Showing papers on "Critical speed published in 2016"
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TL;DR: In this article, a sophisticated 3D finite element (FE) modeling was developed to simulate the dynamic response of ballasted railway tracks subjected to train moving loads, and the critical speed was investigated for various train-track-ground system conditions.
Abstract: Due to recent congestion of highways in many countries around the world, railways have become the most popular means of public transportation, which has increased the demand for heavier and faster trains. High speeds and heavy loads of trains are usually accompanied with large vibrations in the train–track–ground system, especially when train speed reaches its critical value, leading to possible train derailment and track damages. This unwanted scenario makes it important for railway geotechnical engineers to investigate the behaviour of ballasted railway track foundations for high-speed trains, with special reference to critical speed. In the current paper, a sophisticated three-dimensional (3D) finite element (FE) modelling was developed to simulate the dynamic response of ballasted railway tracks subjected to train moving loads, and the critical speed was investigated for various train–track–ground system conditions. The results were presented in terms of the evolution of the coefficient of dynamic amplification of sleeper deflection versus train speed, which have been synthesized into simple sensitivity charts that can be used to determine the critical speed corresponding to the conditions of a particular train–track–ground system.
89 citations
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TL;DR: In this paper, the authors present a method to rapidly predict the speed at which high speed trains travel close to the wave propagation velocity of the supporting track-ground system, which results in a significant increase in track maintenance due to subgrade deterioration.
Abstract: When high speed trains travel close to the wave propagation velocity of the supporting track-ground system, large amplitude track deflections are generated. This has safety implications, and also results in a significant increase in track maintenance due to subgrade deterioration. Thus, this paper presents a method to rapidly predict the speed at which these ‘critical velocity’ effects occur. The method is based upon a dispersion analysis of both the track (either ballast or non-ballasted/slab track) and the underlying ground, which are treated as uncoupled systems. Unlike previous approaches, the new calculation approach is fully automated thus not requiring any post-processing to extract the soil dispersion curve. It also works for soil layers of arbitrary depth, uses minimal computing power and can calculate critical speeds associated with higher soil modes. The dispersion based method can be deployed on new/existing lines via a drop-weight test, or using existing geotechnical data. Its accuracy is tested by comparing the results against an alternative semi-analytical, quasi-static railtrack model, and found to be 97% accurate. The code is useful for railway track infrastructure design and its short run times mean it can be used as a scoping tool for newly proposed high speed railroad lines. To obtain new insights into the key variables effecting critical velocity, a sensitivity analysis is undertaken using 1000 random soil profiles. It is found that on average, for the same track height, slab tracks are less likely to encounter critical velocity issues than ballasted tracks because their critical speed is typically 11% higher. It is also shown that track height plays an important role with increases in slab track thickness and reductions in ballasted track thickness both causing increases in critical velocity. Furthermore, it is found that soil saturation affects critical speed considerably (by up to 12–17% depending on track type) because changes to Poisson’s ratio alter the dispersion characteristics of layered soils in the mid-frequency range, where critical velocity effects occur. Finally, it is shown that railpad stiffness has a low influence, and that increasing the rail bending stiffness on ballasted tracks increases critical speed.
71 citations
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TL;DR: In this article, the rotordynamics of a double-helical gear transmission system with bearing and gyroscopic effect was investigated using the finite element method, in which Timoshenko beam finite element is used to represent the shaft, a rigid mass for the gear.
Abstract: The rotordynamics of a double-helical gear transmission system is investigated. The equation of motion of the system with bearing and gyroscopic effect is derived by using the finite element method, in which Timoshenko beam finite element is used to represent the shaft, a rigid mass for the gear. Natural frequencies, mode shapes and Campbell diagrams are illustrated to indicate the effects of gear input speed and time varying mesh stiffness. Besides, effects of mesh stiffness on the critical speed of the gear transmission system are analyzed. The numerical results show that the axial force has significant influence on the natural frequency and the mode shape of the double-helical gear transmission system, for which the mix whirling motion dominates the natural characteristics. There are two higher critical speed curves which increase with the mesh stiffness, but one of them is related to the gyroscopic effect.
62 citations
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TL;DR: In this article, the impact of axially moving single-layered graphene sheet (SLGS) subjected to magnetic field is investigated and the results indicated that the critical speed of moving SLGS is strongly dependent on the moving speed.
Abstract: In the present research, vibration and instability of axially moving single-layered graphene sheet (SLGS) subjected to magnetic field is investigated. Orthotropic visco-Pasternak foundation is developed to consider the influences of orthotropy angle, damping coefficient, normal and shear modulus. Third order shear deformation theory (TSDT) is utilized due to its accuracy of polynomial functions than other plate theories. Motion equations are obtained by means of Hamilton’s principle and solved analytically. Influences of various parameters such as axially moving speed, magnetic field, orthotropic viscoelastic surrounding medium, thickness and aspect ratio of SLGS on the vibration characteristics of moving system are discussed in details. The results indicated that the critical speed of moving SLGS is strongly dependent on the moving speed. Therefore, the critical speed of moving SLGS can be improved by applying magnetic field. The results of this investigation can be used in design and manufacturing of marine vessels in nanoscale.
58 citations
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TL;DR: In this article, the use of finite element models in the time domain to represent a load moving on a railway track on a flexible ground was investigated, and a systematic study was carried out to compare different sizes and shapes of the finite element mesh, different boundary conditions intended for suppressing reflections from the truncated model boundaries, and different models of soil damping.
58 citations
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TL;DR: In this article, a magnetic bearing (MB) structure with short axial length and small thrust disk is proposed for a high-speed motor with a high speed rigid rotor, which takes into consideration eddy current effects, and leakage effects is developed.
Abstract: With the development of the magnetic bearing (MB) and its application in industry, the demands of high-speed MB are developing, which bring up new problems and requirement for the design. The motor with a high-speed rigid rotor requires that the axial length of MB is minimized to increase the critical speed, and the diameter of thrust disk is minimized to reduce the air friction loss. Also because of the high speed, the eddy currents induced in the iron cores have a significant influence on the dynamic characteristics of the MB. To design a high-performance MB for a high-speed motor, first an MB structure that has the feature of short axial length and small thrust disk is recommended for the high-speed motor. Next, the dynamic model, which takes into consideration eddy current effects, and leakage effects is developed. Then, an optimal design method with multiobjective of minimum length and air friction loss is presented. Design constraints are imposed, which includes the simultaneous consideration of material properties, load capacity, and power amplifier. Finally, a design example is given and a prototype is manufactured. The experimental results demonstrate that the proposed MB structure and the optimization method are feasible and valid in the application of a high-speed motor.
54 citations
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03 Sep 2016TL;DR: In this article, a finite element (FE) model is developed to study the detailed earthwork geometries and then used to investigate the effect of embankment stiffness on ground vibration levels both within and at distance from the track-embankment structure.
Abstract: This paper presents the numerical prediction of ground vibration levels on embankmented high speed rail lines. First, a campaign of field tests is described, focused on the three-dimensional (3D) monitoring of ground vibration on at-grade and embankmented tracks. Next, a finite element (FE) model is developed to study the detailed earthwork geometries. The model is validated and then used to investigate the effect of embankment stiffness on ground vibration levels both within and at distance from the track-embankment structure. It is found that ground vibration levels are sensitive to embankment stiffness, with stiffer embankments resulting in greatly reduced vibration levels. Further, it is found that in general, well constructed embankmented tracks generate lower levels of ground vibration in comparison to at-grade lines.
51 citations
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TL;DR: In this paper, a dynamic analysis model comprising track, embankment and layered ground was presented based on the two-and-half-dimensional (2.5D) finite elements combining with thin-layer elements to predict vibrations generated by train moving loads.
Abstract: High-speed train induced vibrations of track structure and underlying soils differ from that induced by low-speed train. Determining the critical speed of train operation remains difficult due to the complex properties of the track, embankment and ground. A dynamic analysis model comprising track, embankment and layered ground was presented based on the two-and-half-dimensional (2.5D) finite elements combining with thin-layer elements to predict vibrations generated by train moving loads. The track structure is modeled as an Euler–Bernoulli beam resting on embankment. The train is treated as a series of moving axle loads; the embankment and ground are modeled by the 2.5D finite elements. The dynamic responses of the track structure and the ground under constant and vibrating moving loads at various speeds are presented. The results show that the critical speed of a train moving on an embankment is higher than the Rayleigh wave velocity of the underlying soil, attributed to the presence of the track structure and the embankment. It is found that the dynamic response of ground induced by moving constant loads is mostly dominated by train speed. While for the moving load with vibration frequency, the ground response is mostly affected by the vibration frequency instead of train speed. Mach effect appears when the train speed exceeds the critical speed of the track–embankment–ground system.
49 citations
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TL;DR: In this article, a theoretical study on the sudden unbalance and rub-impact caused by blade loss was conducted, in particular the response of the rotor on a rotor test rig with sudden imbalance and a Rub-impact device designed respectively.
46 citations
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TL;DR: In this paper, the free vibrations and stability of rotating micro/nano beams based on modified couple stress theory were investigated, and the effects of geometrical and theoretical parameters on natural frequency and stability were investigated and important outcomes were reported.
40 citations
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TL;DR: An effective phase-leading compensation method is presented to compensate the large phase lag of the switching power amplifier and shows that the optimum compensator can minimize the displacement resonance peak, when the rotor passes the first bending critical speed with low amplifier current.
Abstract: This paper proposes a phase compensator design for the flexible rotor in the high-energy-density magnetically suspended motor (HEDMSM), whose rated rotational speed is above the first bending critical speed. The optimum phase and magnitude of the control system nearby the critical speed are analyzed. Additionally, the coil inductance of the large-power HEDMSM is always large. That will lead to large phase lag and make optimum compensator design more difficult. Therefore, an effective phase-leading compensation method is presented to compensate the large phase lag of the switching power amplifier. Then, the optimum compensator is designed and carried out in the 315-kW HEDMSM. The experimental results show that the optimum compensator can minimize the displacement resonance peak, when the rotor passes the first bending critical speed with low amplifier current.
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TL;DR: In this paper, a parametric dynamic model of prestressed long-span suspension bridges is coupled with a nonlinear quasi-steady aerodynamic formulation to obtain the governing aeroelastic partial differential equations adopted herewith.
Abstract: The limit cycle oscillations (LCOs) exhibited by long-span suspension bridges in postflutter condition are investigated. A parametric dynamic model of prestressed long-span suspension bridges is coupled with a nonlinear quasi-steady aerodynamic formulation to obtain the governing aeroelastic partial differential equations adopted herewith. By employing the Faedo-Galerkin method, the aeroelastic nonlinear equations are reduced to their state-space ordinary differential form. Convergence analysis for the reduction process is first carried out and time-domain simulations are performed to investigate LCOs while continuation tools are employed to path follow the post-critical LCOs. A supercritical Hopf bifurcation behavior, confirmed by a stable LCO, is found past the critical flutter condition. The analysis shows that the LCO amplitude increases with the wind speed up to a secondary critical speed where it terminates with a fold bifurcation. The stability of the LCOs within the range bracketed by the Hopf and fold bifurcations is evaluated by performing parametric analyses regarding the main design parameters that can be affected by uncertainties, primarily the structural damping and the initial wind angle of attack.
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TL;DR: In this article, a lateral mathematical model of a railway wheel-set with two degrees of freedom has been built to study the effect of yaw damper on the stability of high-speed wheelset, the constrain between the wheelset and the bogie is assumed as rigid.
Abstract: In this paper, the lateral mathematical model of a railway wheel-set with two degrees of freedom has been built. To study the effect of yaw damper on the stability of high-speed wheel-set, the constrain between the wheel-set and the bogie is assumed as rigid. In this lateral model, only the nonlinear wheel/rail contact relationship has been taken into consideration in the lateral direction, due to both the nonlinear relationship between lateral displacement and contact angle of the wheel and the rail, and the nonlinear relationship between lateral displacement and the equivalent radius of the wheel at the contact point. And the nonlinear parameters are attained by using polynomial interpolation. By using Center Manifold Theorem, the method of Normal Form and Poincare method, the model is reduced to a planar dynamical system, and the symbolic expression of the first-order fine focus is given, which can be used to determine which kind of bifurcation will occur at the critical speed. The simulation is established by using the parameters of the wheel-set of Chinese high-speed railway vehicle CRH3, and the result is consistent with the determination of the first-order fine focus. At last, the influence of different parameters on the stability of CRH3 wheel-set is investigated.
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TL;DR: In this paper, an out-of-plane vibration of rotating annular disks with a circumferential open crack is investigated, where the cracks are modeled as two sub-disks connected by translational and rotational line springs.
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TL;DR: In this article, an axially moving Timoshenko beam is investigated in the supercritical region and the static equilibrium equation is deduced from the governing equation of transverse vibration of the beam.
Abstract: In this paper, equilibrium bifurcations of an axially moving Timoshenko beam are studied in the supercritical region. For the first time, Timoshenko beam theory is applied to investigate nonlinear dynamics of high-speed axially moving structures. The static equilibrium equation is deduced from the governing equation of transverse vibration of the axially moving Timoshenko beam. Two kinds of boundary conditions are considered. The non-trivial equilibrium solutions are analytically determined. Moreover, the equilibrium equations are discretized by using the finite difference method. Therefore, equilibrium configurations are numerically verified by proposing an iterative scheme. This investigation shows that non-trivial equilibrium solutions of Timoshenko beams bifurcate with axially moving speed. By comparing with Euler–Bernoulli beam theory, this study finds that the critical speed, determined by the Timoshenko beam, is remarkably smaller. Nevertheless, the equilibrium deformation of the moving Timoshenko beam is obviously larger. Furthermore, the present work
derives the critical speed of the axially moving Timoshenko beam. At last, the effects of the system parameters
on the equilibrium bifurcation and the critical speed are presented.
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TL;DR: In this article, a dynamic analysis model comprising track, embankment and layered ground was proposed based on the two-and-half-dimensional (2.5D) finite elements combining with thin-layer elements to predict vibrations generated by train moving loads.
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TL;DR: In this article, a nonlinear analysis is performed on a closed-loop system of articulated heavy vehicles with driver steering control, where an integration method is employed to derive an analytical periodic solution of the system in the neighbourhood of critical speed.
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TL;DR: In this article, a coupled dynamic model of rotor-ball bearing-stator of aero-engine is built by means of the lumped mass method, taking into account the nonlinear rub-impact, bearing failure force and deformation of the casing.
Abstract: Aimed at the vibration of whole aero-engine, a coupled dynamic model of rotor-ball bearing-stator of aero-engine is built. By means of the lumped mass method, taking into account the nonlinear rub-impact, bearing failure force and deformation of the casing, the dynamic equation of the system containing typical rub-impact is derived. The response of the system under different conditions is obtained by using the fourth order Runge-Kutta numerical integration algorithm. By adopting the finite element analysis software ANSYS, the finite element model of the rotor shaft is established and the first six-order natural frequencies of the rotor system are acquired. Taking advantage of the parameters of the signal in time domain and frequency domain, frequency characteristics are extracted as the fault features. The single-point rubbing experiment is carried out in the test rig, and the working speed is higher than the first critical speed, so the rotor shaft is flexible rotor. By the methods of spectrum and cepstrum analysis, the rub-impact characteristics of the casing vibration acceleration time series data are analyzed. The results show that the casing vibration acceleration has obvious impact characteristics; the impact frequency is equal to the product of rotating frequency and number of the blades; the impact frequency component and its multiple-frequencies are demonstrated in the frequency spectrum; the strength of impact is modulated by the rotating frequency, so that there are families of side bands on impact frequency and both sides of frequency doubling, and the interval of sideband equals the rotating frequency. The frequency components of the rotating frequency and its frequency doubling are clearly shown in the cepstrum. By comparing the simulation and experiment, the rubbing characteristics found out in this paper has enough accuracy.
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TL;DR: In this paper, the rotordynamic behavior of a Centrifugal Compressor under wet gas conditions is investigated. And the results show that both vibrations when crossing the rotor first critical speed and stability are not critically affected by the liquid phase.
Abstract: The most recent development in centrifugal compressor technology is toward wet gas operating conditions. This means the centrifugal compressor has to manage a liquid phase which is varying between 0% and 3% liquid volume fraction (LVF) according to the most widely agreed definition. The centrifugal compressor operation is challenged by the liquid presence with respect to all the main aspects (e.g., thermodynamics, material selection, thrust load) and especially from a rotordynamic viewpoint. The main test results of a centrifugal compressor tested in a special wet gas loop (Bertoneri et al., 2014, “Development of Test Stand for Measuring Aerodynamic, Erosion, and Rotordynamic Performance of a Centrifugal Compressor Under Wet Gas Conditions,” ASME Paper No. GT2014-25349) show that wet gas compression (without an upstream separation) is a viable technology. In wet gas conditions, the rotordynamic behavior could be impacted by the liquid presence both from a critical speed viewpoint and stability-wise. Moreover, the major rotordynamic results from the previously mentioned test campaign (Vannini et al., 2014, “Centrifugal Compressor Rotordynamics in Wet Gas Conditions,” 43rd Turbomachinery Symposium, Houston) show that both vibrations when crossing the rotor first critical speed and stability (tested through a magnetic exciter) are not critically affected by the liquid phase. Additionally, it was found that the liquid may affect the vibration behavior by partially flooding the internal annular seals and causing a sort of forced excitation phenomenon. In order to better understand the wet gas test outcomes, the authors performed an extensive computational fluid dynamics (CFD) analysis simulating all the different types of balance piston annular seals used (namely, a tooth on stator (TOS) labyrinth seal and a pocket damper seal (PDS)). They were simulated in both steady-state and transient conditions and finally compared in terms of liquid management capability. CFD simulation after a proper tuning (especially in terms of LVF level) showed interesting results which are mostly consistent with the experimental outcome. The results also provide a physical explanation of the behavior of both seals, which was observed during testing.
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TL;DR: In this article, a mathematical model of a high-speed railway vehicle during curve negotiation with aerodynamic loads is set up, and a program based on the model is written and verified.
Abstract: Aerodynamic loads may have effects on the hunting stability, and the factor of curved track makes it more complicated. Therefore, considering the steady aerodynamic loads generated by crosswind and airflow in the opposite advancing direction of train, the hunting stability of high-speed railway vehicle on a curved track is studied in this paper. The changes of gravitational restoring force and creep coefficients which are caused by aerodynamic loads are considered, and the change of equilibrium position due to aerodynamic loads, centrifugal force and the factor of curved track is also in consideration. A mathematical model of a high-speed railway vehicle during curve negotiation with aerodynamic loads is set up. A program based on the model is written and verified. Using this program, the linear critical speed considering the effects of aerodynamic loads is determined by the eigenvalue analysis. This paper investigates the critical speeds in three aerodynamic conditions. Considering the aerodynamic loads,...
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TL;DR: In this paper, an active gas bearing is proposed to reduce the high vibration levels of a radial gas bearing when crossing resonant areas, and the control action of this active bearing is selected based on two different strategies: a simple proportional integral derivative controller and an optimal controller.
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TL;DR: In this paper, the authors derived the governing linear equations of transverse motion of a spinning disk with a splined inner radius and constrained from lateral motion by guide pads, where the disk is driven by a matching spline arbor that offers no restraint to the disk in the lateral direction.
Abstract: The governing linear equations of transverse motion of a spinning disk with a splined inner radius and constrained from lateral motion by guide pads are derived. The disk is driven by a matching spline arbor that offers no restraint to the disk in the lateral direction. Rigid body translational and tilting degrees-of-freedom are included in the analysis of total motion of the spinning disk. The disk is subjected to lateral constraints and loads. Also considered are applied conservative in-plane edge loads at the outer and inner boundaries. The numerical solution of these equations is used to investigate the effect of the loads and constraints on the natural frequencies, critical speeds, and stability of a spinning disk. The sensitivity of eigenvalues of spline spinning disk to the in-plane edge loads is analyzed by taking the derivative of the spinning disk's eigenvalues with respect to the loads. An expression for the energy induced in the spinning disk by the in-plane loads, and their interaction at the inner radius, is derived by computation of the rate of work done by the lateral component of the edge loads. Experimental idling and cutting tests for a guided spline saw are conducted at the critical speed, super critical speeds, and at the flutter instability speed. The cutting results at different speeds are compared to show that the idling results of a guided spline disk can be used to predict stable operation speeds of the system during cutting.
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TL;DR: In this article, the effects of thermal stress on the vibration characteristics, buckling limit and critical speed of a rotating pretwisted beam clamped to rigid hub at a stagger angle were investigated.
Abstract: Effects of thermal stress on the vibration characteristics, buckling limit and critical speed of a rotating pretwisted beam clamped to rigid hub at a stagger angle were investigated. By considering the work done by thermal stress, the thermal influence on stiffness matrix was introduced in the dynamic model. The motion equations were derived based on Lagrange equation by employing three pure Cartesian deformation variables combined with nonlinear von Karman strain formula. Numerical investigations studied the modal characteristics of the beam. Numerical results calculated from a commercial finite element code and obtained with the present modeling method were in good agreement with the previous results reported in the literature. The combined softening effects due to the thermal stress and the rotation motion were observed. Furthermore, it is shown that the inclusion of thermal stress is necessary for blades operating under a high temperature field. Buckling thermal loads and the critical rotating speed were calculated through solving the corresponding nonlinear equations numerically, and some pertinent conclusions are outlined. It is also found that the peak value position of the first mode shape approaches to the tip of blade with the increment of rotating speed and hub radius. However, the variation in the environment temperature causes only a slight alteration in the mode shape.
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01 Mar 2016TL;DR: In this paper, the lateral dynamic features of a railway vehicle are investigated using two similar wheel/rail contact models: the Vermeulen-Johnson and the Shen-Hedrick-Elkins models.
Abstract: The lateral dynamic features of a railway vehicle are investigated using two similar wheel/rail contact models: the Vermeulen–Johnson and the Shen–Hedrick–Elkins models. The symmetric/asymmetric bifurcation behaviour and chaotic motions of the railway vehicle are investigated in great detail by varying the speed and using the ‘resultant bifurcation diagram’ method. It is found that multiple solution branches exist and they can lead to more steady states in the dynamic behaviour of the railway vehicle. The coexistence of multiple steady states can lead to jumps in the amplitude of oscillations, resulting in problems for safe operation of the vehicle. Therefore, it should be avoided in everyday operation. Furthermore, the creation of multiple solution branches suggests that the critical speed of a vehicle should be determined from a comprehensive analysis of the various kinds of possible excitations and numerous tests.
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01 May 2016
TL;DR: In this article, the authors investigated the vibration stress of a single-stage fan blade influenced by the interactio-theoretic interaction between the vibration and resonance in an aircraft engine, and found that resonance is the main cause of failure in aircraft engine blades.
Abstract: The vibration stress caused by resonance is the main cause of failure in aircraft engine blades. The study investigated the vibration stress of a single-stage fan blade influenced by the interactio...
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TL;DR: In this paper, a relatively simple method was proposed to predict the first critical speed using data from low rotational speeds, which is shown to work well for two standard rotordynamic models and with data from experiments conducted during this study.
Abstract: The prediction of critical speeds of a rotating shaft is a crucial issue in a variety of industrial applications ranging from turbomachinery to disk storage systems. The modeling and analysis of rotordynamic systems is subject to a number of complications, but perhaps the most important characteristic is to pass through a critical speed under spin-up conditions. This is associated with classical resonance phenomena and high amplitudes, and is often a highly undesirable situation. However, given uncertainties in the modeling of such systems, it can be very difficult to predict critical speeds based on purely theoretical considerations. Thus, it is clearly useful to gain knowledge of the critical speeds of rotordynamic systems under in situ conditions. The present study describes a relatively simple method to predict the first critical speed using data from low rotational speeds. The method is shown to work well for two standard rotordynamic models, and with data from experiments conducted during this study.
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TL;DR: In this paper, a flexible double-support rotor with two passive automatic balancers of pendulum, ball or roller type was constructed and the system of differential equations, which describes the motion of a rotor machine, was obtained.
Abstract: A discrete Nmass model of a flexible doublesupport rotor with two passive automatic balancers of pendulum, ball or roller type was constructed. Automatic balancers are placed near supports. The system of differential equations, which describes the motion of a rotor machine, is obtained. The primary (sustained) motions of a system as the motions, in which automatic balancers eliminated displacements of a rotor in supports, were found. It is shown that on the primary motions, the total imbalances of a rotor and AB, reduced to two correction planes (supports), equal zero. It was proposed to examine the stability (of the family) of sustained motions by generalized coordinates, which set the displacement of a rotor in the supports and by dynamic variables that equal total imbalances of a rotor and AB in two correction planes. We obtained differential equations, which describe the change in these variables that describe the process of selfbalancing. By the analysis of differential equations of the motion of a system it was established that: – on the primary motions, AB eliminate rotor deflections and vibrations in elastic viscous supports, but do not remove shaft deflections in nonsupporting points; – on the primary motions elastic viscous supports are conditionally converted into hinge supports; – shaft deflections in nonsupporting points and the primary motions change with the change in angular speed of rotation of the rotor; – primary motions exist at a certain distance of the speed of rotation of the rotor from the critical speeds of flexible rotor rotation with the hinge supports instead of elastic viscous supports; – at the speeds of rotation of a rotor shaft close to any of these speeds, the conditions of existence of the primary motions are disrupted because shaft deflections theoretically grow to infinity and the balancing capacity of AB is not sufficient for the compensation for the imbalances of the rotor; – in practice these deflections are limited and, therefore, proper selection of the balancing capacity of AB can ensure existence of primary motions at all speeds of rotation of rotor.
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TL;DR: In this article, the nonlinear analysis of free vibrations, dynamic stability, and rotational dynamics of rotating annular circular thin plates, made of functionally graded material (FGM), is studied.
Abstract: In this article, the nonlinear analysis of free vibrations, dynamic stability, and rotational dynamics of rotating annular circular thin plates, made of functionally graded material (FGM), is studied. Based on classical plate theory, von Karman’s nonlinear plate theory, and assuming the FGM mechanical properties vary in the radial direction, the governing equations of motion are obtained by direct use of Newton’s laws. A 1-D differential quadrature is used to solve the governing equations determining the natural frequencies, corresponding transverse mode shapes, and the critical speeds of rotation. The accuracy and convergence of the method are studied by comparing the results with the similar results whenever available in the literature. The influence of different parameters such as inner-to-outer radii ratio, thickness-to-outer radius ratio, graded index, angular velocity of the plate, and different boundary conditions on the natural frequencies of FG rotating plate are demonstrated by numerical example...
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TL;DR: In this article, the Galerkin method is employed to discretize the governing equation into a gyroscopic system consisting of a set of coupled nonlinear ordinary differential equations.
Abstract: Nonlinear vibration of a fluid-conveying pipe subjected to a transverse external harmonic excitation is investigated in the case with two-to-one internal resonance. The excitation amplitude is in the same magnitude of the transverse displacement. The fluid in the pipes flows in the speed larger than the critical speed so that the straight configuration becomes an unstable equilibrium and two curved configurations bifurcate as stable equilibriums. The motion measured from each of curved equilibrium configurations is governed by a nonlinear integro-partial-differential equation with variable coefficients. The Galerkin method is employed to discretize the governing equation into a gyroscopic system consisting of a set of coupled nonlinear ordinary differential equations. The method of multiple scales is applied to analyze approximately the gyroscopic system. A set of first-order ordinary differential equations governing the modulations of the amplitude and the phase are derived via the method. In the supercritical regime, the subharmonic, superharmonic, and combination resonances are examined in the presence of the 2 : 1 internal resonance. The steady-state responses and their stabilities are determined. The various jump phenomena in the amplitude-frequency response curves are demonstrated. The effects of the viscosity, the excitation amplitude, the nonlinearity, and the flow speed are observed. The analytical results are supported by the numerical integration.
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24 Feb 2016
TL;DR: In this paper, a raw material plate starting method of a cold rolling mill is presented, which solves the problem of starting difficulty caused when all racks are started at the same time, further reducing the loss due to waste products and improving the yield.
Abstract: The invention discloses a raw material plate starting method of a cold rolling mill. When a first starting signal related to starting in a raw material plate starting mode is received, the roll bending force of each rack in a rack group is loaded to a roll bending force set value, and the compacting and rolling force of the cold rolling mill is loaded to a rolling force set value; when a second starting signal is received after the first starting signal, the real time rolling mill speed of the cold rolling mill is increased at a constant speed to a first set speed value; when that the real time rolling mill speed reaches a preset critical speed value is detected, whether tensions between the racks are stable or not is judged, and whether strip steel thickness deviations are stable or not is judged; and when the tensions between the racks and the strip steel thickness deviations are all stable, the rolling mill speed of the cold rolling mill is increased to a second set speed value from the first set speed value. The raw material plate starting method provided by the invention solves the problem of starting difficulty caused when all racks are started at the same time, further the loss due to waste products is reduced, and the yield is effectively improved.