On the hyperbolicity of 3D plasticity equations in isostatic coordinates

We consider the problem of classifying partial differential equations of the three-dimensional problem of ideal plasticity (for stress states corresponding to an edge of the Tresca prism) and the problem of finding a change of independent variables reducing these equations to the simplest normal Cauchy form. The original system of equations is represented in an isostatic coordinate system and is substantially nonlinear. We state a criterion for the simplest normal Cauchy form and find a coordinate system reducing the original system to the simplest normal Cauchy form. We show that the condition obtained in the present paper for a system to take the simplest normal form is stronger than the Petrovskii t-hyperbolicity condition if t is understood as the canonical isostatic coordinate whose level surfaces in space form fibers normal to the principal direction field corresponding to the maximum (minimum) principal stress.     

Pattern Formation in Turing Systems on Domains with Exponentially Growing Structures

Turing reaction–diffusion systems have been used to model pattern formation in several areas of developmental biology. Previous biomathematical Turing system models employed static domains which failed to incorporate the growth that inherently occurs as an organism develops. To address this shortcoming, we incorporate an exponentially growing domain into a Turing system, allowing one to more realistically model biological pattern formation. This Turing system can generate patterns on an exponentially growing domain in any of the eleven coordinate systems in which the Helmholtz equation is separable, making the system incredibly flexible and giving one the capability to mathematically model pattern formation on a geometrically diverse group of domains. Linear stability analysis is employed to generate mathematical conditions which ensure such a system can generate patterns. We apply the exponentially growing Turing system to a prolate spheroidal domain and conduct numerical simulations to investigate the system’s pattern-generating behavior. We find that the addition of growth to a Turing system causes a significant change in the pattern-generating behavior of the system. While a static domain Turing system converges to a final pattern, an exponentially growing domain Turing system produces transient patterns that continually evolve and increase in complexity over time.     

Integrodifferential equations for plane filtration in the presence of cracks

A system of singular integr Differential equations is derived for the plane problem of steady-state filtration in a plate cut by a system of cracks. We consider an arbitrary set of cracks, and also monoperiodic and biperiodic systems of cracks, in an infinite plane. In the case of a system of infinite parallel rectilinear cracks, the general solution is obtained in explicit form-in quadratures. As an example, we find the complex potential and the formula for the output from a borehole for a linear system of tiered, flooded plates, cut by a system of rectilinear parallel cracks.     

Flutter of spring-mounted flexible plates in uniform flow

A fluid–structure interaction (FSI) system is studied wherein a cantilevered flexible plate aligned with a uniform flow has its upstream end attached to a spring mounting. This allows the entire system to oscillate in a direction perpendicular to that of the flow as a result of the mounting׳s dynamic interaction with the flow-induced oscillations, or flutter, of the flexible plate. We also study a hinged-free rotational-spring attachment as a comparison for the heaving system. This variation on classical plate flutter is motivated by its potential as an energy-harvesting system in which the reciprocating motion of the support system would be tapped for energy production. We formulate and deploy a hybrid of theoretical and computational modelling for the two systems and comprehensively map out their linear-stability characteristics at low mass ratio. Relative to a fixed cantilever, the introduction of the dynamic support in both systems yields lower flutter-onset flow speeds; this is desirable for energy-harvesting applications. We further study the effect of adding an inlet surface upstream of the mount as a means of changing the destabilising mechanism from single-mode flutter to modal-coalescence flutter which is a more powerful instability more suited to energy harvesting. This strategy is seen to be effective in the heaving system. However, divergence occurs in the rotational system for low spring natural frequencies and this would lead to its failure for energy production. Finally, we determine the power-output characteristics for both systems by introducing dashpot damping at the mount. The introduction of damping increases the critical speeds and its variation permits optimal values to be found that maximise the power output for each system. The addition of an inlet surface is then shown to increase significantly the power output of the heaving system whereas this design strategy is not equally beneficial for the rotational system.     

Polynomial and Exponential Spatial Decay Estimates in Elasticity

In this note we investigate the spatial behavior of the solutions of a combination of a hyperbolic system with an elliptic system. We consider a semi-infinite cylinder which is the union of two sub-cylinders. In one of them, we assume an elastodynamical problem and in the other an elastostatic problem. Both are coupled through an interface. It is known that the elastostatic problem and the elastodynamic problem have a fast decay (at least exponential). However, as their spatial behaviors are of different kind, it is not clear how this combination could be controlled in a similar way. We prove that the decay of solutions can be controlled in a polynomial way. We also describe how to obtain an upper bound for the amplitude term. We conclude the paper sketching the exponential decay behavior for the harmonic vibrations. Supported by the project “Qualitative study of thermomechanical problems” (MTM2006-03706). The author thanks Professor Leseduarte for helping to compose the figures of this paper and an anonymous referee for useful criticisms.     

Asymptotic investigation of a weakly nonlinear multifrequency oscillating system

We consider a weakly nonlinear multifrequency autonomous system of differential equations with a small parameter on the right-hand side under the condition that the unperturbed system has a quasiperiodic general solution. The system is reduced to a simpler form by averaging and separation. We establish sufficient conditions for the preservation of an invariant torus under a small perturbation.     

A review of development and implementation of an active nonlinear vibration absorber

Summary We present an account of an implementation of an active nonlinear vibration absorber that we have developed. The control technique exploits the saturation phenomenon that is known to occur in quadratically-coupled multi-degree-of-freedom systems subjected to primary excitation and possessing a two-to-one internal resonance. The technique is based on introducing an absorber and coupling it with the structure through a sensor and an actuator, where the feedback and control signals are quadratic. First, we consider the case of controlling the vibrations of a single-degree-of-freedom system. We develop the equations governing the response of the closed-loop system and use the method of multiple scales to obtain an approximate solution. We investigate the performance of the control strategy by studying its steady-state and transient characteristics. Additionally, we compare the performance of the quadratic absorber with that of a linear absorber. Then, we present theoretical and experimental results that demonstrate the versatility of the technique. We design an electronic circuit to emulate the absorber and use a variety of sensors and actuators to implement the active control strategy. First, we use a motor and a potentiometer to control the vibration of a rigid beam. We develop a plant model that includes Coulomb friction and demonstrate that the closed-loop system exhibits the saturation phenomenon. Second, we extend the strategy to multi-degree-of-freedom systems. We use PZT ceramics and strain gages to suppress vibrations of flexible steel beams when subjected to single- and simultaneous two-mode excitations. Third, we employ Terfenol-D, a nonlinear actuator, and accelerometers to control the vibrations of flexible beams. In all instances, the technique is successful in reducing the response amplitude of the structures. Received 3 May 1999; accepted for publication 3 June 1999     

Bifurcation of periodic solution in a Predator–Prey type of systems with non-monotonic response function and periodic perturbation

A Predator–Prey type of dynamical systems with non-monotonic response function and time-periodic perturbation is considered in this paper. We present a proof for the number of equilibria in the unperturbed system at some parts of the parameter space. The perturbed system is a dynamical system defined by a periodic vector field. We present an alternative proof for a classical result on the period of the periodic solution. By using a numerical continuation method AUTO (Doedel et al., 1986 [9]), we present a bifurcation analysis for periodic solution of the perturbed system where we found fold, cusp and Swallowtail bifurcations.     

Lagrangian Gas Dynamics in Two Dimensions and Lagrangian systems

We propose a new and canonical way of writing the equations of gas dynamics in Lagrangian coordinates in two dimensions as a weakly hyperbolic system of conservation laws. One part of the system is called the physical part and contains physical variables; the other part is the geometrical part. We show that the physical part is symmetrizable. We show that the weak hyperbolicity is due to shear contact discontinuities. Free divergence constraints play an important role in the system. We prove the L² stability of the physical part of the system. Based on this formulation, we derive a new conservative and entropy-consistent finite-volume numerical scheme. We prove the stability of the numerical scheme. Numerical results show the potential interest of this approach. Various examples (Born-Infeld, MHD, 3D lagrangian gas dynamics) can be written using the same abstract formalism.     

Chaotic pitch motion of an asymmetric non-rigid spacecraft with viscous drag in circular orbit

We study the pitch motion dynamics of an asymmetric spacecraft in circular orbit under the influence of a gravity gradient torque. The spacecraft is perturbed by a small aerodynamic drag torque proportional to the angular velocity of the body about its mass center. We also suppose that one of the moments of inertia of the spacecraft is a periodic function of time. Under both perturbations, we show that the system exhibits a transient chaotic behavior by means of the Melnikov method. This method gives us an analytical criterion for heteroclinic chaos in terms of the system parameters which is numerically contrasted. We also show that some periodic orbits survive for perturbation small enough.     

What is the Optimal Shape of a Pipe?

We consider an incompressible fluid in a three-dimensional pipe, following the Navier–Stokes system with classical boundary conditions. We are interested in the following question: is there any optimal shape for the criterion “energy dissipated by the fluid”? Moreover, is the cylinder the optimal shape? We prove that there exists an optimal shape in a reasonable class of admissible domains, but the cylinder is not optimal. For that purpose, we define the first order optimality condition, thanks to the adjoint state and we prove that it is impossible that the adjoint state be a solution of this over-determined system when the domain is the cylinder. At last, we show some numerical simulations for that problem.     

On the dynamics of a spinning top under high-frequency excitation: part I—pivot point under vertical harmonic vibration

We investigate the dynamics of a spinning top whose pivot point undergoes a small amplitude high-frequency vertical vibration. The method of Direct Partition of Motion is used to obtain an autonomous equation governing the leading order slow dynamics of the top’s nutation and to derive an approximate closed form solution for the forced spinning top problem. We show that the fast vibration can lead to the stabilization of the “sleeping top” state and an expression for the minimum amplitude required is given in terms of system parameters. We also show the existence of a degenerate family of special solutions in which the spinning top is locked at constant nutation and precession angles; we refer to those as “skewed sleeping top” states. We derive the conditions under which these states exist and are stable. The results are verified through numerical integration of the full non-autonomous system.     

Classical Solvability of the Coupled System Modelling a Heat-Convergent Poiseuille-Type Flow

We consider the coupled system of two nonlinear scalar parabolic equations modelling a simple uni-directional Poiseuille-type flow of a homogeneous incompressible Newtonian fluid whose viscosity is a temperature-dependent function. The energy balance equation of this system takes into account the phenomena of the viscous energy dissipation. We prove existence of a classical solution to this system on an arbitrary interval of time. The smooth solution turns out to be unique in a wider class of weak solutions.     

Vibrations of a continuous web on elastic supports

We consider an infinite, homogenous linearly elastic beam resting on a system of linearly elastic supports, as an idealized model for a paper web in the middle of a cylinder-based dryer section. We obtain closed-form analytical expressions for the eigenfrequencies and the eigenmodes. The frequencies increase as the support rigidity is increased. Each frequency is bounded from above by the solution with absolutely rigid supports, and from below by the solution in the limit of vanishing support rigidity. Thus in a real system, the natural frequencies will be lower than predicted by commonly used models with rigid supports.     

Suppression of chaos through coupling to an external chaotic system

We explore the behaviour of an ensemble of chaotic oscillators diffusively coupled only to an external chaotic system, whose intrinsic dynamics may be similar or dissimilar to the group. Counter-intuitively, we find that a dissimilar external system manages to suppress the intrinsic chaos of the oscillators to fixed point dynamics, at sufficiently high coupling strengths. So, while synchronization is induced readily by coupling to an identical external system, control to fixed states is achieved only if the external system is dissimilar. We quantify the efficacy of control by estimating the fraction of random initial states that go to fixed points, a measure analogous to basin stability. Lastly, we indicate the generality of this phenomenon by demonstrating suppression of chaotic oscillations by coupling to a common hyper-chaotic system. These results then indicate the easy controllability of chaotic oscillators by an external chaotic system, thereby suggesting a potent method that may help design control strategies.     

Complex response of an oscillating vertical cantilever with clearance

We study the dynamics of an elastic inverted pendulum with amplitude limiters excited horizontally. This particular model corresponds to a class of systems where a clearance is present naturally as an effect of imperfect clamping or it is included to tailor the response. We explore the complex responses of the system for a fixed value of amplitude clearance. The simulation and experimental results are analysed by a 0–1 test, Fourier, and wavelet transforms. The results show that the system can vibrate with subharmonic solution where the main response frequency of a flexible beam is 3 times lower than the excitaion frequency. We claim that an inverted pendulum with imperfect clamping of mechanical resonator can be used in broad frequency band energy harvesting.

     

Nonlinear oscillations of rotor active magnetic bearings system

In this paper, an analytical approximate solution is constructed for a rotor-AMB system that is subjected to primary resonance excitations at the presence of 1:1 internal resonance. We obtain an approximate solution applying the method of multiple scales, and then we conducted the system bifurcation analyses. The stability of the system is investigated applying Lyapunov’s first method. The effects of the different parameters on the system behavior are investigated. The analytical results showed that the rotor-AMB system exhibits a variety of nonlinear phenomena such as bifurcations, coexistence of multiple solutions, jump phenomenon, and sensitivity to initial conditions. Finally, the numerical simulations are performed to demonstrate and validate the accuracy of the approximate solutions. We found that all predictions from analytical solutions are in excellent agreement with the numerical integrations.     

The response of nonlinear systems to modulated high-frequency input

We study the response of a single-degree-of-freedom system with cubic nonlinearities to an amplitude-modulated excitation whose carrier frequency is much higher than the natural frequency of the system. The only restriction on the amplitude modulation is that it contain frequencies much lower than the carrier frequency of the excitation. We apply the theory to different types of amplitude modulation and find that resonant excitation of the system may occur under some conditions.     

Bayesian System Identification using auxiliary stochastic dynamical systems

Bayesian approaches to statistical inference and system identification became practical with the development of effective sampling methods like Markov Chain Monte Carlo (MCMC). However, because the size and complexity of inference problems has dramatically increased, improved MCMC methods are required. Dynamical systems based samplers are an effective extension of traditional MCMC methods. These samplers treat the posterior probability distribution as the potential energy function of a dynamical system, enabling them to better exploit the structure of the inference problem. We present an algorithm, Second-Order Langevin MCMC (SOL-MC), a stochastic dynamical system based MCMC algorithm, which uses the damped second-order Langevin stochastic differential equation (SDE) to sample a posterior distribution. We design the SDE such that the desired posterior probability distribution is its stationary distribution. Since this method is based upon an underlying dynamical system, we can utilize existing work to develop, implement, and optimize the sampler's performance. As such, we can choose parameters which speed up the convergence to the stationary distribution and reduce temporal state and energy correlations in the samples. We then apply this sampler to a system identification problem for a non-linear hysteretic structure model to investigate this method under globally identifiable and unidentifiable conditions.     

On steady rotations of a rigid body bearing a single-axis powered gyroscope whose precession axis is parallel to a principal plane of inertia

The set of steady motions of the system named in the title is represented parametrically via the gyro gimbal rotation angle for an arbitrary position of the gimbal axis.We study the set of steady motions for a system in which the gyro gimbal axis is parallel to a principal plane of inertia as well as for a system with a dynamic symmetry. We determine all motions satisfying sufficient stability conditions. In the presence of dissipation in the gimbal axis, we use the Barbashin-Krasovskii theorem to identify each steady motion as either conditionally asymptotically stable or unstable.