide deep inside into the time variations of the SRB parameters in various nominal and off-nominal regimes, and to build a data base of dynamical signatures of various fault modes.