Low-Dimensional Model

## IV. Low-Dimensional Model of the Case Breach Fault

To derive the LDPM of the fault we integrate Eqs (1) along the rocket axis and add Eqs (3)-(5), (7) to obtain,

Here the following dimensionless variables are used

where subscript m refers to the maximum reference values of the pressure and density. This set of equations together with Eqs (14)-(19) represent the LDPM of the case breach derived in this example.

Important novel features of the case breach LDPM derived above are the following:

• The burning area of the propellant Sb is calculated using a given design curve Sb = f(R);
• The dynamics of the metal flame erosion and nozzle ablation are taken into account;
• The dynamics of the volume of the combustion chamber are taken into account in equation 4 of the system (20).
• With these modifications the LDPM (20) can reproduce very accurately the results of the ground tests. Typical time-series data for the case breach fault obtained using (20) are shown in Figure 9. In these figures the fault occurs at tf = 0.5 sec and metal erosion rate in the regime of constant erosion is 0.15 in/sec. Importantly, the derived model can reproduce very accurately the experimental time-traces of internal ballistics in the observed in the ground tests as will be described in details elsewhere. It allows us to incorporate obtained LDPM into Bayesian inferential framework of the on-board FD&P of SRBs and to use the synthetic data generated by this model to verify the performance of the Bayesian algorithm, as will be described in the next section.

• Click to enlarge image above

Figure 9 Typical synthetic time-series data generated by the LDPM (20). (left) Nominal pressure (blue line) as compared to the fault-induced pressure (solid dashed line). (right) Nominal thrust (blue line) as compared to the fault-induced thrust (dashed black line) and thrust generated by the hole (red dotted line). The fault occurs at tf = 0.5 sec. The flame erosion rate is approximately 0.15 in/sec.