Institute of Fundamental Technological Research

The article presents an investigation of the stabilization of a cantilever pipe discharging fluid using electromagnetic actuators of the transformer type. With the flow velocity reaching a critical value, the straight equilibrium position of the pipe becomes unstable, and self-excited lateral vibrations arise. Supplying voltage to the actuators yields two opposite effects. First, each of the actuators attracts the pipe, thus introduces the effect of negative stiffness which destabilizes the middle equilibrium. Second, lateral vibrations change the gap in magnetic circuits of the actuators, which leads to oscillations of magnetic field in the cores and the electromagnetic phenomena of induction and hysteresis that impede the motion of the pipe. The combination of these two non-linear effects is ambiguous, so the problem is explored both theoretically and experimentally. First, a mathematical model of the system in form of a partial differential equation governing the dynamics of the pipe coupled with two ordinary differential equations of electro-magnetodynamics of the actuators is presented. Then, the equation of the pipe’s dynamics is discretized using the Galerkin procedure, and the resultant set of ordinary equations is solved numerically. It has been shown that the overall effect of actuators action is positive: the critical flow velocity has been increased and the amplitude of post-critical vibrations reduced. These results have been validated experimentally on a test stand.

fluid-structure interaction, pipe, flow, dynamic stability, electromagnetic actuator

The problem of decentralized semi‐active stabilization of vibration of a beam structure is studied. The decentralized controller's architecture is attained by means of optimal system modelling. In this approach, based on a specially designed and optimized set of basis functions, the solution to the continuous Euler-Bernoulli beam equation is approximated by a discrete system, where the mass and stiffness matrices ensure that the assumed stabilizing control law can be operated by using solely the local state information. The performance of the method is examined through numerical experiments for a series of free‐vibration scenarios with comparison to competitive decentralized and centralized control strategies. The performance impact of the selection of the parameters of the optimal system model is also studied. The designed method allows practical modular arrangements of the control system and is applicable to large‐scale structures.

bilinear system, decentralized control, polynomial basis, semi-active control, stabilization

An application of electromagnetic devices of the motional type (i.e. eddy-current dampers) to improve the dynamic stability of a cantilever pipe discharging fluid is proposed. When the flow velocity reaches a critical value, this system loses stability through the flutter. A contactless damping device is used. This actuator is made of a conducting plate attached to the pipe that moves together with it within the perpendicular magnetic field that is generated by the controlled electromagnets. During the motion the eddy currents in the plate and a resultant drag force of a viscous character are generated. First, an optimal control problem that aims to stabilise the system with the optimal rate of decrease of the system’s energy is posed and solved. Then a state-feedback parametrization of the obtained optimal control, which can be used in a closed-loop scheme is proposed. The effectiveness of the designed optimal controller is validated by making a comparison with the corresponding passive solutions on the specially designed and constructed experimental test stand of a pipe conveying air.

fluid–structure interaction, electromagnetic device, eddy-current damper, optimal control, stabilisation, smart structure

A method of vibration reduction based on activation of an MRE block that couples twin cantilever beams at their free ends is investigated. Four types of magnetorheological elastomers have been manufactured, and their rheological properties in a range of magnetic field intensities are established. Free vibrations of several double-beam structures with controllable damping members made of these MREs are investigated, and a method of semiactive control of such structures is proposed. The effects of compression of the elastomers and alignment of the magnets used to activate them are reported. The mathematical modeling of the system is verified experimentally.

This paper deals with the problem of applying electromagnetic devices of the motional type to improve the dynamic stability of a pipe conveying air. When the flow velocity reaches a critical value, the steady equilibrium position becomes unstable, and self-excited lateral vibrations arise. In contrast, electromagnetic devices of the transformer type have been demonstrated to be highly effective in the passive stabilization of such a system, as well as the active stabilization of similar non-conservative systems with a follower force. In the present paper, we apply a pair of motional devices made of a conducting plate which is attached to the pipe and moves together with it within the perpendicular magnetic field generated by the controlled electromagnets. This motion generates eddy currents in the plates and a drag force of a viscous character. In this setting, we first investigate the possibility of designing a stabilizing control within the region of the magnetic field where every passive solution results in an unstable or conservative state. For that purpose, we determine a practical condition justifying the existence of a stabilizing control for a given set of system parameters. Later we pose and solve an optimal control problem aiming at stabilizing the system with the optimal rate of decrease of the system's energy. The solution is examined by means of numerical simulations performed within the three regions of the flow velocity: low subcritical, where the Coriolis acceleration of the conveyed fluid generates the predominate damping force; high subcritical, where the inertia of the fluid begins to dominate the dynamics of the system; and low supercritical,where unstable flutter vibrations start to arise. The effectiveness of the designed optimal controller is validated by comparisons with the corresponding passive solutions.

fluid–structure interaction, electromagnetic device, optimal control, stabilization, smart structure

Reducing displacements of a plate vibrating under a pair of masses traveling in opposite directions can be improved by adding a smart subsoil instead of a classical damping layer. We propose a material that acts according to the instantaneous state of the plate, i.e., its displacements and velocity. Such an intelligent damping layer reduces vertical displacements even by 40%–60%, depending on the type of load and the assumed objective function. Existing materials enable the application of the proposed layer in a semi-active mode. The passive mode can be applied with materials exhibiting direction-dependent viscosity.

plate vibration, moving load, intelligent damping layer, semi-active damping

In this paper, a semi-active method to control the vibrations of twin beams connected at their tips by a smart damping element is investigated. The damping element can be made of a magnetorheological elastomer or a smart material of another type, for instance, vacuum packed particles. What is crucial is the ability to modify the storage and loss moduli of the damping block by means of devices attached directly to the vibrating structure. First, a simple dynamical model of the system is proposed. The continuous model is discretized using the Galerkin procedure. Then, a practical state-feedback control law is developed. The control strategy aims at achieving the best instantaneous energy dissipation of the system. Numerical simulations confirm its effectiveness in reducing free vibrations. The proposed control strategy appears to be robust in the sense that its application does not require any knowledge of the initial conditions imposed on the structure, and its performance is better than passive solutions, especially for the system induced in the first mode.

Vibration control, Double-beam structure, Sandwich beam, Magnetorheological elastomer, Semi-active damping, Stabilization

The dynamics of twin cantilever beams connected at their free ends by a viscoelastic member is investigated. A series of experiments with damping members made of various materials was conducted. A dynamic model of the structure is proposed. It fits the experimental data well, and allows estimating the Kirchhoff modulus and shear damping coefficient of the member. The influence of geometry of the damping element on oscillatory characteristics of the system is investigated.

Beam dynamics, Double-beam system, Shear deformation, Elastomer, Semi-active, Damping

A control method to stabilize vibration of a double cantilever system with a set of smart damping blocks is designed and numerically evaluated. The externally controlled magnetorheological sheared elastomer damping block is considered, but other smart materials can be used as well. The robust bang-bang control law for stabilization the bilinear system is elaborated. The key feature of the closed loop controller is the efficiency for different types of initial excitement. By employing the finite element model, the performance of the controller is validated for strong wind blow load and concentrated impact excitement of the particular point of one of the beams. For each of the excitations, the closed loop control outperforms the optimal passive damping case by over 27% for the considered energy metric.

This article details a two-step method of quantifying eye movement transitions between areas of interest (AOIs). First, individuals' gaze switching patterns, represented by fixated AOI sequences, are modeled as Markov chains. Second, Shannon's entropy coefficient of the fit Markov model is computed to quantify the complexity of individual switching patterns. To determine the overall distribution of attention over AOIs, the entropy coefficient of individuals' stationary distribution of fixations is calculated. The novelty of the method is that it captures the variability of individual differences in eye movement characteristics, which are then summarized statistically. The method is demonstrated on gaze data collected from two studies, during free viewing of classical art paintings. Normalized Shannon's entropy, derived from individual transition matrices, is related to participants' individual differences as well as to either their aesthetic impression or recognition of artwork. Low transition and high stationary entropies suggest greater curiosity mixed with a higher subjective aesthetic affinity toward artwork, possibly indicative of visual scanning of the artwork in a more deliberate way. Meanwhile, both high transition and stationary entropies may be indicative of recognition of familiar artwork.

Eye movement transitions, entropy, Markov chain

A new semi-active damping of vibrations method is presented, which is based on loose granular grains that are placed in a hermetic sleeve that is surrounding a steel beam, forming a complex controllable granular beam structure. Changing the partial pressure inside the structure results in changes of the elastic and dissipative properties of the system resulting from the granules’ jamming mechanism. The paper presents both experimental and theoretical analyses of a vibrating cantilever granular beam system. The linear Bernoulli–Euler beam model with Kelvin–Voigt damping that depends on underpressure is used to describe the behavior of the investigated structure. The proposed strategy of vibrations damping is found to be easy to implement, provides satisfactory damping, and is efficient, but also indicates that a nonlinear beam model may be a better fit.

Special Granular Structures, Semi-active damping of vibrations, Jamming mechanism, Experiments, Modeling, Beam vibrations

The dynamics of an inertially excited steel cantilever beam encapsulated in a hermetic sleeve filled with polypropylene grains is investigated experimentally and theoretically. An electric motor rotating an unbalanced mass is attached at the end of the beam to generate forced vibrations of the system. Changing the underpressure in the sleeve results in changes of the stiffness and dissipative properties of the composite structure, which allow one to control the resonant characteristics of the system. Is is shown that the amplitude of vibrations can be reduced in this way. This innovative semi-active method of vibration damping is based on the jamming mechanism of the granules, and proved to be efficient in the reduction of the free vibrations of the beam, which was studied in the authors' latest paper. In the present work, it is confirmed that the classical Bernoulli–Euler beam model with Kelvin–Voigt damping provides a satisfactory explanation of the observed relation between the vibrations amplitude and underpressure.

Vacuum Packed Particles, semi-active damping, jamming mechanism, inertial excitation

This paper deals with the dynamic stability and near-critical vibrations of Beck׳s column with electromagnetic actuators. The governing equations of the actuators are derived, and the continuous dynamics problem of the column is discretised with the multimodal Galerkin procedure. Firstly, the linear analysis of the stability is carried out. An application of the electromagnetic actuators leads to increase in the critical load and considerably affects the initial flutter frequency. Then the Iooss and Joseph method of construction of the bifurcating solution is used to obtain analytical approximations of the near-critical periodic vibrations. It is shown that the electromagnetic effects lead to a supercritical Hopf bifurcation of the equilibrium state of the column into an orbitally stable limit cycle. Moreover, the actuators allow one to decrease significantly the amplitude of the post-critical vibrations. The effect of magnetic damping strongly depends on the position at which the actuators are attached to the column. Theoretical predictions are verified by numerical simulations of the system.

Beck׳s column, Electromagnetic actuators, Follower load, Hopf bifurcation, Flutter, Dynamic stability

We study the influence of electromagnetic damping on the dynamics of a pipe conveying fluid. Pipes supported at both ends as well as cantilever ones (both discharging and aspirating the fluid) are considered. We assume physical parameters of the systems which allow an experimental verification of results. We develop simple methods of calculation of the internal and external damping coefficients which are based on known models, and do not require experiments. The governing partial equation of the pipe is discretised with Galerkin’s procedure, and the stability of the resultant dynamical system is determined with eigenvalues of its linearization. The actuators destabilise the pipe supported at both ends, but can remarkably improve stability of cantilever ones. The effect of magnetic damping strongly depends on the position at which actuators are attached to the pipe.

pipe conveying fluid, dynamic stability, flutter, electromagnetic actuators

We study the application of electromagnetic actuators for the active stabilization of the Leipholz column. The cases of the compressive and tensional load of the column placed in air and in water are considered. The partial differential equation of the column is discretized by Galerkin’s procedure, and the stability of the obtained control system is evaluated by the eigenvalues of its linearization. Four different methods of active stabilization are investigated. They incorporate control systems based on feedback proportional to the transverse displacement of the column, its velocity and the current in the electromagnets. Conditions in which these strategies are effective in securing safe operation of the column are discussed in detail.

follower load, flutter, closed-loop control, active stabilization, electromagnetic actuators

The paper presents analysis of nonlinear response of a classical mechanical oscillator placed within a magnetic field and driven by a harmonic force. With an appropriate choice of control parameters, the system vibrates chaotically between different equilibrium positions. To prove this result, Lyapunov exponents have been calculated using the algorithm proposed by Rangarajan G., Habib S. and Ryne R. [18]. Moreover, the appropriate time series, phase portrait, Poincaré cross-section and power spectrum are given to support the conclusion.

nonlinear vibrations, oscillator, chaos, stability, magnetic damping

The paper presents a method of damping of mechanical vibrations by application of electromagnetic actuators. The physical phenomena that take place in such elements are related to magnetic induction and hysteresis. Induction, which consists in generation of electric current in a conductor placed within a variable magnetic field, has a precise quantitative description in form of the Faraday law. At the same time, magnetic hysteresis in ferromagnetic materials reflects the relation between magnetization of a given sample and magnitude of the external magnetic field that had been active during its whole history. Unfortunately, there is no at present a single commonly accepted model of this phenomenon. In this paper, some considerations by Bertotti (1998) are incorporated to this subject. The authors use a bilinear magnetization curve, which depicts the phenomenon of magnetic saturation. Analysis of the derived equations of motion of the coupled electromagneto-mechanical model revealed that the effect of hysteresis on reduction of vibration amplitude is rather of minor impact, however noticeable. At the same time, the electromagnetic damping itself is powerful. To support the conclusions, appropriate time histories, graphs of the logarithmic damping decrement as well as resonant characteristics are given here as in the previous authors’ paper.

damping of vibration, electromagnetic actuator, magnetic hysteresis, saturation

Oscylator harmoniczny zawieszony pomiędzy dwoma elektromagnesami może być skutecznie tłumiony magnetycznie. Postać drgań, charakter tłumienia, liczba i stateczność punktów równowagi zależą od napięcia załączonego do elektromagnesów. Drgania wymuszone w pewnych warunkach przypominają drgania chaotyczne. Wpływ histerezy magnetycznej jest pominięty.

siłowniki elektromagnetyczne, tłumienie drgań, oscylator

The paper introduces a two-step method of quantifying eye movement transitions between Areas of Interests (AOIs). First, individuals’ gaze switching patterns, represented by fixated AOI sequences, are modeled as Markov chains. Second, Shannon’s entropy coefficient of the fit Markov model is computed to quantify the complexity of individual switching patterns. To determine the overall distribution of attention over AOIs, the entropy coefficient of individuals’ stationary distribution of fixations is calculated.

The novelty of the method is that it captures the variability of individual differences in eye movement characteristics, which are then summarized statistically. The method is demonstrated on gaze data collected during free viewing of classical art paintings. Shannon’s coefficient derived from individual transition matrices is significantly related to participants’ individual differences as well as to their aesthetic experience of art pieces.

eye movements, transition matrix, Shannon entropy, Markov model, gaze distribution

Pipe Conveying Fluid, Electromagnetic Actuator, Stability, Flutter Experiment

The study deals with a problem of applying electromagnetic devices of motional type to improve the dynamic stability of a pipe conveying air. When the flow velocity reaches its critical value the steady equilibrium position becomes unstable, and self-excited lateral vibrations arise. Electromagnetic devices of another – transformer – type demonstrated to have been highly effective in the passive stabilization of such system as well as active stabilization of similar non-conservative system with a follower force. In the present work, we apply a pair of motional devices made of a conducting plate which is attached to the pipe and moves together with it within the perpendicular magnetic field generated by the controlled electromagnets. This motion generates eddy currents in the plates and a drag force of a viscous character. Under this setting, we firstly investigate the possibility of designing a stabilizing control within the magnetic field's region where every passive solution results in an unstable or conservative state. For that purpose, we constitute a practical condition justifying the existence of a stabilizing control for a given set of system parameters. Later we pose and solve an optimal control problem aiming at stabilizing the system with the optimal rates of decrease of the system's energy. The solution is examined by means of numerical simulations performed within the three regions of the flow velocity: low subcritical where the Coriolis acceleration of the conveyed fluid generates the predominate damping force, high subcritical where the inertia of the fluid begins to dominate dynamics of the system, and low supercritical where unstable flutter vibrations start to arise. The effectiveness of the designed optimal controller is validated by comparisons with the corresponding passive solutions.

The analysis of stability the pipe with electromagnetic coupling in active elements has been done. Application of electromagnetic actuators leads to an increase in the critical flow velocity.The frequency of self-excited vibrations can either change with the volt-age supplied depending on the position of the active elements.

Flutter vibrations, electromagnetic coupling, electromagnetic actuators, EM damper of motional type

The work presents a novel distributed vibration control method of twin cantilever beams system coupled by a set of magnetorheological elastomers. The control is realized by a change of magnetic field influencing elastomers’ mechanical properties. The objective is to stabilize the system with the optimal rates of decrease of the energy. The control is based on an easy for practical implementation distributed state-feedback structure. It employs a set of communication channels to exchange the state information between the neighboring controllers. The performance of the designed method is validated by means of the numerical experiments. In terms of the assumed metrics, the proposed distributed method significantly outperforms the passive case and is competitive to standard centralized control.

Lateral vibrations of two parallel cantilever beams joined at their free ends by a viscoelastic member are studied experimentally and theoretically. A dynamic model of the structure is proposed. It fits the experimental data well and allows to estimate the shear modulus and damping coefficient of the member. The model can be useful for the development of semi-active control strategies for double-beam systems with controllable damping members.

sandwich beam, double-beam system, shear deformation, magnetorheological elastomer, semi-active damping