Particularities of Newtonʹs method in space frame force determination, utilising eigenpair functions

Redundant space frame dynamics are modelled through the structural eigenproblem, which is formulated to account for the geometric stiffness due to self-equilibrating membrane forces. The resulting mathematical eigenvalues and eigenvectors, respectively indicative of natural frequencies and modes of oscillation, are then assumed to be continuous functions of a scalar factor upon a normalised set of forces in equilibrium. Newtonʹs method—or sensitivity analysis—provides a means for minimising the difference between mathematical and physically observed eigenvalues, whereupon the axial forces are inferred from the model. Issues of convergence and root uniqueness are anticipated. Eigenvalue coalescence, in which eigenvalues transitorily coalesce to permute their arbitrarily numerical ordering, is seen to define non-smooth eigenvalue functions and hence encumber Newtonʹs method; mode tracing strategies to overcome this are discussed. The existence of various degrees of eigenvalue degeneracy, owing to frame cyclic symmetry or otherwise, is anticipated in terms of the influence upon Newtonʹs method. If the degree of frame static redundancy is greater than one and there exist a number of linearly independent force distributions, then the required force determination is a multidimensional Newton method. Problems encountered in these manifold dimensions, including those arising from frame spatial periodicity and the adversity of Newtonʹs method, are overcome. Emphasis is placed upon minimising the dimensionality of Newtonʹs method through enforcement of equilibrium constraints. Illustrative numerical simulations are given and some applications of the method are proposed.