Normalization and Simplified Solution of Rayleigh Beam Dynamic Systems

To resolve the repeated solution of transcendental equations in the free vibration analysis of Rayleigh beams caused by dependence on multiple physical parameters, a general solution approach based on nondimensionalization is proposed in this paper. By introducing specific time and space restoring coefficients, the dynamic equation is fully nondimensionalized and reduced to a most concise form, in which all coefficients are equal to 1 or −1, yielding a universal vibration equation applicable to Rayleigh beams with arbitrary values of physical parameters. Based on this equation, the frequency equations are derived for three typical boundary conditions (clamped-clamped, simply supported, and clamped-free), and dimensionless frequency-beam length curves are obtained to systematically describe the frequency characteristics under different boundary conditions and modal orders. Further investigation indicates that the effect of rotary inertia on natural frequencies is uniformly characterized by the dimensionless beam length. Accordingly, the critical dimensionless beam lengths separating the applicability of the Euler-Bernoulli and Rayleigh beam models are identified. Finally, based on the dimensionless frequency-beam length curves, a normalized method is proposed for frequency computation, and its accuracy is verified through comparisons with finite element results. The study demonstrates that the proposed method is independent of the specific physical parameters of the beam; variations in system parameters only affect the time and space restoring coefficients, while the dimensionless frequency-beam length curves retain their general applicability to arbitrary parameter changes. The method involves only the solution of a single-variable transcendental equation and linear transformations, thereby avoiding nonlinear iterative procedures associated with multi-parameter transcendental equation systems. Consequently, computational efficiency and numerical stability are significantly improved, and repetitive calculations caused by parameter variations are eliminated. The proposed approach provides an efficient and general tool for the vibration analysis of Rayleigh beams.