11,996 research outputs found
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 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
The second spectrum of Timoshenko beam theory - Further assessment
No full textA review of contributions and views on the second spectrum of Timoshenko beam theory (TBT) over the past two decades, together with some new results, are presented. It is shown that the Timoshenko frequency equation factorises not solely for hinged–hinged end conditions, as is often claimed, but also for guided–guided and guided–hinged; these new cases may be regarded as portions of a multi-span hinged–hinged beam. A higher-derivative Lagrangian that leads directly to the well-known fourth-order Timoshenko beam equation is reviewed. A simple relationship between the so-called Ostrogradski energy and the mechanical energy is derived for hinged–hinged end conditions. It is shown that the Ostrogradski energy is positive for the first spectrum but negative for the second; within some branches of physics, this would be sufficient to conclude that the second spectrum is “unphysical”. A numerical example presented by Levinson and Cooke is re-examined using both TBT and exact plane stress elastodynamic theory. Agreement is excellent for the first spectrum. However, the second spectrum predictions are not in consistent agreement with any single mode of vibration. For long wavelength it is very close to the second asymmetric mode, but as wavelength shortens, it becomes closer to the second symmetric, then the third asymmetric modes. The conclusion remains unchanged: the second spectrum predictions of TBT should be disregarded
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
Coupled tension-torsion vibration of repetitive beam-like structures
No full textEquivalent continuum stiffness properties, derived from the eigenanalysis of a single cell of a planar beam-like repetitive structure, have previously been employed within well-known dynamic theories, such as Euler–Bernoulli and Timoshenko for flexural vibration, suitably modified, to predict natural frequencies of vibration. Here the approach is applied to two structure types that exhibit tension–torsion coupling. The first is modelled on a NASA deployable structure with a cross-section of equilateral triangular form, but with asymmetric triangulation on the faces. The second is a related, more symmetric, structure but with pre-twist. The simplest tension–torsion dynamic theory due to Di Prima is employed, and this is extended to more general end conditions. This combined periodic structure/substitute continuum approach provides excellent agreement with predictions from the finite element method, especially for the lower modes of vibration; typically, agreement is within ±1% for the lowest 8–10 natural frequencies for the longer, 30-cell structures considered here, the majority of these being torsional modes, and within ±1% for the lowest 4–5 modes for the shorter, ten-cell, structures. This level of accuracy is attainable so long as a single wavelength spans 2–3 cells of the repetitive structure
On the valid frequency range of Timoshenko beam theory
No full textThe frequency equation of Timoshenko beam theory factorises for hinged-hinged end conditions, leading to a first and second spectrum of natural frequencies; the latter is largely inaccurate and can be isolated and disregarded. For the majority of other end conditions, when the frequency equation does not factorise, one may think in terms of pseudo-second spectrum contributions arising when evanescent waves become propagating above the cut-off frequency , and it is conjectured that these may have a corrupting effect on the frequency predictions. Comparisons with measured and simulated frequencies lead to the conclusion that Timoshenko predictions above the cut-off frequency should be disregarded for those end conditions for which the frequency equation does not factorise
Perturbation theory and the Rayleigh quotient
No full textThe characteristic frequencies ? of the vibrations of an elastic solid subject to boundary conditions of either zero displacement or zero traction are given by the Rayleigh quotient expressed in terms of the corresponding exact eigenfunctions. In problems that can be analytically expanded in a small parameter ?, it is shown that when an approximate eigenfunction is known with an error O(?N), the Rayleigh quotient gives the frequency with an error O(?2N), a gain of N orders. This result generalizes a well-known theorem for N=1. A non-trivial example is presented for N=4, whereby knowledge of the 3rd-order eigenfunction (error being 4th order) gives the eigenvalue with an error that is 8th order; the 6th-order term thus determined provides an unambiguous derivation of the shear coefficient in Timoshenko beam theor
Transfer matrix analysis of the elastostatics of one-dimensional repetitive structures
No full textTransfer 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
On the Riccati transfer matrix method for repetitive structures
No full textThe Riccati transfer matrix method is employed in the elastostatic analysis of a repetitive structure subject to various loadings; the eigenvalues of particular terms featuring in the recursive relationships show why the method is numerically stabl
On the Ostrogradski instability for higher-order derivative theories and a pseudo-mechanical energy
No full textA two-degree of freedom spring-mass system, described by two coupled second-order ODEs, is considered from a higher-order one-particle (HOOP) viewpoint. A Lagrangian leading to the single fourth-order differential equation now contains time derivatives greater than the first order. The Hamiltonian constructed according to Ostrogradski's method, traditionally regarded as the conserved energy of the system, is negative for one of the modes of vibration, an attribute that would lead to a conclusion in some branches of physics that the mode is unphysical. Reversing this process, a given fourth-order equation, with no apparent underlying second-order structure, is cast into a coupled second-order form allowing one to construct a pseudo-mechanical energy which, unlike the Ostrogradski Hamiltonian, is always positive. The introduction of a viscous damping element leads to velocity and jerk (third derivative)-dependent terms within the HOOP description. In contrast, a physical realisation leading to an isolated velocity-dependent term in the HOOP description shows the instability to be flutter caused by external excitation at the higher natural frequency, rather than an exchange of energy between positive and negative energy modes
- …
