The invariant manifold of beam deformations

Abstract

The subcentre invariant manifold of elasticity in a thin rod may be used to give a rigorous and appealing approach to deriving one-dimensional beam theories. Here I investigate the analytically simple case of the deformations of a perfectly uniform circular rod. Many, traditionally separate, conventional approximations are derived from within this one approach. Furthermore, I show that beam theories are convergent, at least for the circular rod, and obtain an accurate estimate of the limit of their validity. The approximate evolution equations derived by this invariant manifold approach are complete with appropriate initial conditions, forcing and, in at least one case, boundary conditions.

Modelling the Dynamics of Turbulent Floods

“…We have proven: if the boundary conditions (36)(37)(38)(39)(40)(41)(42)(43) are satisfied, then, except for the four-fold eigenvalue zero whose eigenfunctions span M 0 , the eigenvalues of L are negative and bounded away from 0. Thus we expect the manifold (35) is locally attractive.…”

On the low-dimensional modelling of Stratonovich stochastic differential equations

“…One of the remarkable features of the preceding analysis is the appearance of many noise processes in the low-dimensional model (18). In this example there are 8 new noises up to fifth order; in interesting physical examples there would be many more corresponding to each of the neglected transient modes (see (28-29) in [6] for example).…”

“…Centre manifold theory is increasingly recognised as providing a rational route to the low-dimensional modelling of high-dimensional dynamical systems. Applications of the techniques have ranged over, for example, triple convection [1], feedback control [3], economic theory [5], shear dispersion [11], nonlinear oscillations [22], beam theory [18], flow reactors [2], and the dynamics of thin fluid films [19]. New insights given by the centre manifold picture enable one to not only derive the dynamical models, but also to provide accurate initial conditions [15,8], boundary conditions [17], and, particularly relevant to this paper, the treatment of forcing [7].…”

Low-dimensional modelling of dynamics via computer algebra

“…Such low-dimensional dynamical models are significantly easier to analyse, simulate and understand. Applications of the techniques have ranged over, for example, triple convection [1], feedback control [3], economic theory [7], shear dispersion [17,18], nonlinear oscillations [26], beam theory [23], flow reactors [2], and the dynamics of thin fluid films [24]. New insights given by the centre manifold picture enable one to not only derive the dynamical models, but also to provide accurate initial conditions [21,10], boundary conditions [22], and the treatment of forcing [9].…”

“…The only places where we need to explicitly consider these rules are in the terms in the equations for the corrections, u ′ , v ′ , and p ′ . However, these only involve y derivatives, see (23)(24)(25)(26)(27), which simply transform ∂ ∂y → 1 η ∂ ∂ζ . Thus, we multiply the residuals on the right-hand sides of these equations by the appropriate power of η, as seen in lines 40, 47 and 53 of the following reduce program.…”