In this extended note a critical discussion of an extension of the Lorentz transformations for velocities faster than the speed of light given recently by Hill and Cox [1] is provided. The presented approach reveals the connection between faster-than-light speeds and the issue of isotropy of space. It is shown if the relative speed between the two inertial frames v is greater than the speed of light, the condition of isotropy of space cannot be retained. It further specifies the respective transformations applying to -∞<v<-c and c<v<+∞. It is proved that such Lorentz-like transformations are improper transformations since the Jacobian is negative. As a consequence, the wave operator, the light-cone and the volume element are not invariant under such Lorentz-like transformations. Also it is shown that such Lorentz-like transformations are not new and already known in the literature.

The investigations fulfilled in this article are founded on two results. The first is experiments of M. Beteno, Y. Duboshinsky. The description of these experiments is adduced in [1]. In these experiments the low frequency oscillations of iron ball suspended on the thin string [1] were obtained (frequency of these oscillations is order to eigen frequency of pendulum). And the second is theoretical conclusions about possible stabilizations of early (without alternating magnetic field) unstable equilibrium positions. This theoretical result was obtained by the asymptotical solution of Lagrange-Maxwell equations of dynamic of electromechanical systems suspended in alternating magnetic field [2].

The phenomenon of brake squeal, which is a type of friction-induced vibration, is analyzed using a pin-on-disc system. For this purpose, a finite element model is derived and its parameters are updated on the basis of experiments. The FEM analysis includes the complex eigenvalue analysis and the transient analysis. As the brake-squeal phenomenon is very sensitive with respect to parametric uncertainty, the two numerical analyses are combined with an uncertainty analysis, which in this study is based on fuzzy arithmetic. The uncertainty analysis enables the determination of both the overall uncertainty of the considered output quantity and the influence of each individual uncertain model parameter on the overall uncertainty of the output. With this information about propagation and influence of parametric uncertainty in the system, the methods of complex eigenvalue analysis and transient analysis can be compared with respect to their appropriateness for predicting the tendency of the brake to squeal.

The manufacturing process of paper machines consists of several steps to produce high quality papers. This is done by sequentially lined-up machines including the head box, the drying sections, the finishing part and the wrapping systems. In the finishing part, the rollers of the paper calender compress the fibrous material involving viscoelastic and plastic deformations. Modern calenders are composed of several roller pairs, each consisting of a soft and a hard roller. The homogenization of the paper density and the refinement of the paper surface is achieved by the compression in the roller pairs. While very high values for the plastic strain occur in the first roller pair, the plastification decreases for the subsequent pairs. Therefore, the force distribution and the occurring vibrations can deviate significantly between the roller pairs. Two main vibration problems are observed in paper calenders caused by the contact and the orthotropic behavior of the paper: wear-induced corrugation on the surface of the soft rollers and sudden instabilities going along with high vibration amplitudes. In this paper the main focus is placed on a simplified modeling of the paper plastification during the calendering process. The restoring forces are non-smooth due to the plasticity and additional considerations have to be included for the derivation of the stability problem.

The AUFS scheme by Sun and Takayama is a flux splitting scheme without breakdown of discrete shock profiles, usually called carbuncle, but still with a good resolution of entropy waves. Unfortunately, numerical tests with this scheme yield that the viscosity on entropy waves is too small while the viscosity on shear waves is too large. In this paper, we prove that both deficiencies are inherent to the construction of the scheme and provide fixes to overcome them.

The paper is devoted to the life and work of Alexander Mikhailovich Ertel, the founder of elastohydrodynamics. He was the first to solve problems of hydrodynamic lubrication accounting for effects of elastic deformation of the bodies in contact as well as for the dependence of the viscosity of the lubricant on pressure and temperature, thereby opening a new branch of tribology – elastohydrodynamics. However, due to complicated historical entanglements, the real authorship remained unknown over many years. On occasion of the 100th anniversary of Alexander Mohrenstein-Ertel, we investigate the circumstances of the creation of elastohydrodynamics and provide a short sketch of the main ideas of the early works of Mohrenstein-Ertel. The biography of Mohrenstein-Ertel is like a criminal novel: the 20th century history is reflected in the hard fate of this scientist and Russian history up to the time of Pushkin can be traced in his genealogy.

We study the stationary interaction between a 2D viscous fluid, governed by the Stokes equation, and a rigid structure that can move following rigid displacements. The displacements of the structure are determined using an algebraic equation. A slip boundary condition of friction type is used on the fluid–solid interface. An existence result is proved and numerical tests are presented.

The linearized equations of motion of finite dimensional autonomous mechanical systems are normally written as a second order system and are of the **MDGKN** type, where the different n × n matrices have certain characteristic properties. These matrix properties have consequences for the underlying eigenvalue problem. Engineers have developed a good intuitive understanding of such systems, particularly for systems without gyroscopic terms (**G**-matrix) and circulatory terms (**N**-matrix, which may lead to self-excited vibrations). A number of important engineering problems in the linearized form are described by this type of equations. It has been known for a long time, that damping (**D**-matrix) in such systems may either stabilize or destabilize the system depending on the structure of the matrices. Here we present some new results (using a variety of methods of proof) on the influence of the damping terms, which are quite general. Starting from a number of conjectures, they were jointly developed by the authors during recent months.

Degradation mechanisms in Li-ion batteries such as SEI formation, isolation of active material and reduction in electronic conductivity appear to correlate with the increase in surface area of electrode particles caused by particle fracture during charge/discharge cycles. The focus of this study is on the surface cracking of an electrode particle, as large tensile stresses operate on the surface during the delithiation process of charge/discharge cycling. The pre-existing surface flaws act as crack initiators under this scenario and we discuss the extension of these cracks under two different operating conditions. Approximate analytical expressions for the propensity for surface crack growth is derived in terms of stress intensity factor and fracture toughness of the material. Utilizing dimensional analysis, we arrive at fracture limit diagrams to determine fracture-free conditions as design guidelines for prescribed electrode particle size or diffusion boundary conditions related to the charging/discharging process. Another significant result of the fracture analysis is that smaller particles can withstand a wider range of fluctuations in concentration or flux at the boundary for surface fracture-free conditions. This result supplements the conventional understanding that smaller particles show higher structural integrity because of fewer pre-existing defects.

The application of high-frequency vibration processes for intensification of machining requires a control technique for identification, excitation and stabilisation of the nonlinear resonant mode in machining systems with unpredictable variation of processing loads. Such a technique was developed with the use of a self-exciting mechatronic system. This method of control is known as autoresonance. Autoresonant control of ultrasonically assisted drilling machine intended to improve machining process is thoroughly analysed and the simulation results of analysis for both mechanical feedback and electrical feedback are presented together with the application of different filters.

We show that the linear water wave problem in a bounded liquid domain may have continuous spectrum, if the interface of a two-layer liquid touches the basin walls at zero angle. The reason for this phenomenon is the appearance of cuspidal geometries of the liquid phases. We calculate the exact position of the continuous spectrum. We also discuss the physical background of wave propagation processes, which are enabled by the continuous spectrum. Our approach and methods include constructions of a parametrix for the problem operator and singular Weyl sequences.

The classical method of separation of variables in elliptical coordinates in conjunction with the translational addition theorems for Mathieu functions are used to investigate free transverse vibrations of an elastic membrane of elliptical planform with an arbitrarily located elliptical perforation. Subsequently, the elaborated method of eigenfunction expansion is employed to obtain an exact time-domain series solution, in terms of products of angular and radial Mathieu functions, for the forced transverse oscillations of the eccentric membrane. The analytical solution is illustrated through numerical examples including circular/elliptical membranes with a circular perforation or with an elliptical perforation of selected geometric, orientation, and location parameters. The first five natural frequencies are tabulated, and selected vibration mode shapes are presented in graphical form. Also, the displacement responses of representative membranes in a practical loading configuration (i.e., a uniformly distributed step load) are calculated. The accuracy of solutions is ensured through proper convergence studies, and the validity of results is demonstrated with the aid of a commercial finite element package as well as by comparison with the existing data. The set of data reported herein is believed to be the first rigorous attempt on the free/forced vibrational characteristics of eccentric elliptical membranes for a wide range of geometric parameters.

The anti-plane strain elastodynamic problem for a continuously inhomogeneous half-plane with free-surface relief subjected to time-harmonic SH-wave is studied. The computational tool is a boundary integral equation method (BIEM) based on analytically derived Green's function for a quadratically inhomogeneous in depth half-plane. To show the versatility of the proposed BIE method, it is considered SH-wave propagation in an inhomogeneous half-plane with free surface relief presented by a semi-circle, semi-elliptic and triangle canyon. The inhomogeneous in depth half-plane is modeled in two different ways: (i) the material properties vary continuously in depth and BIEM based on Green's function is used; (ii) the material properties vary in a discrete way and the half-plane is presented by a set of homogeneous layers with horizontal interfaces and a hybrid technique based on wave number integration method (WNIM) and BIEM is applied. The equivalence of these two different models is shown. The simulations reveal a marked dependence of the wave field on the material inhomogeneity and the potential of the BIEM based on the Green's function for half-plane to produce highly accurate results by using strongly reduced discretization mesh in comparison with the conventional boundary element technique using fundamental solution for the full plane.

Based upon a one-dimensional nonlinear Klein-Gordon equation with a perturbed one-gap periodic potential, this paper deals with the question as to whether spatially localized structures in periodic media can exist for all times. As it turns out that, given our model equation, the latter question cannot be answered in the affirmative, we show the asymptotic stability of the vacuum state in appropriate dispersive norms and provide an upper bound for the temporal decay rates of the corresponding solutions. This is done by using the dispersive estimates proved in [23]. More precisely, if the perturbed Hill operator associated to our problem has no eigenvalue, we add a power nonlinearity u^{p} with p ∈ {6, 7, 8, …}. In this setting, the convergence to the trivial solution w. r. t. the L^{∞} norm is shown in a canonical way. We obtain the corresponding linear rate. In contrast, if the spatially localized potential creates an eigenvalue in the band gap of the continuous spectrum, then we multiply u^{p} by a spatial weight function and prove an asymptotic stability result w. r. t. a weighted L^{2} norm for p ∈ {3, 4, 5, …}. Now, in the presence of an eigenvalue, there is a strongly reduced decay compared to the associated linearized problem. It is due to the component that belongs to the discrete spectral subspace of L^{2} w. r. t. the perturbed Hill operator. As in [29], this phenomenon is referred to as metastability of the corresponding solutions.

We review the concept of well-posedness in the context of evolutionary problems from mathematical physics for a particular subclass of problems from elasticity theory modeling solids with micro-structure. The complexity of physical phenomena appears as encoded in so called material laws. The usefulness of the structural perspective developed is illustrated by showing that many initial boundary value problems in the theory of such elastic solids share the same type of solution theory. Moreover, interconnections of the respective models are discussed via a previously introduced mother/descendant mechanism.

We consider the convergence to (global) equilibrium of a partially dissipative hyperbolic system using entropy methods. An entropy functional is determined and exponential decay for this functional is proven. Moreover, an explicit estimate for the decay rate is given.

Error estimates of finite element methods for reaction-diffusion problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the H^{1} seminorm leads to a balanced norm which reflects the layer behavior correctly. We prove error estimates in balanced norms and investigate also stability questions. Especially, we propose a new C^{0} interior penalty method with improved stability properties in comparison with the Galerkin FEM.

The damage acting on a structure can lead to disproportionate consequences, i.e., the global collapse. This extreme situation has to be avoided and, thus, structural monitoring is requested in those structures where human losses are possible and large economic consequences are expected. Static measurement devices are the most economic instrumental set-ups able to highlight the presence of progressive damages. However, this monitoring system suffers from the structural behaviour under the external loads. In many situations, alternate load paths shown the non-effectiveness of the measurement system since the instrumentation are installed on elements not relevant for the response under the given loads. The same problems occur when the structure is compartmentalized, i.e. the structural responses of the single parts dependent on the loads acting almost only on the same single component. In order to measure the degree of compartmentalization, different novel metrics based on stiffness matrix properties are proposed and their effectiveness discussed. The new idea of this paper is to connect compartmentalization of structures with a sort of distance of the stiffness matrix from the set of diagonal matrices. Few examples are illustrated.

In this contribution the eigenfrequencies of a special linear vibration system are investigated. Based on the properties of the corresponding mass and stiffness matrices of the chain structured mass-spring vibration system with arbitrary n degrees of freedom an algebraic proof for the determination of the eigenfrequencies is given.

This paper deals with the eigenvalues of the Neumann Laplacian on simply-connected Lipschitz planar domains with some rotational symmetry. Our aim is to continue the investigations from Enache and Philippin [7] and derive new isoperimetric estimates for eigenvalues of higher order.

The cover picture shows a typical block diagram for a networked control system, it is taken from the contribution by Molin and Hirche.

The advent of networked control systems urges the digital control design to incorporate communication constraints efficiently. In order to accommodate this requirement, this article studies the joint design of controller and event-trigger for linear stochastic systems in the presence of a resource-limited communication channel which exhibits packet dropouts and time-delay. The event-trigger situated at the sensor decides at every sampling instance, whether to send information over the communication channel to the controller. The design approach is formulated as a stochastic average-cost optimization problem, where the communication constraints are reflected as an additional cost penalty of the average transmission rate. Different conditions on the communication model are given where the joint optimal design can be split into a separate control and event-trigger design. Based on these results, two suboptimal design approaches are developed. By using drift criteria, stability guarantees of the closed-loop system for both approaches are derived in terms of bounded moment stability. Numerical simulations illustrate the efficacy of the event-triggered approach compared with optimal time-triggered controllers.

This article studies the joint design of controller and event-trigger for linear stochastic systems in the presence of a resource-limited communication channel which exhibits packet dropouts and time-delay. The design approach is formulated as a stochastic average-cost optimization problem, where the communication constraints are reflected as an additional cost penalty of the average transmission rate. Different conditions on the communication model are given where the joint optimal design can be split into a separate control and event-trigger design. Based on these results, two suboptimal design approaches are developed and stability of the closed-loop system is shown in terms of bounded moment stability. Numerical simulations illustrate the efficacy of the event-triggered approach compared with optimal time-triggered controllers.

This paper proposes a novel event-based control method for nonlinear systems that are input-output linearizable. A method for the design of an event-based state-feedback controller is proposed which approximates the disturbance rejection behavior of a continuous state-feedback system, referred to as reference system, with adjustable accuracy, while simultaneously reducing the feedback communication effort. The event-based control loop is shown to be input-to-state stable. A bound for the deviation between the behavior the event-based control loop and the reference system is derived. This bound is shown to be a function of the event threshold, which is a design parameter that can be freely chosen. Moreover, the deviation between the outputs of both systems is also proven to be bounded and can be made arbitrarily small by appropriately scaling the event threshold. The minimum time in between consecutive events is shown to be bounded from below by some bound which is explicitly determined. The efficiency of the new control method is demonstrated for a bioreactor example.

This paper proposes an event-based control method for nonlinear systems that are input-output linearizable. The method approximates the disturbance rejection behavior of a continuous state-feedback system, referred to as reference system, with adjustable accuracy, while simultaneously reducing the feedback communication effort. The event-based control loop is shown to be input-to-state stable. A bound for the deviation between the behavior the event-based control loop and the reference system is derived. This bound is shown to be a function of the event threshold, which is a design parameter that can be freely chosen. The minimum time in between consecutive events is shown to be bounded from below by some bound which is explicitly determined. The efficiency of the new control method is demonstrated for a bioreactor example.

This paper is concerned with distributed model predictive control (DMPC) for distributed linear time invariant systems with decoupled dynamics but coupling via a common cost function and common convex state constraints. In the proposed scheme, local model predictive controllers exchange state measurements and predicted input sequences with bounded uncertainty via possibly delayed communication. A robust model predictive control (RMPC) problem is solved locally under consideration of uncertain information received from other subsystems. The local RMPCs separately optimize over nominal inputs as well as feedback policies for the delayed information and can be implemented as a tractable quadratic program. Under appropriate assumptions, the proposed scheme satisfies the coupling constraints and renders the system practically stable. Simulation results for the control of a platoon of autonomous vehicles relative to a lead vehicle illustrate the effectiveness of the approach.

This paper is concerned with model predictive control for distributed linear time invariant systems with decoupled dynamics but coupling via a common cost function and common convex state constraints. In the proposed scheme, local model predictive controllers exchange state measurements and predicted input sequences with bounded uncertainty via possibly delayed communication. Under appropriate assumptions, the proposed scheme satisfies the coupling constraints and renders the system practically stable. Simulation results for the control of a platoon of autonomous vehicles relative to a lead vehicle illustrate the effectiveness of the approach.

A typical bottleneck of model predictive control algorithms is the computational burden in order to compute the receding horizon feedback law which is predominantly determined by the length of the prediction horizon. Based on a relaxed Lyapunov inequality we present techniques which allow us to show stability and suboptimality estimates for a reduced prediction horizon. In particular, the known structural properties of suboptimality estimates based on a controllability condition are used to cut the gap between theoretic stability results and numerical observations.

A typical bottleneck of model predictive control algorithms is the computational burden in order to compute the receding horizon feedback law which is predominantly determined by the length of the prediction horizon. Based on a relaxed Lyapunov inequality we present techniques which allow us to show stability and suboptimality estimates for a reduced prediction horizon. In particular, the known structural properties of suboptimality estimates based on a controllability condition are used to cut the gap between theoretic stability results and numerical observations.

For continuous-time linear control systems invariance entropy of controlled invariant subspaces is introduced. It is shown that it coincides with a variant of topological entropy for linear flows which we call subspace entropy. Using this characterization, upper bounds in terms of eigenvalues of an induced flow are derived. Under additional assumptions (diagonalizability, single inputs) these bounds are improved.

We consider weakly coupled LQ optimal control problems and derive estimates on the sensitivity of the optimal value function in dependence of the coupling strength. In order to improve these sensitivity estimates a “coupling adapted” norm is proposed. Our main result is that if a weak coupling suffices to destabilize the closed loop system with the optimal feedback of the uncoupled system then the value function might change drastically with the coupling. As a consequence, it is not reasonable to expect that a weakly coupled system possesses a weakly coupled optimal value function. Also, for a known result on the connection of the separation operator and the stability radius a new and simpler proof is given.

The authors consider weakly coupled LQ optimal control problems and derive estimates on the sensitivity of the optimal value function in dependence of the coupling strength. In order to improve these sensitivity estimates a “coupling adapted” norm is proposed. Their main result is that if a weak coupling suffices to destabilize the closed loop system with the optimal feedback of the uncoupled system then the value function might change drastically with the coupling. As a consequence, it is not reasonable to expect that a weakly coupled system possesses a weakly coupled optimal value function. Also, for a known result on the connection of the separation operator and the stability radius a new and simpler proof is given.

We investigate controllability properties of bilinear control systems, defined by classes of Toeplitz matrices. It is shown that the Lie algebra of all pseudo-circulant matrices coincides with the full matrix Lie algebra. This implies the controllability of associated bilinear control systems. As a by-product we deduce that every complex invertible matrix is the finite product of invertible Toeplitz matrices; moreover, every complex unitary matrix is shown to be a finite product of complex unitary Toeplitz matrices.