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NASTRAN nonlinear vibration analysis of beam and frame structures
A capability for the nonlinear vibration analysis of beam and frame structures suitable for use with NASTRAN level 15.5 is described. The nonlinearity considered is due to the presence of axial loads induced by longitudinal end restraints and lateral displacements that are large compared to the beam height. A brief discussion is included of the mathematical analysis and the geometrical stiffness matrix for a prismatic beam (BAR) element. Also included are a brief discussion of the equivalent linearization iterative process used to determine the nonlinear frequency, the required modifications to subroutines DBAR and XMPLBD of the NASTRAN code, and the appropriate vibration capability, four example problems are presented. Comparisons with existing experimental and analytical results show that excellent accuracy is achieved with NASTRAN in all cases
Categorified Symplectic Geometry and the String Lie 2-Algebra
Multisymplectic geometry is a generalization of symplectic geometry suitable
for n-dimensional field theories, in which the nondegenerate 2-form of
symplectic geometry is replaced by a nondegenerate (n+1)-form. The case n = 2
is relevant to string theory: we call this 2-plectic geometry. Just as the
Poisson bracket makes the smooth functions on a symplectic manifold into a Lie
algebra, the observables associated to a 2-plectic manifold form a "Lie
2-algebra", which is a categorified version of a Lie algebra. Any compact
simple Lie group G has a canonical 2-plectic structure, so it is natural to
wonder what Lie 2-algebra this example yields. This Lie 2-algebra is
infinite-dimensional, but we show here that the sub-Lie-2-algebra of
left-invariant observables is finite-dimensional, and isomorphic to the already
known "string Lie 2-algebra" associated to G. So, categorified symplectic
geometry gives a geometric construction of the string Lie 2-algebra.Comment: 16 page
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