Bahra, A.S.; (2006) Newton's method in static force inference from redundant space frame dynamics. Doctoral thesis , University of London.

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Abstract

This thesis is concerned with inferring static, self-equilibrating, axial forces in redundant space frames from knowledge of their natural frequencies and associated mode shapes. Accordingly, it is necessary to have a mathemati cal description of the physical frame in an eigenproblem parameterised with variables accounting for load. Newton's method provides an iterative means of minimising the difference between the eigenvalues and eigenvectors and the measured frequencies and mode shapes they respectively represent forces are thus inferred from the converged eigenproblem. Rather than updating all member forces, models are formulated on force distributions and scalars re lating to the extent of loading form the updating parameters. Enforcing such equilibrium constraints beneficially minimises the order of Newton's method. For multiply redundant frames, it is necessary to formulate the model on a number of force distributions and any state of equilibrium can be described by their linear superposition. The ways in which load affects the dynamic characteristics are investigated thoroughly. Frequencies are shown to coa lesce and exchange places in the spectrum, leading to non-smooth functions since the eigenvalues are numerically ordered. Mode tracing strategies, which utilise eigenvector consistency across coalescence points to conserve function smoothness, are investigated. This consistency, however, is observed to dete riorate if the eigenvalues exhibit veering. Measures facilitating mode tracing when consistency is deficient are explored. Special treatment is required at eigenvalue degeneracy in order to observe eigenpair differentiability, which is necessary for Newton's method. Numerical simulations demonstrate success of force identification in a variety of contexts. Newton's method is effectively applied to identify load in actual, physical frames with single and multiple force distributions. Offset and length parameters supplement load to sta bilise and improve the accuracy of solution. For complicated frames, it is shown that starting iteration in the eigenvector, as well as eigenvalue, neigh bourhood is crucial for convergence to result.

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