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An outline is presented of the analysis, engineering study, and theoretical investigation of the steady-state vibrations and the sudden-impulse-shock dynamic characteristics of a new military airborne gimbaled system. A mathematical model is described which consists of three mathematical regimes--each defining the distribution of dynamic masses and motions for one plane of excitation. Comprehensive recursion tables for the three regimes were filled out by the high-speed No. 704 Digital Computer. Dynamic-equation matrix coefficients were determined for each frequency. Where the error function was zero, the recursion tables were filled out to determine true mode shapes and distributions. Additional programming determined forcing-function responses and, through the use of open parameters, determined the effects of modifying masses and other elements of the system. Aerodynamic loads for extreme flight conditions were rigorously investigated, and margins-of-safety were determined. It is reported that this mathematical method led to recommendations for weight savings, as well as improvement in the dynamic characteristics of the system.