A hierarchy of dynamic equations for solid isotropic circular cylinders

Abstract

This work considers homogeneous isotropic circular cylinders adopting a power series expansion method in the radial coordinate. Equations of motion together with consistent sets of end boundary conditions are derived in a systematic fashion up to arbitrary order using a generalized Hamilton's principle. Time domain partial differential equations are obtained for longitudinal, torsional, and flexural modes, where these equations are asymptotically correct to all studied orders. Numerical examples are presented for different sorts of problems, using exact theory, the present series expansion theories of different order, and various classical theories. These results cover dispersion curves, eigenfrequencies and the corresponding displacement and stress distributions, as well as fix frequency motion due to prescribed end displacement or lateral distributed forces. The results illustrate that the present approach may render benchmark solutions provided higher order truncations are used, and act as engineering cylinder equations using low order truncation.

Dynamic equations for solid isotropic radially functionally graded circular cylinders

“…The convergence rate resembles the torsional case, where the eigenfrequencies are slightly higher for the axisymmetric case as expected [40,41]. The pattern that a certain truncation order is needed in order to influence the eigenfrequencies as the power order p is increased is also seen here.…”

“…It should be noted that it is possible to handle other boundary conditions numerically, e.g. free or clamped ends [41], albeit the simply supported case is the simplest.…”

“…It is clear from these two tables that rather high truncation orders are needed to render accurate results; this differ considerably from the homogeneous case where accurate results are obtained already for the first few truncations orders [41]. As the power order p is increased, more terms are needed to render accurate results due to the more complex material variation.…”

“…Consider prescribed end conditions where either displacements {û r ,û θ ,û z } or tractions {t r ,t θ ,t z } or combinations thereof are to be stated on z = {0, L}. Variationally consistent end boundary conditions are obtained adopting a generalized Hamilton principle described previously for homogeneous cylinders in [40,41]. The method is based on stating virtual displacements and tractions, and using integrations over the end cross sections for each Fourier mode as…”

“…The present work on FG cylinders is an extension to earlier work adopting a power series approach for homogeneous cylinders [38,40,41]. The method is based on the three dimensional elastodynamic theory for a cylinder with radially varying material properties.…”