In this paper, an approach based on transfer matrix method of linear multibody systems (MS-TMM) is developed to analyze the free vibration of a multilevel beam, coupled by spring/dashpot systems attached to them in-span. The Euler-Bernoulli model is used for the transverse vibration of the beams, and the spring/dashpot system represents a simplified model of a viscoelastic material. MS-TMM reduces the dynamic problem to an overall transfer equation which only involves boundary state vectors. The state vectors at the boundaries are composed of displacements, rotation angles, bending moments, and shear forces, which are partly known and partly unknown, and end up with reduced overall transfer matrix. Nontrivial solution requires the coefficient matrix to be singular to yield the required natural frequencies. This paper implements two novel algorithms based on the methodology by reducing the zero search of the reduced overall transfer matrix’s determinate to a minimization problem and demonstrates a simple and robust algorithm being much more efficient than direct enumeration. The proposal method is easy to formulate, systematic to apply, and simple to code and can be extended to complex structures with any boundary conditions. Numerical results are presented to show the validity of the proposal method against the published literature. 1. Introduction The vibration problem of beam-type structures is of particular urgent issue in many branches of modern aerospace, mechanical, and civil engineering. Natural vibration frequencies and modes are one of the most important dynamic characteristics of these kinds of systems. For example, the precision in manufacturing can be highly influenced by vibrations. If the vibration characteristics cannot be solved or preestimated exactly when designing a mechanism system, it is often hard to obtain a good dynamic performance of the mechanism system and consequently hard to control its vibration. There are different types of beam models. One of the well-known models is the Euler-Bernoulli beam theory that works well for slender beams. According to the Euler-Bernoulli beam theory, the length of each beam section is much greater than the height of each section and the shear and rotary inertia effects are ignored. The vibration theory of single-beam systems is well developed and studied in detail in hundreds of contributions. The vibration of systems composed of uniform double-beam coupled by translational springs or elastic layers have been studied extensively in the literature. Inceo？lu and Gürg？ze [1] studied the bending
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