Supercritical Speed Research Articles

Multiple attractors and critical parameters and how to find them numerically: the right, the wrong and the gambling way

In recent years, several authors have proposed ‘easier numerical methods’ to find the critical speed in railway dynamical problems. Actually, the methods do function in some cases, but in most cases it is really a gamble. In this article, the methods are discussed and the pros and contras are commented upon. I also address the questions when a linearisation is allowed and the curious fact that the hunting motion is more robust than the ideal stationary-state motion on the track. Concepts such as ‘multiple attractors’, ‘subcritical and supercritical bifurcations’, ‘permitted linearisation’, ‘the danger of running at supercritical speeds’ and ‘chaotic motion’ are addressed.

Gravity-driven granular free-surface flow around a circular cylinder

AbstractSnow avalanches and other hazardous geophysical granular flows, such as debris flows, lahars and pyroclastic flows, often impact on obstacles as they flow down a slope, generating rapid changes in the flow height and velocity in their vicinity. It is important to understand how a granular material flows around such obstacles to improve the design of deflecting and catching dams, and to correctly interpret field observations. In this paper small-scale experiments and numerical simulations are used to investigate the supercritical gravity-driven free-surface flow of a granular avalanche around a circular cylinder. Our experiments show that a very sharp bow shock wave and a stagnation point are generated in front of the cylinder. The shock standoff distance is accurately reproduced by shock-capturing numerical simulations and is approximately equal to the reciprocal of the Froude number, consistent with previous approximate results for shallow-water flows. As the grains move around the cylinder the flow expands and the pressure gradients rapidly accelerate the particles up to supercritical speeds again. The internal pressure is not strong enough to immediately push the grains into the space behind the cylinder and instead a grain-free region, or granular vacuum, forms on the lee side. For moderate upstream Froude numbers and slope inclinations, the granular vacuum closes up rapidly to form a triangular region, but on steeper slopes both experiments and numerical simulations show that the pinch-off distance moves far downstream.

External and internal resonances of the pipe conveying fluid in the supercritical regime

Nonlinear forced vibration of a viscoelastic pipe conveying fluid around the curved equilibrium configuration resulting from the supercritical flow speed is investigated with the emphasis on dynamics of external and internal resonance. The governing equation for the pipe system, a nonlinear integro-partial-differential model with variable coefficients, is truncated into a discrete perturbed gyroscopic system via the Galerkin method. A condition for the two-to-one internal resonance is established, and the condition implies that internal resonance is possible in the supercritical regime. The method of multiple scales is developed to present the solvability condition of approximate solutions. The modulation equations of the amplitude and the phase are derived from the condition. The first two primary resonances in the presence of the two-to-one internal resonance are examined. Steady-state solutions and their stabilities are determined. Some typical steady-state responses are demonstrated via the amplitude–frequency curves. In addition to jumping, hysteresis and saturation, other dynamical behaviors are observed to highlight the effects of the modal interaction in the internal resonance. The analytical results are confirmed by the numerical integrations.

Post-buckling bifurcations and stability of high-speed axially moving beams

The nonlinear forced dynamics of an axially moving beam in the supercritical speed regime is numerically investigated in this paper, with special consideration to the case possessing an internal resonance between the first two modes. At critical speed, the initial equilibrium configuration of the beam becomes unstable and a new stable non-trivial equilibrium solution together with its symmetric counterpart arises. A closed-form solution for the post-buckling configuration is introduced and the system is assumed to be subject to a transverse harmonic excitation load at its buckled state. The equation of motion is cast into new coordinates which gives the equation governing the motion of the beam about the buckled state. This equation is discretized via the Galerkin method which yields a set of nonlinear ordinary differential equations (NODEs) with quadratic and cubic nonlinear terms. The set of NODEs are solved either via the pseudo-arclength continuation technique or by means of direct time integration so as to obtain frequency–response curves as well as bifurcation diagrams of Poincaré sections about the buckled state. Results are shown through time histories, phase-plane diagrams, and Poincaré sections.

A numerical study of traveling waves in a viscoelastic cylinder cover under rolling contact

Polymer-covered cylinders are widely used in high-speed industrial rolling contact machines. In this paper, traveling waves which appear at high speeds in the viscoelastic polymer cover of a cylinder due to rolling contact are studied using a 2D finite element (FE) model for a two-cylinder system. Through eigenmode analysis of the polymer cover, it is found that an infinite number of natural mode families exist for the cylinder cover due to its finite thickness. A critical speed below which the traveling waves do not appear can be calculated on the basis of a resonance condition using the modal information. In dynamic analyses, traveling waves in the cover are identified as modified quasi-elastic Rayleigh waves composed of the eigenmodes of the polymer cover. The symbiosis of the eigenmode and wave propagation aspects allows the formation of a coherent overall picture of the traveling wave phenomenon. The critical speed according to the resonance condition is also the minimum phase velocity of the waves propagating in the cover. At the critical speed and above, there is a spectrum of resonant speeds due to wave dispersion. It is found that the traveling wave phenomenon is best described as a Rayleigh wave resonance in which contact-induced modified quasi-elastic Rayleigh waves arising at critical and supercritical speeds superpose to form a strong traveling shock wave.

Modelling the Floating Ladder Track Response to a Moving Load by an Infinite Bernoulli-Euler Beam on Periodic Flexible Supports

Abstract.An infinite Bernoulli-Euler beam (representing the “combined rail” consisting of the rail and longitudinal sleeper) mounted on periodic flexible point supports (representing the railpads) has already proven to be a suitable mathematical model for the floating ladder track (FLT), to define its natural vibrations and its forced response due to a moving load. Adopting deliberately conservative parameters for the existing FLT design, we present further results for the response to a steadily (uniformly) moving load when the periodic supports are assumed to be elastic, and then introduce the mass and viscous damping of the periodic supports. Typical support damping significantly moderates the resulting steady deflexion at any load speed, and in particular substantially reduces the magnitude of the resonant response at the critical speed. The linear mathematical analysis is then extended to include the inertia of the load that otherwise moves uniformly along the beam, generating overstability at supercritical speeds – i.e. at load speeds notably above the critical speed predicted for the resonant response when the load inertia is neglected. Neither the resonance nor the overstability should prevent the safe implementation of the FLT design in modern high speed rail systems.

Nonlinear dynamics of axially moving viscoelastic beams over the buckled state

This paper investigates the nonlinear forced dynamics of an axially moving viscoelastic beam in the supercritical speed regime, when the system is beyond the first instability. Both cases with and without internal resonances are examined. The beam configurations beyond the first bifurcation as well as the critical speeds are obtained analytically. The equation of motion about the buckled state is obtained by substituting the buckled configuration into the main equation of motion. A distributed external excitation load is then applied to the buckled system. The partial differential equation of motion about the buckled state is then discretized by means of the Galerkin method, yielding a set of second-order nonlinear ordinary differential equations with cubic and quadratic terms. The nonlinear resonant responses as well as bifurcation diagrams of Poincaré maps of the system are analyzed using the pseudo-arclength continuation technique along with direct time integration.

Supercritical forced response of coupled motion of a nonlinear transporting beam

Steady-state periodic responses of nonlinear coupled planar motions are investigated for transporting beams in the supercritical transport speed ranges. The straight equilibrium configuration bifurcates into multiple equilibrium positions in the supercritical regime. The finite-difference schemes are developed to calculate the non-trivial static equilibrium and the steady-state response under simply supported or clamped boundary conditions. The forced vibration is assumed to be spatially uniform and temporally simple harmonic. Based on the long time series, the steady-state transversal amplitudes of nonlinear planar motions are recorded with changing load frequencies. A resonance exists if the external load frequency approaches the fundamental frequency. The effects of material parameters and vibration amplitude on the resonance responses are investigated. The coupled planar model can be reduced to two nonlinear models on transversal vibrations, an integro-partial–differential equation and a partial–differential one. Numerical examples are displayed for the pros and cons between the two transversal models. It is also revealed that the increased axial speed converts the hardening-type behavior into the softening-type one.

Optimisation of hybrid high-modulus/high-strength carbon fibre reinforced plastic composite drive shafts

This study deals with the optimisation of hybrid composite drive shafts operating at subcritical or supercritical speeds, using a genetic algorithm. A formulation for the flexural vibrations of a composite drive shaft mounted on viscoelastic supports including shear effects is developed. In particular, an analytic stability criterion is developed to ensure the integrity of the system in the supercritical regime. Then it is shown that the torsional strength can be computed with the maximum stress criterion. A shell method is developed for computing drive shaft torsional buckling. The optimisation of a helicopter tail rotor driveline is then performed. In particular, original hybrid shafts consisting of high-modulus and high-strength carbon fibre reinforced epoxy plies were studied. The solutions obtained using the method presented here made it possible to greatly decrease the number of shafts and the weight of the driveline under subcritical conditions, and even more under supercritical conditions. This study yielded some general rules for designing an optimum composite shaft without any need for optimisation algorithms.

Research on Wash Wave in Shallow Water Caused by Ship Moving at High Speed

Based on the theory of shallow-water wave, the theoretical model was established for calculating wash wave caused by ship moving at subcritical and supercritical speed. Wave elevation and pressure variation were obtained by numerical simulation, and their features were analyzed. A measuring system of wash wave and pressure variation was developed, and wave elevation and pressure variation induced by a towed ship model were measured. A good agreement existed between the calculated with experimental results.

Isolation of Train-Induced Ground-Borne Vibration by Trenches on a Poroelastic Half-Space

The screening effects of trenches on train-induced ground vibration are investigated by an analytical method. The embankment and soil banks beside the trench are modeled as elastic layers with appropriate widths. A fully saturated poroelastic half-space is first utilized in the investigation of the isolation of train-induced ground vibration. Biot’s theory is applied to characterize the saturated poroelastic half-space, considering the coupling between the soil skeleton and the fluid. Based on Fourier transforms and Fourier series, the governing equations are solved. The various performances of the trenches on the single-phase elastic half-space and saturated poroelastic half-space as screening barriers to mitigate the ground vibration are specially investigated. It is found for the vibration generated by the axle train load with supercritical speed that the discrepancy between the screening efficiency of the trenches on the poroelastic half-space and elastic half-space is significant; the single-phase elastic half-space model may significantly underestimate the screening effect of the trench. In addition, an excellent screening effect can be achieved by an open trench to the ground vibrations generated by the dynamic wheel-rail load.

Forced Vibrations of Supercritically Transporting Viscoelastic Beams

This study focuses on the steady-state periodic response of supercritically transporting viscoelastic beams. In the supercritical speed range, forced vibrations are investigated for traveling beams via the multiscale analysis with a numerical confirmation. The forced vibration is excited by the spatially uniform and temporally harmonic vibration of the supporting foundation. A nonlinear integro-partial-differential equation is used to determine steady responses. The straight equilibrium configuration bifurcates in multiple equilibrium positions at supercritical translating speeds. The equation is cast in the standard form of continuous gyroscopic systems via introducing a coordinate transform for nontrivial equilibrium configuration. The natural frequencies and modes of the supercritically traveling beams are analyzed via the Galerkin method for the linear standard form with space-dependent coefficients under the simply supported boundary conditions. Based on the natural frequencies and modes, the method of multiple scales is applied to the governing equation to determine steady-state responses. To confirm results via the method of multiple scales, a finite difference scheme is developed to calculate steady-state response numerically. Quantitative comparisons demonstrate that the approximate analytical results have rather high precision. Numerical results are also presented to show the contributions of foundation vibration amplitude, viscoelastic damping, and nonlinearity to the response amplitude for the first and the second mode.

Stability and Instability of the KDV Solitary Wave Under the KP-I Flow

We consider the KP-I and gKP-I equations in $\mathbb{R}\times (\mathbb{R}/2\pi \mathbb{Z})$. We prove that the KdV soliton with subcritical speed $0<c<c^*$ is orbitally stable under the global KP-I flow constructed by Ionescu and Kenig \cite{IK}. For supercritical speeds $c>c^*$, in the spirit of the work by Duyckaerts and Merle \cite{DM}, we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity. This last result also holds for the gKP-I equation.

Vibration Characteristics of Rotating Thin Disks—Part II: Analytical Predictions

This paper is concerned with the geometric nonlinear analysis of the lateral displacement of thin rotating disks when subjected to a space fixed stationary force. Of particular interest is the development of the stationary wave and the effect of this wave on the frequency response of the disk as a function of its rotational speed. The predictions of this analysis are compared with experimental data obtained in a companion paper (Khorasany and Hutton, “Vibration Characteristics of Rotating Thin Disks—Part I: Experimental Results,” ASME J. Appl. Mech., 79(4), p. 041006). The governing equations are based on Von Kármán plate theory. A Galerkin solution of the governing non linear equations is developed. The eigenfunctions derived from the linear analysis of a stationary disk are used as approximations to the spatial response of the disk, and the eigenfunctions of the biharmonic equation as approximations for the stress function. Using the developed solution, the equilibrium configuration of the disk under the application of a space fixed force is found. In order to facilitate the prediction of the frequency response, as a function of disk rotational speed, the governing nonlinear equations are linearized around the equilibrium solution. The linearized equations are then used to find the eigenvalues of the spinning disk under the application of a space fixed force. The effect of different levels of nonlinearity on the disk frequencies is studied and compared with experimental results. The analysis is shown to produce an accurate representation of the measured response. Of particular interest is the disk response at speeds close to and above the linear critical speed. In this region, both the analysis and the experimental results display frequency “lock-in” behavior in which the frequency of backward travelling waves becomes constant for supercritical speeds. No speed exists for which backward travelling waves have zero frequency. Thus, critical speeds do not exist in the presence of geometric nonlinearities.

The Water Depth Effect on Ship Pressure Distribution and Surface Wave

Applying linear marine hydrodynamics theory, the Kelvin wave source Green function of steady motion in finite depth was decomposed to three parts: array of simple Rankine sources, local distribution and wave part. Both at subcritical and supercritical speed, the singularity of integral function were eliminated by different integral path. The Kelvin source was distributed on the surface of ship by panel method, and the ship pressure distribution and wave pattern of ship in finite depth were calculated. The difference and connection between finite and infinite depth results were compared. It can provide the theoretical arithmetic base of improving the ship seaworthiness, increasing the speed and optimization of ship in river and offshore strip.

Linear parameter varying control design for rotating systems supported by journal bearings

Linear parameter varying (LPV) control is a model-based control technique that takes into account time-varying parameters of the plant. In the case of rotating systems supported by lubricated bearings, the dynamic characteristics of the bearings change in time as a function of the rotating speed. Hence, LPV control can tackle the problem of run-up and run-down operational conditions when dynamic characteristics of the rotating system change significantly in time due to the bearings and high vibration levels occur. In this work, the LPV control design for a flexible shaft supported by plain journal bearings is presented. The model used in the LPV control design is updated from unbalance response experimental results and dynamic coefficients for the entire range of rotating speeds are obtained by numerical optimization. Experimental implementation of the designed LPV control resulted in strong reduction of vibration amplitudes when crossing the critical speed, without affecting system behavior in sub- or super-critical speeds.

The optimal mass transport problem for relativistic costs

In this paper, we study the optimal mass transportation problem in \({\mathbb{R}^{d}}\) for a class of cost functions that we call relativistic cost functions. Consider as a typical example, the cost function c(x, y) = h(x − y) being the restriction of a strictly convex and differentiable function to a ball and infinite outside this ball. We show the existence and uniqueness of the optimal map given a relativistic cost function and two measures with compact support, one of the two being absolutely continuous with respect to the Lebesgue measure. With an additional assumption on the support of the initial measure and for supercritical speed of propagation, we also prove the existence of a Kantorovich potential and study the regularity of this map. Besides these general results, a particular attention is given to a specific cost because of its connections with a relativistic heat equation as pointed out by Brenier (Extended Monge–Kantorovich Theory. Optimal Transportation and Applications, 2003).

Efficiency Improvement of an Automatic Ball Balancer

An automatic ball balancer is a unique vibration suppression device for rotor systems. Theoretically, two balls in a cylindrical chamber of the rotor are located at the optimal positions on the opposite side to the unbalance and cancel the unbalance automatically in the super-critical speed range. However, this device is not used widely due to two malfunctions. One is the influence of friction. Due to the inevitable friction between the balls and the inside wall of the channel, the balls stop near the optimal positions and do not balance the rotor perfectly. The other is the self-excited oscillation which occurs near and above the major critical speed. The objectives of the present paper are to clarify the fundamental characteristics of a ball balancer and to introduce some simple methods to eliminate these malfunctions.

A Description of the Stationary Wave Pattern of a Marine Craft in Shallow and Intermediate Water Depths

A stationary phase model is used to study supercritical waves generated by high speed ferries. Some general relationships in terms of wave angle, propagation direction, dispersion relationship and depth wavelength relationship are explored and discussed. In particular, it is shown that the wave pattern generated by high speed craft at supercritical speeds depends mainly on the relationship of water depth and ship speed and that the wave patterns are similar in terms of location of crests and troughs for a given depth Froude number. In addition it is found that the far field wave pattern can be described adequately using a single moving point source. The theoretical model compares well with towing tank measurements and full scale data over a range of parameters and hull shapes. The paper also demonstrates that the far field wave pattern at supercritical speeds should be non-dimensionalised by water depth and not hull length unlike it is usually done for subcritical speeds.

Dynamics of a Milkshaker – Passage Through Resonance and Frequency Transformation

AbstractRotor dynamics is a fascinating subject both from an experimental and a theoretical point of view. Most often experiments on various phenomena require more or less sophisticated test rigs, either because the experiments are dangerous or because the effects are difficult to reproduce. In teaching it is, however, beneficial to have experiments that are simple and can be performed by the students themselves. One of these examples is a standard milkshaker, which at a closer look, exhibits rich dynamical phenomena. The first phenomenon that can be observed and studied is the passage through resonance. Depending on the eccentricity of the rotor, the driving torque of the motor is strong enough or not to reach supercritical speeds. If for a given torque the eccentricity is too large, the system gets stuck in the resonance with a rather large amplitude. A second phenomenon that can be observed is the following: When the milkshaker is placed on an even surface it starts to move on the surface. The movement is caused by a wobbling motion of the system due to the eccentricity. Although the angular velocity of the rotor is high, the motion on the surface is quite slow in comparison. This is an interesting phenomenon that can be related to mechanical frequency transformation which occurs in the contact between the milkshaker and the ground. Depending on whether the rotor is running in a supercritical range or stuck below the resonance frequency, different motions can be observed. The system can be analyzed with a relatively simple nonlinear rigid body model. In this paper we study both phenomena mentioned above from a theoretical point of view. The equations of motion are derived in analytical form and their nonlinear behavior is investigated. Due to its relatively simple nature, the system has been used in lectures as a demonstrator and for student tutorial projects.