Journal articles on the topic 'Non-classical linear damping'

The presence of non-classical dissipation in a general discrete dynamic system is investigated through a perturbation method for the eigenvalues and vectors. Results accurate to second-order are obtained, with corrections to the base solution being expressed in terms of readily-calculated quadratic forms. Exact solutions, and the derived asymptotic ones, are compared with the predictions of the so-called method of approximate decoupling, in which certain non-classical dissipative terms are omitted from calculations in the eigenvalue problem. The perturbation method is discussed through its application in several examples, indicating circumstances in which a non-classically damped system can be well-approximated by an “equivalent” classically damped one. Somewhat surprisingly, the addition of non-classical damping does not necessarily increase the stability of all vibration modes, and the perturbation method is shown to be useful in identifying those critical modes.

This Technical Brief presents a new method for vibration analysis of a non-classically damped system. The basic idea is to introduce a transformation, which bears clear physical meaning, so that the original non-classical damped system is transformed into a new 2nd-order system that does not have the damping term. The transformed system not only provides an alternative of calculating response, but also reveals more clearly vibration behaviors of the original system.

Analysis of non-linear systems is an essential part of engineering structural dynamics. A number of methods have been developed in recent years. Classical Fourier-based methods have been extended to the use of phase plane, combined time-frequency, time-scale approaches and multidimensional spectra. This paper is an attempt to collate in one place some of the recent advances in wavelet analysis for the study of non-linear systems. This includes methods related to system identification based on wavelet ridges and skeletons, damping estimation procedures, wavelet-based frequency response functions, cross-wavelet analysis, self-similar signals, coherent structures and chaos.

This paper considers the linear dynamics of a single-degree-of-freedom non-viscously damped oscillator. It is assumed that the non-viscous damping force depends on the history of velocity via a convolution integral over an exponentially decaying kernel function. Classical qualitative dynamic properties known for viscously damped oscillators have been generalized to such non-viscously damped oscillators. The following questions of fundamental interest have been addressed: (i) under what conditions can a non-viscously damped oscillator sustain oscillatory motions? (ii) how does the natural frequency of a non-viscously damped oscillator compare with that of an equivalent undamped oscillator? and (iii) how does the decay rate compare with that of an equivalent viscously damped oscillator? Introducing two non-dimensional factors, namely, the viscous damping factor and the non-viscous damping factor, we provide answers to these questions. Wherever possible, attempts are made to relate the new results with equivalent classical results for a viscously damped oscillator.

The damping system characterizes the spatial distribution of structural energy dissipation. Identifying a damping system is the premise for determining the dynamic analysis method. Concepts of nonlinear processes and non-classical damping systems are often confused without theoretical primary and experimental verification. This paper proposes an identification method of damping systems based on motion states at the interface by analyzing the correlation between the damping system and dynamic characteristics. The relationship between the change of damping system type and relative motion states at the interface is studied by investigating multiple material properties through shaking table model tests of a large-scale soil-structure interaction (SSI) system. The results show that a nonlinear system can demonstrate the characteristics of the classical damping system as soon as there is no mutation of motion states at the interface of the system. The identification method of damping system based on motion states at the interface can reflect the change of dynamical characteristics of the system under linear and nonlinear processes.

Purpose The purpose of this paper is to study the effects of viscoelastic links between two adjacent buildings for pounding mitigation under white-noise seismic input. Design/methodology/approach A formulation is first extracted for the effective modal damping ratios of the system. Then, two single DOF linear buildings connected by viscoelastic links are considered with both classical and non-classical damping schemes. The inelastic behavior is also taken into account by using equivalent natural frequencies and damping ratios of the buildings. The effect of ground dominant frequency and damping on the displacement response is also investigated by using Kanai‒Tajimi filtered white noise as the random input. Findings The difference between classical and non-classical damping is shown to be less than 20 percent, implying the permission in using the simpler classical damping scheme. Finally, the problem is extended to two-storey buildings, where using viscoelastic links only at the top story level of the buildings is shown to be sufficient for controlling individual, as well as relative, motions of the structures. Originality/value Results demonstrate that the use of link with a moderate stiffness may reduce the stiffer building displacement up to approximately 20 percent in comparison to the free displacement, while the seismic pounding of the adjacent buildings is effectively controlled. Further, an upper limit of link stiffness is obtained for preventing the increase in the stiffer building displacement, which may be exceeded by the minimum link stiffness necessary for pounding prevention if small gap size exists.

Datum are the key of “Digital Earth”.In measurement, dealing with nonlinear models of observation datum, we may take their approximate values at observation values by Taylor series expansion, say, taking first-order item as a linear function of classical adjustment. But requirements of observation data, processing and accuracy assessment are higher and higher with today's fast-growing of high-tech mapping and surveying. So study on nonlinear least squares adjustment has been paid more and more attention. Damping least squares, as a modified algorithm of Gauss-Newton’s algorithm, is necessary to add a damping factor to improve the nature of a coefficient matrix. But it is difficult to choose a suitable damping factor, and needs to solve a group of linear equations repeatedly. In this paper, an improved damping least square was utilized for the non-linear processing of measurement datum in order to reduce a lot of computational workload.

AbstractIn this paper the concept of generalized form of proportional damping is proposed. Classical modal analysis of non-conservative continua is extended to multi DOF linear dynamic systems with asymmetric matrices. Mode orthogonality relationships have been generalized to non-conservative systems. Several discretization methods of continua are presented. Finally, an expression for derivatives of eigenvalues and eigenvectors of non-conservative system is presented. Examples are provided to illustrate the proposed methods.

This paper deals with the active control of a non-linear active landing gear system equipped with oleo pneumatic shock absorber. Runway induced vibration can cause reduction of pilot’s capability of control the aircraft and results the safety problem before take-off and after landing. Moreover, passenger–crew comfort is adversely affected by vertical vibrations of the fuselage. The active landing gears equipped with oleo pneumatic shock absorber are highly non-linear systems. In this study, uncertain polytopic state space representation is developed by modelling the pneumatic shock absorber dynamics as a mechanical system with non-linear stiffness and damping properties. Then, linear matrix inequalities-based robust linear quadratic regulator controller having pole location constraints is designed, since the classical linear quadratic regulator control design is dealing with linearized state space models without considering the non-linearities and uncertainties. Thereafter, numerical simulation studies are carried out to analyse aircraft response during taxiing. Bump- and random-type runway irregularities are used with various runway class and wide range of longitudinal speed. Simulation results revealed that neglecting the non-linear dynamics associated with oleo pneumatic shock absorber results significant performance degradation. Consequently, it is demonstrated that proposed robust linear quadratic regulator controller has a superior performance in terms of passenger–crew comfort and operational safety when compared to classical linear quadratic regulator.

A comparative research has been carried out for obtaining the time-consuming exact solution (state-space) and approximate solution (mode superposition) of transient and steady-state vibrations of linearly damped linear frame buildings. In the mode superposition method, the proportional damping matrix has been constructed by different approaches such as modal combination of mass and stiffness matrixes (Rayleigh) and disregarding the off-diagonal elements of the non-classical damping matrix, while in the state-space method the non-proportional damping matrix is constructed in exact situation. These observations are individually investigated, which the most suitable parameter to render the approximate results as close as possible to the exact results. Harmonic forces are applied on the different storeys of three and five storey frame buildings, and the responses are displayed in comparative tables and figures. The maximum responses are calculated by square root of sum of the squares (SRSS) method. A MATLAB code is generated and the equations of exact and approximate methods are solved.

In many engineering applications, the vibration analysis of a structure requires the set up of a large number of sensors. These studies are mostly performed in post processing and based on linear modal analysis. However, many studied devices highlight that modal parameters depend on the vibration level non linearities and are performed with sensors as accelerometers that modify the dynamics of the device. This work proposes a significant evolution of modal testing based on the real time identification of non linear parameters (natural frequencies and damping) tracked with a linear modal basis. This method, called Kinematic-SAMI (for multiSensors Assimilation Modal Identification) is assessed firstly on a numerical case with known non linearities and secondly in the framework of a classical cantilever beam with contactless measurement technique (high speed and high resolution cameras). Finally, the efficiency and the limits of the method are discussed.

In the present paper classical flutter phenomena in LP steam turbine rotor is studied . A reduced order aeroelastic model (ROAM) with acceptable accuracy and fast execution is developed for this purpose. The aerodynamics damping (AD) is estimated using Traveling wave mode (TWM) method. Flow field is modeled using Panel method. For the structural part ROM non-linear beam element method (BEM) based FEM structural solver is used. Partitioned based ( loose) coupling approach is adopted to perform aeroelastic (flutter) cosimulation. Both 2D cascade flow and 3D cascade are modeled.The estimated stability parameters are compared with experimental data. Moreover, present ROAM shows significant reduction in computational time.

The Voigt-type of dynamic vibration absorber is a classical model and has attracted considerable attention in many years because of its simple design, high reliability and useful applications in the fields of civil and mechanical engineering. Recently, a non-traditional type of dynamic vibration absorber was proposed. Unlike the traditional damped absorber configuration, the non-traditional absorber has a linear viscous damper connecting the absorber mass directly to the ground instead of the main mass. There have been some studies on the design of the non-traditional dynamic vibration absorber in the case of undamped primary structures. Those studies have shown that the non-traditional dynamic vibration absorber has better performance than the traditional dynamic vibration absorber. However, when damping is present at the primary system, there are very few studies on the design of non-traditional dynamic vibration absorber. This article presents a simple approach to determine the approximate analytical solutions for the [Formula: see text] optimization of the non-traditional dynamic vibration absorber attached to the damped primary structure subjected to force excitation. The main idea of the study is based on the dual criterion suggested by Anh in order to replace approximately the original damped structure by an equivalent undamped structure. Then the approximate analytical solution of dynamic vibration absorber’s parameters is given by using known results for undamped structure obtained. The comparisons have been done to verify the effectiveness of the obtained results.

SUMMARY We present a 2-D thermomechanical computational framework for simulating earthquake sequences in a non-linear visco-elasto-plastic compressible medium. The method is developed for a plane-strain problem and incorporates an invariant formulation of the classical rate- and state-dependent friction law and an adaptive time-stepping, which allows the time step to vary by many orders of magnitude during a simulation. Long-term tectonic convergence is imposed by displacing a boundary at a constant rate, whereas temperature-dependent viscosity is solved using a rapidly converging Newton–Raphson scheme. The 2-D volume is discretized using finite differences on a fully staggered grid and marker-in-cell techniques. An adaptive free-surface approximation is used to modulate the air viscosity with the time step, which allows stresses to vanish on the free surface during the propagation of fast slipping events. We present a set of increasingly complex models in which we investigate how inertia, radiation damping, thermally activated non-linear rheology and off-megathrust splay-fault events affect sequences of seismic and aseismic slip on a simplified subduction megathrust. The new method provides a unique computational framework to analyse earthquake sequences and to connect forearc deformation with the dynamic properties of the megathrust, thus providing a physical link between observations spanning from slow interseismic strain accumulation to localized coseismic slip of individual earthquakes and post-seismic viscoelastic relaxation.

Vibration equations of discrete multi-degrees-of-freedom (MDOF) structural systems is system of differential equations. In linear systems, the differential equations are also linear. Various analytical and numerical methods are available for solving the vibration equations in structural dynamics. In this paper modified differential transform method (MDTM) as a semi-analytical approach is generalized for the system of differential equations and is utilized for solving the vibration equations of MDOF systems. The MDTM is a recursive method which is a hybrid of Differential Transform Method (DTM), Pade' approximant and Laplace Transformation. A series of examples including forced and free vibration of MDOF systems with classical and non-classical damping are also solved by this method. Comparison of the results obtained by MDTM with exact solutions shows good accuracy of the proposed method; so that in some cases the solutions of the vibration equation that found by MDTM are the exact solutions. Also, MDTM is less expensive in computational cost and simpler with compare to the other available approaches.

This work deals with the application of a nature-inspired optimization technique to solve an inverse problem represented by the identification of an aircraft landing gear model. The model is described in terms of the landing gear geometry, internal volumes and areas, shock absorber travel, tire type, and gas and oil characteristics of the shock absorber. The solution to this inverse problem can be obtained by using classical gradient-based optimization methods. However, this is a difficult task due to the existence of local minima in the design space and the requirement of an initial guess. These aspects have motivated the authors to explore a nature-inspired approach using a method known as LifeCycle Model. In the present formulation two nature-based methods, namely the Genetic Algorithms and the Particle Swarm Optimization were used. An optimization problem is formulated in which the objective function represents the difference between the measured characteristics of the system and its model counterpart. The polytropic coefficient of the gas and the damping parameter of the shock absorber are assumed as being unknown: they are considered as design variables. As an illustration, experimental drop test data, obtained under zero horizontal speed, were used in the non-linear landing gear model updating of a small aircraft.

Fractional-order derivative has been shown an adequate tool to the study of so-called "anomalous" social and physical behaviors, in reflecting their non-local, frequency- and history-dependent properties, and it has been used to model practical systems in engineering successfully, including the famous Bagley-Torvik equation modeling forced motion of a rigid plate immersed in Newtonian fluid. The solutions of the initial value problems of linear fractional differential equations are usually expressed in terms of Mittag-Leffler functions or some other kind of power series. Such forms of solutions are not good for engineers not only in understanding the solutions but also in investigation. This paper proves that for the linear SDOF oscillator with a damping described by fractional-order derivative whose order is between 1 and 2, the solution of its initial value problem free of external excitation consists of two parts, the first one is the 'eigenfunction expansion' that is similar to the case without fractional-order derivative, and the second one is a definite integral that is independent of the eigenvalues (or characteristic roots). The integral disappears in the classical linear oscillator and it can be neglected from the solution when stationary solution is addressed. Moreover, the response of the fractionally damped oscillator under harmonic excitation is calculated in a similar way, and it is found that the fractional damping with order between 1 and 2 can be used to produce oscillation with large amplitude as well as to suppress oscillation, depending on the ratio of the excitation frequency and the natural frequency.

This study is concerned with a new generalized mathematical model for single degree-of-freedom bistable oscillators with harmonic excitation of low-frequency, linear viscous damping and a restoring force that contains a negative linear term and a positive nonlinear term which is a power-form function of the generalized coordinate. Comprehensive numerical mapping of the range of bifurcatory behaviour shows that such non-autonomous systems can experience mixed-mode oscillations, including bursting oscillations (fast flow oscillations around the outer curves of a slow flow), and relaxation oscillations like a classical (autonomous) van der Pol oscillator. After studying the global system dynamics the focus of the investigations is on cubic oscillators of this type. Approximate techniques are presented to quantify their response, i.e. to determine approximations for both the slow and fast flows. In addition, a clear analogy between the behaviour of two archetypical oscillators—the non-autonomous bistable oscillator operating at low frequency and the strongly damped autonomous van der Pol oscillator—is established for the first time.

This paper is interested in solution of two-dimensional aeroelastic problems. Two mathematical models are compared for a benchmark problem. First, the classical approach of linearized aerodynamical forces is described to determine the aeroelastic instability and the aeroelastic response in terms of frequency and damping coefficient. This approach is compared to the coupled fluid-structure model solved with the aid of finite element method used for approximation of the incompressible Navier-Stokes equations. The finite element approximations are coupled to the non-linear motion equations of a flexibly supported airfoil. Both methods are first compared for the case of small displacement, where the linearized approach can be well adopted. The influence of nonlinearities for the case of post-critical regime is discussed.

ABSTRACT Galactic bars are unstable to a vertical buckling instability which heats the disc and in some cases forms a boxy/peanut shaped bulge. We analyse the buckling instability as an application of classical Euler buckling followed by non-linear gravitational Landau damping in the collisionless system. We find that the buckling instability is dictated by the kinematic properties and geometry of the bar. The analytical result is compared to simulations of isolated galaxies containing the disc and dark matter components. Our results demonstrate that violent buckling does not destroy bars while a less energetic buckling can dissolve the bar. The discs that undergo gentle buckling remain stable to bar formation which may explain the observed bar fraction in the local Universe. Our results align with the results from recent surveys.

Context. Rapidly decaying slow magnetoacoustic waves are regularly observed in the solar coronal structures, offering a promising tool for a seismological diagnostics of the coronal plasma, including its thermodynamical properties. Aims. The effect of damping of standing slow magnetoacoustic oscillations in the solar coronal loops is investigated accounting for field-aligned thermal conductivity and a wave-induced misbalance between radiative cooling and some unspecified heating rates. Methods. The non-adiabatic terms were allowed to be arbitrarily large, corresponding to the observed values. The thermal conductivity was taken in its classical form, and a power-law dependence of the heating function on the density and temperature was assumed. The analysis was conducted in the linear regime and in the infinite magnetic field approximation. Results. The wave dynamics is found to be highly sensitive to the characteristic timescales of the thermal misbalance. Depending on certain values of the misbalance, timescales three regimes of the wave evolution were identified, namely the regime of a suppressed damping, enhanced damping in which the damping rate drops down to observational values, and acoustic over-stability. The specific regime is determined by the dependences of the radiative cooling and heating functions on thermodynamical parameters of the plasma in the vicinity of the perturbed thermal equilibrium. Conclusions. The comparison of the observed and theoretically derived decay times and oscillation periods allows us to constrain the coronal heating function. For typical coronal parameters, the observed properties of standing slow magnetoacoustic oscillations could be readily reproduced with a reasonable choice of the heating function.

A theory is developed for the relaxation of two-dimensional non-radiative (Coulomb) plasmon-excitons in thin closely located layers of a metal and a semiconductor. In the framework of classical electrodynamics, the equations of motion are formulated for the polarization waves of non-radiative plasmons and excitons with taking into account the Coulomb coupling and the near-field of external polarization. In the model of coupled harmonic oscillators represented by the polarization fields of excitations, the problem of relaxation is solved for Coulomb plasmons, excitons and plasmon-excitons. It is shown that the two dispersion branches of normal plasmon-exciton modes undergo anticrossing (mutual repulsion) at the resonance between plasmon and exciton. With dissipative damping and power interchange between the excitations taken into account, the process of plasmon-exciton relaxation depending on time is investigated. The theory displays the principal analogies between dynamics of plasmon-excitons and of excitations in other objects of linear vibration theory, such as mechanical oscillators, resonant electric chains, etc.

In this paper, we consider an initial-boundary value problem for a homogeneous string (or wave) equation. One end of the string is assumed to be fixed and the other end of the string is attached to a spring-mass-dashpot system, where the damping generated by the dashpot is assumed to be small. This problem can be regarded as a simple model describing oscillations of flexible structures such as overhead power transmission lines. A semigroup approach will be used to show the well-posedness of the problem as well as the asymptotic validity of formal approximations of the solution on long timescales. A multiple timescales perturbation method will be used to construct asymptotic approximations of the solution. Although the problem is linear the construction of these approximations is far from being elementary because of the complicated, non-classical boundary condition.

The complexity of the internal dynamics of a modular multi-level converter (MMC) has raised severe issues for designing corresponding controllers. The existing MMC cascaded control strategies, based on classical linear control theory, require a relatively complex structure to achieve control objectives and the parameter tuning processes during the corresponding controller design are normally difficult to solve for the highly non-linear systems with highly coupled states in MMC. On account of this, advanced controllers are required for the regulation tasks of MMC. Passivity is introduced into the MMC control system by the passive control (PC) proposed in this paper. PC can provide an extra damping effect to help save energy through utilizing passivity in the system. A controllable de-coupled form is achieved by passivation of the output calculation. Hence, well-tuned controllers can be designed and employed to effectively regulate the output current and inner differential currents of the MMC under system operating point variation. Simulation results yield numerical data that show significantly improved steady-state and transient-state performances with greatly reduced control costs.

This paper introduced analysis and improvement of power networks stability. It focused on the impact of flexible AC transmission systems (FACTS) device interaction with the other components of the network. It investigated the impact of dynamic charges on the ability of FACTS to eliminate power oscillations problems. A small-signal analysis, frequency analysis and non-linear time simulations using EUROSTAG made it possible to study these problems. Other, research has shown that the damping loops of the power oscillations effects using classical techniques of sensibility are not robust in relation to the variations of load models. Thus, this paper proposed a method based on the sensitivity of the eigenvalues and it takes into account the variations of the load models. The method calculates an optimal phase compensator based on a weighted average of the sensitivity of the target mode. It considers the variations of sensitivity as a function of the uncertainty in the load model. According to the obtained results, this method is effective in most stability problems.

This paper presents the development of a mathematical approach targeting the modelling and analysis of coupled flap-lag-torsion vibration characteristics of non-uniform continuous rotor blades. The proposed method is based on the deployment of Lagrange’s equation of motion to the three-dimensional kinematics of rotor blades. Modal properties derived from classical-beam and torsion theories are utilized as assumed deformation functions. The formulation, which is valid for hingeless, freely hinged and spring-hinged articulated rotor blades, is reduced to a set of closed-form integral expressions. Numerical predictions for mode shapes and natural frequencies are compared with experimental measurements, non-linear finite element analyses and multi-body dynamics analyses for two small-scale hingeless rotor blades. Excellent agreement is observed. The effect of different geometrical parameters on the elastic and inertial coupling is assessed. Additionally, the effect of the inclusion of gyroscopic damping is evaluated. The proposed method, which is able to estimate the first seven coupled modes of vibration in a fraction of a second, exhibits excellent numerical stability. It constitutes a computationally efficient alternative to multi-body dynamics and finite element analysis for the integration of rotor blade flexible modelling into a wider comprehensive rotorcraft tool.

Purpose The purpose of this paper is to investigate the evolution of the permeability of spherical packing during cold compaction by pore-scale modeling. Design/methodology/approach The discrete element method (DEM) is used to generate spherical packing structure under different compressive pressures and the Lattice Boltzmann method (LBM) is adopted to calculate the permeability of each spherical assembly. Findings It is found that the decrease of the porosity is the main reason of the reduction in permeability in the initial compression stage, but its influence becomes insufficient in the late compression stages. Besides, two empirical formulas are obtained, which describe the relation between the permeability and the equivalent mean diameter and the variation of normalized permeability with compressive pressure, respectively. Research limitations/implications In this study, the authors study the spherical particles and ignore the non-spherical effects. Besides, the classical contact model, the linear-spring-damping model, is used in DEM, so the plastic deformation cannot be considered. Originality/value The DEM and the LBM are well combined to study the compaction effects on permeability of spherical packing. Two simple expressions of the spherical packing structure with uniform diameter distribution are given for the first time.

The non-linear vibration of a squeeze film damper (SFD) supported rotor assembly is closely linked to the presence of air bubbles in the lubricant, due to cavitation, where the discrete gas phase influences the squeeze film pressure profile and gives rise to a non-linear stiffness force. The aim of this paper is to assess the ability of a computational model for homogeneous bubbly oil to predict the influence of air bubbles on the film reaction forces under various operating parameters. The numerical model which considers the solubility of gas and the growth of gas bubbles was developed using the finite-element software, FEMLAB™. Parametric studies of eccentricity ratio, whirling frequency, and supply pressure were conducted to evaluate the influences of air bubbles on the pressure field and hence the squeeze film forces. Results show that an increase in eccentricity ratio and whirling frequency enhances the growth of air content in lubricant and hence increases the radial (stiffness) to tangential (damping) force ratio in an SFD, whereas an opposite effect is gained by applying higher supply pressure. Compared with the classical theoretical half-film model predictions, the bubbly oil model provides a more realistic estimation with respect to different damper operating conditions. From the simulation findings, it can be concluded that the homogeneous two-phase flow model reasonably describes the bubbly oil behaviour in SFDs and effectively shows the rise in stiffness force due to the growth of air bubbles. The homogeneous model can be easily applied to the well-established lubrication equation and solved with efficiency. However, any possible interfacial effects between the liquid and gas phases are inevitably concealed. The success of the current model allows its subsequent coupling with a structural rotor to form a multi-disciplinary model for unbalance analysis.

The improvement of heat transfer conditions in liquid-metal magnetohydrodynamic (MHD) flows is of prime importance for self-cooled fusion blanket design concepts. For many years the research was based on stationary inertialess assumptions since it was expected that time-dependent inertial flows would be suppressed by strong electromagnetic damping, especially in the extreme range of fusion relevant parameters. In the present analysis the stationary inertialess assumptions are abandoned. Nevertheless, the classical ideas usually used to obtain inertialess asymptotic solutions are drawn on. The basic inertial equations are reduced to a coupled two-dimensional problem by analytical integration along magnetic field lines. The magnetic field is responsible for a quasi-two-dimensional flow; the non-uniform distribution of the wall conductivity creates a wake-type profile, the MHD effect reducing to a particular forcing and friction. The solution for the two-dimensional variables, the field aligned component of vorticity, the stream function, and the electric potential are obtained by numerical methods. In a flat channel with non-uniform electrical wall conductivity, time-dependent solutions similar to the Kármán vortex street behind bluff bodies are possible. The onset of the vortex motion, i.e. the critical Reynolds number depends strongly on the strength of the magnetic field expressed by the Hartmann number. Stability analyses in viscous hydrodynamic wakes often use the approximation of a unidirectional flow which does not take into account the spatial evolution of the wake. The present problem exhibits a wake-type basic flow, which does not change along the flow path. It represents, therefore, an excellent example to which the simple linear analysis on the basis of Orr-Sommerfeld-type equations applies exactly. Once unstable, the flow first exhibits a regular time periodic vortex pattern which is rearranged further downstream. One can observe an elongation, pairing, or sometimes more complex merging of vortices. All these effects lead to larger flow structures with lower frequencies. The possibility for a creation and maintenance of time-dependent vortex-type flow pattern in MHD flows is demonstrated.

The solution of the so-called problem of speed of response in one coordinate, which has important theoretical and practical importance, is investigated. It is formulated with reference to linear one-dimensional high-order control objects described by a system of ordinary differential equations in a certain phase space. The transient time tnn of the system designed is understood in a sense of the classical control theory in reference to one (output) coordinate of the object and is determined by using the zone Δ = σ* = 4.321 %, which equals the given (desirable) value of the overshoot of the system synthesized. This overshoot corresponds with the speed of response oscillating second-order element with a damping coefficient ζ= = 2 2 0,7071 / . It is indispensable to mention here that the equation Δ = σ is one of the necessary conditions for the maximum speed of response of the system with the oscillating character of transient processes. In accordance to this the task of the speed of response by one coordinate can be described by the following generalized formulation: one must find the linear algorithm of the feedback signal, which provides a preset order of the astatism na for the closed-loop control system and converts the control object from a zero state into a final state, which is determined by the constant signal of the input, with a minimal time value of the transient processes of the system tnn and the preset value of the overshoot σ m σ* while fulfilling the constraint of the control signal |u(t)| m umax. Nowadays the task mentioned is approximately solved by the algebraic method of the synthesis of linear control systems with the determination of a desirable transfer function of the designed closed-loop system based on model normalized transfer functions (NTF). In the works by Kim D. P. there was carried out the analysis of four types of normalized transfer functions characterized by the increased speed of response. In this work two additional types of normalized transfer functions are suggested, in comparison with mentioned NTF they have the increased speed of response in case of the preset value of the overregulation σ* = 4.321 %. On their basis and using the methodology of the modal control the method of the synthesis of the controller is suggested; this method ensures the transient time of the designed system to be close to the minimum in case of the preset constraint of the overregulation and the value of the control signal. It needs to be emphasized that in contrast to the algebraic method of the synthesis, this method is applied to a wider range of control objects: as to minimal-phased objects as to non-minimum-phased ones; as to the objects containing zeros as to those without them. The method is illustrated by an example of synthesis of control system speed of response of the fourth order, containing the results of its modeling.

Many dynamic simulations of a rotor with a journal bearing employ non-linear fluid-film lubrication models and calculate the bearing coefficients at each time step. However, calculating such a simulation is tedious and computationally expensive. This paper presents a simplified dynamic simulation model of a vertical rotor with tilting pad journal bearings under constant and variable (transient) rotor spin speed. The dynamics of a four-shoes tilting pad journal bearing are predefined using polynomial equations prior to the unbalance response simulations of the rotor-bearing system. The Navier–Stokes lubrication model is solved numerically, with the bearing coefficients calculated for six different rotor speeds and nine different eccentricity amplitudes. Using a MATLAB inbuilt function (poly53), the stiffness and damping coefficients are fitted by a two-dimensional polynomial regression and the model is qualitatively evaluated for goodness-of-fit. The percentage relative error (RMSE%) is less than 10%, and the adjusted R-square (Radj2) is greater than 0.99. Prior to the unbalance response simulations, the bearing parameters are defined as a function of rotor speed and journal location. The simulation models are validated with an experiment based on the displacements of the rotor and the forces acting on the bearings. Similar patterns have been observed for both simulated and measured orbits and forces. The resultant response amplitudes increase with the rotor speed and unbalanced magnitude. Both simulation and experimental results follow a similar trend, and the amplitudes agree with slight deviations. The frequency content of the responses from the simulations is similar to those from the experiments. Amplitude peaks, which are associated with the unbalance force (1 × Ω) and the number of pads (3 × Ω and 5 × Ω), appeared in the responses from both simulations and experiments. Furthermore, the suggested simulation model is found to be at least three times faster than a classical simulation procedure that used FEM to solve the Reynolds equation at each time step.

The behavior of acoustic waves in a rarefied high-temperature plasma is studied; as an example, the plasma of the solar corona is considered. Effects of thermal conductivity and a heating/radiative loss are taken into account; data on a temperature distribution of a radiation intensity obtained from the CHIANTI 10 code are used. The classical Spitzer expression for a full-ionized plasma is used for the thermal conductivity. Based on the found values of the radiation-loss function, the cubic spline method is used to construct an approximate analytical expression necessary for studying linear waves. A dispersion relation is obtained, and a frequency, a phase speed, and a damping coefficient are found. Dispersion and damping properties are considered for a temperature of about 106 K and a particle density of about 1015m−3, which are typical for the coronal plasma. In sum, superiority in the dispersion and damping of the thermal conduction is shown; the heating and radiation loss manifest themselves at large wavelengths. In accordance with general results by Field, a condition was found under which the acoustic oscillations become unstable. It is shown that at certain values of the temperature and density, the wave damping is dominated by the heating/radiative loss misbalance. Thus, the earlier results on mechanisms of damping of observed acoustic waves in the solar corona are refined here.

We study the dynamics of a linear, uniform, undamped string under harmonic base excitation, with an attachment consisting of either a spring–dashpot system or a vibration absorber. Mode complexity caused by the local damping of the attachment can lead to coexistence of vibrations and waves in the string. We consider either identical harmonic base motions at both ends or harmonic base excitation at one end. In the case of double harmonic base excitation, it is possible to choose the parameters of the attachment, so that the mode complexity is maximal in one part of the string (leading to travelling waves and elimination of vibrations) and almost zero in the other part (with standing waves or vibration modes). Similarly, for single base excitation, we analytically predict the parameters of the attachment that maximize mode complexity and enhance the interplay of vibrations and travelling waves in the string. Under such conditions, the system acts as a passive vibration confinement device, with induced energy being transmitted through travelling waves to a region where it is confined in the form of standing waves. Our results can be used for predictive design and reveal an unexpected new application of the classical linear vibration absorber.

In this paper, the effects of the field damping and Kerr-like medium on a system of a four-level atom in the [Formula: see text]-type interacting nonlinearly with a two-mode field are studied. Analytical solution of the system of differential equations resulting from the Schrödinger equation is obtained. The effects of detuning parameters, Kerr-like parameter, coupling function and damping rate are discussed for atomic population inversion, the second-order correlation function, Shannon information entropy, and linear entropy. Under justifiable all conditions, extreme entanglement states may appear periodically with the evolution of time. The arrival of the quantum system to a pure state can be controlled by variations of the Kerr medium and the damping parameters. The results also exhibit that the periods of strong and weak entanglement between the parts of the system are achieved during the periods of collapses and revivals of the atomic population inversion. Moreover, the photon distribution changes from non-classical to classical by increasing the damping rate.

We examine the properties of linear electrostatic waves in unmagnetised quantum and classical plasmas consisting of one or two populations of electrons with analytically tractable distribution functions in the presence of a stationary neutralising ion background. Beginning with the kinetic quantum plasma longitudinal susceptibility, we assess the effects due to increasing complexity of the background distribution function. Firstly, we analyse dispersion and Landau damping in one-component plasmas and consider distribution functions with a variety of analytical properties: the Dirac delta function, the Cauchy profile with two complex first-order poles, the squared Cauchy profile with two second-order poles and the inverse-quartic profile with four first-order poles; we also briefly discuss the non-meromorphic totally and arbitrarily degenerate Fermi–Dirac distribution. In order to study electrostatic instabilities, we then turn to plasmas with two populations of electrons streaming relative to each other in two cases: a symmetric case of two counter-streaming identical populations and a bump-on-tail case with a primary population and a delta-function beam. We obtain the corresponding linear kinetic dispersion relations and evaluate the properties of instabilities when the electron distribution functions are of the delta function, Cauchy, squared Cauchy or inverse-quartic types. In agreement with other studies, we find that in general quantum effects reduce the range of wavelengths for unstable modes at long wavelengths. We also find a second window of instability at shorter wavelengths and elucidate its nature as being due to quantum recoil. We note the possible implications for studies of laboratory and astrophysical quantum plasmas.

AbstractFifty years ago Kostin (J Chem Phys 57(9):3589–3591, 1972. https://doi.org/10.1063/1.1678812) proposed a description of damping in quantum mechanics based on a nonlinear Schrödinger equation with the potential being governed by the phase of the wave function. We show for the example of a moving Gaussian wave packet, that the deceleration predicted by this equation is the result of the same non-dissipative, homogeneous but time-dependent force, that also stops a classical particle. Moreover, we demonstrate that the Kostin equation is a special case of the linear Schrödinger equation with three potentials: (i) a linear potential corresponding to this stopping force, (ii) an appropriately time-dependent parabolic potential governed by a specific time dependence of the width of the Gaussian wave packet and (iii) a specific time-dependent off-set. The freedom of the width opens up the possibility of engineering the final state by the time dependence of the quadratic potential. In this way the Kostin equation is a precursor of the modern field of coherent control. Motivated by these insights, we analyze in position and in phase space the deceleration of a Gaussian wave packet due to potentials in the linear Schrödinger equation similar to those in the Kostin equation.

Time-dependent flows are notoriously challenging for classical linear stability analysis. Most progress in understanding the linear stability of these flows has been made for time-periodic flows via Floquet theory focusing on time-asymptotic stability. However, little attention has been given to the transient intracyclic linear stability of periodic flows since no general tools exist for its analysis. In this work, we explore the potential of using the recent framework of the optimally time-dependent (OTD) modes (Babaee & Sapsis, Proc. R. Soc. Lond. A, vol. 472, 2016, 20150779) to extract information about both the transient and the time-asymptotic linear stability of pulsating Poiseuille flow. The analysis of the instantaneous OTD modes in the limit cycle leads to the identification of the dominant instability mechanism of pulsating Poiseuille flow by comparing them with the spectrum and the eigenmodes of the Orr–Sommerfeld operator. In accordance with evidence from recent direct numerical simulations, it is found that structures akin to Tollmien–Schlichting waves are the dominant feature over a large range of pulsation amplitudes and frequencies but that for low pulsation frequencies these modes disappear during the damping phase of the pulsation cycle as the pulsation amplitude is increased beyond a threshold value. The maximum achievable non-normal growth rate during the limit cycle was found to be nearly identical to that in plane Poiseuille flow. The existence of subharmonic perturbation cycles compared with the base flow pulsation is documented for the first time in pulsating Poiseuille flow.

Using our non-local and time-dependent theory of convection and a fixed set of convective parameters (C1, C2/C1 , C3)= (0.70, 0.50, 3.0) calibrated against the Sun, the linear non-adiabatic oscillations for evolutionary models with masses 1–20 M⊙ are calculated. The results show that almost all the classical instability strips can be reproduced. The theoretical instability strips of δ Scuti and γ Doradusvariables agree well with Kepler spacecraft observations. There is no essential difference in the excitation mechanism for δ Scuti and γ Doradus stars. They are excited by the combined effects of the radiative κ-mechanism and coupling between convection and oscillations. They represent two subgroups of a broader type of δ Scuti and γ Doradus stars, located in the lower part of the Cepheid instability strip. δ Scuti is the p-mode subgroup and γ Doradus is the g-mode subgroup. The luminous variable red giants observed by MACHO and OGLE are low-order radial pulsators among low-mass red giant and asymptotic giant branch stars. The excitation and damping mechanism of oscillations for low-temperature stars is studied in detail. Convective flux and turbulent viscosity are consistent damping mechanisms. The damping effect of the convective enthalpy flux is inversely proportional to the frequency of the modes, so it plays an important role in stabilizing the low-order modes and defining the red edge of the Cepheid instability strip. The damping effect of turbulent viscosity reaches its maximum at 3ωτc/16∼1, where τc is the dynamic time scale of turbulent convection and ω is the angular frequency of the modes. Turbulent viscosity is the main damping mechanism for stabilizing the high-order modes of low-temperature variables. The turbulent pressure is, in general, an excitation mechanism; it reaches maximum at 3ωτc/4∼1, and it plays an important role for the excitation of red variables. Convection is not, in fact, a pure damping effect for stellar oscillations. The relative contributions of turbulent pressure, turbulent viscosity, and convective enthalpy flux for excitation and damping effects change with stellar parameters (mass, luminosity, effective temperature) and with the radial order and spherical harmonic degree of the oscillation mode; therefore, the combined effect of convection is sometimes damping, and sometimes the excitation of oscillations. Our research shows that, for low-luminosity red giants, the low-order modes are pulsationally stable, while the intermediate- and high-order modes are unstable. Toward higher luminosity, the range of unstable modes shifts gradually toward the lower order. All of the intermediate- and high-order modes become stable, and a few low-order modes become unstable for high-luminosity red giants. They show the typical pulsational characteristics of Mira-like variables. The variable red giants are, at least for the high-luminosity RGs, self-excited. For red giants, the frequency of the maximally unstable modes predicted by our theory is similar to that given by the semi-empirical scaling relation.

These lecture notes were presented by Allan N. Kaufman in his graduate plasma theory course and a follow-on special topics course (Physics 242A, B, C and Physics 250 at the University of California Berkeley). The notes follow the order of the lectures. The equations and derivations are as Kaufman presented, but the text is a reconstruction of Kaufman’s discussion and commentary. The notes were transcribed by Bruce I. Cohen in 1971 and 1972, and word processed, edited and illustrations added by Cohen in 2017 and 2018. The series of lectures is divided into four major parts: (i) collisionless Vlasov plasmas (linear theory of waves and instabilities with and without an applied magnetic field, Vlasov–Poisson and Vlasov–Maxwell systems, Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) eikonal theory of wave propagation); (ii) nonlinear Vlasov plasmas and miscellaneous topics (the plasma dispersion function, singular solutions of the Vlasov–Poisson system, pulse-response solutions for initial-value problems, Gardner’s stability theorem, gyroresonant effects, nonlinear waves, particle trapping in waves, quasilinear theory, nonlinear three-wave interactions); (iii) plasma collisional and discreteness phenomena (test-particle theory of dynamic friction and wave emission, classical resistivity, extension of test-particle theory to many-particle phenomena and the derivation of the Boltzmann and Lenard–Balescu equations, the Fokker–Planck collision operator, a general scattering theory, nonlinear Landau damping, radiation transport and Dupree’s theory of clumps); (iv) non-uniform plasmas (adiabatic invariance, guiding-centre drifts, hydromagnetic theory, introduction to drift-wave stability theory).

This paper will compare the metaphoric structuring of the ecological concept of resilience—with its roots in Holling's 1973 paper; with psychological concepts of resilience which followed from research—such as Werner, Bierman, and French and Garmezy and Streitman) published in the early 1970s. This metaphoric analysis will expose the difference between complex adaptive systems models of resilience in ecology and studies related to resilience in relation to climate change; compared with the individualism of linear equilibrium models of resilience which have dominated discussions of resilience in psychology and economics. By examining the ontological commitments of these competing metaphors, I will show that the individualistic concept of resilience which dominates psychological discussions of resilience is incompatible with the ontological commitments of ecological concepts of resilience. Because the ontological commitments of the concepts of ecological resilience on the one hand, and psychological resilience on the other, are so at odds with one another, it is important to be clear which concept of resilience is being evaluated for its adequacy as a concept. Having clearly distinguished these competing metaphors and their ontological commitments, this paper will show that it is the complex adaptive systems model of resilience from ecology, not the individualist concept of psychological resilience, that has been utilised by both the academic discussions of adaptation to climate change, and the operationalisation of the concept of resilience by social movements like the permaculture, ecovillage, and Transition Towns movements. Ontological Metaphors My analysis of ontological metaphors draws on insights from Kuhn's (114) account of gestalt perception in scientific paradigm shifts; the centrality of the role of concrete analogies in scientific reasoning (Masterman 77); and the theorisation of ontological metaphors in cognitive linguistics (Gärdenfors). Figure 1: Object Ontological commitments reflect the shared beliefs within a community about the sorts of things that exist. Our beliefs about what exists are shaped by our sensory and motor interactions with objects in the physical world. Physical objects have boundaries and surfaces that separate the object from not-the-object. Objects have insides and outsides, and can be described in terms of more-or-less fixed and stable “objective” properties. A prototypical example of an “object” is a “container”, like the example shown in Figure 1. Ontological metaphors allow us to conceive of “things” which are not objects as if they were objects by picking “out parts of our experience and treat them as [if they were] discrete entities or substances of a uniform kind” (Lakoff and Johnson 25). We use ontological metaphors when we imagine a boundary around a collection of things, such as the members of a team or trees in a forest, and conceive of them as being in a container (Langacker 191–97). We can then think of “things” like a team or forest as if they were a single entity. We can also understand processes and activities as if they were things with boundaries. Whether or not we characterise some aspect of our experience as a noun (a bounded entity) or as a verb (a process that occurs over time) is not determined by the nature of things in themselves, but by our understanding and interpretation of our experience (Langacker 233). In this paper I employ a technique that involves examining the details of “concrete images” from the source domains for metaphors employed in the social sciences to expose for analysis their ontological commitments (Harrison, “Politics” 215; Harrison, “Economics” 7). By examining the ontological metaphors that structure the resilience literature I will show how different conceptions of resilience reflect different beliefs and commitments about the sorts of “things” there are in the world, and hence how we can study and understand these “things.” Engineering Metaphors In his discussion of engineering resilience, Holling (“Engineering Vs. Ecological” 33) argues that this conception is the “foundation for economic theory”, and defined in terms of “resistance to disturbance and the speed of return to the equilibrium” or steady state of the system. Whereas Holling takes his original example of the use of the engineering concept of resilience from economics, Pendall, Foster, & Cowell (72), and Martin-Breen and Anderies (6) identify it as the concept of resilience that dominates the field of psychology. They take the stress loading of bridges to be the engineering source for the metaphor. Figure 2: Pogo stick animation (Source: Blacklemon 67, CC http://en.wikipedia.org/wiki/File:Pogoanim.gif). In order to understand this metaphor, we need to examine the characteristics of the source domain for the metaphor. A bridge can be “under tension, compression or both forces at the same time [and] experiences what engineers define as stress” (Matthews 3). In order to resist these forces, bridges need to be constructed of material which “behave much like a spring” that “strains elastically (deforms temporarily and returns to its original shape after a load has been removed) under a given stress” (Gordon 52; cited in Matthews). The pogostick shown in Figure 2 illustrates how a spring returns to its original size and configuration once the load or stress is removed. WGBH Educational Foundation provides links to simple diagrams that illustrate the different stresses the three main designs of bridges are subject to, and if you compare Computers & Engineering's with Gibbs and Bourne's harmonic spring animation you can see how both a bridge under live load and the pogostick in Figure 2 oscillate just like an harmonic spring. Subject to the elastic limits of the material, the deformation of a spring is proportional to the stress or load applied. According to the “modern theory of elasticity [...] it [is] possible to deduce the relation between strain and stress for complex objects in terms of intrinsic properties of the materials it is made of” (“Hooke’s Law”). When psychological resilience is characterised in terms of “properties of individuals [that] are identified in isolation” (Martin-Breen and Anderies 12); and in terms of “behaviours and attributes [of individuals] that allow people to get along with one another and to succeed socially” (Pendall, Foster, and Cowell 72), they are reflecting this engineering focus on the properties of materials. Martin-Breen and Anderies (42) argue that “the Engineering Resilience framework” has been informed by ontological metaphors which treat “an ecosystem, person, city, government, bridge, [or] society” as if it were an object—“a unified whole”. Because this concept of resilience treats individuals as “objects,” it leads researchers to look for the properties or characteristics of the “materials” which individuals are “made of”, which are either elastic and allow them to “bounce” or “spring” back after stress; or are fragile and brittle and break under load. Similarly, the Designers Institute (DINZ), in its conference on “Our brittle society,” shows it is following the engineering resilience approach when it conceives of a city or society as an object which is made of materials which are either “strong and flexible” or “brittle and fragile”. While Holling characterises economic theory in terms of this engineering metaphor, it is in fact chemistry and the kinetic theory of gases that provides the source domain for the ontological metaphor which structures both static and dynamic equilibrium models within neo-classical economics (Smith and Foley; Mirowski). However, while springs are usually made out of metals, they can be made out of any “material [that] has the required combination of rigidity and elasticity,” such as plastic, and even wood (in a bow) (“Spring (device)”). Gas under pressure turns out to behave the same as other springs or elastic materials do under load. Because both the economic metaphor based on equilibrium theory of gases and the engineering analysis of bridges under load can both be subsumed under spring theory, we can treat both the economic (gas) metaphor and the engineering (bridge) metaphor as minor variations of a single overarching (spring) metaphor. Complex Systems Metaphors Holling (“Resilience & Stability” 13–15) critiques equilibrium models, arguing that non-deterministic, complex, non-equilibrium and multi-equilibrium ecological systems do not satisfy the conditions for application of equilibrium models. Holling argues that unlike the single equilibrium modelled by engineering resilience, complex adaptive systems (CAS) may have multi or no equilibrium states, and be non-linear and non-deterministic. Walker and Salt follow Holling by calling for recognition of the “dynamic complexity of the real world” (8), and that “these [real world] systems are complex adaptive systems” (11). Martin-Breen and Anderies (7) identify the key difference between “systems” and “complex adaptive systems” resilience as adaptive capacity, which like Walker and Salt (xiii), they define as the capacity to maintain function, even if system structures change or fail. The “engineering” concept of resilience focuses on the (elastic) properties of materials and uses language associated with elastic springs. This “spring” metaphor emphasises the property of individual components. In contrast, ecological concepts of resilience examine interactions between elements, and the state of the system in a multi-dimensional phase space. This systems approach shows that the complex behaviour of a system depends at least as much on the relationships between elements. These relationships can lead to “emergent” properties which cannot be reduced to the properties of the parts of the system. To explain these relationships and connections, ecologists and climate scientists use language and images associated with landscapes such as 2-D cross-sections and 3-D topology (Holling, “Resilience & Stability” 20; Pendall, Foster, and Cowell 74). Figure 3 is based on an image used by Walker, Holling, Carpenter and Kinzig (fig. 1b) to represent possible states of ecological systems. The “basins” in the image rely on our understanding of gravitational forces operating in a 3-D space to model “equilibrium” states in which the system, like the “ball” in the “basin”, will tend to settle. Figure 3: (based on Langston; in Walker et al. fig. 1b) – Tipping Point Bifurcation Wasdell (“Feedback” fig. 4) adapted this image to represent possible climate states and explain the concept of “tipping points” in complex systems. I have added the red balls (a, b, and c to replace the one black ball (b) in the original which represented the state of the system), the red lines which indicate the path of the ball/system, and the black x-y axis, in order to discuss the image. Wasdell (“Feedback Dynamics” slide 22) takes the left basin to represents “the variable, near-equilibrium, but contained dynamics of the [current] glacial/interglacial period”. As a result of rising GHG levels, the climate system absorbs more energy (mostly as heat). This energy can force the system into a different, hotter, state, less amenable to life as we know it. This is shown in Figure 3 by the system (represented as the red ball a) rising up the left basin (point b). From the perspective of the gravitational representation in Figure 3, the extra energy in the basin operates like the rotation in a Gravitron amusement ride, where centrifugal force pushes riders up the sides of the ride. If there is enough energy added to the climate system it could rise up and jump over the ridge/tipping point separating the current climate state into the “hot earth” basin shown on the right. Once the system falls into the right basin, it may be stuck near point c, and due to reinforcing feedbacks have difficulty escaping this new “equilibrium” state. Figure 4 represents a 2-D cross-section of the 3-D landscape shown in Figure 3. This cross-section shows how rising temperature and greenhouse gas (GHG) concentrations in a multi-equilibrium climate topology can lead to the climate crossing a tipping point and shifting from state a to state c. Figure 4: Topographic cross-section of possible climate states (derived from Wasdell, “Feedback” 26 CC). As Holling (“Resilience & Stability”) warns, a less “desirable” state, such as population collapse or extinction, may be more “resilient”, in the engineering sense, than a more desirable state. Wasdell (“Feedback Dynamics” slide 22) warns that the climate forcing as a result of human induced GHG emissions is in fact pushing the system “far away from equilibrium, passed the tipping point, and into the hot-earth scenario”. In previous episodes of extreme radiative forcing in the past, this “disturbance has then been amplified by powerful feedback dynamics not active in the near-equilibrium state [… and] have typically resulted in the loss of about 90% of life on earth.” An essential element of system dynamics is the existence of (delayed) reinforcing and balancing causal feedback loops, such as the ones illustrated in Figure 5. Figure 5: Pre/Predator model (Bellinger CC-BY-SA) In the case of Figure 5, the feedback loops illustrate the relationship between rabbit population increasing, then foxes feeding on the rabbits, keeping the rabbit population within the carrying capacity of the ecosystem. Fox predation prevents rabbit over-population and consequent starvation of rabbits. The reciprocal interaction of the elements of a system leads to unpredictable nonlinearity in “even seemingly simple systems” (“System Dynamics”). The climate system is subject to both positive and negative feedback loops. If the area of ice cover increases, more heat is reflected back into space, creating a positive feedback loop, reinforcing cooling. Whereas, as the arctic ice melts, as it is doing at present (Barber), heat previously reflected back into space is absorbed by now exposed water, increasing the rate of warming. Where negative feedback (system damping) dominates, the cup-shaped equilibrium is stable and system behaviour returns to base when subject to disturbance. [...]The impact of extreme events, however, indicates limits to the stable equilibrium. At one point cooling feedback loops overwhelmed the homeostasis, precipitating the "snowball earth" effect. […] Massive release of CO2 as a result of major volcanic activity […] set off positive feedback loops, precipitating runaway global warming and eliminating most life forms at the end of the Permian period. (Wasdell, “Topological”) Martin-Breen and Anderies (53–54), following Walker and Salt, identify four key factors for systems (ecological) resilience in nonlinear, non-deterministic (complex adaptive) systems: regulatory (balancing) feedback mechanisms, where increase in one element is kept in check by another element; modularity, where failure in one part of the system will not cascade into total systems failure; functional redundancy, where more than one element performs every essential function; and, self-organising capacity, rather than central control ensures the system continues without the need for “leadership”. Transition Towns as a Resilience Movement The Transition Town (TT) movement draws on systems modelling of both climate change and of Limits to Growth (Meadows et al.). TT takes seriously Limits to Growth modelling that showed that without constraints in population and consumption the world faces systems collapse by the middle of this century. It recommends community action to build as much capacity as possible to “maintain existence of function”—Holling's (“Engineering vs. Ecological” 33) definition of ecological resilience—in the face of failing economic, political and environmental systems. The Transition Network provides a template for communities to follow to “rebuild resilience and reduce CO2 emissions”. Rob Hopkins, the movements founder, explicitly identifies ecological resilience as its central concept (Transition Handbook 6). The idea for the movement grew out of a project by (2nd year students) completed for Hopkins at the Kinsale Further Education College. According to Hopkins (“Kinsale”), this project was inspired by Holmgren’s Permaculture principles and Heinberg's book on adapting to life after peak oil. Permaculture (permanent agriculture) is a design system for creating agricultural systems modelled on the diversity, stability, and resilience of natural ecosystems (Mollison ix; Holmgren xix). Permaculture draws its scientific foundations from systems ecology (Holmgren xxv). Following CAS theory, Mollison (33) defines stability as “self-regulation”, rather than “climax” or a single equilibrium state, and recommends “diversity of beneficial functional connections” (32) rather than diversity of isolated elements. Permaculture understands resilience in the ecological, rather than the engineering sense. The Transition Handbook (17) “explores the issues of peak oil and climate change, and how when looked at together, we need to be focusing on the rebuilding of resilience as well as cutting carbon emissions. It argues that the focus of our lives will become increasingly local and small scale as we come to terms with the real implications of the energy crisis we are heading into.” The Transition Towns movement incorporate each of the four systems resilience factors, listed at the end of the previous section, into its template for building resilient communities (Hopkins, Transition Handbook 55–6). Many of its recommendations build “modularity” and “self-organising”, such as encouraging communities to build “local food systems, [and] local investment models”. Hopkins argues that in a “more localised system” feedback loops are tighter, and the “results of our actions are more obvious”. TT training exercises include awareness raising for sensitivity to networks of (actual or potential) ecological, social and economic relationships (Hopkins, Transition Handbook 60–1). TT promotes diversity of local production and economic activities in order to increase “diversity of functions” and “diversity of responses to challenges.” Heinberg (8) wrote the forward to the 2008 edition of the Transition Handbook, after speaking at a TotnesTransition Town meeting. Heinberg is now a senior fellow at the Post Carbon Institute (PCI), which was established in 2003 to “provide […] the resources needed to understand and respond to the interrelated economic, energy, environmental, and equity crises that define the 21st century [… in] a world of resilient communities and re-localized economies that thrive within ecological bounds” (PCI, “About”), of the sort envisioned by the Limits to Growth model discussed in the previous section. Given the overlapping goals of PCI and Transition Towns, it is not surprising that Rob Hopkins is now a Fellow of PCI and regular contributor to Resilience, and there are close ties between the two organisations. Resilience, which until 2012 was published as the Energy Bulletin, is run by the Post Carbon Institute (PCI). Like Transition Towns, Resilience aims to build “community resilience in a world of multiple emerging challenges: the decline of cheap energy, the depletion of critical resources like water, complex environmental crises like climate change and biodiversity loss, and the social and economic issues which are linked to these. […] It has [its] roots in systems theory” (PCI, “About Resilience”). Resilience.org says it follows the interpretation of Resilience Alliance (RA) Program Director Brian Walker and science writer David Salt's (xiii) ecological definition of resilience as “the capacity of a system to absorb disturbance and still retain its basic function and structure.“ Conclusion This paper has analysed the ontological metaphors structuring competing conceptions of resilience. 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