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FIGURE 13-6. Example of mode frequency display; also called a "waterfall" diagram of a motor firing. Only four complete time-frequency curves are shown; for easy visualization the other time lines are partly omitted except near the resonating frequencies. The height of the wave is proportional to pressure. As the cavity volume increases, the frequencies of the transverse modes decrease. (Adapted from E. W. Price, Chapter 13 of Ref. 13-1, with permission of AIAA.)

FIGURE 13-6. Example of mode frequency display; also called a "waterfall" diagram of a motor firing. Only four complete time-frequency curves are shown; for easy visualization the other time lines are partly omitted except near the resonating frequencies. The height of the wave is proportional to pressure. As the cavity volume increases, the frequencies of the transverse modes decrease. (Adapted from E. W. Price, Chapter 13 of Ref. 13-1, with permission of AIAA.)

broken-off insulation or unburned propellant flows through the nozzle and temporarily blocks all or a part of the nozzle area (causing a momentary pressure rise).

The shifting balance between amplifying and damping factors changes during the burning operation and this causes the growth and also the abatement of specific modes of vibration. The response of a solid propellant describes the change in the gas mass production or energy release at the burning surface when it is stimulated by pressure perturbations. When a momentary high pressure peak occurs on the surface, it increases the instantaneous heat transfer and thus the burning rate, causing the mass flow from that surface to also increase. Velocity perturbations along the burning surface are also believed to cause changes in mass flow. Phenomena that contribute to amplifying the vibrations, or to gains in the acoustic energy (see Ref. 13-1, Chapter 13 by Price), are:

1. The dynamic response of the combustion process to a flow disturbance or the oscillations in the burning rate. This combustion response can be determined from tests of T-burners as described on pages 533 and 534. The response function depends on the frequency of these perturbations and the propellant formulation. The combustion response may not be in a phase with the disturbance. The effects of boundary layers on velocity perturbations have been investigated in Ref. 13-15.

2. The interactions of flow oscillation with the main flow, similar to the basis for the operation of musical wind instruments or sirens (see Ref. 13-16).

3. The fluid dynamic influence of vortexes.

Phenomena that contribute to a diminishing of vibration or to damping are energy-absorbing processes; they include the following:

1. Viscous damping in the boundary layers at the walls or propellant surfaces.

2. Damping by particles or droplets flowing in an oscillating gas/vapor flow is often substantial. The particles accelerate and decelerate by being "dragged" along by the motion of the gas, a viscous flow process that absorbs energy. The attenuation for each particular vibration frequency is an optimum at a particular size of particles; high damping for low-frequency oscillation (large motors) occurs with relatively large solid particles (8 to 20 |im); for small motors or high-frequency waves the best damping occurs with small particles (2 to 6 |im). The attenuation drops off sharply if the particle size distribution in the combustion gas is not concentrated near the optimum for damping.

3. Energy from longitudinal and mixed transverse/longitudinal waves is lost out through the exhaust nozzle. Energy from purely transverse waves does not seem to be damped by this mechanism.

4. Acoustic energy is absorbed by the viscoelastic solid propellant, insulator, and the motor case; its magnitude is difficult to estimate.

The propellant characteristics have a strong effect on the susceptibility to instability. Changes in the binder, particle-size distribution, ratio of oxidizer to fuel, and burn-rate catalysts can all affect stability, often in ways that are not predictable. All solid propellants can experience instability. As a part of characterizing a new or modified propellant (e.g., determining its ballistic, mechanical, aging, and performance characteristics), many companies now also evaluate it for its stability behavior, as described below.

Analytical Models and Simulation of Combustion Stability

Many interesting investigations have been aimed at mathematical models that will simulate the combustion behavior of solid propellants. This was reviewed by T'ien in Chapter 14 of Ref 13-1.

Using complex algorithms and computers it has been possible to successfully simulate the combustion for some limited cases, such as for validating or extrapolating experimental results or making limited predictions of the stability of motor designs. This applies to well-characterized propellants, where empirical constants (such as propellant response or particle-size distribution) have been determined and where the range of operating parameters, internal geometries, or sizes has been narrow. The analytical methods used to date have by themselves not been satisfactory to a motor designer. It is unlikely that a reliable simple analysis will be found for predicting the occurrence, severity, nature, and location of instability for a given propellant and motor design. The physical and chemical phenomena are complex, multidimensional, unsteady, nonlinear, influenced by many variables, and too difficult to emulate mathematically without a good number of simplifying assumptions. However, theoretical analysis gives insight into the physical phenomena, can be a valuable contributor to solving instability problems, and has been used for preliminary design evaluation of grain cavities.

Combustion Stability Assessment, Remedy, and Design

In contrast with liquid rocket technology, an accepted combustion stability rating procedure does not now exist for full-scale solid rockets. Undertaking stability tests on large full-scale flight-hardware rocket motors is expensive, and therefore lower-cost methods, such as subscale motors, T-burners, and other test equipment, have been used to assess motor stability.

The best known and most widely used method of gaining combustion stability-related data is the use of a T-burner, an indirect, limited method that does not use a full-scale motor. Figure 13-7 is a sketch of a standard T-burner; it has a 1.5-in. internal diameter double-ended cylindrical burner vented at its midpoint (see Refs. 13-17 to 13-19). Venting can be through a sonic nozzle to the atmosphere or by a pipe connected to a surge tank which maintains a constant level of pressure in the burner cavity. T-burner design and usage usually concentrate on the portion of the frequency spectrum dealing with the transverse oscillations expected in a full-scale motor. The desired acoustical frequency, to be imposed on the propellant charge as it burns, determines the burner length (distance between closed ends).

The nozzle location, midway between the ends of the burner, minimizes attentuation of fundamental longitudinal mode oscillations (in the propellant grain cavity). Theoretically, an acoustic pressure node exists at the center and antinodes occur at the ends of the cavity. Acoustic velocity nodes are out of phase with pressure waves and occur at the ends of the burner. Propellant charges are often in the shape of discs or cups cemented to the end faces of the burner. The gas velocity in the burner cavity is kept intentionally low (Mach 0.2 or less) compared with the velocity in a full-scale motor. This practice minimizes the influence of velocity-coupled energy waves and allows the influence of pressure-coupled waves to be more clearly recognized.

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