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Repetitive beam-like structures: Distributed loading and intermediate support
No full textTheory is developed for the elastostatic transfer matrix analysis of a repetitive beam-like structure subject to point-wise distributed loading, with and without an intermediate support; this complements previous work in which loading is applied at one end only and reacted at the other. State-vectors are
expressed in terms of participation coefficients of the eigen- and principal vectors, leading to the simplest description of the evolution of nodal displacement and force components as one moves along the structure, in terms of powers of the Jordan canonical form. Inaccuracy due to ill-conditioning is explained in
terms of powers of greater than unity eigenvalue
On state-space elastostatics within a plane stress sectorial domain: the wedge and the curved beam
No full textThe plane stress sectorial domain is analysed according to a state-space formulation of the linear theory of elasticity. When loading is applied to the straight radial edges (flanks), with the circular arcs free of traction, one has the curved beam; when loading is applied to the circular arcs, with the flanks free of traction, one has the elastic wedge. A complete treatment of just one problem (the elastic wedge, say) requires two state-space formulations; the first describes radial evolution for the transmission of the stress resultants (force and moment), while the second describes circumferential evolution for determination of the rates of decay of self-equilibrated loading on the circular arcs, as anticipated by Saint-Venant’s principle. These two formulations can be employed subsequently for the curved beam, where now radial evolution is employed for the Saint-Venant decay problem, and circumferential evolution for the transmission modes. Power-law radial dependence is employed for the wedge, and is quite adequate except for treatment of the so-called wedge paradox; for this, and the curved beam, the formulations are modified so that ln r takes the place of the radial coordinate r. The analysis is characterised by a preponderance of repeating eigenvalues for the transmission modes, and the state-space formulation allows a systematic approach for determination of the eigen- and principal vectors. The so-called wedge paradox is related to accidental eigenvalue degeneracy for a particular angle, and its resolution involves a principal vector describing the bending moment coupled to a decay eigenvector. Restrictions on repeating eigenvalues and possible Jordan canonical forms are developed. Finally, symplectic orthogonality relationships are derived from the reciprocal theorem
On a check on the accuracy of Timoshenko's beam theory
Full text linkIn a recent article, Rneton (1) presented a comparison between the standing wave natural frequency predictions of Timoshenko beam theory (TBT) for a long beam of thin rectangular cross-section, when the flexural mode is sinusoidal in the axial co-ordinate x, and those of a plane stress elastodynamic solution; the latter may be regarded as the exact benchmark. Renton's main concern was the accuracy of the lower of the two frequency predictions of TBT (TBT1), although the results presented in Figure 1 of reference (1) also show the frequency prediction of the so-called second spectrum (TBT2), and re-presents evidence that this mode should be disregarded
Transfer matrix analysis of the elastostatics of one-dimensional repetitive structures
Full text linkTransfer matrices are used widely for the dynamic analysis of engineering structures, increasingly so for static analysis, and are particularly useful in the treatment of repetitive structures for which, in general, the behaviour of a complete structure can be determined through the analysis of a single repeating cell, together with boundary conditions if the structure is not of infinite extent. For elastostatic analyses, non-unity eigenvalues of the transfer matrix of a repeating cell are the rates of decay of self-equilibrated loading, as anticipated by Saint-Venant's principle. Multiple unity eigenvalues pertain to the transmission of load, e.g. tension, or bending moment, and equivalent (homogenized) continuum properties, such as cross-sectional area, second moment of area and Poisson's ratio, can be determined from the associated eigen- and principal vectors. Various disparate results, the majority new, others drawn from diverse sources, are presented. These include calculation of principal vectors using the Moore–Penrose inverse, bi- and symplectic orthogonality and relationship with the reciprocal theorem, restrictions on complex unity eigenvalues, effect of cell left-to-right symmetry on both the stiffness and transfer matrices, eigenvalue veering in the absence of translational symmetry and limitations on possible Jordan canonical forms. It is shown that only a repeating unity eigenvalue can lead to a non-trivial Jordan block form, so degenerate decay modes cannot exist. The present elastostatic analysis complements Langley's (Langley 1996 Proc. R. Soc. A452, 1631–1648) transfer matrix analysis of wave motion energetics
How long is a curved beam?
Full text linkFor a straight or thin curved beam, the expression for strain energy due to bending is U = M2 L/(2EI); for this to be applicable to a thick curved beam, the requisite length is slightly greater than the centre-line length
On energy harvesting from ambient vibration
No full textFuture MEMS devices will harvest energy from their environment. One can envisage an autonomous condition monitoring vibration sensor being powered by that same vibration, and transmitting data over a wireless link; inaccesible or hostile environments are obvious areas of application. The base excitation of an elastically mounted magnetic seismic mass moving past a coil, considered previously by several authors, is analysed in detail. The amplitude of the seismic mass is limited in any practical device and this, together with the magnitude and frequency of the excitation define the maximum power that can be extracted from the environment. The overall damping coefficient (part of which is mechanical) is associated with the harvesting and dissipation of energy and also the transfer of energy from the vibrating base into the system
On veering of eigenvalue loci
No full textEigenvalue veering is studied in the context of two simple oscillators coupled by a (presumed weak) spring, variants of which have been considered by several authors. The concept of a center of veering is introduced, leading to a coordinate translation; a subsequent coordinate rotation, dependent on the degree of asymmetry of the system, reduces the frequency equaiton to a standard north-south opening hyperbola. Thus veering occurs even when coupling is strong, and may be characterized by these coordinate transformations and geometric features of the hyperbola, rather than eigenvalue and eigenvector derivatives
Macaulay's method for a Timoshenko beam
No full textThe Macaulay bracket notation is familiar to many engineers for the deflection analsyis of a Euler-Bernoulli beam subject to multiple or discontinuous loads. An expression for the internal bending moment, and hence curbature, is valid at all locations along the beam, and the deflection curve can be calculated by integrating twice with respect to the axial coordinate. The notation obviates the need for matching of multiple constants of integration for the various sections of the beam. Here, the method is extended to a Timoshenko beam, which includes the additional deflection due to shear. This requires an additional expression for the shearing force, also valid at all locations along the beam
Letter to the editor. On the vibration of one-dimensional periodic structures
No full textA structure is said to be periodic, or repetitive, when its construction takes the form of a spatially repeated cell; a honeycomb sandwich panel is a good example of a two-dimensional (plate-lie) periodic structure, whilst examples of one-dimensional (beam-like) periodic structures include rail track supported on equi-spaced sleepers, and trusses employed to provide a large span. A review of the various approaches to their analysis was given by Noor (1) in 1988; more recently Mead (2) has provided an overview of the contributions made by researchers at the University of Southampton
Discussion: "Shear Coefficients for Timoshenko Beam Theory" (Hutchinson, Jr.R., 2001, ASME J. Appl. Mech., 68, pp.87-92)
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