Search

This paper presents the development of theoretical and experimental models for the study of rotodynamic behavior of a multistage rotor. The transfer matrix method, which includes the characteristics of stiffness and damping for the supports and the stages respectively as well as the characteristics of unbalance in the stages, is used for the theoretical model. The data from a physical model was employed as a way of validating the theoretical results. The first two critical speeds were determined with the theoretical model and they differ in a low percentage with respect to the values measured experimentally. Moreover, the vibration level recorded in the physical model rises 2.5 times when the multistage rotor approaches the first two critical speeds. In addition to this, significant displacements of the lateral critical speeds are noticeable when an increase in mass imbalance is induced in several of the rotor impellers. RESUMEN Este artículo presenta el desarrollo de modelos, teórico y experimental, para el estudio del comportamiento rotodinámico de un rotor multietapas. El modelo teórico se obtiene a partir de las ecuaciones de Lagrange y se resuelve empleando el método de la matriz de transferencia. En el análisis se incluyen las características de rigidez y amortiguamiento de los apoyos y de las diferentes etapas del rotor, así como el desbalance másico de estas últimas. Como vía de validación de los resultados teóricos se emplearon los datos provenientes de un modelo físico rotodinámico, especialmente desarrollado para la investigación. Las dos primeras velocidades críticas fueron determinadas con el modelo teórico y difieren en un bajo porcentaje con respecto a los valores medidos experimentalmente. Por otro lado, el nivel de vibración registrado en el modelo físico se eleva 2,5 veces cuando el rotor multietapas se aproxima a cualquiera de las primeras dos velocidades críticas. Adicionalmente, se describe un corrimiento apreciable en la magnitud de las velocidades críticas laterales cuando se aumenta el desbalance másico en varios de los impulsores del rotor.

In this article, a nonlinear dynamic model with five degrees-of-freedom for a four-wheel-drive vehicle driveline connected by a cardan joint, including dynamic intersection angle, nonconstant velocity, and additional moment caused by the cardan joint, is established by using the Lagrange method to analyze the driveline coupling vibration in both torsional and lateral directions. High-order Runge–Kutta algorithm is applied to solve the differential equations and to calculate transient responses of the driveline rotors under acceleration condition. The color maps and second-order vibration of the driveline are acquired by frequency spectrum analysis and order tracking analysis, respectively. The second-order vibration and noise of the driveline and vehicle interior caused by the cardan joint is validated by vehicle experimental results and reduced effectively by decreasing intersection angle of the cardan joint under the operational condition. Moreover, application of a flexible coupling instead of the cardan joint significantly reduces the second-order vibration but simultaneously generates low-level third-order vibration.

This paper presents a study on the interaction of lateral vibration and friction vibration of a marine water-lubricated bearing when the bearing and stern shaft journal contact. The coupled vibration modes were modeled innovatively to analyze their interactions, where the lateral vibration response of the shaft journal into the normal load was applied to the dynamic friction force. The regular patterns were analyzed utilizing the theoretical derivation, numerical simulation and tests whose results show that the contacted motion statuses are relatively complicated and the feedback mechanism due to interference expressed as the term of KbC0r′ accelerates the instability of the coupled system in the negative area of the friction–speed curve.

Reduction of vibration in passively supported lateral directions by varying stiffness control is discussed in a vertically active magnetic suspension system. In the target system, one pair of electromagnets is arranged in differential driving mode to actively control the vertical motion of the floator. Usually the floator is prone to vibrate in the lateral direction because it is passively supported by virtue of the edge effect of the electromagnets. In this work, such vibrations are reduced by incrementing or decrementing the currents simultaneously during vibration without changing the vertical position of the floator. This control strategy is implemented in a developed apparatus where an iron ball is suspended by differentially operated electromagnets without any mechanical contact. Experiments are carried out, and the results show the reduction of lateral vibrations without changing the vertical position of the floator.

This work aims to improve the vibrational behaviors of the rotor-active magnetic bearings (AMB) system via proposing a new control methodology for such lateral vibrations. Accordingly, the rotor-AMB system model is modified by integrating the conventional proportional–derivative (PD) controller along with the positive position feedback (PPF) controller to the considered system. The control currents in the rotor-AMB system are designed to become a combination of two control signals: One of them is generated via the PD controller, and the other one comes from the PPF controller. Based on the suggested control method, the whole dynamical system model is derived and then analyzed by the multiple time-scale perturbation method. Then, the bifurcation diagrams of both the spinning speed and rotor eccentricity response curves are investigated before and after control. The obtained analytical results showed that the integrating of the PPF controller to the considered system could mitigate the system lateral vibration close to zero. Moreover, the system quasiperiodic motions have been stabilized after control and the sensitivity to the initial conditions has eliminated. Finally, numerical validations for the obtained analytical results are performed that illustrated an excellent agreement with the analytical ones.

This paper presents an active control method for reducing the lateral vibration transmission in a shaft-hull coupled system of submerged vehicles under the excitation of propeller forces. In this method, a 6-DOF (degree of freedom) active stern support and the local velocity feedback control strategy are employed to suppress vibration transmission from the shaft to the hull. The dynamic model of the shaft-hull system embedded with the active stern support is established with the FRF (frequency response function)-based synthesis method and the FEM/BEM (finite element method/boundary element method). The six-channel local velocity feedback control strategy is employed to suppress vibrations of the strut-hull interface. Numerical results indicate that a noticeable attenuation of vibration and sound radiation can be achieved by the active stern support. In addition, the combination control at the stern support and the thrust bearing is able to obtain better attenuation at some critical frequencies. Suppression of the mean square velocity and radiated sound power of the hull surface are observed.

Modern rotating machineries require vibration suppression devices with simple structure and wide band effective vibration suppression frequency range, so a grounded nonlinear energy sink (GNES) with piecewise linear stiffness is developed to satisfy the requirements. Firstly, the structure and working principles of the GNES is explained. Then, the piecewise linear stiffness characteristic of elastic rods is analyzed and the dynamic equations for the rotor-GNES system are established, in which the rotor system is assumed to be either isotropic or anisotropic; numerical simulations are then carried out to study the vibration suppression effect of the GNES, and the wide-band vibration suppression capability of the GNES is assessed; comparison is made between the vibration suppression effect of the GNES and that of the dynamic vibration absorbers. Finally, experiments are carried out on a rotor-GNES system test platform to verify the suppression effects. The results show that the GNES is effective at suppressing vibrations in terms of both transient and steady-state vibrations within a rotor system. In the numerical simulation an optimal vibration suppression rate of 78% is achieved and in the test a rate of 68% is achieved.

Industrial robot is widely used to accomplish the drilling work in aviation industry due to its flexibility and relatively low cost. However, the low stiffness property of robot makes it easy to induce lateral vibration in the drilling process and restricts further improvement of drilling quality and efficiency. It is urgent to propose a new method for robotic drilling which is able to reduce the cutting force and suppress the lateral chatter effectively. In this study, robotic rotary ultrasonic drilling (RRUD) technology is proposed firstly. And the dynamic model to compute the stability lobe for RRUD is developed. The stability problem is simplified as a two degrees of freedom model. After that, the axial chip thickness is achieved on base of kinematic characteristics analysis. Furthermore, exponential method is applied to develop the dynamic cutting force model. Then, the stability lobe diagrams are obtained by frequency domain analysis. It is found that the stability region of RRUD gets a larger extension compared with robotic conventional drilling (RCD). Finally, pilot experiments are conducted and the results indicate that these stability lobe diagrams agree well with the experimental results.

Many driveline systems are designed to accommodate angular misalignment by the use of flexible couplings or Universal Joints (U-Joints) which link individual shaft segments. The Sommerfeld effect is a nonlinear phenomenon observed in some rotor systems being driven through a critical speed when there is not enough power to accelerate the rotor through resonance. Previous studies have shown that rotor speed can become captured when transitioning through natural frequencies due to nonlinear interactions between a non-ideal driving input and rotor imbalance. This paper, for the first time, shows that this type of rotor speed capture phenomena can also be induced by driveline misalignment. During rotor spinup under constant motor torque, it is found that misalignment-induced rotor speed capture phenomena can occur as the shaft speed approaches ½ the first elastic torsional natural frequency. Depending on misalignment level and motor torque, the shaft speed will either dwell near this speed and then pass through, or the speed will become trapped. Here, a nonlinear rotordynamics model of a segmented driveshaft connected by two U-joints including effects of angular misalignment and load torque is developed for the study. This analysis also determines the minimum driveline misalignment angle for which the shaft speed capture phenomena will occur for a given motor torque and load torque condition.

Fatigue crack in a rotary shaft is a common failure observed in rotor systems. Since vibration of the shaft causes alternating fatigue loads, the crack propagates slowly. Meanwhile, the propagating crack may cause nonlinear or unstable vibration of the rotor system. In fact, growth of the crack and vibration of the shaft are coupled with each other. Hence, it is necessary to study the fatigue degradation behavior of the cracked rotor accounting for this coupling effect. In this paper, a coupling model of rotor vibration and crack growth is established through dynamic theory and fracture mechanics, and a sequential iterative procedure is proposed to solve the coupling model. Then, the competing degradation failure mode of the cracked rotor is analyzed with regard to the rapid crack growth failure and the unstable vibration failure. And degradation measures are proposed based on the competing degradation failure criterion. At last, degradation behaviors with the coupling effect of nonlinear vibration behavior and multiple parameters including rotation speed, unbalance eccentricity and orientation angle, and damping are investigated by numerical simulation. The results indicate that nonlinear vibration behavior and multiple parameters have considerable influence on the degradation behaviors, which present complex regularity. The findings are of significance to guide the safety design of the rotor system for long time operation and help to the further research on prognostics and lifetime prediction.

Since there are manufacturing and assembling inaccuracies rotors are always slightly imbalanced. Their rapid angular acceleration leads to an increase in the magnitude of the tangential component of the imbalance forces. The slow acceleration enables the development of an almost steady state vibration. In both cases the amplitude of their oscillation is high. The frequency tuning achieved by changing the stiffness of the rotor supports represents one approach to its attenuation. Stiffness decreases shortly before reaching the critical speed, and after crossover, the stiffness increases again. As almost all publications and computer procedures from the field of rotor dynamics deal only with the case when the rotor turns at a constant angular speed, derivation of the equations of motion of the rotor rotating at variable velocities and investigation of the operating conditions and individual parameters of the supports on the vibration of the rotor passing critical speeds represent the fully new contribution of this paper. The critical revolutions are determined from the Campbell diagram. To solve the equations of motion the Newmark method has been proposed and tested.

At the initial stage of propulsion shaft design, in line with shaft alignment, an intensified consideration of lateral vibration is needed to verify its operational safety. Recently the alignment problem affecting the lateral vibration has been becoming issues. However, the theoretical method of forced lateral vibration analysis is not cleary established so far and it is about to simply limited among the classification societies and international standards to avoid the blade natural frequency resonance cpm outside of of engine rpm at MCR. On the other hand, longer center distance between each support bearing shows an affirmative result normally in shaft alignment analysis whereas the blade order resonance speed may cause lowering near the limitation in the aspect of lateral vibration. Therefore, it is required careful attention to engineers as described above. As a method to solve the problem, it is mainly considered that remove forward stern tube bearing. In this paper, based on a medium size container ship case, theoretical study was carried out in the context of the forward stern tube bearing. The various effects and functions of forward stern tube bearing are reviewed and evaluated. Then an guidance note to lead the conclusion is proposed.

An active stern support is proposed to suppress lateral vibration transmission in the shaft-hull system of underwater vehicles subjected to the excitation of propeller forces. The active stern support uses six active struts instead of a common bearing support to suppress vibration transmission from the shaft to the hull. The dynamic model of a shaft-hull system embedded with the active stern support is established on the Timoshenko beam theory and the FRF (frequency response function)-based synthesis method. Based on this model, the characteristics of vibration transmission in the system and the feasibility of active control are analyzed. An adaptive control algorithm is constructed and the performance of vibration suppression is evaluated. An experimental system is built as well to verify the effectiveness of the control method. Numerical and experimental results show that the active stern support is able to suppress the transmission of lateral vibration induced by the propeller forces, and the attenuation of the normal vibration of the hull structure can be achieved.

The lack of damping of radial gas bearings leads to high vibration levels of a rotor supported by this type of bearing when crossing resonant areas. This is even more relevant for flexible rotors, as studied in this work. In order to reduce these high vibration levels, an active gas bearing is proposed. The control action of this active bearing is selected based on two different strategies: a simple proportional integral derivative controller and an optimal controller. Both controllers are designed based on a theoretical model previously presented. The dynamics of the flexible rotor are modelled aided by the finite element method and the rotor–fluid interaction in the gas bearing is included using the solution of a modified version of the Reynolds equation for compressible fluids, taking into account the piezoelectrically controlled jet action. Performance and accuracy of both model-based controllers are compared against experimental results, showing good agreement. Theoretical and experimental results show a significant increase in the damping ratio of the system, enabling the flexible rotor to run safely across the critical speeds and up to 12,000 rev/min, i.e. 50 percent over the second critical speed without any instability problems.

A new flexible-pinion-shaft rotor model is presented to predict the response of the pinion shaft of an integrally geared compressor (IGC). The motion of the pinion shaft is driven by: (1) its own imbalance and (2) the axial reaction force developed by relative axial motion at the thrust collar (TC) that connects it to the pinion. The axial reaction force at the TC arises because of the relative axial motion between the pinion shaft and the bull gear (BG) at the overlapping area of the TC. The relative axial motion arises because of: (1) absolute axial motion of the pinion (assumed to be a rigid body), (2) pitch and yaw motion of the pinion at the TC, and (3) absolute axial motion of the BG at the TC overlap area. Because the axial reaction force acts at a radial distance from the mass center of the TC disk, it creates moments that couple the relative axial motion of the pinion and BG shafts to the radial motion of the pinion. The present model includes the local flexural stiffness of the BG. Excitation for the model is provided by: (1) runout from the BG at its running speed, Ω, acting through the axial reaction force of the TC, (2) runout from the pinion at its running speed, ω, acting through the axial reaction force of the TC, and (3) pinion TC mass imbalance at ω. Measured axial runouts of the BG and TC were taken from a test rig at the authors’ laboratory. Predictions for the TC oil-film axial stiffness and damping come from a proprietary Reynolds equation solution to the TC oil-film. The local axial stiffness of the BG at the overlap area was obtained from a finite element analysis of the authors’ test rig. The base rotordynamic model for the pinion was provided for a production IGC pinion by an IGC manufacturer including the bearings and structural dynamics model. Waterfall plots are presented from the model’s predictions of radial motion at the IGC’s Stage 1 compressor impeller. The response is dominated by synchronous response at the pinion speed, ω, and tracking subsynchronous response at the BG speed, Ω. The response at ω comes from the pinion’s imbalance, not the pinion runout at the TC. The response at Ω comes from the BG runout acting across the TC. The IGC manufacturer’s representatives state that predictions from the model are consistent with measurements from real IGCs, particularly in regard to the presence (and amplitudes) of tracking subsynchronous response amplitudes at the BG frequency. Obviously, more detailed models can be developed for the rotordynamics of IGCs, but the authors feel that this relatively simple, one-rotor model, is adequate to predict the observed tracking phenomena in IGCs. The analysis can be produced by modifying an existing rotor code or by simply downloading the rotor’s [M], [K], and [C] matrices over a range of speeds and then using MATLAB or similar codes.

The feedback-controlled lubrication regime, based on a model-free designed proportional–derivative controller, is experimentally investigated in a flexible rotor mounted on an actively-lubricated tilting-pad journal bearing. With such a lubrication regime, both the resulting pressure distribution over the pads and hence the bearing dynamic properties are dynamically modified. The control strategy is focused on reducing the lateral vibrations of the system around its operational equilibrium within a wide frequency range. To synthesize the proportional–derivative controller gains, an objective function is optimized in the stabilizing gain domain and then chosen from a subdomain imposed by servovalve restrictions. This work demonstrates enhancements of the dynamic response of flexible rotor-bearing systems supported by an active tilting-pad journal bearing by means of the feedback-controlled lubrication regime featured via proportional–derivative controllers.

This paper studies the efficiency of a number of smooth non-linear energy sinks (NESs) on the vibration attenuation of a rotor system under mass eccentricity force. The non-linear energy sinks have a linear damping, linear stiffness and a cubic stiffness. To reduce the number of equations of motion, the modal coordinates and complex transformations are used. For analytical solution, the Multiple Scales-Harmonic Balance Method (MSHBM) is used. The most important parameters to examine NES efficiency is the range of happening of the SMR in the detuning parameter range. For different parameters of the system, the SMR, the WMR, the low amplitude periodic motion, and the high amplitude periodic motion happen in the system. It is shown that, the linear stiffness of the NES is canceled out by the stiffness which is created by the centrifugal force and therefore, by changing the linear stiffness of the NESs, the collection of the NESs to the desired rotational speed of the rotor can be tuned. It should be noted that after this cancellation, the remained stiffness is essentially non-linear (non-linearizable) stiffness. In addition, when the external force reaches its medium magnitude, the area of the occurrence of the SMR in the domain of the system parameters would be larger and the collection of the NESs demonstrates an impressive effect.

The paper shows a mathematical rotordynamic model for lateral vibration analysis of induction motors with dynamic eccentricities regarding start-up. With this mathematical model, different kinds of dynamic eccentricities—mass eccentricity, magnetic eccentricity and bent rotor deflection—can be analyzed. Beside the mechanical influences—e.g. the rotor mass, rotor shaft stiffness, oil film coefficients of the sleeve bearings, structural stiffness of the sleeve bearing housings and end shields—also electromagnetic influence—e.g. the electromagnetic field and magnetic eccentricity—can be analyzed for the start-up. The paper is based on the well-known theory of accelerated resonance passing and is an enhancement of the theory for induction motors, considering also electromagnetism during start-up. Based on this rotordynamic model, the absolute displacements of the shaft centre, the shaft journals and the sleeve bearing housings can be analyzed as well as the relative displacements between the shaft journals and the sleeve bearing housings. Additionally, also the vibration velocities at the sleeve bearing housings can be derived. The aim of the paper is not to replace a detailed finite-element-model by a simplified analytical model, but to show the mathematical coherences between rotordyamics and electromagnetics for a start-up of an induction motor, considering different dynamic eccentricities.

Dynamic fibre-optic curvature sensing is demonstrated by interrogating chains of fibre segments, separated by broadband Bragg grating reflectors, using range-resolved interferometry (RRI). Four fibre strings, containing four fibre segments each of gauge length 20 cm, are attached to the opposing sides of a support structure and the resulting differential strain measurements allow inference of lateral displacements of a cantilever test object. Dynamic tip displacement resolutions in the micrometre range at an interferometric bandwidth of 21 kHz demonstrate the suitability of this approach for highly sensitive and cost-effective fibre-optic directional vibration measurements of smart structures. © (2017) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only

Analytic study of rotor vibration plays an important role in design, and fault detection and diagnosis of rotating machines. In this paper, vibration characteristics of a flexible rotor due to a central mass unbalance were analysed numerically under two cases. In Case I, bearings have symmetric, isotropic stiffness and damping matrices, while in Case II they have asymmetric, non-isotropic stiffness and damping matrices. It was found in Case I that critical speed occurs at 1690 rpm and the orbital trajectories of rotor mass stations are circular and corotational. In Case II, response orbit of either the bearing or the disk is elliptic and the size, shape, and orientation of the elliptical orbit changes with shaft speed, and also from the bearing to the disk.

The aft part of large scale vessel propulsion shaft is a cantilever with a propeller. As the shaft doing the lateral vibration, the trail of propeller shaft center-line, which appears together with the movement, is the cone surface with the shaft center as the symmetrical axis, at the same time the propeller will vibrate with the shaft. In order to investigate the influence of propeller effect to shaft lateral vibration, the main work in the paper includes modeling the finite element of a large scale vessel propulsion shaft, studying the factors of water mass adhered to propeller, pin softening and gyroscopic moment to the shaft lateral vibration. The calculation shows that: if the propeller's gyroscopic moment is positive, the free frequency of shaft will increase, while added water mass and spin softening will obviously decrease the free vibration frequency of shaft.

For an electromagnetic suspension (EMS)-type urban Maglev train using U-shaped electromagnets, both the vertical and the lateral air gaps for levitation are maintained only by the electromagnet. The train can run over curved rails without active lateral air gap control because the U-shaped electromagnet simultaneously produces both a levitation force and a guidance force, which is dependent on the levitation force. Owing to the passive control of the lateral air gap, the lateral vibration could exceed the limits of the lateral air gap and acceleration. In this study, dynamic analysis of a Maglev train is carried out, and the effectiveness of a lateral damper for vibration reduction is investigated. To more accurately predict the lateral vibration, a Maglev vehicle multibody model including air-sparing, guideway irregularities, electromagnets, and their controls is developed.

In this paper, the vehicle-bridge lateral vibration mechanism was analyzed; the vehicle-bridge vibration model was built and the lateral reinforcement schemes of open steel plate bridges were designed. Numbers of analysis were carried out for the lateral vibration of 40m deck steel plate bridges before and after reinforcement, under input of random artificial hunting waves and track irregularity. The results showed that, the frequency of hunting motion is approaching loaded frequency of the girder. The larger lateral amplitude appears on the bridge when the hunting wavelength is around 8~9m and the velocity of the train is around 55~70km/h. The wavelength is longer, the resonant velocity of the bridge is higher.

According to the structure and stress trait of bearing bolts, a lateral-vibration mechanics model was established for them, and the relation between lateral-vibration frequency and axial load was analyzed; then, lateral-vibration trait of bearing bolts was studied through laboratory simulation test. The results indicate that vibration frequency of bolt support system increases as well as axial force, the detection on axial load of bolts can be made by generating lateral vibration of bearing bolts. Theoretical and experimental research results show that frequency method is effective for detecting the axial force of bolt support system.

In magnetic tape drives, lateral in-plane vibration of the tape leads to misalignment between data tracks and read/write head's position resulting in reduced storage capacity. To attenuate this lateral tape motion (LTM), surface guides-which include grooved, porous or roughened rollers-are used. The axial motion of the tape over the roller surface switches between two states: a sticking state when the axial force is smaller than the static frictional force; and a slipping state when the axial force is larger than the frictional force. A good understanding of the physical phenomena involved in this frictional interaction between the magnetic tape and surface friction guides will allow the appropriate and optimal choice of roller characteristics. We conduct a parametric study of frictional interaction between roller surface and the traveling magnetic tape by systematically varying the axial tension and transport velocity of the tape. An experimental setup is used to independently control these parameters and obtain lateral vibration measurements at two equidistant points-upstream and downstream-from the tape-roller interface. Techniques from spectral analysis are applied to the two signals to analyze and isolate the effect of stick-slip friction on LTM. It is noticed that the coherence function between the LTM signals provides valuable insight into the nature of stick-slip friction at the interface. We subsequently use it as a metric to construct and understand the “dynamic phase diagram,” i.e., to demarcate regions in the tension-velocity phase-space where predominance of stick or slip occurs.

The prestressed concrete (PC) stress ribbon footbridge is a type of suspension bridge without towers, which has been applied in Japan and all over the world for years in light of its low construction cost and aesthetic merit. It generally consists of the precast concrete slabs with embedded cables. However, the walking-induced lateral vibration trouble of the Millennium Bridge in London in 2000 gave a lesson to the engineers that the lateral vibration feature must be taken into consideration for the footbridge vibration evaluation. In this sense, the field investigations on the lateral vibration features of 14 pre-stressed concrete stress ribbon footbridge in Japan was carried out by artificial impact and damping free vibration tests. According to the investigations, the larger the bridge span, the lower the frequencies of lateral-related vibration modes. In addition, based on the damping-free vibration field tests, there was a tendency toward the damping constant degradation when bridge span became larger.

This paper focuses on the size dependency of lateral vibration of a micro overhung rotor-disk system subjected to an axial load based on the modified strain gradient theory. The governing equations of motions as well as the boundary conditions are derived from Hamilton’s principle. The assumed modes approach is employed to transform the governing partial differential equations into a set of infinite ordinary differential equations. The first two natural frequencies and associated instability rotational speeds of the system are analytically determined. The effects of variation in the parameters, such as the rotational speed, material length scale parameters, axial load and rotor length, on the natural frequencies and instability speeds are discussed and presented. The results obtained reveal that each of the material length scale parameters has a significant impact on the vibration response characteristics of the system.

In tilting pad bearing design process, the selection of the proper configuration type of either a load-between-pad(LBP) or load-on-pad(LOP) as well as preload and pivot offset conditions is to be carefully considered. Also the bearing needs to be designed in order to be suited for the rotor-bearing system and operating condition. In this paper, it is observed that the dynamic characteristics in a five pad tilting pad bearing for the LBP and the LOP configurations are influenced by the variation of preload and pivot offset. In this context, rotor dynamic analysis of the 5 MW industrial gas turbine supported by the tilting pad bearing at the front and roller bearing at the rear is carried out based on the dynamic coefficients of the tilting pad bearing investigated. The result shows that two rigid body critical modes experience various changes according to the influence of the tilting pad bearing uniquely applied to one side of this machine. Mainly, the second critical speed, the rigid body mode of conical shape with high whirling in the tilting pad bearing, is significantly changed by preload and pivot offset regardless of the LBP and LOP configurations. And, the first critical mode, the rigid body mode of conical shape with high whirling in the roller bearing, is sensitively affected by preload applied to the LOP configuration and by its asymmetric dynamic properties.

A method is presented to reduce the lateral vibration amplitude of an unbalanced rotor accelerating or decelerating through its critical speed, by means of modulating the response phase with varying acceleration rate. Theoretical and numerical analysis on the amplitude and the phase characteristics of lateral vibration of a rotor model are made. Numerical results show that when the rotor passes through its critical speed with a modulated phase, the response amplitude can be reduced by 20% or so, compared with the nonmodulated (constant) acceleration case.

The drive screw is simplified as a rotating Tinoshenko beam under the pre-tension force. Using boundary conditions and continuity conditions, the frequency equation of the screw with the elastic supports is derived considering the effect of gyroscope, the pre-tension force, the bearing stiffness, and the contact between the worktable and the screw. The effect of the system parameters on the lateral natural frequency is studied to supply a base for designing the drive system.

The resonant modes of sessile water drops on a hydrophobic substrate subjected to a small-amplitude lateral vibration are investigated using computational fluid dynamic (CFD) modeling. As the substrate is vibrated laterally, its momentum diffuses within the Stokes layer of the drop. Above the Stokes layer, the competition between the inertial and Laplace forces causes the formation of capillary waves on the surface of the drop. In the first part of this paper, the resonant states of water drops are illustrated by investigating the velocity profile and the hydrostatic force using a 3d simulation of the Navier-Stokes equation. The simulation also allows an estimation of the contact angle variation on both sides of the drop. In the second part of the paper, we investigate the effect of vibration on a water drop in contact with a vertical plate. Here, as the plate vibrates parallel to gravity, the contact line oscillates. Each oscillation is, however, rectified by hysteresis, thus inducing a ratcheting motion to the water droplet vertically downward. Maximum rectification occurs at the resonant states of the drop. A comparison between the frequency-dependent motion of these drops and the variation of contact angles on their both sides is made. The paper ends with a discussion on the movements of the drops on a horizontal hydrophobic surface subjected to an asymmetric vibration.

Having established the measurement technology to measure accessory belt behavior using laser displacement measurement methods, we were able to clarify the mechanism of the belt lateral vibration, which is one of the causes of a belt running off the pulley, by analyzing the correlation between the belt's behavior and fluctuations in input to the belt and by performing belt behavior simulation. We confirmed that the belt vibrates laterally if the accessory belt's natural frequency is equal to 1 2 of the frequency of belt vibration length fluctuation caused by crankshaft torsional and the value of the belt lateral vibration becomes larger in proportion to the value of the belt length fluctuation. We also found that lateral vibration occurs due to the existence of a natural vibrational mode in the lateral motion of the belt itself.

The coupled torsional-lateral model of the rotor system is set up and governing equations are derived using the Lagrange approach with six degrees of freedom. The gyroscopic effect and gravity are included. Two kinds of unbalance, namely static unbalance and dynamic unbalance, are considered in the rotor system. Torsional and lateral motions are subjected to external excitations. Through numerical simulation, the developed model is used and the coupled torsional-lateral vibrations of the unbalanced rotor system under external excitations are investigated. Some new phenomena are found and discussed.

The novel transient dynamic vibration model of pump-turbine’s shaft system considers nonlinear seal fluid forces, in addition to the coupled effects of unbalanced magnetic pull (UMP), guide bearing force, unbalanced mass force and hydraulic force. The calculations are based on the finite element method (FEM) and Lagrange equation. Node automation division method (NADM) is developed in-house for quick and accurate node division. The differential equations of motion and the dynamic response of the unit are resolved by implicit Newmark method. The nonlinear dynamic characteristics of crown seal force and UMP are researched. The transient and steady lateral vibrations of three bearings are also analyzed using the periodic response diagrams, steady axis orbits and waterfall diagrams. The calculated results demonstrate that the mounted eccentricity, crown seal channel clearance and air-gap length have significant effects on the dynamic vibration of the shaft system. When the mounted eccentricity is 0mm, clearance of crown seal is 3mm, air-gap length is 35mm, the stability of the system is the highest. Both quantitative and qualitative results are relevant for engineering use in pump-turbine’s shaft system design.

歩行者により生じたM吊橋の水平振動を計測した結果, 歩行者の分布状況により桁は逆対称水平3次モード (固有振動数0.88Hz) または対称水平4次モード (固有振動数1.02Hz) で振動し, 最大応答振幅は約45mmに達していた. 桁と歩行者の水平変位は正弦波形を示し, 歩行者は桁の水平変位に同調していることが確認された. 歩行者の位相は, 桁の位相より120°から160°進んでおり, 歩行者の水平力は桁応答振幅を増大させる起振力となっていた. 常時微動計測により, 桁固有振動数が0.6Hz以下では歩行者と桁の共振は生じないことが明確となった. 橋の質量および減衰定数が小さいほど桁の応答変位は大きくなること, 桁振幅が45mm程度 (速度250mm/s, 加速度1,350mm/s2) になると自然に歩行することは困難であることを見出した.

This paper investigates the effect of shaft rotation on its natural frequency. Apart from gyroscopic effect, the axial force originated from centrifugal force and the Poisson effect results in change of shaft natural frequency. D'Alembert principle for shaft in cylindrical co-ordinate system, along with the stress–strain relation, gives the non-homogenous linear differential equation, which can be used to calculate axial stress in the shaft. Numerical results of this study show that axial stress produced by shaft rotation has a major effect on the natural frequency of long high-speed shafts, while shaft diameter has no influence on the results. In addition, change in lateral natural frequency due to gyroscopic effect is compared with the results of this study.

For a rotating machine with one or more tilting pad journal bearings (TPJBs), it is a common state that pad-to-journal assembling clearances are uneven in one bearing, or inconsistent relationship of assembling clearances are encountered among all the bearings. Considering the structural characteristics of a TPJB, a parameter named as ‘pad pair pre-load coefficient’ corresponding to bearing pre-load coefficient is proposed. First, the influence of bearing pre-load coefficient and pad pair pre-load coefficient on shaft vibration in a Jeffcott rotor system is analyzed. Then taking No.1 unit with two TPJBs in Guangzhou pumped storage power plant (GPSP) as an example, the influence of inconsistent relationship of the two bearing pre-load coefficients on shaft vibration is analyzed. The results show that variety of one bearing pre-load coefficient will only influence vibration amplitude, not involving the shape of shaft center orbit. But variety of one or more pad pair pre-load coefficient will influence not only vibration amplitude, but also the shape of orbit — which will be turned from a circle to an ellipse. By investigating variation characteristics of the orbits' shape and vibration amplitude at the TPJBs, some theoretical bases will be supplied for evaluating the condition of the unit bearings.

In this paper, a new model is proposed to describe the phenomena of the excessive lateral vibration induced by pedestrians walking on a flexible footbridge. The lateral vibration of footbridges is considered, in most existing literature, to be caused by the directly lateral forces induced by crowds on the footbridge. However, the vertical vibration of the mass centre of human bodies may also affect its lateral vibration because of the internal resonant between lateral and vertical modes. The lateral vibration of most footbridges is like a restricted pendulum, and the vertical vibration of human bodies excites the footbridge to vibrate laterally in a way of parametric excitation. This model was obtained by taking into account the parametric excitation between pedestrians and the footbridge. The model shows that when the number of pedestrians walking on the footbridge reaches a threshold, the equilibrium position of lateral vibration of the footbridge loses its stability and the amplitude of the lateral vibration increases rapidly very quickly, which coincides with the observations of the London Millennium Bridge.

Hydrodynamic tilting-pad bearings with electromagnetic actuators are designed to take advantage of the load-carrying capacity of the hydrodynamic bearing together with the actuation capacity of the electromagnets. Hence, actuators can be downsized because they do not work as bearings as it is the case of magnetic bearings. In this work, one presents the numerical and experimental analysis of rotor vibration attenuation using a tilting-pad journal bearing with embedded electromagnetic actuators when proportional-derivative (PD) control is implemented. Experimental results in the frequency domain show reductions in shaft-vibration amplitude of 11% for the rotating speed of 600 rpm and 18% for the rotating speed of 1100 rpm, with good agreement of the mathematical model. The actuation capacity of the bearing in study is compared to the capacity of other active bearings in literature, showing that the present bearing is competitive in terms of specific control capacity.

Observation of human-induced large-amplitude lateral vibration of an actual pedestrian bridge in an extremely congested condition is reported. Walking motions of pedestrians recorded by a video camera are analysed. It is found that walking among 20 per cent or more of the pedestrians on the bridge was synchronized to the girder lateral vibration. With this synchronization, the total lateral force from the pedestrians to the girder is evidently increased and it acts as a resonant force on the girder lateral vibration.

An extensive hydro-generator rotordynamic model is presented along with a technique for determining imbalances and shaft misalignments using the model and mechanical run vibration measurements. The hydro-generator model combines a finite element rotor model with a finite difference guide bearing model, generator magnetic forces and turbine hydraulic forces to calculate natural frequencies, stability and steady-state response. The complete model allows a great number of inputs including imbalances, rotation speed, guide bearing misalignments, coupling misalignments, bearing clearances, bearing temperatures, generator stator and rotor center misalignments, generator average air gap, generator power output and turbine blade tip clearance, all of which are described in this paper. Finally, the numerical results are compared with vibration data from the mechanical run and load test of an actual hydro generator unit.

Magnetic tape is a flexible mechanical structure having dimensions that are orders of magnitude different in its thickness, width, and length directions. In order to position the tape relative to the read/write head, guides constrain the tape’s lateral motion, but even the modest forces that develop during guiding can cause wear and damage to the tape’s edges. This paper presents a tensioned axially-moving viscoelastic Euler‐Bernoulli beam model used to simulate the tape’s lateral dynamics, the guiding forces, and the position error between the data tracks and the read/write head. Lateral vibration can be excited by disturbances in the form of pack runout, flange impacts, precurvature of the tape in its natural unstressed state, and spiral stacking as tape winds onto the take-up pack. The guide model incorporates nonlinear characteristics including preload and deadbands in displacement and restoring force. A tracking servo model represents the ability of the read/write head’s actuator to track disturbances in the tape’s motion, and the actuator’s motion couples through friction with the tape’s vibration. Low frequency excitation arising from pack runout can excite high frequency position error because of the nonlinear characteristics of the guides and impacts against the pack’s flanges. The contact force developed between the tape and the packs’flanges can be minimized without significantly increasing the position error by judicious selection of the flanges’ taper angle. DOI: 10.1115/1.4000665 In the field of computer data storage, tape libraries are used for long-term archive, backup, and restoration of financial information, medical records, geophysical data, satellite imagery, and electronic intelligence. Such data are stored for decades, often under regulatory requirements, and its value increases with time. During the past fifty years, the volumetric storage density for this technology has grown by six orders of magnitude, partly due to advances in the mechanical design of the path, guides, cartridge, and servo mechatronics 1. Technical roadmaps foresee that the tape’s thickness will decrease to 5 m, the bit length will reduce to 33 nm, and the data track will narrow to 2 m over the next decade. At the same time, the tape’s tension is expected to decrease while the transport speed doubles 2. Such trends place demands on the design of guides, servos, and actuators for the control of lateral vibration and the reduction of position error between the tape and read/write head. Magnetic tape itself is a flexible layered polymeric material with dimensions that differ by orders of magnitude in the thickness, width, and machine directions. The nonlinear forcedeflection characteristics of guides and flange impacts 3 enable low frequency disturbances e.g., from bearing runout to excite high frequency lateral vibration of the tape. Although the read/ write head is actively positioned in order to follow the tape’s vibration, its bandwidth is limited to the lower frequencies. The objective of this paper is to develop a model to support path, guide, media, and servo design in order to reduce the position error between the tape’s data tracks and the read/write head. The vibration of magnetic tape is related to the mechanics of axially-moving materials 4,5 and web transport systems 6,7. Vibration models for magnetic tape include traveling strings and tensioned beams and exact closed form expressions for the response of such systems to arbitrary excitation are available through modal analysis and Green’s function methods 8. Solutions also are available for the displacement of an axially-moving Timoshenko beam having defects in its natural shape, where preformed warpage in the beam’s natural shape can drive undesirable lateral motion 9. The discretization of moving media models and the influence of nonlinearity are discussed in Refs. 10‐12. Edge and surface guiding are two approaches for constraining the lateral vibration of magnetic tape and other moving media. Edge guides establish a lateral constraint on the narrow edge of the tape by its contact against the guide’s flanges—termed buttons—but those forces can lead to excessive heating and wear of the tape’s edge 13. The judicious positioning and choice of a guide’s engagement length are useful approaches to reduce steady-state vibration amplitude 14,15. The propagation of boundary runout disturbances from packs and rollers can be reduced through the use of tilted 16 and deadband 3 guides, the latter of which have finite clearance between the guide’s flanges and the edge of the tape while centered. The high frequency vibration associated with impacts in deadband guides is particularly problematic because of the limited frequency response of the track following servo 17. In the alternative approach of surface guiding, edge contact is avoided entirely, and lateral vibration is controlled by distributing friction forces over the relatively broad face of the tape’s width. Surface guides are particularly useful at higher frequencies while edge guides are useful for lower frequency higher amplitude disturbances. The guides, layout of the transport path, and actuator-servo system synergistically contribute to the alignment of the tape’s data tracks and the magnetic head’s read/write elements. Figure 1 depicts an overview of the model that is developed in Sec. 2 for coupled lateral tape vibration and actuator-servo dynamics. Disturbances from the packs, guides, and imperfections in the tape media itself each excite lateral tape vibration. A general representation of the restoring force provided by a nonlinear edge guide is developed for elements that have a deadband in displacement 1

Switching stiffness control is applied to attenuate vibration in the lateral directions in an active magnetic suspension system with electromagnets operated in differential mode. The magnetic suspension system using the attractive force between magnetized bodies is inherently unstable in the normal direction so that feedback control is necessary to achieve stable suspension. In contrast, it can be stable in the lateral directions due to the edge effects in the magnetic circuits. In several applications, such passive suspension is used in combination with the active one to reduce cost and space. However, damping in the lateral directions is generally small. As a result, induced vibrations in these directions are hardly attenuated. To suppress such vibration without any additional actuator (electromagnet), switching stiffness control is applied to an magnetic suspension system operated in the differential mode. The stiffness in the lateral direction is adjusted by varying the bias currents of an opposed pair of electromagnets located in the normal direction simultaneously according to the motion of the suspended object. When the varied bias currents are adjusted for the additive normal forces cancel each other, such control does not affect the suspension in the normal direction. The effectiveness of the proposed control methods is confirmed experimentally.

In tilting pad bearing design process, the selection of the proper configuration type of either a Load-Between-Pad (LBP) or Load-On-Pad (LOP) as well as preload and pivot offset conditions is to be carefully considered. Also the bearing needs to be designed in order to be suited for the rotor-bearing system and operating condition. In this paper, it is observed that the static and dynamic characteristics of a five pad tilting pad bearing for the LBP and LOP configurations are influenced by the variation of preload and pivot offset. In this context, rotor dynamic analysis of the 5 MW industrial gas turbine supported by the tilting pad bearing at the front and roller bearing at the rear is carried out based on the dynamic coefficients of the tilting pad bearing investigated. The result shows that two rigid body critical modes experience various changes according to the influence of the tilting pad bearing uniquely applied to one side of this machine. Mainly, the second critical speed, the rigid body mode of conical shape with high whirling in the tilting pad bearing, is significantly changed by preload and pivot offset regardless of the LBP and LOP configurations. And the first critical mode, the rigid body mode of conical shape with high whirling in the roller bearing, is sensitively affected by preload applied to the LOP configuration and by its asymmetric dynamic properties.Copyright © 2011 by ASME

免震積層ゴムは, 薄いゴム層と鉄板とを交互に重ね合わせた柱状の複合構造体で, 地震動に対する建物の揺れを防ぐために開発されたものである. 筆者らは, この固有振動数と振動モード, 及び地震動に対する建物の振幅比に関する伝達関数について解析を行った.この解析では, 免震積層ゴムの構成を均質な材料特性をもつ短柱と見做し, 一様な曲げ-せん断剛性及び回転慣性を考慮した. 一般に固有振動数は, 軸圧縮力の増加するにつれて減少する. 座屈荷重は1次固有振動数がゼロとなったときの圧縮力として求められる. 実験例について計算した1次固有振動数と圧縮力との関係は実験結果とよく一致した.

A component modal synthesis technique for calculating the system dynamics of a rotor system is presented. In this method, the rotor system is divided into bladed disks, rotor shafts, and the nonrotating subsystems. The subsystem frequency transfer function matrices, geometric compatibility, and force equilibrium equations are combined to solve the critical speeds of the rotor system. The degrees of freedom in the final critical speed equation are reduced to the number of physical connections between subsystems. It is much smaller than the total number of substructure modes used in the synthesis. The proposed technique is capable of handling the rotor system with frequency-dependent properties, such as bearing stiffness. The method applies equally well for the synchronous and nonsynchronous rotor whirl. Numerical results indicate that the proposed method can efficiently and accurately determine the dynamic characteristics of a rotor system.