A new approach to design a control system for a FGR furnace using the combination of the CFD and linear system identification techniques

A new approach to design control systems for an industrial furnace with flue gas recirculation (FGR) is presented. To facilitate the control system design, a linear dynamic model is needed for the furnace. Full-scale computational fluid dynamics (CFD) simulations are used to generate the required small signal input and output data sets. Subsequently, a least squares based system identification technique is used to obtained the linear dynamic models. After model validation, feedback controller is designed based on these linear dynamic models. Finally, the performance of the designed closed-loop control system is also evaluated using both linear dynamic model and full-scale nonlinear CFD model. The comparison shows that the control system designed using the proposed approach can minimize the deviation of nitric oxides (NO) emission from the design point by minimize the dynamic NO formation, hence to prevent any excessive NO formation in the combustion process when the system subjects to disturbances.     

A CFD assisted control system design with applications to NO x control in a FGR furnace

In this paper, a novel technique to design control systems for industrial processes with non-linear distributed parameters is proposed. The technique utilizes computational fluid dynamics (CFD) simulation to extract the most essential characteristics from the non-linear industrial process, and then represent them as a set of linear dynamic models around a specific operating point. Based on the linear dynamic representation, a closed-loop feedback linear control system can be designed to maintain the desired performance for the system around the chosen operating point. To illustrate such a design process, an industrial reheating furnace with flue gas recirculation (FGR) is selected herein. The method involves the numerical solution of the partial differential equations describing the fluid flow, heat transfer and combustion process in the furnace. The resulting dynamic relations between the furnace inputs and outputs can then be represented in terms of a multi-input and multi-output transfer function matrix. The objective of the control system is then to maintain the optimally selected furnace operating conditions and compensate for any deviations caused by disturbances to minimize the nitric oxides (NO _x ) emission through feedback mechanisms. The performance of the closed-loop controlled furnace is evaluated not only in the linear domain, but also with the detailed full-scale non-linear CFD model. The results have shown that the proposed method is viable and the designed control system can indeed minimize the deviation of the furnace from the desired operating conditions and hence to prevent any excessive NO _x formation in the combustion process.     

Range of applicability of the linear fluid slosh theory for predicting transient lateral slosh and roll stability of tank vehicles

An analytical model is developed to study the transient lateral sloshing in horizontal cylindrical containers assuming inviscid, incompressible and irrotational flows. The model is derived by implementing the linearized free-surface boundary condition and bipolar coordinate transformation, resulting in a truncated system of linear ordinary differential equations, which is numerically solved to determine the fluid velocity potentials followed by the hydrodynamic forces and moment. The model results are compared with those obtained from the multimodal solution. The free-surface elevation and hydrodynamic coefficients are also compared with the reported experimental and analytical data as well as numerical simulations to establish validity of the model. The capability of the model for predicting non-resonant slosh is also evaluated using the critical free-surface amplitude. The model validity is further illustrated by comparing the transient liquid slosh responses of a partially filled tank subject to steady lateral acceleration characterizing a vehicle turning maneuver with those obtained from fully nonlinear CFD simulations and pendulum models. It is shown that the linear slosh model yields more accurate prediction of dynamic slosh than the pendulum models and it is significantly more computationally efficient than the nonlinear CFD model. The slosh model is subsequently applied to roll plane model of a suspended tank vehicle to study the effect of dynamic liquid slosh on steady-turning roll stability limit of the vehicle under constant and variable axle load conditions. The results suggest that the roll moment arising from the dynamic fluid slosh yields considerably lower roll stability limit of the partly-filled tank vehicle compared to that predicted from the widely reported quasi-static fluid slosh model.     

Nonlinear structural identification by the “Reverse Path” spectral method

When dealing with nonlinear dynamical systems, it is important to have efficient, accurate and reliable tools for estimating both the linear and nonlinear system parameters from measured data. An approach for nonlinear system identification widely studied in recent years is “Reverse Path”. This method is based on broad-band excitation and treats the nonlinear terms as feedback forces acting on an underlying linear system. Parameter estimation is performed in the frequency domain using conventional multiple-input–multiple-output or multiple-input–single-output techniques. This paper presents a generalized approach to apply the method of “Reverse Path” on continuous mechanical systems with multiple nonlinearities. The method requires few spectral calculations and is therefore suitable for use in iterative processes to locate and estimate structural nonlinearities. The proposed method is demonstrated in both simulations and experiments on continuous nonlinear mechanical structures. The results show that the method is effective on both simulated as well as experimental data.     

Anti-optimisation for modelling the vibration of locally nonlinear structures: An exploratory study

Modelling the vibration of complex structures with uncertain nonlinearities is a significant challenge. However, nonlinearities are often spatially localised: this enables efficient linear methods to describe the behaviour of the majority of the structure and reduces the size of the nonlinear problem. This paper explores anti-optimisation as an approach to modelling uncertain nonlinearities for this class of system. The ‘worst-case’ output metric is sought by considering nonlinear forces as an external input subject to constraints that capture what is known about the nonlinearity. A systematic sequence of tests is carried out using a mass on spring system within a pair of end-stops: the results show how the anti-optimised solutions become less conservative as the constraints are increasingly restrictive. The method is applied to bending vibration of a beam within a pair of local end-stops. Anti-optimised solutions are found as a function of frequency and are compared with a Monte Carlo set of benchmark simulations. Almost all anti-optimised solutions over-predict the simulations and the overall trend of the simulations is also clearly captured. The method shows significant potential and motivates further research.     

Design considerations for an active feedback optical system

The optical processor proposed here consists basically of a coherent light source and two liquid crystal light valves, of which the first converts incoherent input patterns into coherent ones; the second operates entirely on coherent signals. It is shown that, if the valve operating point is correctly chosen and its transfer characteristic linearized, negative feedback is implemented around the second valve. The valves perform as active system components, because they modulate the source power by their driving signals.The linear dynamic range, modulation transfer function, and time transient behaviour of the system are briefly studied.     

High nonlinearities in Langevin transducer: a comprehensive model

The design and simulation of power transducers are difficult since piezoelectric, dielectric and elastic properties of ferroelectric materials differ from linear behavior when driven at large levels. This paper is devoted to modeling of a resonant power transducer at a high level of dynamic mechanical stress. The power transducer is subjected to a sine electrical field E of varying frequency which was considered as the excitation of the transducer.The mechanical equation of the piezoelectric element is written using electrostriction. The dielectric part is written as a nonlinear function of an equivalent electric field including stress influence (scaling relationship between electric field and mechanical stress). Using various simulations, we show then that typical resonance nonlinearities are obtained, such as jump phenomenon of transducer speed amplitude and phase, resonance peak that become asymmetric, and diminution of mechanical quality factor. As a consequence, we state that those typical nonlinearities are only due to dielectric nonlinearities, in good correlation with typical ferroelectric behavior. Moreover, this demonstrates the usefulness of scaling relationships in ferroelectrics, which explain static depoling under stress and butterfly strain hysteresis loop. The same scaling law gives here several nonlinearities for resonant transducers as well.     

Establishment of infinite dimensional Hamiltonian system of multilayer quasi-geostrophic flow & study on its linear stability

A multilayer flow is a stratified fluid composed of a finite number of layers with densities homogeneous within one layer but different from each other. It is an intermediate system between the single-layer barotropic model and the continuously stratified baroclinic model. Since this system can simulate the baroclinic effect simply, it is widely used to study the large-scale dynamic process in atmosphere and ocean. The present paper is concerned with the linear stability of the multilayer quasi-geostrophic flow, and the associated linear stability criteria are established. Firstly, the nonlinear model is turned into the form of a Hamiltonian system, and a basic flow is defined. But it cannot be an extreme point of the Hamiltonian function since the system is an infinite-dimensional one. Therefore, it is necessary to reconstruct a new Hamiltonian function so that the basic flow becomes an extreme point of it. Secondly, the linearized equations of disturbances in the multilayer quasi-geostrophic flow are derived by introducing infinitesimal disturbances superposed on the basic flows. Finally, the properties of the linearized system are discussed, and the linear stability criteria in the sense of Liapunov are derived under two different conditions with respect to certain norms.     

Stabilization of Lagrange points in circular restricted three-body problem: A port-Hamiltonian approach

Current station keeping strategies target periodic orbits around the unstable Lagrange points. These control strategies are based on the Circular Restricted Three-Body Problem (CRTBP) linearized around an equilibrium point and cannot ensure global stability. In this paper, we use the port-Hamiltonian approach to reformulate the CRTBP with input, which preserves the original nonlinear dynamics. Designing a control strategy based on energy shaping and dissipation injection, we obtain the closed-loop Hamiltonian as the candidate of Lyapunov function, which guarantees asymptotic stability. The control strategy designed here is successfully applied to the stabilization of Lagrange points in CRTBP. Furthermore, the designed control approach shows global stability within the application region of CRTBP model, and it is applicable to set arbitrary equilibrium points. The current framework is also stable against error in the thrust and still works when the third body moves beyond the region of applicability of the linearized dynamics, where the linear controller may fail. Finally, this method has potential to be extended to the three-dimensional CRTBP, where both the perturbation and the thrust out of the plane are considered.     

Improving nonlinear performance of the HEPS baseline design with a genetic algorithm

A baseline design for the High Energy Photon Source has been proposed, with a natural emittance of 60pm·rad within a circumference of about 1.3 kilometers. Nevertheless, the nonlinear performance of the design needs further improvements to increase both the dynamic aperture and the momentum acceptance. In this study, genetic optimization of the linear optics is performed, so as to find all the possible solutions with weaker sextupoles and hence weaker nonlinearities, while keeping the emittance at the same level as the baseline design. The solutions obtained enable us to explore the dependence of nonlinear dynamics on the working point. The result indicates that with the same layout, it is feasible to obtain much better nonlinear performance with a delicate tuning of the magnetic field strengths and a wise choice of the working point.     

On the nonlinear primary resonances of a piezoelectric laminated micro system under electrostatic control voltage

In this article, a comprehensive nonlinear analysis for a piezoelectric laminated micro system around its static deflection is presented. This static deflection is created by an electrostatic DC control voltage through an electrode plate. The micro system beam is assumed as an elastic Euler-Bernoulli beam with clamped-free end conditions. The dynamic equations of this model have been derived by using the Hamilton method and considering the nonlinear inertia, curvature, piezoelectric and electrostatic terms. The static and dynamic solutions have been achieved by using the Galerkin method and the multiple-scales perturbation approach, respectively. The results are compared with numerical and other existing experimental results. By studying the primary resonance excitation, the effects of different parameters such as geometry, material and excitations voltage on the system?s softening and hardening behaviors are evaluated. In a piezoelectrically actuated micro system it was showed that because of existence of curvature and inertia nonlinear terms a small change in excitation amplitude can lead to the formation and expansion of nonlinear response. In this paper, it is demonstrated that by applying an electrostatic DC control voltage, these nonlinearities can be controlled and altered to a linear domain. This model can be used to design a nano or micro-scale smart device.     

Prediction condensed models adapted to the nonlinear structures in time domain

This paper deals with a method to study closely the stationary solution of nonlinear dynamic systems in time domain. This method is based on the exploitation of Karhunen-Loève decomposition with or without parametric modifications as well as on the characteristics of localized nonlinearities. With the application of this method at first on linear models initially condensed by Karhunen-Loève, the predictions of nonlinear responses can be obtained rapidly. This method is adapted to a condensed linear model used in the first optimization procedure of the nonlinear dynamic behaviour. This robust basis will be used as condensation basis of the modified model local per zone, which leads to a prediction of vibratory responses of complex structures modified and affected by localized nonlinearities.     

Uncertainty analysis near bifurcation of an aeroelastic system

Variations in structural and aerodynamic nonlinearities on the dynamic behavior of an aeroelastic system are investigated. The aeroelastic system consists of a rigid airfoil that is supported by nonlinear springs in the pitch and plunge directions and subjected to nonlinear aerodynamic loads. We follow two approaches to determine the effects of variations in the linear and nonlinear plunge and pitch stiffness coefficients of this aeroelastic system on its stability near the bifurcation. The first approach is based on implementation of intrusive polynomial chaos expansion (PCE) on the governing equations, yielding a set of nonlinear coupled ordinary differential equations that are numerically solved. The results show that this approach is capable of determining sensitivity of the flutter speed to variations in the linear pitch stiffness coefficient. On the other hand, it fails to predict changes in the type of the instability associated with randomness in the cubic stiffness coefficient. In the second approach, the normal form is used to investigate the flutter (Hopf bifurcation) boundary that occurs as the freestream velocity is increased and to analytically predict the amplitude and frequency of the ensuing LCO. The results show that this mathematical approach provides detailed aspects of the effects of the different system nonlinearities on its dynamic behavior. Furthermore, this approach could be effectively used to perform sensitivity analysis of the system's response to variations in its parameters.     

The evolution of the magnetic moment in a corrugated magnetic field

In the first part, the equations of motion in a weakly corrugated, periodic magnetic field are linearized and solved by using paraxial approximation, to describe the model and the associated resonance condition. In the second part, the nonlinear evolution of the magnetic moment of resonant particles, in connection with their axial displacement is investigated analytically by using the multiple scale method. It is seen that the linear evolution is converted into a slow and periodic oscillation around the unperturbed value, with a considerable amplitude. The analytic expressions for the period and amplitude of the oscillations are derived and compared with the numerical simulations, which are also presented. Finally, the limitations of the paraxial approximation are concluded by investigating the numerical simulations, with actual field expressions. (c) 1997 American Institute of Physics.     

On the orthogonalised reverse path method for nonlinear system identification

The problem of obtaining the underlying linear dynamic compliance matrix in the presence of nonlinearities in a general multi-degree-of-freedom (MDOF) system can be solved using the conditioned reverse path (CRP) method introduced by Richards and Singh (1998 Journal of Sound and Vibration, 213(4): pp. 673–708). The CRP method also provides a means of identifying the coefficients of any nonlinear terms which can be specified a priori in the candidate equations of motion. Although the CRP has proved extremely useful in the context of nonlinear system identification, it has a number of small issues associated with it. One of these issues is the fact that the nonlinear coefficients are actually returned in the form of spectra which need to be averaged over frequency in order to generate parameter estimates. The parameter spectra are typically polluted by artefacts from the identification of the underlying linear system which manifest themselves at the resonance and anti-resonance frequencies. A further problem is associated with the fact that the parameter estimates are extracted in a recursive fashion which leads to an accumulation of errors. The first minor objective of this paper is to suggest ways to alleviate these problems without major modification to the algorithm. The results are demonstrated on numerically-simulated responses from MDOF systems. In the second part of the paper, a more radical suggestion is made, to replace the conditioned spectral analysis (which is the basis of the CRP method) with an alternative time domain decorrelation method. The suggested approach – the orthogonalised reverse path (ORP) method – is illustrated here using data from simulated single-degree-of-freedom (SDOF) and MDOF systems.     

Nonlinear equations with superposition principles and the theory of transitive primitive Lie algebras

The classification of systems of nonlinear ordinary differential equations with superposition principles is reduced to a classification of transitive primitive Lie algebras. Each system can be associated with the transitive primitive action of a Lie group G on a homogenous space G/H, where H is a maximal subgroup of G. The equations can have specific polynomial or rational nonlinearities.     

Wave Scattering by Defects in Media with Spatial Dispersion and Nonradiative Dynamic Solitons

Specific features of dynamic solitons in a nonlinear system described by a differential equation with a fourth-order spatial derivative are discussed. Solutions of the linearized equation and the problem of scattering of a double-partial wave by a point defect are analyzed. Conditions of existence of a nonradiative soliton are formulated and demonstrated in the case in which the natural soliton frequency falls within the continuous spectrum of harmonic oscillations of the examined system. These conditions are determined by the dispersion law of linear oscillations.     

Vibration control of two degrees of freedom system using variable inertia vibration absorbers: Modeling and simulation

Variable inertia vibration absorbers (VIVA) are previously used for the vibration control of single degree of freedom (dof) primary systems. The performance of such absorbers is studied in many investigations. This paper presents the dynamic modeling and simulation of a proposed modified design of such VIVA’s for the vibration control of two dof primary systems. Lagrange formulation is used to obtain its dynamic model in an analytical form. This model, which is highly nonlinear, is used to develop a computational algorithm to study the absorber performance characteristics. This algorithm is programmed and simulated in Matlab. The obtained results are numerically verified using SAMS2000 software. The effect of mass and stiffness of the proposed VIVA on its performance and tuning is discussed. An optimization algorithm is developed to select the best absorber parameters for vibration suppression of a specific primary system. The obtained results show a good agreement with those obtained using similar techniques. In addition, a linearized model of VIVA dynamics is developed, tested and simulated for the same data used in its nonlinear model. The relative deviation between results of the linear and nonlinear models is less than 1%, which confirms the realistic use of this linearized model. The experimental testing and verification of the simulation results of the proposed VIVA is the subject of another paper.     

Adaptive active control of periodic vibration using maglev actuators

In this paper, active control of periodic vibration is implemented using maglev actuators which exhibit inherent nonlinear behaviors. A multi-channel feedforward control algorithm is proposed to solve these nonlinear problems, in which maglev actuators are treated as single-input–single-output systems with unknown time-varying nonlinearities. A radial basis function network is used by the algorithm as its controller, whose parameters are adapted only with the model of the linear system in the secondary path. Compared with the strategies in the conventional magnetic-levitation system control as well as nonlinear active noise/vibration control, the proposed algorithm has the advantage that the nonlinear modeling procedure of maglev actuators and the usage of displacement sensors could be both avoided. Numerical simulations and real-time experiments are carried out based on a multiple-degree-of-freedom vibration isolation system. The results show that the proposed algorithm not only could efficiently compensate for the actuators’ time-varying nonlinearities, but also has the ability to greatly attenuate the energy of periodic vibration.     

On the propagation of second-sound in linear and nonlinear media: Results from Green-Naghdi theory

We investigate thermal wave propagation in one-dimensional media according to Green-Naghdi's heat conduction theory. Under the linearized theory, the dynamic propagation of a Heaviside input signal in a half-space is examined. Exact analytical solutions are derived for the three cases (i.e., types I-III) of this theory. We then numerically compare the evolution of the linear and nonlinear type-II temperature profiles, and track the finite-time blow-up of the latter's temperature rate wave, in the setting of an initial-boundary value problem involving a sudden sinusoidal input signal. Lastly, an exact traveling wave solution of a lossless, nonlinear equation, which arises under type-II theory, is determined and analyzed.