A theoretical and experimental approach was used to investigate the motion and effectiveness of a Self-Compensating Dynamic Balancer (SCDB). This is a device intended to minimize the effects of rotor imbalance and vibratory forces on a rotating system during normal operation. The basic concept of an automatic dynamic balancer has been described in many U.S. patents. The SCDB is composed of a circular disk with a groove containing massive balls and a low viscosity damping fluid. The objective of this research is to determine the motion of the balls and how this ball motion is related to the vibration of the rotating system using both theoretical and experimental methods. The equations of motion the balls were derived by the Lagrangian method. Static and dynamic solutions were derived from the analytic model. To consider dynamic stability of the motion, perturbation equations were investigated by two different methods: Floquet theory and direct computer simulation. On the basis of the results of the stability investigation, ball positions which result in a balance system are stable above the critical speed and unstable at critical speed and below critical speed. To determine the actual critical speed of the rotating system used in the experimental work, a modal analysis was conducted. Experimental results confirm the predicted ball positions. Based on the theoretical and experimental results, when the system operates below and near the first critical speed, the balls do not balance the system. However, when the system operates above the first critical speed the balls can balance the system.

The acoustic boundary-layer structure is investigated in a cylindrical tube where steady sidewall injection is imposed upon an oscillatory ow. Culick's steady, rotational, and inviscid solution is assumed for the mean ow. The time-dependent velocity is obtained by superimposing the acoustic (compressible, inviscid, irrotational) and the vortical (incompressible, viscous, rotational) velocity vectors. A multiplescales perturbation technique that utilizes proper scaling coordinates is applied to the axial momentum equation by retaining the viscous terms and ignoring the axial convection of vorticity. A closed-form expression for the time-dependent axial velocity is derived that agrees well with the corresponding numerical solution, cold-ow experimental data, and Flandro's near-wall analytic expression. A similarity parameter that controls the thickness of the rotational region is identi ed. The role of the Strouhal number in controlling the wavelength of rotational waves is established. An accurate assessment of the amplitude and phase relation between unsteady velocity and pressure components is obtained. Increasing viscosity is found to reduce the depth of penetration of the rotational region. Nomenclature a 0 = mean chamber speed of sound, m/s f = oscillation mode frequency, Hz k = dimensionless wave number or frequency, vR/a 0 L = internal tube length M b = blowing Mach number, V b/ a 0 P 0 = mean chamber pressure p = dimensionless pressure, p*/P 0 R = effective radius, volume/half of porous area, m Re = Reynolds number based on sound speed, a 0 R/n Re a = acoustic Reynolds number, k/d 2 = vR 2 /n = 2 2 R /d s r = dimensionless radial position, r*/R r 1 = radial scale, magni ed or compressed S p = penetration number, = 3 2 2 2 2 3 2 2 2 1 2 1St = Strouhal number, k/M b = vR/Vb t = dimensionless time, t*a 0/ R =t/k U = Culick's steady ow velocity vector, (U r , U z ) U r = Culick's steady radial velocity, 2r 2 1 sin u u = dimensionless velocity, u*/a0 u 9 z = acoustic velocity, sin(kz)exp(ikt) V b = injection velocity at the porous boundary, m/s Y = penetration control parameter, S p

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A model of the time-dependent velocity field in solid rocket motors is derived analytically for an oscillatory field that is subject to steady sidewall injection. The oscillatory pressure amplitude is assumed to be small by comparison to the mean pressure. The mathematical approach includes solving the momentum equation governing the rotational flow using separation of variables and multiple-scales. This requires identifying scales at which unsteady inertia, convection, and diffusion are significant. A composite scale is obtained that combines three disparate scales. The timedependent axisymmetric solution obtained incorporates the effects of unsteady inertia, viscous diffusion, and the radial and axial convection of unsteady vorticity by Culick's mean flow components.The resulting agreement with the numerical solution to the momentum equation is remarkable. The uncertainty in a short analytical expression is found to be smaller than the injection Mach number, which represents the error associated with the mathematical model itself. The multiple-scales solution agrees extremely well with Flandro's recent flowfield solution. The present solution has the advantage of being shorter, more manageable in extracting quantities of interest, and capable of showing the significance of physical parameters on the solution. Nomenclature a Q = mean stagnation sound speed, ^/p 0 / p 0 A p = dimensional oscillatory pressure amplitude k m = wave number, mnR IL = 0) 0 R I a 0 L = internal chamber length M h -wall injection Mach number, V b I a 0 Po = mean chamber pressure, p 0

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