Matrix Analysis of Longitudinal and Torsional Vibrations in Nonuniform Multibranch Beams

Since longitudinal modes and frequencies provide basic data for dynamic analyses of arbitrary beam-like structures and since closed-form solutions for the modes are generally not feasible to obtain, an approximate method is developed for computing the natural frequencies and the corresponding mode shapes for a variable-section, unconstrained multibranch beam. A lumped mass analogy employing influence coefficients is used to represent the beam. The simultaneous equations of motion for the lumped mass system are derived in matrix form and algebraically manipulated to yield a classical eigenvalue equation solvable by standard procedures. The orthogonality relationship of the natural modes is derived and used to form the basis of an orthogonal sweeping process for determination of modes above the fundamental. Numerical examples including an application to a solid-fuel launch system are presented. Also, a detailed discussion is devoted to the theoretical verifications of the approximate modes and frequencies.