Frequency Model of Fixed-Ends Collinear System with Two Flexible Members and One Rigid Connector by Lumped-Parameter, Compliance-Based Matrix Method

A new lumped-parameter matrix method is proposed to model the decoupled, in-plane longitudinal and transverse free undamped vibrations of a collinear system with fixed ends and formed of two end flexible and prismatic members linked by a middle rigid connector. The method calculates the natural frequencies associated with the system’s three degrees of freedom by solving a linear algebraic characteristic equation related to the dynamic matrix, which is obtained from the system compliance and mass matrices. The linear, small-displacement model characterizes either long or short beams by adequately formulating the compliance and mass matrices. The lumped-parameter model is comprehensively validated by two separate distributed-parameter models, which determine the system’s longitudinal-vibration and long-beam, bending-vibration natural frequencies. Numerical simulations are performed with the lumped-parameter model to identify the sensitivity of the natural frequencies to system parameters variations and model variants. The system’s matrices are also utilized to perform frequency-domain analysis of the three-member system in a displacement/acceleration sensing application. The method can be adapted and expanded to describe more complex configurations with multiple, non-collinear, and non-prismatic members.