Design of optimum beam flexural damping in a missile by application of root-locus techniques

The control of a missile in free-fall in the presence of external disturbances requires that the control system be stable when various body bending modes are excited. It is an inevitable consequence of the internal and external missile environment that all body bending modes are excited. The general method of stabilizing a missile is to choose a minimum of inertial sensors such as rate and attitude gyros, and through the use of passive electrical networks in the information paths, generate stabilizing poles and zeros in the control system open-loop transfer functions. Generally, the electrical shaping networks are approximations to "maximally" flat functions with the cutoff frequency below the first bending mode resonant frequency, and in extreme cases, a notch filter is needed to eliminate body bending information. Tae basic purpose of the electrical shaping networks is to attenuate the high signal frequencies circulating in the control loop without the loss of important low frequency data. In some applications the use of passive shaping networks becomes very difficult because of the large attenuation required near the first body bending resonant frequency and the required amplitude flatness at lower frequencies. If passive shaping cannot be used, then active networks must be substituted. This is an undesirable complexity. This paper presents an optimum design procedure to utilize the characteristic properties of completed airframe and missile structures to acquire a maximum increase in modal damping without auxiliary equipment and further autopilot complexity by the proper choice of the subassembly attaching fixtures. Completed airframes and missiles are composed of a very lightweight integral structure in which a large number of equipment sub-assemblies are attached. The sub-assemblies or equipment modules are generally massive and of small volume so that they act as a rigid mass attached to the lightweight integral structure via an equivalent lateral spring constant and viscous damper. For example, the guidance package and the rocket engine of a ballistic missile are usually attached fore and aft of the missile integral tank structure. Each of these elements can be quite rigid and will act as a separate mass from the integral tank structure. The proper choice of the equivalent lateral spring constant and viscous damper of the attaching fixtures that hold the sub-assemblies to the basic lightweight missile or airframe integral structure will result in an optimum increase in modal damping for a particular beam-flexural mode of the completed structure. To determine the values of the proper equivalent spring constant and viscous coefficient, the dynamic principles underlying the Frahm damper are used by extending the analysis of the Frahm damper concept to include a general linear flexural beam which satisfies a linear partial differential equation which can be solved by the product of the solutions of two ordinary, linear differential equations. The extension of these principles is accomplished by deriving the solution of the attached sub-assembly to the integral structure through normal servomechanism analysis. This analysis uses the analytical relationships of the Evans or Root-Locus method.