The effects of distribution of in-span pinned supports and various concentrated elements on the dynamic characteristics of the Timoshenko beam are also studied.
Any trial value of ωj that renders the value of the determinant ∣B∣ vanishes denoting one of the eigenvalues of the uniform circular shaft carrying multiple concentrated elements, and the substitution of the associated integration constants into the relevant eigenfunctions determines the corresponding mode shape.
Second, based on the closed form solution and using the mode superposition technique, the eigenvalue equation of the "constrained" plate (with any number of concentrated elements attached) is derived.
The advantage of the proposed method is that the resulting frequency equation for torsional vibration of multi-step non-uniform rods with arbitrary number of concentrated elements can be conveniently determined from a homogeneous algebraic equation.
An analytical approach and exact solutions for the torsional vibration of a multi-step non-uniform rod carrying an arbitrary number of concentrated elements such as rigid disks and with classical or non-classical boundary conditions is presented.
The boundary (supported) conditions of the plate and the magnitudes and locations of the concentrated elements are arbitrary.
The graph of dynamic response amplitudes versus various exciting frequencies gives the frequency response curve for any point of a multiple-span beam carrying a number of various concentrated elements.
In addition to the major elements identification, e.g. those entering into the composition of pigments [37], it allows probing the localisation of less concentrated elements such as environmental contaminants, degradation products and trace elements that could bring important information on the pigment's history (provenance, production, etc).
This paper employs the numerical assembly method (NAM) to determine the "exact" frequency response amplitudes of a multiple-span beam carrying a number of various concentrated elements and subjected to a harmonic force, and the exact natural frequencies and mode shapes of the beam for the case of zero harmonic force.
The purpose of this paper is to utilize the numerical assembly method (NAM) to determine the exact natural frequencies and mode shapes of the multispan Timoshenko beam carrying a number of various concentrated elements including point masses, rotary inertias, linear springs, rotational springs and spring mass systems.
