Longitudinal vibration frequencies of steadily whirling rods

Abstract

Past researchers suggested that "static instabilities" exist at certain rotational speeds of whirling rods. This paper shows these instabilities are an artefact of the material constitutive laws that are being used well outside their range of applicability. An alternative approach is developed where strains due to rotation are separated from the superimposed vibration. This enables the generally predicted lowering of longitudinal natural frequencies with rotational speed shown to be simply a result of the bulk changes in the geometry of whirling rods. Steady-state equations of whirling rods are formulated in Lagrangian coordinates. Due to the nonlinear nature of the governing equations, an original numerical method is applied to solve the problem. Numerical results are compared with analytical results obtained from the linearized uniaxial model. There is close agreement between these two models at low angular velocities. However, at high angular velocities, discrepancies between them arise, confirming that the nonlinear strain-displacement relationship has significant effect on the results and the inferred "static instabilities." This approach first solves the "static" problem of the deformed geometry of a highly strained whirling rod before longitudinal natural modes are determined by classical methods. Furthermore, conditions for existence and uniqueness of solutions are derived.

Dynamic motion of whirling rods with Coriolis effect

“…[1], although their proof did not mention about the subintervals division when a Taylor series representation is not valid in the whole domain of interval ½0; L.…”

“…However, only a small number of researchers had attempted to formulate and solve rotating rods or beams problems coupled with Coriolis effect. Recently, results obtained by Shum and Entwistle [1] showed that natural frequencies of axial vibration were decreasing for an increasing angular velocity of steadily whirling rods without Coriolis effect by applying receptance approach. More previously, Bhuta and Jones [2] analysed the axial vibration of a rotating bar problem under the small strain assumption.…”

Static resonance in rotating nanobars

“…In those analyses (Bhuta and Jones, 1963;Brunelle, 1971) Lagrangian coordinates were used. Shum and Entwistle (2006) reported that the linear uniaxial model is not representative for the situation at larger strains due to higher angular velocity. The axial deformation of rotating rods was investigated (Hodges and Bless, 1994) by using two simpler nonlinear strain energy models.…”

Complete function collocation method for solving preliminary cable-stayed bridge pretension forces

“…The present study attempts to solve the more complicated practical engineering problem of calculating the reference pretension forces of a cable-stayed bridge. A simple two-dimensional cable-stayed bridge is set up and a collocation method based on the complete function solution previously developed by Shum and Entwistle (2006) is used to calculate the stay cable forces.…”