A mathematical model for horizontal axis wind turbine blades

A mathematical model describing the nonlinear vibration of horizontal axis wind turbine (HAWT) blades is proposed in this paper. The system consists of a rotating blade and four components of deformation including longitudinal vibration (named axial extension), out-of-plane bend (named flap), in-plane/edgewise bend (named lead/lag) and torsion (named feather). It is assumed that the center of mass, shear center and aerodynamic center of a cross section all lie on the chord line, and do not coincide with each other. The structural damping of the blade, which is brought about by materials and fillers is taken into account based on the Kelvin–Voigt theory of composite materials approximately. The equivalent viscosity factor can be determined from empirical data, theoretical computation and experimental test. Gravitational loading and aerodynamic loading are considered as distributed forces and moments acting on blade sections. A set of partial differential equations governing the coupled, nonlinear vibration is established by applying the generalized Hamiltonian principle, and the current model is verified by previous models. The solution of equations is discussed, and examples concerning the static deformation, aeroelastic stability and dynamics of the blade are given.     

A new sensitivity analysis for structural optimization of composite rotor blades

This paper presents the mathematical details of the formulation for the sensitivity derivatives for the structural dynamic, aeroelastic stability, and response characteristics of a rotor blade in hover and forward flight. The formulation is denoted by the term semi-analytical approach, because certain derivatives are evaluated by a finite difference scheme. Using this formulation, sensitivity derivatives for structural dynamic and aeroelastic stability characteristics were evaluated for a composite rotor blade. Useful conclusions regarding the relative merits of the semi-analytical approach, when compared to a pure finite difference approach, are obtained. In addition, influence of ply orientation angles on the vibratory hub loads in forward flight is also considered.     

Reduced hybrid model of moving and rotating continua

The paper introduces the method of the model reduction of systems that experience a Coriolis acceleration or gyroscopic effect component. In such causes that corresponding system equations are non-self-adjoined. Modal reduced model is built up for the system without Coriolis or gyroscopic effect terms. These phenomena are next included by application of any lumping technique. Hence, the final reduced model is a hybrid one, obtained by both lumping and modal methods of modeling. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A triangular shell element for vibration analysis of cambered and twisted fan blades

A ten-node triangular shell element of thirty degrees of freedom was successfully developed and applied to plate and shell static analysis, and free vibrations of rectangular plates with different boundary conditions. The element is also suitable for the vibration analysis of pretwisted and cambered fan blades. Natural frequencies of cambered and untwisted fan blades having a rectangular planform with constant thickness are determined for different shallowness ratio and blade thickness. For different blade-tip twist angles, the element can be applied to obtain the frequencies of twisted blades with constant thickness. Frequencies, resulting from the applied element, are compared to those of experimental results which are available in literature, and to those of shallow shell theory and other finite elements. Good correlations and reasonably accurate results are obtained by using a coarse mesh size.     

A fully adaptive hybrid optimization of aircraft engine blades

A new fully adaptive hybrid optimization method (AHM) has been developed and applied to an industrial problem in the field of the aircraft engine industry. The adaptivity of the coupling between a global search by a population-based method (Genetic Algorithms or Evolution Strategies) and the local search by a descent method has been particularly emphasized. On various analytical test cases, the AHM method overperforms the original global search method in terms of computational time and accuracy. The results obtained on the industrial case have also confirmed the interest of AHM for the design of new and original solutions in an affordable time.     

A class of stable energy minimisers for the rotating drop problem

We implement variational techniques and an implicit function theorem to derive constraints on angular velocity under which we may verify the existence, boundary regularity, and stability of an energy-minimising family of rotating liquid drops in a neighbourhood of the closed unit ball in Rn+1.     

A note on the rotating chain problem

For a non-linear boundary value problem describing the shape of a heavy rotating chain a priori estimates of the maximal deflection are obtained. Also, for the transformed chain problem an extremum Variational principle and error estimates are presented.

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Zusammenfassung Für eine nichtlineare Differentialgleichung, die die Form einer schwer rotierenden Kette beschreibt, werden a priori Abschätzungen der maximalen Ausbiegung erhalten. Dabei werden für ein transformiertes Ketten-Problem ein extremales Variationsprinzip und die Feller Abschätzung angegeben.

     

A note on quantization in rotating co-ordinates

The quantization of scalar fields in a uniformly rotating frame is reconsidered. It is pointed out that the coordinates usually used do not represent a frame describing uniform circular motion and the metric in the correct coordinates has a singularity. The result obtained earlier with the wrong coordinates, that there is no radiation, nevertheless holds.

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Zusammenfassung Die Quantisierung skalarer Felder in einem gleichförmig rotierenden Koordinatensystem wird erneut betrachtet. Dabei wird hervorgehoben, daß die gewöhnlich verwendeten Koordinaten die gleichförmige Kreisbewegung nicht beschreiben und daß die Metrik in den richtigen Koordinaten eine Singularität aufweist. Trotzdem bleibt das früher mit den falschen Koordinaten abgeleitete Resultat erhalten, daß nämlich ein strahlungsfreier Zustand existiert.

     

Wellposedness for anisotropic rotating fluid equations

The wellposedness problem for an anisotropic incompressible viscous fluid in R3,rotating around a vector B(t,x):=(b1(t,x),b2(t,x),b3(t,x)),is studied.The global wellposedness in the homogeneous case (B...     

A criterion on instability of cylindrical rotating surfaces

We consider a column of a stationary rotating surface in Euclidean space. In this paper we obtain a value l ₀ > 0 in such a way that if the length l of the column satisfies l > l ₀, then the surface is instable. This extends previous results due to Plateau and Rayleigh for columns of surfaces with constant mean curvature.     

Asymptotic estimates for slowly rotating Newtonian stars

This work is mainly concerned with the rotating Newtonian stars with prescribed angular velocity law. For general compressible fluids, the existence of rotating star solutions was proved by using concentration-compactness principle. In this paper, we establish the asymptotic estimates on the diameters of the stars with small rotation. The novelty of this paper is that a direct and concise definition of slowly rotating stars is given, different from the case with given angular momentum law, and the most general fluids are considered.     

Asymptotic estimates for an axisymmetric rotating fluid

A variational method is used to study the shape of a self-gravitating fluid mass rotating with a prescribed angular momentum per unit mass. The model is of primary interest in astrophysics, where it is taken as a model of a star. Sharp estimates are obtained on the diameter of the rotating body in terms of the total mass M and parameter $\text{Q =}\text{T}\text{W}\text{; T}$ and W represent, respectively, the kinetic and potential energy. The cases of Q small (slowly rotating body) and Q large (rapidly rotating body) are dealt with separately.     

Stall-delay modelling of wind turbine blades

Most aerodynamic design tools for horizontal-axial wind turbines are based on the blade-element momentum theory (BEM). Due to the nature of this theory, the design tools need 2-D steady sectional lift and drag curves as an input. In practice, flow over a wind turbine rotor blade is neither two-dimensional nor steady, and is affected by rotation. Pioneering experiments have identified a consequence: at inboard rotor blade sections stall is delayed. This so-called Himmelskamp effect [1] gives a larger lift than predicted and, as a result, a higher power and loading than expected. Consequently, an aerodynamic model is needed to explain and predict sectional lift and drag under rotating conditions. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A stochastic model for solitons

The soliton physics for the propagation of waves is represented by a stochastic model in which the particles of the wave can jump ahead according to some probability distribution. We demonstrate the presence of a steady state (stationary distribution) for the wavelength. It is shown that the stationary distribution is a convolution of geometric random variables. Approximations to the stationary distribution are investigated for a large number of particles. The model is rich and includes Gaussian cases as limit distribution for the wavelength (when suitably normalized). A sufficient Lindeberg‐like condition identifies a class of solitons with normal behavior. Our general model includes, among many other reasonable alternatives, an exponential aging soliton, of which the uniform soliton is one special subcase (with Gumbel's stationary distribution). With the proper interpretation, our model also includes the deterministic model proposed in Takahashi and Satsuma [A soliton cellular automaton, J Phys Soc Japan 59 (1990), 3514–3519]. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

A model for glioma growth

Glioblastoma Multiforme (GBM) is the most invasive form of primary brain tumor. We propose a mathematical model that describes such tumor growth and allows us to describe two different mechanisms of cell invasion: diffusion (random motion) and chemotaxis (directed motion along the gradient of the chemoattractant concentration). The results are in a quantitative agreement with recent in vitro experiments. It was observed in experiments that the outer invasive zone grows faster than the inner proliferative region. We argue that this feature indicates transient behavior, and that the growth velocities tend to the same constant value for larger times. A longer‐time experiment is needed to verify this hypothesis and to choose between the two basic mechanisms for tumor growth. © 2005 Wiley Periodicals, Inc. Complexity 11: 53–57, 2005     

A mathematical model for unemployment

In this paper we have proposed and analyzed a non-linear mathematical model for unemployment by considering three variables, namely the numbers of unemployed, temporarily employed and regularly employed persons. The model is studied using the stability theory of differential equations. It is found that the model has only one equilibrium, which is non-linearly stable under certain conditions. Numerical simulation of the model has been carried out to confirm the analytical results.