Supercritical Speed Research Articles

Efficiency considerations for Sommerfeld effect attenuation

In rotating machinery with a flexible foundation used for vibration isolation, the drive (motor) is designed to provide maximum power at the critical speed so that the rotor can be coasted up to a supercritical operating speed. Over the time, rotors develop some degree of unbalance and it may be impossible to balance the rotors in-situ, such as in a wind turbine. Therefore, oversized motors are used in anticipation of some unbalance. To promote passage through resonance, without altering the motor, various kinds of additional hardware and/or control mechanisms have been proposed in the past. On the other hand, simply increasing the foundation (or structural) damping reduces the amplitude of unbalance induced vibrations and the motor power requirement at the critical speed. This appears to be the simplest possible solution and has been proposed in relevant mathematical research. However, it is shown in this article that increasing foundation damping significantly reduces the overall system efficiency at the supercritical operating speed and it should not be at all considered as a meaningful engineering solution. In fact, the loss of efficiency due to the increased foundation damping is so large that it may not be compensated by a redesigned motor.

Equilibrium solutions of axially moving Timoshenko beam with a supercritical speed

Analytical solution of the equilibrium configuration of an axially moving Timoshenko beam in supercritical regime is studied. Three kinds of classical boundary conditions are considered. An analytical solution of the equilibrium configuration in terms of the axial velocity in supercritical regime is determined. But most of all, for an axially moving Timoshenko beam with the fixed ends in supercritical regime, an anti-symmetric configuration is firstly detected. And, its analytical solution is solved. Besides, numerical example shows that the solution of the equilibrium configuration bifurcates with axially velocity. And the critical velocity derived from the Timoshenko beam theory is smaller than that derived from Euler-Bernoulli beam theory. The amplitude of the solution of equilibrium configuration derived from Timoshenko beam is larger.

Dynamic modelling and vibration characteristics analysis for the bolted joint with spigot in the rotor system

The bolted flange joints with spigot (short for bolted joints) are widely used in the rotor system of aero-engine. The current models of the bolted joint usually ignore the spigot, therefore cannot account for the effect of spigot on joint's mechanical characteristics. In this paper, a new analytical model of bolted joint is proposed to describe its nonlinearity under bending moment. The model takes into account the sticking and sliding statuses of spigot by introducing a Jenkins element, therefore is able to describe the joint's damping effectively. By performing a comparison study with the finite contact element model, the accuracy of the proposed model is validated. Meantime, the comparison study highlights the important role of spigot on the hysteresis nonlinearity of the bolted joint. After that, a dynamical model of the bolted rotor is developed, in which the proposed bolted joint model is introduced by considering the influence of rotor's whirl motion. Based on the dynamical model, the dynamic characteristics of the bolted rotor are investigated in detail. The results show that the bolted joint mainly produces stiffness loss and damping nonlinearity into the rotor system. The stiffness loss occurs in any loading conditions and leads to a decrease of the rotor's critical speed. While the damping nonlinearity is mainly excited under large unbalance, it may induce a self-excited components in rotor's response at super-critical speed. The influence of the bolted joint on rotor dynamics can be reduced by decreasing flange length, friction coefficient and interference fit of the spigot.

Effects of geometric nonlinearity on the response of a long beam on viscoelastic foundation to a moving mass

We investigate the effect of the nonlinear terms arising from exact geometry on the dynamic response of the mass-beam-foundation system. In particular, we are interested in the case when the moving speed of the point mass exceeds the critical speed. We replace the infinitely long beam with a sufficiently long finite beam and use a harmonic expansion method to discretize the partial differential equation of motion. The feasibility of this technique is verified by comparing our numerical results for the linear stationary case under a point force against existing analytical solutions. By solving the eigenvalues of the linear problem, one can find the linear critical speed for a specified mass. When the moving speed of the mass is greater than the critical speed, the dynamic response of the nonlinear system eventually settles to a steady state of periodic motion as seen by an observer travelling along with the mass. This is because the beam behaves like a hardening spring, i.e., the magnitude of the resisting bending moment is always larger than its linear counterpart. In order to maintain the uniform speed of the mass during the periodic vibration, the pushing force must change with time. The amplitude of the periodic vibration increases with the moving speed of the mass in the super-critical speed range. The periodic vibration undergoes a Hopf super-critical bifurcation at a nonlinear critical speed, which is slightly higher than its linear counterpart.

Modified ball-type automatic balancer for rotating shafts: Analysis and experiment

The automatic balancer is a passive device with masses that are free to move in a shaft-concentric race. At supercritical shaft speeds, the masses automatically re-align to counteract imbalance and reduce vibration. This phenomenon is caused by the phase shift of the shaft bending response that occurs through resonance. The conventional automatic balancer (dual-ball, single race) reduces or eliminates vibration at supercritical speeds given a frictionless race, but increases vibration at subcritical and critical speed transitions. In this work, a fully passive, partitioned-track, automatic balancer with centrifugal clamps is proposed to improve performance during suboptimal operation. Both analysis and experiment are presented to support the novel design. Experimentally, a partitioned-race balancer reduced vibration by 7% during spin up, 83% at supercritical steadystate, and 65% during spin down – a complete improvement over conventional balancers. Model predictions are within 14% of experimental data at steadystate. Additionally, the use of centrifugal clamps reduced vibrations by up to 95% during critical speed transitions in analysis. The vibration reduction capability presented in this research outside of supercritical steadystate operation is a necessary advancement for the practical application of passive balancing devices.

2.5D formulation and analysis of a half-space subjected to internal loads moving at sub- and super-critical speeds

The 2.5D dynamic response of a half-space to an internal point load moving at sub- and super-critical speeds is studied in Cartesian coordinates both analytically and numerically. Firstly, the partial differential equations of waves are converted to the ordinary differential equations by the Fourier transformation. Then, a multiplying factor is derived and added to the stiffness matrix to account for the dissipation of the load moving at different velocities, with or without self-frequency. Finally, the displacements of the half-space induced by the waves propagating upward and downward are obtained analytically for the specified boundary conditions. For comparison, the half-space is also analyzed by the 2.5D finite/infinite element method. The findings of the paper include: (1) the 2.5D solutions for a point load moving extremely fast are same as those for the line load by the 2D approach, (2) the dissipation of the moving load can be considered by adjusting the multiplying factor, (3) the attenuation of the half-space above the load level is reduced by the dissipation of the moving load, and (4) the displacement of the half-space increases with the self-frequency of the moving load.

Traveling Waves for a Discrete Diffusion Epidemic Model with Delay

This paper is concerned with traveling wave solutions in a discrete diffusion epidemic model with delayed transmission. Employing the way of contradictory discussions and the bilateral Laplace transform, we obtain the nonexistence of nontrivial positive bounded traveling wave solutions. Utilizing the super-/sub-solutions method and the fixed point theory, we derive the existence of nontrivial positive traveling wave solutions with both super-critical and critical speeds. Our results indicate that the critical speed is the minimal speed.

Parametric resonances of Timoshenko pipes conveying pulsating high-speed fluids

Parametric resonances of pipes caused by pulsating fluids have received much attention. However, the main concern is the pulsation of subcritical flow. Moreover, it is usually based on the Euler-Bernoulli pipe model. This paper focuses on revealing the characteristics of parametric resonances of the Timoshenko pipe with pulsation of supercritical high-speed fluids. Supercritical flow causes the linear instability of the pipe. The coupled partial differential equations with varying parameters are derived for governing the vibration of the pipe around the non-trivial static equilibrium configuration. A direct multi-scale method is developed to analytically obtain parametric resonance responses from coupled partial differential equations with varying parameters. For the first time, the nonlinear parametric response of the Timoshenko pipe is verified by using the finite difference method (FDM). Some interesting phenomena are demonstrated. For example, unlike parametric resonance at subcritical speeds, the steady-state response of velocity pulsation excitations at supercritical speeds is not monotonic with changes in some physical parameters. For another example, when pulsation occurs at supercritical speeds, the smaller steady-state response and the instability region can be obtained through the Timoshenko pipe model. In addition, the relationship between the instability threshold of the velocity pulsation amplitude and the slenderness ratio is non-monotonic, suggesting that the Euler-Bernoulli pipe model may simplify some vibration characteristics. Due to these different phenomena, the necessity of studying the velocity pulsation in the supercritical high-speed range and the necessity of the Timoshenko model are demonstrated.

Traveling wave solutions for a predator–prey system with two predators and one prey

We study a predator–prey model with two alien predators and one aborigine prey in which the net growth rates of both predators are negative. We characterize the invading speed of these two predators by the minimal wave speed of traveling wave solutions connecting the predator-free state to the co-existence state. The proof of the existence of traveling waves is based on a standard method by constructing (generalized) upper-lower-solutions with the help of Schauder’s fixed point theorem. However, in this three species model, we are able to construct some suitable pairs of upper-lower-solutions not only for the super-critical speeds but also for the critical speed. Moreover, a new form of shrinking rectangles is introduced to derive the right-hand tail limit of wave profile.

Sommerfeld effect in a single-DOF system with base excitation from motor driven mechanism

Sommerfeld effect is a nonlinear phenomenon observed in rotating machinery driven by a non-ideal source. It is characterised by capture at resonance for a finite range of drive power followed by a sudden release. This paper investigates the dynamics of a direct current (DC) motor driven mechanism which excites the base of a vibrating structure. A slotted cam follower mechanism and a scotch yoke mechanism are chosen for the base excitations of a single degree of freedom oscillator. It is shown that the Sommerfeld effect leads to capture-and-escape (jump) through multiple resonance zones, and as a result, some intermediate speed ranges are missed. The critical power/voltage input to escape through all the resonances may exceed that required to escape the primary resonance. Moreover, while increased structural damping makes it easier to escape resonance capture, it reduces the system efficiency at super-critical operating speeds. These are important design considerations for sizing the motor drives of these mechanisms. Analytical predictions for jumps are obtained from a steady-state power balance whereas the transient analysis is performed with the help of Bond Graph (BG) models and MSC-ADAMS software. The transient responses from the numerical models are used to verify the analytical results.

A New Three-Dimensional Moving Timoshenko Beam Element for Moving Load Problem Analysis

Abstract A new three-dimensional moving Timoshenko beam element is developed for dynamic analysis of a moving load problem with a very long beam structure. The beam has small deformations and rotations, and bending, shear, and torsional deformations of the beam are considered. Since the dynamic responses of the beam are concentrated on a small region around the moving load and most of the long beam is at rest, owing to the damping effect, the beam is truncated with a finite length. A control volume that is attached to the moving load is introduced, which encloses the truncated beam, and a reference coordinate system is established on the left end of the truncated beam. The arbitrary Lagrangian–Euler method is used to describe the relationship of the position of a particle on the beam between the reference coordinate system and the global coordinate system. The truncated beam is spatially discretized using the current beam elements. Governing equations of a moving element are derived using Lagrange’s equations. While the whole beam needs to be discretized in the finite element method or modeled in the modal superposition method (MSM), only the truncated beam is discretized in the current formulation, which greatly reduces degrees-of-freedom and increases the efficiency. Furthermore, the efficiency of the present beam element is independent of the moving load speed, and the critical or supercritical speed range of the moving load can be analyzed through the present method. After the validation of the current formulation, a dynamic analysis of three-dimensional train–track interaction with a non-ballasted track is conducted. Results are in excellent agreement with those from the commercial software simpack where the MSM is used, and the calculation time of the current formulation is one-third of that of simpack. The current beam element is accurate and more efficient than the MSM for moving load problems of long three-dimensional beams. The derivation of the current beam element is straightforward, and the beam element can be easily extended for various other moving load problems.

ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ ДИНАМИКИ ГИБКОГО РОТОРА С ДВУМЯ ШАРОВЫМИ АВТОБАЛАНСИРАМИ

Ball auto-balancing devices can to compensate changes of unbalance "on the move" only for rotors operating at supercritical speeds. For automatic balancing of such rotors, classified as flexible rotors, several auto-balancers located in different cross sections of the shaft are necessary. This makes it necessary to account bending fluctuations on studies of dynamics of the rotor with auto-balancers, that is especially important in the design of the real rotors. In view of the complexity of experimental studies of such rotors in the article the method of direct numerical simulation of the dynamics of the flexible rotor system – supports – auto-balances is considered. The methodological basis of this method is the use of a discrete multi-mass rotor model, which is equivalent in dynamic characteristics to a real rotor, and also the equations of dynamics of the system discrete rotor – supports – auto-balancers, obtained in the direct form of recording. For definition of discrete masses and a matrix of coefficients of influence of stiffness of rotor cross-sections it is supposed to use calculations for finite-element model of a real rotor by existing software complexes of the engineering analysis. The mathematical model of the system dynamics obtained by the Lagrange method takes into account the non-stationarity of the rotor rotation speed, the influence of gravity and the rolling friction of the balls in the auto-balancer cages. Verification of the mathematical model was performed by reproducing the published data using a computational model for a two-support single-disk three-mass rotor with a two-ball auto-balancer. For a four-mass rotor with two two-ball auto-balancers, the results of numerical simulation of dynamics for the modes of acceleration, steady-state rotation and deceleration are presented. It is shown that for the system under consideration, only partial auto-balancing takes place in the steady rotation mode, including after a stepwise increase of the imbalance.

Coupled bending torsional vibrations of non-ideal energy source rotors under non-stationary operating conditions

In the automotive industry, high-speed elecrical engines will be used in a very large range of speeds, leading to non-stationary operating conditions. Thus more critical speeds may be crossed a very large number of times during the whole life-time of the engine. Therefore, estimating accurately the non-stationary loads and deformations during these transient regimes is of first importance for a correct design. In this paper, we propose a novel dynamic model for unbalanced high-speed rotors with less restrictive assumptions. The finite element model accounts for flexion, torsion and traction-compression leading to six degrees of freedom on each node. The non-ideal energy source is considered and the rotor is running under non-stationary operating conditions and crossing supercritical speeds. The angular displacement is defined in such a way that it combines simultaneously the intrinsic nominal rotation and the torsional deformation. The comparison of the proposed model with other models under different assumptions on the energy source and on the bending-torsion coupling shows that our model offers more accurate prediction for both lateral and torsional vibrations when crossing critical speeds. The ability of the proposed model to account for the mutual influence between lateral and torsional behavior is highlighted and is exhibited through time-frequency analyses.

Motion modes of the nonlinear mechanical system of the rotor autobalancer

The paper analyzes the possible motion modes of an unbalanced rotor with an auto-balancing device. It is shown that for given parameters of the system, there are several qualitatively different stable motion modes. Previously unknown types of motion modes are detected. It is found that for a given supercritical working rotor speed, the system can have at least three possible stable motion modes depending on the initial conditions of its motion. A qualitative analysis of the stability ranges of all possible motion modes of the system was performed.

Wave propagation of a discrete SIR epidemic model with a saturated incidence rate

This paper is concerned with the traveling wave solutions for a discrete SIR epidemic model with a saturated incidence rate. We show that the existence and non-existence of the traveling wave solutions are determined by the basic reproduction number [Formula: see text] of the corresponding ordinary differential system and the minimal wave speed [Formula: see text]. More specifically, we first prove the existence of the traveling wave solutions for [Formula: see text] and [Formula: see text] via considering a truncated initial value problem and using the Schauder’s fixed point theorem. The existence of the traveling wave solutions with speed [Formula: see text] is then proved by using a limiting argument. The main difficulty is to show that the limit of a decreasing sequence of the traveling wave solutions with super-critical speeds is non-trivial. Finally, the non-existence of the traveling wave solutions for [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] is proved.

Characteristics analysis of rotor model with support–foundation looseness fault considering elastic–plastic contact

In this paper, the nonlinear response characteristics of a single degree of freedom lumped mass model with a new support looseness fault are analyzed. In this model, a new nonlinear contact stiffness model between the support and foundation is considered, which considers the elastic–plastic deformation of the contact parts. The nonlinear response of the system is obtained by numerical integration methods. The nonlinear results are compared with an asymmetric model and a piece-wise contact model. By comparing three different looseness fault models, the mechanism of generating multi-frequencies and fractional frequencies is analyzed. The result shows that plastic deformation makes the displacement amplitude increase rapidly; odd harmonics appear in symmetric stiffness model due to the modulation of symmetric shock forces; however, harmonic response appears in asymmetric stiffness model; when the contact number is odd, the frequency components contain odd or even order fractional frequencies at supercritical speeds due to the incompatibility of contacts on the upper and lower surfaces; when the contact number is even, multiple frequency components appear at subcritical frequencies.

Existence of travelling waves with the critical speed for an influenza model with treatment

The main purpose of this paper is to study the existence of travelling waves with a critical speed for an influenza model with treatment. By using some analysis techniques that involve super-critical speeds and an approximation method, the existence of travelling waves with the critical speed is proved.

Wave Motion in an Ice Sheet with Crack under Uniformly Moving Load

The analytical solution to the problem of the behavior of an ice sheet with rectilinear crack under the action of uniformly moving rectangular external pressure zone simulating an air-cushion vehicle is obtained using the Wiener–Hopf technique. The ice cover is simulated by thin elastic semi-infinite plates of constant thickness floating on the surface of an incompressible fluid of finite depth. Two configurations are considered: 1) two semi-infinite plates with free edges (whose thicknesses can be different) are separated by a crack; 2) fluid is bounded by the vertical wall and the ice cover edge can be both free and frozen to the wall. In the case of the contact of plates of the same thickness, as well as in the presence of the wall, the solution is obtained in the explicit form. It is shown that in the case of the contact of identical plates with free edges, edge waveguide modes traveling along the crack are excited when the load moves at a supercritical speed. Both the wave forces acting on the moving body and the deflections of the plates are investigated for various values of the plate thicknesses and the load velocity in the sub- and supercritical regimes.

Sommerfeld effect at forward and backward critical speeds in a rigid rotor shaft system with anisotropic supports

A rotor dynamic system requires high power/torque to accelerate through its resonance conditions or critical speeds to operate at a super-critical speed. Most practical drive sources are non-ideal in nature, i.e. they can only provide a limited amount of power to the rotor system. Thus, if there is insufficient power to overcome a resonance then the rotor speed may get arrested at that resonance or take a long time to escape from the resonance; and thereby damage the system. For the present analysis, a Direct Current (DC) motor is considered as the non-ideal prime mover for a rigid rotor shaft with unbalanced disk which is supported by flexible anisotropic supports/bearings. Due to the anisotropy in the supports, both forward and backward whirl motions of the rotor get excited. This causes a multi-Sommerfeld effect where two non-linear jump phenomena are manifested at the first forward and backward critical speeds during rotor coast up and coast down. When the two critical speeds are close to each other, a complex speed capture and jump phenomena occurs. Accordingly, the transitions through both resonance conditions are examined. At first, the analytical solutions are obtained under steady state assumption and then transient simulations are performed with the help of a bond graph (BG) model.

Marine Vessel Wave Wake: Transient Effects When Accelerating or Decelerating

It is well known that the waves generated by marine vessels, often referred to as wave wake or wash, can cause many issues when operating in sheltered waterways, including, but not limited to, erosion of shorelines and damage to maritime structures, and present a danger to other waterway users. Much research has been undertaken to understand the characteristics of these waves and their effects better, especially for high-speed vessels that operate in shallow water where particularly large and energetic waves are generated. However, in general, all previous studies have considered only steady-state conditions in which vessel speed remains constant; however, many vessel operations, particularly those of commuter ferries, in which regular passages through the transcritical zone to supercritical speeds (in terms of depth Froude number) are required. The present study describes a novel series of model-scale experiments used to quantify the waves during both acceleration and deceleration phases. Notable transient effects were found to occur during the acceleration phase that significantly increased both the height and period of the maximum wave compared to height and period of the maximum wave occurring at the corresponding steady-state speed. The wave characteristics at constant speed were used when assessing whether a particular vessel met wash criteria, and such criteria were likely significantly exceeded when a vessel accelerated to a supercritical speed, which could lead to the occurrence of wave wake issues. In an interesting finding, the study also found no such increase in wave characteristics when the same vessel decelerated back through the transcritical speed zone.