Modeling and analysis of nonlinear rotordynamics due to higher order deformations in bending

A mathematical model incorporating the higher order deformations in bending is developed and analyzed to investigate the nonlinear dynamics of rotors. The rotor system considered for the present work consists of a flexible shaft and a rigid disk. The shaft is modeled as a beam with a circular cross section and the Euler Bernoulli beam theory is applied with added effects such as rotary inertia, gyroscopic effect, higher order large deformations, rotor mass unbalance and dynamic axial force. The kinetic and strain (deformation) energies of the rotor system are derived and the Rayleigh–Ritz method is used to discretize these energy expressions. Hamilton’s principle is then applied to obtain the mathematical model consisting of second order coupled nonlinear differential equations of motion. In order to solve these equations and hence obtain the nonlinear dynamic response of the rotor system, the method of multiple scales is applied. Furthermore, this response is examined for different possible resonant conditions and resonant curves are plotted and discussed. It is concluded that nonlinearity due to higher order deformations significantly affects the dynamic behavior of the rotor system leading to resonant hard spring type curves. It is also observed that variations in the values of different parameters like mass unbalance and shaft diameter greatly influence dynamic response. These influences are also presented graphically and discussed.     

Solving Hankel matrix approximation problem using semidefinite programming

Positive semidefinite Hankel matrices arise in many important applications. Some of their properties may be lost due to rounding or truncation errors incurred during evaluation. The problem is to find the nearest matrix to a given matrix to retrieve these properties. The problem is converted into a semidefinite programming problem as well as a problem comprising a semidefined program and second-order cone problem. The duality and optimality conditions are obtained and the primal–dual algorithm is outlined. Explicit expressions for a diagonal preconditioned and crossover criteria have been presented. Computational results are presented. A possibility for further improvement is indicated.     

The dynamics of cyclic ventilation systems

Ventilation and air conditioning modelling methods, for large scale, spatially dispersed systems are considered. The limitations of existing techniques are discussed and proposals for the application of novel analysis approaches are outlined. The use of hybrid, distributed–lumped parameter procedures enabling the incorporation of the relatively concentrated and significantly dispersed, system elements, is advocated. A dynamic model for a ventilation arrangement comprising inlet and exhaust fans, with filtered air re-circulation and air conditioning units is presented. Pressure, flow and temperature predictions within the system are computed following input and disturbance changes. The employment of the model for automatic, multivariable controller design purposes is commented upon.     

The Mackey–Glass model of respiratory dynamics: Review and new results

Since the celebrated Mackey–Glass model of respiratory dynamics was introduced in 1977, many results on its qualitative behavior have been obtained, including oscillation, stability and chaos. The paper reviews some known properties and presents new results for more general models: equations with time-dependent parameters, several delays, a positive periodic equilibrium and distributed delays. The problems considered in the paper involve existence, positivity and permanence of solutions, oscillation and global asymptotic stability. In addition, some general approaches to the study of nonlinear nonautonomous scalar delay equations are outlined. The paper generalizes and unifies existing results and provides an outlook on further studies.     

Computation of dilute two-phase flow in a pump

This paper is a report on a joint project between academia and industry which is concerned with computation of dilute two-phase flow through a pump in turbulent condition. The flow field for the continuous phase is computed using the Reynolds averaged Navier–Stokes equations together with mixing length turbulence modeling. The dispersed phase is treated using the Lagrangian approach by tracking it's trajectory along which the information is passed. It is found that the bubbles and small solid particles flow out of the chamber (between the rotating impeller and the casing wall) with the conveying fluid. The solid particles of relatively bigger sizes accumulate at the low pressure zones near the cashing wall or the rotating shaft.     

On the antiplane shear deformation of finite wedges

Antiplane shear deformation of finite wedges is considered under different boundary conditions. First, the assertions and results of a recent paper, namely Chue and Liu [C.H. Chue, W.J. Liu, Comments on “Analysis of an isotropic finite wedge under antiplane deformation”, Int. J. Solids Struct. 41 (2004) 5023–5034] are invalidated. Then, closed form solutions are extracted for the stress distribution in the wedge. These closed forms have the advantages of showing the possible geometric stress singularity as well as the load singularity explicitly, in addition to the continuity or discontinuity as well as the convergence of the results in the entire region. Finally, the stress intensity factors are extracted in the special case of a circular shaft containing an edge crack under different boundary conditions.     

Dual Orlicz–Brunn–Minkowski theory

In this paper, a dual Orlicz–Brunn–Minkowski theory is presented. An Orlicz radial sum and dual Orlicz mixed volumes are introduced. The dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality are established. The variational formula for the volume with respect to the Orlicz radial sum is proved. The equivalence between the dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality is demonstrated. Orlicz intersection bodies are defined and the Orlicz–Busemann–Petty problem is posed.     

CFD modelling of the flow and reactions in the Olympic Dam flash furnace smelter reaction shaft

The performance of flash furnace burners can be evaluated quickly and efficiently using CFD modelling. Gas flows are modelled using the conventional Eulerian approach, while Lagrangian particle tracking is used to model the flow of solid feed through the burner and into the reaction shaft. A composite particle model has been developed that considers the solid feed to be made up of single particles containing appropriate quantities of concentrate, flux and dust. Solid fuels (such as coal) can also be included in the composite particle. Reactions between the solids and gas are then modelled using standard heat and mass transfer relationships. Results from the modelling process are shown for BHP-Billiton’s Olympic Dam copper flash smelter with the burner that was used from 1998–2003. Flow patterns, temperature and gas composition distributions, particle dispersion and residence time, and overall extent of sulphur removal are predicted and used to evaluate furnace performance. However, results are sensitive to the assumed size of the composite particles, and plant measurements are required to determine the appropriate composite particle size to predict quantitative data.     

A determinantal approach to Sheffer–Appell polynomials via monomiality principle

In this article, the Sheffer and Appell polynomials are combined to introduce the family of Sheffer–Appell polynomials by using operational methods. The determinantal definition and other properties of the Sheffer–Appell polynomials are established. As particular cases of these polynomials, the Sheffer–Bernoulli and Sheffer–Euler polynomials are introduced and their determinantal definitions are obtained. The operational correspondence between the Appell and Sheffer–Appell polynomials is used to derive the results for the Sheffer–Appell polynomials. Certain results for the Hermite–Appell and Laguerre–Appell polynomials are also obtained.     

A deterministic particle method for the Vlasov–Fokker–Planck equation in one dimension

The Vlasov–Fokker–Planck equation is a model for a collisional, electrostatic plasma. The approximation of this equation in one spatial dimension is studied. The equation under consideration is linear in that the electric field is given as a known function that is not internally consistent with the phase space distribution function. The approximation method applied is the deterministic particle method described in Wollman and Ozizmir [Numerical approximation of the Vlasov–Poisson–Fokker–Planck system in one dimension, J. Comput. Phys. 202 (2005) 602–644]. For the present linear problem an analysis of the stability and convergence of the numerical method is carried out. In addition, computations are done that verify the convergence of the numerical solution. It is also shown that the long term asymptotics of the computed solution is in agreement with the steady state solution derived in Bouchut and Dolbeault [On long time asymptotics of the Vlasov–Fokker–Planck equation and of the Vlasov–Poisson–Fokker–Planck system with coulombic and Newtonian potentials, Differential Integral Equations 8(3) (1995) 487–514].     

An alternative approach to solve the mixed AKNS equations

The algebraic–geometric solutions of the mixed AKNS equations are investigated through a finite-dimensional Lie–Poisson Hamiltonian system, which is generated by the nonlinearization of the adjoint equation related to the AKNS spectral problem. First, each mixed AKNS equation can be decomposed into two compatible Lie–Poisson Hamiltonian flows. Then the separated variables on the coadjoint orbit are introduced to study these Lie–Poisson Hamiltonian systems. Further, based on the Hamilton–Jacobi theory, the relationship between the action-angle coordinates and the Jacobi-inversion problem is established. In the end, using Riemann–Jacobi inversion, the algebraic–geometric solutions of the first three mixed AKNS equations are obtained.     

Effect of material in-homogeneity on electro-thermo-mechanical behaviors of functionally graded piezoelectric rotating shaft

In this article, a hollow circular shaft made from functionally graded piezoelectric material (FGPM) such as PZT_5 has been studied which is rotating about its axis at a constant angular velocity ω. This shaft subjected to internal and external pressure, a distributed temperature field due to steady state heat conduction with convective boundary condition, and a constant potential difference between its inner and outer surfaces or combination of these loadings. All mechanical, thermal and piezoelectric properties except for the Poisson’s ratio are assumed to be power functions of the radial position. The governing equation in polarized form is shown to reduce to a system of second-order ordinary differential equation for the radial displacement. Considering six different sets of boundary conditions, this differential equation is analytically solved. The electro-thermo-mechanical stress and the electric potential distributions in the FGPM hollow shaft are discussed in detail for the piezoceramic PZT_5. The presented results indicate that the material in-homogeneity has a significant influence on the electro-thermo-mechanical behaviors of the FGPM rotating shaft and should therefore be considered in its optimum design.     

Direct solution of Navier–Stokes equations by radial basis functions

The pressure–velocity formulation of the Navier–Stokes (N–S) equation is solved using the radial basis functions (RBF) collocation method. The non-linear collocated equations are solved using the Levenberg–Marquardt method. The primary novelty of this approach is that the N–S equation is solved directly, instead of using an iterative algorithm for the primitive variables. Two flow situations are considered: Couette flow with and without pressure gradient, and 2D laminar flow in a duct with and without flow obstruction. The approach is validated by comparing the Couette flow results with the analytical solution and the 2D results with those obtained using the well-validated CFD-ACE™ commercial package.     

High-order approximation of Pearson diffusion processes

This paper focuses on Pearson diffusions and the spectral high-order approximation of their related Fokker–Planck equations. The Pearson diffusions is a class of diffusions defined by linear drift and quadratic squared diffusion coefficient. They are widely used in the physical and chemical sciences, engineering, rheology, environmental sciences and financial mathematics. In recent years diffusion models have been studied analytically and numerically primarily through the solution of stochastic differential equations. Analytical solutions have been derived for some of the Pearson diffusions, including the Ornstein–Uhlenbeck, Cox–Ingersoll–Ross and Jacobi processes. However, analytical investigations and computations for diffusions with so-called heavy-tailed ergodic distributions are more difficult to perform. The novelty of this research is the development of an accurate and efficient numerical method to solve the Fokker–Planck equations associated with Pearson diffusions with different boundary conditions. Comparisons between the numerical predictions and available time-dependent and equilibrium analytical solutions are made. The solution of the Fokker–Planck equation is approximated using a reduced basis spectral method. The advantage of this approach is that many models for pricing options in financial mathematics cannot be expressed in terms of a stochastic partial differential equation and therefore one has to resort to solving Fokker–Planck type equations.     

Lattice Boltzmann model for the bimolecular autocatalytic reaction–diffusion equation

A lattice Boltzmann model for the bimolecular autocatalytic reaction–diffusion equation is proposed. By using multi-scale technique and the Chapman–Enskog expansion on complex lattice Boltzmann equation, we obtain a series of complex partial differential equations, complex equilibrium distribution function and its complex moments. Then, the complex reaction–diffusion equation is recovered with higher-order accuracy of the truncation error. This equation can be used to describe the bimolecular autocatalytic reaction–diffusion systems, in which a rich variety of behaviors have been observed. Based on this model, the Fitzhugh–Nagumo model and the Gray–Scott model are simulated. The comparisons between the LBM results and the Alternative Direction Implicit results are given in detail. The numerical examples show that assumptions of source term can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the complex reaction–diffusion equation.     

Non-stationary analysis of a rotating shaft with geometrical nonlinearity during passage through critical speeds

In this paper, analysis of a rotating shaft with stretching nonlinearity during passage through critical speeds is investigated. In the model, the rotary inertia and gyroscopic effects are included, but shear deformation is neglected. The nonlinearity is due to large deflection of the shaft. First, nonlinear equations of motion governing the flexural–flexural–extensional vibrations of the rotating shaft with non-constant spin are derived by the Hamilton principle. Then, the equations are simplified using stretching assumption. To analyze the non-stationary vibration of the rotating shaft, the asymptotic method is applied to the equations expressed in normal coordinates. Two analytical expressions, as function of system parameters that describe the amplitude and phase of motion during passage through critical speeds are derived. The effects of angular acceleration, stretching nonlinearity, eccentricity and external damping on maximum amplitude of the shaft are investigated. It is shown that the nonlinearity has important effect on maximum amplitude when the rotating shaft passing through critical speeds, especially in low angular acceleration. To validate the results of the perturbation method, numerical simulation is applied.     

A generalized Fitzhugh–Nagumo equation

In this paper we investigate mappings of the classical Fitzhugh–Nagumo equation to a generalized Fitzhugh–Nagumo equation. These mappings are invertible and transform the solutions of the classical Fitzhugh–Nagumo equation into solutions of the generalized Fitzhugh–Nagumo equation considered here. These mappings are found by considering the Lie point symmetries admitted by the classical Fitzhugh–Nagumo equation and the generalized Fitzhugh–Nagumo equation considered here. A particular example of a generalized Fitzhugh–Nagumo equation that satisfies the boundary conditions of the classical Fitzhugh–Nagumo equation is considered. Numerical solutions of the generalized Fitzhugh–Nagumo equation that do not satisfy the boundary conditions of the classical Fitzhugh–Nagumo equation are obtained by implementing the Method of Lines.     

Chaos control of fractional order Rabinovich–Fabrikant system and synchronization between chaotic and chaos controlled fractional order Rabinovich–Fabrikant system

In this article the local stability of the Rabinovich–Fabrikant (R–F) chaotic system with fractional order time derivative is analyzed using fractional Routh–Hurwitz stability criterion. Feedback control method is used to control chaos in the considered fractional order system and after controlling the chaos the authors have introduced the synchronization between fractional order non-chaotic R–F system and the chaotic R–F system at various equilibrium points. The fractional derivative is described in the Caputo sense. Numerical simulation results which are carried out using Adams–Boshforth–Moulton method show that the method is effective and reliable for synchronizing the systems.     

General solutions of three-dimensional problems for two-dimensional quasicrystals

Three-dimensional problems are systematically investigated for the coupled equations in two-dimensional hexagonal quasicrystals, and two new general solutions, which are called generalized Lekhnitskii–Hu–Nowacki (LHN) solutions and generalized Elliott–Lodge (E–L) solutions, are presented, respectively. By introducing two higher-order displacement functions, an operator analysis technique is applied in a novel way to obtain generalized LHN solutions. For further simplification, a decomposition and superposition procedure is taken to replace the higher-order displacement functions with five quasi-harmonic displacement functions, and then generalized E–L solutions are simplified in terms of these functions. In consideration of different cases of characteristic roots, generalized E–L solutions take different forms, but all are in simple forms that are conveniently applied. To illustrate the application of the general solutions obtained, the closed form solution is obtained for an infinite quasicrystal medium subjected to a point force at an arbitrary point.     

Multivariate Exponential Distributions with Constant Failure Rates

In this paper a multivariate failure rate representation based on Cox's conditional failure rate is introduced, characterizations of the Freund–Block and the Marshall–Olkin multivariate exponential distributions are obtained, and generalizations of the Block–Basu and the Friday–Patil bivariate exponential distributions are proposed.