Refined Analytical Approximations to Limit Cycles for Non-Linear Multi-Degree-of-Freedom Systems

This paper presents analytical higher order approximations to limit cycles of an autonomous multi-degree-of-freedom system based on an integro-differential equation method for obtaining periodic solutions to nonlinear differential equations. The stability of the limit cycles obtained was then investigated using a method for carrying out Floquet analysis based on developments to extensions of the method for solving Hill's Determinant arising in analysing the Mathieu equation, which have previously been reported in the literature. The results of the Floquet analysis, together with the limit cycle predictions, have then been used to provide some estimates of points on the boundary of the domain of attraction of stable equilibrium points arising from a sub-critical Hopf bifurcation. This was achieved by producing a local approximation to the stable manifold of the unstable limit cycle that occurs. The integro-differential equation to be solved for the limit cycles involves no approximations. These only arise through the iterative approach adopted for its solution in which the first approximation is that which would be obtained from the harmonic balance method using only fundamental frequency terms. The higher order approximations are shown to give significantly improved predictions for the limit cycles for the cases considered. The Floquet analysis based approach to predicting the boundary of domains of attraction met with some success for conditions just following a sub-critical Hopf bifurcation. Although this study has focussed on cubic non-linearities, the method presented here could equally be used to refine limit cycle predictions for other non-linearity types.Peer reviewedFinal Accepted Versio

A PERSPECTIVE ON STRUCTURAL HEALTH AND USAGE MONITORING IN AEROSPACE APPLICATIONS

There is great interest in the benefits of Structural Health and Usage Monitoring in the Aerospace Industry both from a safety point of view and because of the possibility of extending the life of aerospace structural components. Although fail-safe and damage tolerance approaches to design are extensively used and have great advantages, there are never the less components and circumstances where a safe life approach remains appropriate. This leads to an approach to fatigue clearance whereby a component will be taken out of service after a certain number of hours usage irrespective of the environment it has experienced having been cleared based on very conservative loading assumptions. If the actual loads experienced by critical parts of a structure can be derived from a Structural Health and Usage Monitoring System (SHUMS), this then leads to the possibility of extending the time for which the component can remain in service with consequent cost savings. In this paper, a number of fundamental approaches to loads prediction using data available from a Structural Health and Usage Monitoring Systems are reviewed, with the particular application in mind being that of an air-carried guided weapon. Approaches considered will include time-domain and frequency-domain based methods making use of a structural model, together with machine learning based approaches. Their different strengths, weaknesses and pitfalls will be highlighted together with ways to overcome them. Practical aspects of their possible implementation will also be addressed.Non peer reviewe

The Geometry of the Gibbs-Appell Equations and Gauss' Principle of Least Constraint

We present a generalisation of the Gibbs-Appell equations which is valid for general Lagrangians. The general form of the Gibbs-Appell equations is shown to be valid in the case when constraints and external forces are present. In the case when the Lagrangian is the kinetic energy with respect to a Riemannian metric, the Gibbs function is shown to be related to the kinetic energy on the tangent bundle of the configuration manifold with respect to the Sasaki metric. We also make a connection with the Gibbs-Appell equations and Gauss' principle of least constraint in the general case

A Symmetric Product for Vector Fields and its Geometric Meaning

We introduce the notion of geodesic invariance for distributions on manifolds with a linear connection. This is a natural weakening of the concept of a totally geodesic foliation to allow distributions which are not necessarily integrable. To test a distribution for geodesic invariance, we introduce a symmetric, vector field valued product on the set of vector fields on a manifold with a linear connection. This product serves the same purpose for geodesically invariant distributions as the Lie bracket serves for integrable distributions. The relationship of this product with connections in the bundle of linear frames is also discussed. As an application, we investigate geodesically invariant distributions associated with a left-invariant affine connection on a Lie group

Managing on-farm environmental impact using EMA.

There have been many initiatives recently aimed towards delivering policies relating to agricultural sustainability, particularly with respect to minimising environmental impact without compromising profitability. The final objective of these is normally to protect food and environmental quality, and preserve biodiversity and the natural heritage. The agricultural industry is in a time of significant change, and managing change can be difficult not least when trying to ensure awareness, response to new opportunities and compliance with legislation. For changes of this type to be practical and attractive, farmers need considerable guidance on what is inevitably a more labour, resource and time demanding process. The Environmental Management for Agriculture (EMA) software provides one mechanism by which this support can be distributed. It contains a library, decision support tools, databases, planning aids and audits for farm use. It has been available for several years in England and Wales and is now available for Scotland.Non peer reviewedFinal Published versio

A Computational Comparison of Evolutionary Algorithms for Water Resource Planning for Agricultural and Environmental Purposes

The use of water resources for agricultural purposes, particularly in arid and semi-arid regions, is a matter of increasing concern across the world. Optimisation techniques can play an important role in improving the allocation of land to different crops, based on a utility function (such as net revenue) and the water resources needed to support these. Recent work proposed a model formulation for an agricultural region in the Murrumbidgee Irrigation Area of the Murray-Darling River basin in Australia, and found that the well-known NSGA-II technique could produce sensible crop mixes while preserving ground and surface water for environmental purposes. In the present study we apply Differential Evolution using two different solution representations, one of which explores the restricted space in which no land is left fallow. The results improve on those of the prior NSGA-II and demonstrate that a combination of solution representations allows Differential Evolution to more thoroughly explore the multiobjective space of profit versus environment