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Abstract This paper investigates numerically and experimentally the rotor drop dynamics when unexpected Active Magnetic Bearings (AMBs) shut down occurs. In such an event, the rotor behaviour drops on two touchdown bearings (TDBs) composed of a ball bearing and a ribbon damper providing stiffness and damping to the overall system. The aim of this paper is to establish and to validate the rotor-drop system model. A first experimental set up is used to identify the dynamic characteristics of the corrugated ribbon damper and to test the Kelvin-Voigt model and the generalized Dahl model. Then, three TDB models are proposed including either the first or the second ribbon damper models. The second experimental set-up is devoted to the rotor drop response of an industrial scale rotor-AMB system equipped with two TDBs. Rotor drop simulations in the time domain are carried out by using the Finite Element method and the three TDB models. Predicted and measured rotor drop responses are compared regarding the displacements as well as transmitted loads and permits evaluating the model efficiency. [ABSTRACT FROM AUTHOR]

The pulleys of an automotive front engine accessory drive are driven in rotation by a poly-V belt itself driven by the crankshaft pulley. This driving pulley is often used as a torsional vibration damper (TVD) for the crankshaft. Three elements compose the pulley: the hub, a rubber ring and an inertia steel ring with v-ribs on its outer diameter. Although the crankshaft torsional vibrations are dampen, they are transmitted to the belt transmission and therefore to the driven accessories. Hence, recent developments have conducted to add a decoupling function to these pulleys. The decoupling is realized by an other rubber ring. The TVD pulley components are designed in order to dampen crankshaft vibrations over a given frequency range, i.e. stiffness and damping characteristics are determined for the rubber ring. These expected characteristics have to be checked after manufacturing for product certification but also to give some real measured data input for simulation models. An experimental characterization method is presented and discussed for the determination of the stiffness and damping coefficients of the rubber rings used in such pulley. The test rig developed is temperature controlled in order to show the influence of the temperature. [ABSTRACT FROM AUTHOR]

In order to predict the lateral rotordynamics of a high-speed induction motor, an optimization procedure is used for identifying the dynamic behavior of the magnetic core made of a lamination stack, tie rods, and short-circuit rods. Modal parameters predicted by a finite-element model based on beam elements and measured on induction motors are included in modal error functions contained in a functional. The minimization of this functional by using the Levenberg-Marquardt algorithm permits-extracting the equivalent constitutive properties of the lamination stack for several rotors of different sizes. Finally, the size effect on the constitutive properties identified is discussed. [ABSTRACT FROM AUTHOR]

In this paper we investigate the nonlinear dynamic behavior of a rotor with an architecture targeting space turbopumps. Depending on the rotor speed of rotation, these pumps can be supported by ball bearings with a loose fit, i.e. having an outer-race housing clearance, and by annular seals providing radial fluid forces thanks to the Lomakin effect. Moreover, such turbopumps often operate in supercritical regime, which means that they must pass through critical speeds while undergoing the lowest vibration levels to ensure pump safety. In this context, the aim of the paper is to analyze the case of an academic rotor test bench designed to present a dynamic bending behavior similar to that of space turbopumps. The investigation focuses on the test bench equipped with two hydrodynamic bearings and two ball bearings with loose fit. The rotor response when passing through its first critical speed and triggering nonlinear bearing-housing contact phenomena is assessed experimentally and numerically. The hydrodynamic bearings are located at the two shaft ends and the ball bearings at the middle of the shaft. The rotor finite element model is detailed first and the test bench presented. Then, the nonlinear dynamic response of the rotor subject to a mass unbalance and a permanent bow is obtained. The numerical results are computed using either a full transient step-by-step time integration technique or an arc-length continuation method. The main objective is to assess how the nonlinearity stemming from the loose fit of the ball bearings alters the original rotordynamics. Moreover, a parametric analysis based on mass unbalance and on outer-race housing clearance is conducted. Major nonlinear phenomena are highlighted such as the resonance peak flattening at the critical speed, and the numerous harmonics triggered by the hydrodynamic bearing and contact forces. The results obtained and the model proposed here are of considerable relevance for the design of space turbopumps and other pumps having a similar architecture. [ABSTRACT FROM AUTHOR]

In the transportation domain, on-board rotors in bending are subjected not only to rotating mass unbalance but also to several movements of their base. The main objective of this article is to predict the dynamic behavior of a rotor in the presence of base excitations. The proposed on-board rotor model is based on the Timoshenko beam finite element. It takes into account the effects corresponding to rotary inertia, gyroscopic inertia, and shear deformation of shaft as well as the geometric asymmetry of disk and/or shaft and considers six types of deterministic motions (rotations and translations) of the rotor’s rigid base. The use of Lagrange’s equations associated with the finite element method yields the linear second-order differential equations of vibratory motion of the rotating rotor in bending relative to the moving rigid base which forms a non-inertial frame of reference. The linear equations of motion highlight periodic parametric terms due to the geometric asymmetry of the rotor components and time-varying parametric terms due to the rotational motions of the rotor rigid base. These parametric terms are considered as sources of internal excitation and can lead to lateral dynamic instability. In the presented applications, the rotor is excited by a rotating mass unbalance combined with constant rotation and sinusoidal translation of the base. Quasi-analytical and numerical solutions for two different rotor configurations (symmetric and asymmetric) are analyzed by means of stability charts, Campbell diagrams, steady-state responses as well as orbits of the rotor. [ABSTRACT FROM AUTHOR]

Abstract: The major purpose of this study is to predict the dynamic behavior of an on-board rotor mounted on hydrodynamic journal bearings in the presence of rigid support movements, the target application being turbochargers of vehicles or rotating machines subject to seismic excitation. The proposed on-board rotor model is based on Timoshenko beam finite elements. The dynamic modeling takes into account the geometric asymmetry of shaft and/or rigid disk as well as the six deterministic translations and rotations of the rotor rigid support. Depending on the type of analysis used for the bearing, the fluid film forces computed with the Reynolds equation are linear/nonlinear. Thus the application of Lagrange's equations yields the linear/nonlinear equations of motion of the rotating rotor in bending with respect to the moving rigid support which represents a non-inertial frame of reference. These equations are solved using the implicit Newmark time-step integration scheme. Due to the geometric asymmetry of the rotor and to the rotational motions of the support, the equations of motion include time-varying parametric terms which can lead to lateral dynamic instability. The influence of sinusoidal rotational or translational motions of the support, the accuracy of the linear 8-coefficient bearing model and the interest of the nonlinear model for a hydrodynamic journal bearing are examined and discussed by means of stability charts, orbits of the rotor, time history responses, fast Fourier transforms, bifurcation diagrams as well as Poincaré maps. [Copyright &y& Elsevier]

This paper addresses rotor–stator contact issues, especially the dry-whip phenomenon, as caused by base excitations. So far, this particular instability in the case of direct rub between shaft and stator was only shown by experiments to occur in response to mass unbalance excitation during run-up or run-down and/or to external disturbance such as direct impact applied to the rotor. The present paper aims to demonstrate that it can also be triggered by external forces coming from the rigid base motion. To this end, both numerical and experimental approaches are employed. A rotor finite element model combining on-board motions with shaft-stator rub impact is first proposed. This model is validated with a rotor test-rig supported by hydrodynamic bearings and mounted on a multi-axial hydraulic shaker and subject to a harmonic base motion. The dry-whip is then shown experimentally by using a base shock excitation consisting of two transverse translations, which is later confirmed numerically. The influence of important parameters on the dynamic behavior of the system is also assessed, such as the hydrodynamic bearing nonlinearities, the amplitude and direction of the base translations and the dry friction coefficient. • FEM of on-board rotor on journal bearings with rotor–stator contacts. • Experimental validation of the model. • Evidencing of the dry-whip phenomenon due to shock base translations. • Influence of the base motion amplitude and direction on the dry-whip occurrence. • Video of the dry-whip occurrence performed by a high-frequency camera. [ABSTRACT FROM AUTHOR]

• A multiphysics model comprising an on-board FE rotor, AMBs and TDBs is developed. • Shock and harmonic tests on an on-board rotor-AMB kit permit validating the model. • Rotor-TDB interactions are investigated using time response, orbit plots and STFT. • Multi-harmonic response, peak flattening, particular orbits are investigated. • Fleeting instabilities and short period of backward whirl are exhibited. Active magnetic bearing (AMB) technology provides to rotating machinery, high operating speeds, direct couplings, almost frictionless support and a reduced footprint. In case of strong base excitations, the rotor may contact its touchdown bearings (TDBs), which are used as landing and safety bearings, while AMBs still operate. The paper presents, experimentally and numerically, this particular operating conditions where sliding friction-induced vibrations or AMB instabilities might be triggered. For this purpose, the developed finite-element model of the rotor-AMB-TDB system is experimentally validated using a lab-scale rotor test rig mounted on shaker. The validation is first conducted with no base excitation by analysing the mass unbalance responses and drop onto TDBs. In the presence of harmonic then impulse base excitations, the mass unbalance response are analysed in time and frequency domains. Considering harmonic tests, particular combinations of base motions and rotordynamics are observed depending on the frequency ratio between the base motion and the speed of rotation. TDB interactions produce rotor orbits flattening. The impulse tests highlight the friction effects and short periods of backward whirl are observed just after the contact. Despite some transient instabilities, the controller was always able to maintain the system stable by bringing damping. At each of these steps, the model reliability is assessed and predictions are close to the measured phenomena. [ABSTRACT FROM AUTHOR]

• Determination of multi-axial shaker motion with only six measured accelerations. • Direct access to the three angular and three translational displacements. • Experimental validation with multi-axial sine- and random- excitations. • Original combination of numerical integration and data processing tools. • Method suited to large rotations making the problem nonlinear. Qualification tests on the vibrations of on-board equipment depend on the ability of the shaker used to reproduce as precisely as possible the excitation profiles stipulated by standards, given that real operational conditions are generally multiaxial. Using a set of six measured linear accelerations the original method proposed permits determining real six-axial shaker displacements combining three translations and three rotations. Thus, this method, named REDEAT (REal Displacement from Experimental Acceleration with inverse Technique), is useful for introducing these real displacements in the numerical simulation in view of experimentally validating the dynamic behavior model of on-board equipment. Moreover, the REDEAT method evaluates the deviation between real and intended forced displacements. Its principle resides in solving a system of nonlinear time-differential equations expressed, by resorting to numerical integration combining Newmark and Newton–Raphson schemes. In order to overcome the classical divergence issue inherent to the use of numerical integration, data processing tools are employed, consisting of the application of successive high pass filters and window functions. The REDEAT method is applied to a hydraulic 6-DOF shaker equipped with accelerometers and gyroscopes, through a combination of two harmonic rotations and a full 6-DOF random motion. In addition, the influence of the method's parameters and of the six input accelerograms chosen from twelve is investigated. [ABSTRACT FROM AUTHOR]

In order to overcome the loss of performances issue when scaling resonant sensors clown to NEMS, it proves extremely useful to study the behavior of resonators up to large displacements and hence high nonlinearities. A comprehensive nonlinear multiphysics model based on the Euler-Bernoulli equation which includes both mechanical and electrostatic nonlinearities in the case of a capacitive doubly clamped beam is presented. This purely analytical model captures all the nonlinear phenomena present in NEMS resonators electrostatically actuated including bistability, multistability which can lead to several physical limitations such as noise mixing, frequency stability deterioration as well as dynamic pull-in. Moreover, close-form expressions of the critical amplitudes and pull-in domain initiation amplitude are provided which can potentially serve for NEMS designers as quick design rules. [ABSTRACT FROM AUTHOR]

Abstract: Serpentine belt drives are often used in front end accessory drive of automotive engine. The accessories resistant torques are getting higher within new technological innovations as stater-alternator, and belt transmissions are always asked for higher capacity. Two kind of tensioners are used to maintain minimum tension that insure power transmission and minimize slip: dry friction or hydraulic tensioners. An experimental device and a specific transmission error measurement method have been used to evaluate the performances of a generic transmission by determining the pulley–belt slip for these two kinds of tensioner. Transmission error between pulleys, pulley/belt slip are deduced from pulley rotation angle measurements. Results obtained show that: the use of tensioner limits belt slip on pulleys, pulley–belt slip is reachable from transmission error measurement, belt non uniform characteristics are responsible of low frequency modulations of transmission error. [Copyright &y& Elsevier]

The objective of this paper is to describe the modified Dahl and Masing models used for predicting hysteretic behavior, and tested on a belt tensioner for automotive engines. An experimental study with deflection imposed on the tensioner is first carried out to identify hysteresis loop parameters for the two models. The models are implemented in the general motion equations which govern the behavior of a belt–tensioner–mass system. Particular attention is paid to the use of numerical schemes. The numerical and experimental investigations show the reliability of the modified Dahl model. [Copyright &y& Elsevier]

• A new FE model is proposed to assess the role of support motions in on-board rotordynamics. • This model is based on the Timoshenko beam theory and six DOFs. • A parametric influence is generated by the rotations and the translations of the support. • This influence creates single-frequency and multi-frequency parametric excitations. • The dynamic stability of the rotor is evaluated through the Floquet theory. In the transportation domain such as automotive turbochargers and aircraft turbines, the vibrations of on-board rotors are induced not only by the mass unbalance excitation but also by various movements of their support. The dynamics of an on-board rotor mounted on hydrodynamic finite-length bearings is investigated in the presence of support motions which create multi-frequency parametric excitations. The developed on-board rotor model is based on the gyroscopic Timoshenko beam finite element with two nodes and six degrees of freedom per node for 3D motions (transverse and axial displacements as well as rotations due to the bending and to the torsion). The equations of motion highlight time-varying parametric terms due to the mass unbalance, the support rotations, the coupling between both phenomena and the combination of mass unbalance and support translations. These parametric terms can yield a dynamic instability because they contribute as generators of internal excitation. In the presented applications, single-frequency and multi-frequency parametric excitations are used. Namely, the rotor is excited either by simple and combined sinusoidal support rotations or by a rotating mass unbalance combined with sinusoidal support translations to examine the stability of the static equilibrium point through the Floquet theory. [ABSTRACT FROM AUTHOR]

• The initial position of the drillstring on the neutral line of a 3D-geometry well (Co-rotational formulation). • The static equilibrium position of the drillstring taking into account the load and the well interactions (Newton Raphson scheme). • The iterative technique in the time domain with an adaptive time step (a variant of the RK4 scheme) in order to reduce the computation time with the full-model. • The reduction technique proposed by Craig -Bampton to save the computational time for modelling the drillstring dynamics in 3D curved wells. One test case is considered. Two criteria are defined to study the CB convergence and the reduction of computational time are investigated. The drilling operations for oil or geothermic extraction use a slender structure introduced inside the drill well, hanging from a derrick and driven by a rotary table at the surface. The drilling structure consists in a series of drill-pipes and some heavy pipes at the well bottom. The drilling process involves nonlinear dynamic phenomena such as bit-bounce, stick-slip due to the well-drillstring multi-contacts and the pulsating mud flow. The drillstring vibrations may yield, the rate of penetration decrease, the premature wears and damages of drilling equipment. Many numerical models have been proposed to study the dynamics of drillstring to improve the reliability of drilling operations. However, the numerical models of drilling structures representing several kilometers length require a huge amount of computer memory storage and yield a too long computational time. The reduction technique proposed by Craig-Bampton (CB) has been developed for modelling the nonlinear dynamics of rotating machines to save the computational time but still limited in the context of rotordynamics. The paper focuses on the implementation of the CB method in the case of long drillstring assembly modelled by beam finite elements. The pre-loaded states of the drillstring due to the well curvature, well-structure contacts and fluid-structure interactions are determined and taken into account in the dynamic computation. The drillstring transient dynamics is simulated and the orbital motion of several nodes are analyzed. The result convergence and the reduction of computational time obtained by the CB method are investigated and discussed. [ABSTRACT FROM AUTHOR]

In rotary drilling, a drillstring is an assembly of slender pipes. It is used to transmit the driving torque of a motor at the drilling surface to the drill bit at the bottom hole of a 3D well. Numerous vibratory phenomena are induced during the drilling: whirling, stick-slip, bit-bouncing, lateral instability, inducing in particular reduction of the rate of penetration and mean time between failures. For the rotordynamics prediction of such a structure, the drillpipes are modelled with Timoshenko beam elements, containing 12° of freedom, equipped with distributed radial stop-ends. The rotary motion is assumed to have a constant speed of rotation imposed at the top of the drillstring. The drilling mud is taken into account by using a fluid-structure interaction model. The numerical simulations concern a real 3D-borehole and a parametric analysis is carried out for determining the role of the mud density and of the flows rate on the drillstring dynamics. It is shown that increasing the flow rate and densifying the drilling fluid reduce the fluid damping effect that increases drillstring lateral vibrations. • Finite Element Modelling of drillstrings immersed in 3D curved well. • Fluid–drillstring interaction accounting for the drillstring curvature and its deflection making the annular space eccentric. • Forces due to fluid flows inside and outside the drillstring are considered. • Drillstring/wellbore contacts are modeled with regularized and damped contact laws. • Application concerns a part of a field 3D borehole, with realistic data. [ABSTRACT FROM AUTHOR]