Exact natural frequencies of multi-level elastically connected taut strings and related problems

The dynamics of a family of simple, but extremely useful structural elements is governed by a second order Sturm-Liouville equation. This equation allows for the uniform distribution of mass and stiffness and enables the motion of strings and shear beams, together with the axial and torsional motion of bars to be described exactly. As a result, each member type in this family has been treated exhaustively when considered as a single member or when joined contiguously to others. However, when such members are linked in parallel by uniformly distributed elastic interfaces, their complexity becomes significantly more intractable and it is this class of structures that has led to renewed interest and which forms the basis of the work that follows. Initially, differential equations governing the coupled motion of the system are developed from first principles. These are organised into the form of a generalised linear symmetric eigenvalue problem, from which a family of uncoupled differential operators can be established. These operators define a series of exact substitute systems that together describe the complete motion of the original structure. These equations can then be used in either of two ways. In their most powerful form they can be developed into exact dynamic stiffness matrices that enable all the powerful features of the finite element method to be utilised. This subsequently enables sets of members carrying point masses and subject to point spring supports to be analysed easily. Alternatively, the equations are able to yield an exact relational model that links any uncoupled frequency of an original member to the corresponding set of coupled system frequencies. This approach enables ‘back of the envelope calculations’ to be undertaken quickly and efficiently. The exact mode shapes of the original structure can be recovered in either case. Due to space limitations, only aspects of the first technique are described briefly herein, but both are covered exhaustively elsewhere [1]

Eigenvalues and eigenvectors of a system of Bernoulli Euler beams connected together in a tree topology

Eigenvalues and eigenvectors of a system of Bernoulli Euler beams connected together in a tree topolog

Exact eigensolution of a class of multi-level elastically connected members

Attention is given to determining the exact natural frequencies and modes of vibration of a class of structures comprising any number of related parallel members that are connected to each other, and possibly also to foundations, by uniformly distributed elastic interfaces of unequal stiffness. The members themselves are considered to have a uniform distribution of mass and stiffness and account can be taken of additional point masses and spring supports. The formulation is general and applies to any structure in which the motion of the component members is governed by a second order Sturm-Liouville equation. Closed form solution of the governing differential equations leads either, to a series of exact substitute systems that are easy to solve through a stiffness approach and which together yield the complete spectrum of natural frequencies and corresponding mode shapes of the original structure, or to simple exact relationships between the natural frequencies corresponding to coupled and uncoupled motion that enable hand solution of the more standard problems to be achieved. An appropriate form of the Wittrick-Williams algorithm is presented for converging on the required natural frequencies to any desired accuracy with the certain knowledge that none have been missed. Examples are given to confirm the accuracy of the approach and to indicate its range of application

On the eigensolution of elastically connected columns

On the eigensolution of elastically connected column

Homogeneous trees of second order Sturm-Liouville equations: a general theory and program

Quantum graph problems occur in many disciplines of science and engineering and they can be solved by viewing the problem as a structural engineering one. The Sturm–Liouville operator acting on a tree is an example of a quantum graph and the structural engineering analogy is the axial vibration of an assembly of bars connected together with a tree topology. Using the dynamic stiffness matrix method the natural frequencies of the system can be determined which are analogous to the eigenvalues of the quantum graph. Theory is presented that yields exact solutions to the Sturm–Liouville problem on homogeneous trees. This is accompanied by an extremely efficient and compact computer program that implements the theory. An understanding of the former is enhanced by recourse to a structural mechanics analogy, while the latter program is fully annotated and explained for those who might wish to extend its capability. In addition, the use of the program as a ‘black box’ is fully described and a small parametric study is undertaken to confirm the accuracy of the approach and indicate its range of application including the computation of negative eigenvalues

On the provenance of hinged-hinged frequencies in Timoshenko beam theory

An exact differential equation governing the motion of an axially loaded Timoshenko beam supported on a two parameter, distributed foundation is presented. Attention is initially focused on establishing the provenance of those Timoshenko frequencies generated from the hinged-hinged case, both with and without the foundation being present. The latter option then enables an exact, neo-classical assessment of the ‘so called’ two frequency spectra, together with their corresponding modal vectors, to be undertaken when zero, tensile or compressive static axial loads are present in the member. An alternative, ‘precise’ approach, that models Timoshenko theory efficiently, but eliminates the possibility of a second spectrum, is then described and used to confirm the original eigenvalues. This leads to a definitive conclusion regarding the structure of the Timoshenko spectrum. The ‘precise’ technique is subsequently extended to allow, either the full foundation to be incorporated, or either of its component parts individually. An illustrative example from the literature is solved to confirm the accuracy of the approach, the nature of the Timoshenko spectrum and a wider indication of the effects that a distributed foundation can have

A small temperature rise may contribute towards the apparent induction by microwaves of heatshock gene expression in the nematode Caenorhabditis elegans

We have previously reported that low-intensity microwave exposure (0.75-1.0 GHz CW at 0.5 W; SAR 4-40 mW kg-1) can induce an apparently non-thermal heat-shock response in Caenorhabditis elegans worms carrying hsp16-1::reporter genes. Using matched copper TEM cells for both sham and exposed groups, we can detect only modest reporter induction in the latter (15-20% after 2.5 h at 26°C, rising to ~50% after 20 h). Traceable calibration of our copper TEM cell by the National Physical Laboratory (NPL) reveals significant power loss within the cell (8.5% at 1.0 GHz), accompanied by slight heating of exposed samples (~0.3°C at 1.0 W). Thus exposed samples are in fact slightly warmer (by ≤0.2°C at 0.5 W) than sham controls. Following NPL recommendations, our TEM cell design was modified with the aim of reducing both power loss and consequent heating. In the modified silver-plated cell, power loss is only 1.5% at 1.0 GHz, and sample warming is reduced to ~ 0.15°C at 1.0 W (i.e. ≤ 0.1°C at 0.5 W). Under sham:sham conditions, there is no difference in reporter expression between the modified silverplated TEM cell and an unmodified copper cell. However, worms exposed to microwaves (1.0 GHz and 0.5 W) in the silver-plated cell also show no detectable induction of reporter expression relative to sham controls in the copper cell. Thus the 20% “microwave induction” observed using two copper cells may be caused by a small temperature difference between sham and exposed conditions. In worms incubated for 2.5 h at 26.0, 26.2 and 27.0°C (with no microwave field), there is a consistent and significant increase in reporter expression between 26.0 and 26.2°C (by ~20% in each of 6 independent runs), but paradoxically expression levels at 27.0°C are similar to those seen at 26.0°C. This surprising result is in line with other evidence pointing towards complex regulation of hsp16-1 gene expression across the sub-heat-shock range of 25-27.5°C in C. elegans. We conclude that our original interpretation of a non-thermal effect of microwaves cannot be sustained; at least part of the explanation appears to be thermal

MicroRNA-29 specifies age-related differences in the CD8+ T cell immune response

MicroRNAs (miRNAs) have emerged as critical regulators of cell fate in the CD8+ T cell response to infection. Although there are several examples of miRNAs acting on effector CD8+ T cells after infection, it is unclear whether differential expression of one or more miRNAs in the naive state is consequential in altering their long-term trajectory. To answer this question, we examine the role of miR-29 in neonatal and adult CD8+ T cells, which express different amounts of miR-29 only prior to infection and adopt profoundly different fates after immune challenge. We find that manipulation of miR-29 expression in the naive state is sufficient for age-adjusting the phenotype and function of CD8+ T cells, including their regulatory landscapes and long-term differentiation trajectories after infection. Thus, miR-29 acts as a developmental switch by controlling the balance between a rapid effector response in neonates and the generation of long-lived memory in adults