Quentin Labourey

Olivier Aycard

Denis Pellerin

Michele Rombaut

Audiovisual data fusion for successive speakers tracking

Keywords: Audiovisual Data Fusion, Skin Detection, Sound Source Tracking, Talking Face Tracking

In this paper, a human speaker tracking method on audio and video data is presented. It is applied to conversation tracking with a robot. Audiovisual data fusion is performed in a two-steps process. Detection is performed independently on each modality: face detection based on skin color on video data and sound source localization based on the time delay of arrival on audio data. The results of those detection processes are then fused thanks to an adaptation of bayesian filter to detect the speaker. The robot is able to detect the face of the talking person and to detect a new speaker in a conversation.

INTRODUCTION

Over the last decades, robotics has taken a growing importance, in our society and imagination as well as in Science. Until a few years ago, "robots" often only amounted to sensors placed in the infrastructure. Those types of sensors can make people feel ill-atease because they are intrusive. Companions robots that share the environment with humans are a way to remedy the problem. They are equipped with sensors and are able to perceive the behavior of people and interact with them. There are various kinds of sensors, such as video and audio sensors. However the data acquired from those sensors is most of the time uncertain, noisy or partial. That is why multi-sensor data fusion is now an extremely prolific field of research, as it allows to obtain more complete information from partial data in a robust way.

This work presents a method based on audiovisual data fusion for tracking the positions of the successive speakers of a human conversation with a robot equipped with visual and audio sensors. The robot must be able to detect the faces around him and choose which one is the speaker. If the speaker is not in the visual scene, he must be able to track his position.

As the audio and video modalities are very different, a late fusion seems adequate for the present work [START_REF] Snoek | Early versus late fusion in semantic video analysis[END_REF], where detection is performed on each modalities and a final decision is taken as to where the speaker is by fusing the detection results. It implies that this work is based on three main axis: FACE DETECTION ON VISUAL DATA: The literature on face detection is quite abundant. In their survey of the state-of-the-art of face detection, Zhang [START_REF] Zhang | A survey of recent advances in face detection[END_REF] show that most of the recent works on face detection are feature-based and appearance-based methods, where a classifier is trained either with direct positive examples of faces or with visual features extracted from faces (patterns, edges, etc...). Although a lot of work with different methods exist, such as Support Vector Machines [START_REF] Osuna | Training support vector machines: an application to face detection[END_REF] , Neural Networks [START_REF] Vaillant | Original approach for the localisation of objects in images[END_REF], one of the principle contribution of the last decades on face contributions is Viola & Jones algorithm [START_REF] Viola | Robust real-time face detection[END_REF].

AUDIO SOUND SOURCE LOCALIZATION: Various methods exist to localize a sound source thanks to microphone array. Brandstein [START_REF] Brandstein | A practical methodology for speech source localization with microphone arrays[END_REF] divides those methods into 3 broad parts: steered-beamformer based methods, which are probabilistic methods [START_REF] Valin | Localization of simultaneous moving sound sources for mobile robot using a frequency-domain steered beamformer approach[END_REF] and are most of the time quite time-costly, high-resolution spectral-estimation based methods, which are based upon a spatiospectral correlation matrix, and timedifference of arrival methods [START_REF] Gustafsson | Positioning using time-difference of arrival measurements[END_REF], which use the delay of arrival between microphones and are widely spread on various complexity levels.

FUSION OF THE RESULTS FROM BOTH MODAL-ITIES: Various types of fusion methods exist in different fields, a large part of them being based either on classifiers and machine learning [START_REF] Rao | Multimodal information fusion using the iterative decoding algorithm and its application to audio-visual speech recognition[END_REF], or on estimation of the state of objects [START_REF] Nguyen | Audio-visual data fusion for tracking the direction of multiple speakers[END_REF]. Hospedales [START_REF] Hospedales | Structure inference for bayesian multisensory scene understanding[END_REF] worked on a similar problem with unsupervised learning methods.

The method of the present work is a 2-steps pro-cess (figure 1): In the perception step, faces are detected on visual data thanks to a skin detection algorithm based on naive bayesian classifiers while sound source localization is performed on audio data thanks to the cross-correlation method. In the decision step, the results of the perception step are fused thanks to an adaptation of bayesian filter. Finally the choice of the position of the speaker is performed by the robot. This method provides with a real-time and robust way to keep track of the speaker in a conversation, while being adapted to low-quality data (independent from morphological patterns) and low-quality microprocessors (only one scanning of each frame).

This article is elaborated as follows: Section 2 describes the features of the robot used in this work, section 3 details the method for the detection on audio and video modalities independently and section 4 describes the decision process thanks to the audiovisual data fusion method. Finally, results are presented and discussed in section 5.

EXPERIMENTAL PLAFTFORM

This section details the specifications of the robot chosen to illustrate the speaker tracking method and the features on which this work has been performed.

Reeti R (figure 2) is a "humanoid" robot produced by Robopec (www.reeti.fr/), in France. It is equipped with low-cost sensors for perception (two cameras in the eyes and a stereophonic microphone in the base), and with servo-motors for action (turning the head and face expressions).

The two cameras can be accessed simultaneously and produce a 640x480 pixels color RGB video, at 10 frames per seconds. Only one camera is used instead of steroscopic vision to be faster with a less complex algorithm.The sound is acquired in stereo, with the two channel inputs at 15 centimeters apart.

An important constraint is that the field of audio localization covers the whole angular field in front of The global objective here is to track the angular position α of the current speaker in a horizontal plane with respect to the robot (figure 3).

PERCEPTION STEP

This section details our algorithms for face detection from video data and sound source localization from audio data.

Visual Data: Face and Skin Detection

The goal of the video processing is the frame-byframe detection of the angular positions of every face present in the visual scene with respect to the robot, i.e. the process should give out a number N of angles

[α 1 , • • •, α N ]
, each of them being the estimated angular position of a candidate face with respect to the robot in a 2D horizontal plan. Some particular cases can happen such as occlusion (the faces can be fully or partially hidden by an object like a hand, handkerchief, other person...), position changes (the faces may move during a conversation and their angle and position change), lighting changes (e.g. a person standing in front of a window moves, etc...).

Viola & Jones algorithm [START_REF] Viola | Robust real-time face detection[END_REF]) performs very well on separate images with a high rate of detection and a low rate of false alarms. However it is shown that the particular cases cited above related to the dynamicity of human conversation make the exclusive use of this algorithm unadapted to this work. Besides, the needed algorithm must be able to process a video stream in real-time, on a low-quality microprocessor. In such context, Viola & Jones algorithm cannot be used online and for each frame for tracking as it is too time-costly, although it can be used to obtain examples of face pixels. This is why skin detection [START_REF] Chai | Locating facial region of a head-and-shoulders color image[END_REF]) is used, based on naive bayesian classifiers (Schnei- TRAINING OF THE NAIVE BAYESIAN CLASSI-FIERS: 3 classifiers are trained, one for each color channel. Let R be the event "The considered pixel has a red value of R", with R ∈ [0; 255] and C be the event "The considered pixel is a positive sample". From the training set, the aim is to learn the probability distributions P(R|C) and P(R|¬C). This is done by computing the histograms of the positive and negative samples.

Let H C,R (i), i ∈ [0; 255], be the histogram computed on all the positive samples of the training set on the red channel, and H ¬C,R (i) be the histogram computed on all the negative samples of the training set on the red channel. Then the probability distributions P(R|C) and P(R|¬C) are computed as follows:

P(R|C) = H C,R (R) 255 ∑ i=0 H C,R (i) , P(R|¬C) = H ¬C,R (R) 255 ∑ i=0 H ¬C,R (i) (1)
A similar reasoning is performed to obtain the probability distributions on the blue channel P(B|C) and P(B|¬C), and on the green channel P(G|C) and P(G|¬C) .

CLASSIFICATION OF INCOMING PIXELS: The classification is made with the assumption that the three color channels are independent. A score between 0 and 1 is given to each pixel. The higher the score obtained is, the higher the chance that this pixel is skin:

Score = Score skin Score skin + Score ¬skin (2)
where

Score skin = η 1 • P(r|C) • P(g|C) • P(b|C) Score ¬skin = η 2 • P(r|¬C) • P(g|¬C) • P(b|¬C) (3)
with η 1 and η 2 as normalization parameters, and r,g,b respectively the red, green and blue values of the pixel.

Each incoming frame is processed with the naive bayesian classifiers. At the end of the classification, 0 or more skin regions are detected. Each detected skin region with a number of pixels under an arbitrary number of pixels (decided depending on the minimal size of head to detect) is removed: it enables to get rid of noise in the detection and to avoid keeping track of very small regions of the images for the detection.

Video Angular Conversion

At the end of the frame processing, the centers of gravity of the N detected skin regions are obtained, with N ≥ 0.

The goal of the angular conversion is to transform the x-value of each center of gravity into an angle between 0 and 180 degrees in a horizontal plane in the robot coordinate system. The robot reference, defined as the center of the head of the robot is used as the origin of the coordinate system.

In this work, it is considered that the camera reference is the same as the robot reference (figure 3). The angular position of the center of gravity with respect to the robot is computed as follows:

α = β + φ (4)
• φ is the angular position of the camera obtained by linear conversion: 1 step between two camera positions is equivalent to 1.27 degrees.

• β is the angular position of the pixel in the image, obtained by linear conversion: 1 degree contains 13 pixels.

The skin detection process thus gives out a number N of angles [α 1 , • • •, α N ] with respect to the robot at which candidates for faces are detected.

Audio Data: Sound Source Localization

The goal of the audio data processing is to produce a unique angle δ corresponding to the estimated angular position of the current speaker with respect to the robot.

The hypotheses on the framework for sound source localization are: one person is talking at a time, in a silent environment (the noise is inherent to the microphones and the robot's motors and fan). For this work, the detection does not have to be extremely precise as it will be fused with the video data. Because only two microphones are used and real-time processing is required, it seemed thus logical to choose a state-ofthe-art cross-correlation method (C.C.M.) thanks to the Time Difference of Arrival (T.D.O.A.) (Gustafsson and Gunnarsson, 2003) method.

DECISION STEP: AUDIOVISUAL DATA FUSION

The perception step is performed on visual & audio data at a period of 100 ms. The set of N faces extracted at time t, called the set of video observations is noted

O v t = α 1 t , • • •, α N t , with N ≥ 0.
In the same manner, the unique audio observation at time t is noted δ t . All the observations lie within a [0; 180] range.

From those data, a decision has to be taken as to where the speaker in the conversation is located. Those observations are uncertain, and there is high variety of possible visual and audio configurations. That is why an adaptation of a bayesian filter is used [START_REF] Thrun | Probabilistic Robotics (Intelligent Robotics and Autonomous Agents)[END_REF] with a differentiation of cases similar to a model rupture. The bayesian filter fits with the problem as it is a method to predict/estimate the state of a system equipped with sensors from the observations made by those sensors.

In the present case, the state S t of the system at time t is the angular location of the speaker. As the angular measures are uncertain, the angular field is divided into 18 possible states, each of them being an angular region 10 degrees wide (figure 4). The set of states is noted S = (S 0 , • • •, S 17 ). In the present section, the field of visual detection will indifferently be referred to as field of vision (F.O.V.) or field of visual detection.

Dynamic Model

The dynamic model corresponds to the model of the natural evolution of the conversation, i.e. the probability distribution P(S t |S t-1 ). Because the speaker in a conversation can change at any time, it is difficult to estimate the position at time t+1 of a speaker when it is known at time t.

In first approximation, the evolution model is built just to track the current speaker: it is considered that from a time t to a time t+1, or in 100 ms interval, the same person is still talking, even though this person might have moved in between. The probability distribution P(S t |S t-1 ) is thus a discrete "gaussian-like" form around S t-1 . Figure 5 is an example for S t-1 = S 6 . The case where the speaker changes between t and t+1 is not handled by this dynamic model. This is considered thanks to a simplified version of interacting multiple models (IMM) [START_REF] Farmer | Interacting multiple model kalman filters for robust high speed human motion tracking[END_REF]: when the information from both sensors or only the audio sensor is in complete conflict with the previous estimate, the information coming from the previous estimate is nullified and a uniform distribution is used. This case is detailed a bit more later on in this section.

Audio Sensor Model

The sensor model represents the trust put into the sensor. Let P(δ t |S t ) be the audio sensor model. The audio measure is not highly precise. The probability of δ t knowing S t should be a gaussian which mean-value is the mean angular value of the angular interval S t and which variance σ δ is high. Figure 6 shows an example of possible audio sensor model with σ δ = 20 for S t = S 7 . In this case the mean-value of the gaussian is 75.

Video Sensor Model

As the F.O.V. of the camera is not 180 wide, the observations depend on Po t , and the sensor model must be computed at each iteration: The video sensor model is in fact P(α t |S t , Po t ). 2 separate cases must be reviewed for the video sensor model: if the speaker is in the visual scene then the probability of detecting a face is centered on the state in which the speaker is. If he is not, the probability of detecting a face is uniform.

P(α t |S t , Po t ) = U(α t ), if S t ∩ F.O.V. = / 0 N (µ α , σ α ) if S t ∩ F.O.V. = / 0 ( 5
)
where U is the uniform distribution over α t , N (µ α σ α ) is a normal distribution of mean value µ α equal to the mean value of S and of variance σ α = 10.

Estimation of P(S

t |O v t , δ t , Po t )
Once the dynamic model and the video models are defined, the estimate of S t can be computed. Two main cases have to be differentiated here: THE SPEAKER IS IN THE VISUAL SCENE: If all the present faces are well detected, one or more video observations are available. There is a choice to be made as to what visual observation is the best. This is done by taking the video observation the closest to the audio observation. The video observation α t used for fusion is chosen as follows:

α t = arg min α i t ∈O v t ( α i t -δ t ) (6)
The fusion process is then the state-of-the-art prediction/estimation process.

The prediction step is:

P(S t |α 0:t-1 , δ 0:t-1 , Po 1:t ) = ∑ S t-1 (P (S t |S t-1 ) • P (S t-1 |α 1:t-1 , δ 0:t-1 , Po 1:t-1 )) (7)
The estimation step is computed with the hypothesis that the video and the audio sources are independent:

P(S t |α t , δ 0:t , Po 1:t ) = η • P(δ t |S t ) • P(α t |S t ) • P (S t |δ 1:t-1 , α 0:t-1 , Po 1:t ) (8)
with η a normalization factor.

Between the iteration t and the iteration t+1, if the speaker has changed, P(δ t |S t ) and P(α t |S t ) will overweight P (S t-1 |α 1:t-1 , δ 0:t-1 , Po 1:t-1 ) and modify the model in a few iterations, corresponding to the delay.

Once the estimation is over, the probability distribution of the speaker location is known. The most likely face video observation is chosen. Let ML be the maximum of the location probability distribution.

ML = max S∈S (P(S t |α 0:t , δ 0:t , Po 1:t )) (9)
The chosen video observation O Final is the closest observation to ML.

THE SPEAKER IS NOT IN THE VISUAL FIELD: The model has to change. A test has to be made on the audio: if the audio observation is outside the video, it is a model rupture, then the weight of the previous estimate and the video observations are nullified and only the audio observation is considered: the final observation is δ t .

RESULTS

Some results obtained from data extracted from the robot can be observed on figure 7 and an example of full video is accessible on the internet at the following URL: https://www.youtube.com/channel/UCl4mcfTKzB8orarl-SC7qgg.

It is difficult to evaluate the method as there are a lot of parameters that are linked to the experimental platform. However, some global remarks can be made:

• The full perception-decision chain is performing in real-time.

• The particular cases described in section 3.1 are fixed by the skin detection.

• Even if the audio detection is rough, when fused with the face detection it allows us to easily differentiate between two persons side by side as long as the space between their faces is bigger than the resolution of skin detection, so that two video observations are made. • The time of response when the speaker change is no more than a few frames (around 4) before the detection of the new speaker (figure 8).

• In the different videos that were tested the speaker is always detected after some time

CONCLUSIONS

This work focuses on three main issues: produce detection methods on low-quality visual and audio data from low-cost sensors, elaborate a robust audiovisual data fusion method adapted to the situation, and make the method follow a real-time constraint.

A full processing chain has been elaborated: it relies on the independent processing of the audio stream (sound source localization) and the video stream (face detection), fused in a late fusion process to create the decision. One of the main advantages of this processing chain is that each of its links can be modified and upgraded in future works. There are perspectives to improve this processing chain: the robot should be able to differentiate two speakers speaking at the same time at this stage, and the detection process can be improved by dealing with the bias introduced in the coordinate system. A lot of improvement could come from the inclusion of new sensors such as depth sensors or laser sensors to discriminate between region of interests to be explored directly instead of exploring the whole frame at each time. Improvement can also be made from using motion in the video, instead of frame-by-frame processing. This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025).

Figure 1 :

 1 Figure 1: Speaker detection: the two steps of perception and decision

Figure 2 :

 2 Figure 2: Reeti R the robot

  derman and Kanade, 1998) on the 3 video color channels. The proposed algorithm relies on three steps: ONLINE CREATION OF THE TRAINING DATASET: The training set is composed of skin (positive samples) and non-skin pixels (negative samples). To create the actual training set, the Viola & Jones algorithm is performed on each frame until some faces are detected. The pixels that are detected as part of faces are positive examples while those that are not are negative examples. If necessary, to increase the number of positive samples, it is possible to perform Viola & Jones on several successive frames.

Figure 3 :

 3 Figure 3: Angular definition: view from above

Figure 4 :

 4 Figure 4: State repartition: 10 degrees for each state As the visual field is narrower than the field of audio localization, the position of the head has an impact on the model. There are 93 possible head positions Po 1 , • • •, Po 93 . The position of the head at time t is

Figure 5 :

 5 Figure 5: Dynamic model: P(S t |S 6 )

Figure 6 :

 6 Figure 6: Dynamic model: P(δ t |S 7 ), σ δ = 20

Figure 7 :

 7 Figure 7: Results of audiovisual data fusion for a conversation involving 2 speakers

Figure 8 :

 8 Figure 8: Detection of a new speaker among 3 persons after a small delay


Mzaki Zaki Dakel

Sébastien Baguet

Régis Dufour

Steady-state dynamic behavior of an on-board rotor under combined base motions

Keywords: Rotordynamics, asymmetric rotor, on-board rotor, finite elements, base excitation, parametric excitation, transient motion, dynamic instability

Steady-state dynamic behavior of an on-board rotor

under combined base motions.

Introduction

Rotating machines play a paramount role in many modern industrial applications. Most of them can be considered as on-board machines affected mainly by mass unbalance and support excitations. Automotive turbocharger, ship and aircraft turbines as well as locomotive electrical generators are examples of rotors on moving base. The undesirable mass unbalance is due to the eccentricity of the center of mass along the rotor axis and can be caused by material non-homogeneities, manufacturing defaults, assembly and service conditions. The rotor balancing aims at minimizing the mass unbalance but generally does not lead to a complete cancellation, see, for example, [START_REF] Kang | An accuracy improvement for balancing crankshafts[END_REF] and [START_REF] Levecque | Vibration reduction of a single cylinder reciprocating compressor based on multi-stage balancing[END_REF]. A rotor can also be excited by the movement of its base which can increase the lateral vibration of the rotor and create a dynamic instability phenomenon. Therefore a rotor dynamic model taking into account multi-source excitations is of prime interest at the design stage in order to product rotating machines as reliable as possible and avoid catastrophic failure during the operation phase.

The literature comprises numerous books concerning the prediction of dynamics of symmetric/asymmetric rigid/flexible rotor systems mounted on linear/non-linear elastic bearings in the case of a fixed base [START_REF] Rao | Rotor Dynamics[END_REF][START_REF] Lalanne | Rotordynamics Prediction in Engineering[END_REF][START_REF] Genta | Dynamics of Rotating Systems[END_REF]. The asymmetry due to the disk, the shaft and/or the bearing was treated in some works. [START_REF] Guilhen | Instability and unbalance response of dissymmetric rotorbearing systems[END_REF] noted that differential equations without periodic coefficients could be obtained either in the inertial coordinate system or in the frame moving with the rotor. The dynamic stability was investigated using the Floquet theory and the mass unbalance response was computed with the Newmark scheme. [START_REF] Kang | Investigation on the steady-state responses of asymmetric rotors[END_REF] employed the Timoshenko beam finite elements (FEs) for modeling rotor-bearing systems by taking into account the asymmetry of disk, shaft and/or bearing. They calculated the mass unbalance response using the harmonic balance method. According to their work, the resonant speeds could change due to various angles between major axes of disk and shaft, the shaft asymmetry as well as the bearing characteristics. [START_REF] Oncescu | Investigation of the stability and steady state response of asymmetric rotors, using finite element formulation[END_REF] generalized a FE procedure for rotor-bearing systems to include the effects of the asymmetry of shaft and/or bearing. They gave the matrices corresponding to the asymmetric shaft FE and applied the time-transfer matrix method based on the Floquet theory to the linear equations of motion with periodic coefficients expressed in the inertial frame of reference. As shown in their study, the indicated method was a convenient way for predicting the dynamic behavior of the asymmetric rotor-bearing systems. [START_REF] Sabuncu | The dynamic stability of a rotating asymmetric cross-section Timoshenko beam subjected to lateral parametric excitation[END_REF] studied the motion stability of an asymmetric blade induced by a lateral parametric excitation using the Timoshenko beam FEs. They discussed the effects of the shear coefficient, the beam length variation, the rotational speed and the coupling due to the center of flexure distance from the centroid on the dynamic instability regions. [START_REF] Beley-Sayettat | Effet des dissymétries et effet sismique en dynamique des rotors[END_REF] studied the influence of the shaft asymmetry and the earthquake on the dynamic behavior of a rotor. In the case of an asymmetric shaft, the equations of motion are developed in the Galilean frame (fixed in space) and in the rotating frame using the Rayleigh-Ritz method and the FE method. Employing the Floquet theory for the stability and the Newmark method for the dynamic response, the behavior of a turbogenerator of aircraft carriers was investigated in the cases of a construction asymmetry (winding) and an accidental asymmetry (crack). In the case of a seismic motion, the equations of motion were derived for the rigid support (mono-excitation) and the flexible support (multi-excitation). A spectral method with complex modes was employed for solving the equations of motion.

Many studies concentrated on the dynamic behavior of a rotor under seismic or shock excitations of its base. [START_REF] Hori | Earthquake-induced instability of a rotor supported by oil film bearings[END_REF] examined a seismic response of a Jeffcott rotor supported by oil film bearings to a real seismic wave and investigated its stability by calculating the locations of the disk and journal centers (using the Runge-Kutta-Gill method). [START_REF] Suarez | Seismic response of rotating machines[END_REF] presented without any derivation the equations of motion of a FE rotor model subjected to six components of base excitations and studied its seismic response. They observed that even for strong rotational inputs, the parametric terms in these equations could be neglected without affecting the rotor response. [START_REF] Gaganis | Modal analysis of rotor on piecewise linear journal bearings under seismic excitation[END_REF] considered the non-linear bearing coefficients to be piecewise constant and modeled the excitation due to the earthquake as a movement of the base of the rotor-bearing systems using the El Centro earthquake data. In their work, the response and the variation of the dynamic properties of the rotorbearing systems were investigated. [START_REF] Bachelet | Rotors under seismic excitation: A spectral approach[END_REF] predicted the dynamic behavior of an asymmetric rotor excited by a seismic translational base excitation and developed an original spectral approach to approximate the rotor response. They discussed the results of the spectral approach through a comparison with the Monte Carlo simulation results. [START_REF] Lee | A finite element transient response analysis method of a rotor-bearing system to base shock excitations using the state-space Newmark scheme and comparisons with experiments[END_REF] proposed a FE rotor model considering a shock base excitation with the state-space Newmark method based on the average velocity concept. They focused also on the experimental behavior of a rotor under this shock base excitation and compared it with the numerical results.

Some other works incorporated the foundation flexibility into the rotordynamics prediction. [START_REF] Kang | An investigation in stiffness effects on dynamics of rotorbearing-foundation systems[END_REF] studied the effect of the flexible support (foundation) on the dynamic characteristics of the rotor-bearing systems using a FE modeling. [START_REF] Edwards | Experimental identification of excitation and support parameters of a flexible rotor-bearings-foundation system from a single run-down[END_REF] verified experimentally a method which identifies both the excitation and the flexible support parameters of a rotor-bearing-foundation system. [START_REF] Cavalca | An investigation on the influence of the supporting structure on the dynamics of the rotor system[END_REF] estimated experimentally the modal parameters of the foundation from frequency response functions and studied the effect of the flexible foundation on the dynamic characteristics of the rotor-bearing systems. The experimental tests of [START_REF] Feng | Rotor-model-based identification of foundations in rotating machinery using modal parameters[END_REF] showed that even with input data truncated to two significant digits, satisfactory identification were possible for a flexibly supported undamped rigid block foundation in rotating machinery.

A few number of works deal with the investigation of dynamic behavior of a rotor during a harmonic motion of its support. [START_REF] Tan | Dynamic stability of a radially rotating beam subjected to base excitation[END_REF] developed the equations of motion of a rotating cantilever beam induced by a base excitation by employing the Euler-Bernoulli beam theory. They used the method of multiple scales for determining the instability regions and presented numerical results to illustrate the influence of the hub radius to length ratio, the steady-state rotational speed and the base excitation frequency on the dynamic stability of the system. [START_REF] Duchemin | Dynamic behavior and stability of a rotor under base excitation[END_REF] developed the equations of motion by employing the Rayleigh-Ritz (RR) method to study the basic phenomena of a simple rotor model under a sinusoidal support rotation. They applied the method of multiple scales to observe the motion stability of the rotor and presented experimental results to validate the quasi-analytical study. [START_REF] Driot | Theoretical and experimental analysis of a base-excited rotor[END_REF] obtained two gyroscopic and parametric coupled equations of motion of a rotor induced by a harmonic rotational base movement using the RR method. They described the orbits using two numerical methods (the seventh-eighth-order continuous Runge-Kutta method and the normal form method). From their work, the comparison between the numerical and experimental results was quite satisfactory. [START_REF] El-Saeidy | Dynamics of a rigid rotor linear/nonlinear bearings system subject to rotating unbalance and base excitations[END_REF] derived the equations of motion of a rigid rotor linear/non-linear bearing system subjected to rotating mass unbalance plus harmonic base excitations along or around lateral directions. In the case of linear bearings, they presented analytical solutions and obtained the system natural frequencies as a function of the speed of rotation of the rotor in addition to the frequency response (maximum amplitude of vibrations with respect to the base excitation frequency). In the case of a bearing cubic nonlinearity, they found results from a time integration scheme and discussed them with regards to the time domain, the fast Fourier transform as well as the Poincaré map. [START_REF] Das | Active vibration control of unbalanced flexible rotor-shaft systems parametrically excited due to base motion[END_REF] investigated the active vibration control of a flexible rotor system excited by a mass unbalance and a periodic rotational motion of the base. They performed the control with electromagnetic control force provided by an actuator placed on the stator in a suitable plane around the rotor. As shown in their paper, a numerical simulation of the flexible rotor system modeled by Rayleigh beam FEs was performed and the active control was successful for avoiding the lateral parametric instability due to the sinusoidal rotation of the base.

Most of the works mentioned above dealing with base-excited rotor systems are based on a simplified modeling. Thus they limit the satisfactory adequacy between the theoretical rotor models and the realistic ones under operating conditions. The most common simplifications concern either the geometry of the rotor (for example, the Jeffcott rotor [START_REF] Hori | Earthquake-induced instability of a rotor supported by oil film bearings[END_REF]), the use of the RR method instead of the FE method for the spatial discretization of the rotor model [START_REF] Duchemin | Dynamic behavior and stability of a rotor under base excitation[END_REF][START_REF] Driot | Theoretical and experimental analysis of a base-excited rotor[END_REF] or the type of base motion (simple rotation or translation).

The present paper proposes an original contribution to improve an on-board rotor model. Namely, a rotor with geometric asymmetry due to the disk and/or the shaft is discretized using the FE method based on the Timoshenko beam theory and excited by combined deterministic base motions (three rotations and three translations). The vibration analysis focuses on the case of bending of the flexible rotor. Eulerian angles are thus used to describe the orientation of disk and shaft with respect to the moving rigid base. The kinetic and strain energies of the rotating flexible rotor components are calculated. Timoshenko beam FEs including the effects relative to the rotary inertia, the gyroscopic inertia, the shear deformation of shaft as well as the geometric asymmetry of disk and/or shaft are considered for the dynamic modeling of the on-board rotor. Applying the Lagrange's equations, the linear second-order differential equations of motion due to the flexural vibration of the rotor relative to the rigid base (which forms a non-inertial coordinate system) are derived. They contain periodic parametric coefficients relative to the geometric asymmetry of some rotor components (disk and shaft) and timevarying parametric coefficients corresponding to the three rotations of the rotor base. These parametric coefficients can be a source of lateral dynamic instability. In the proposed numerical examples, symmetric and asymmetric rotating rotors, whose rigid support is subjected to combined constant angular velocity and harmonic translation, are studied. Their dynamic behavior is analytically and numerically investigated by discussing the influence of the base motions on stability charts, Campbell diagrams, steady-state responses and orbits of the rotor.

Basic description and assumptions

Let Figure 1 sketch the basic components of the rotor: disk (1), shaft (2), mass unbalance (3), bearing (4) and support (5). The mass unbalance is distributed continuously as well as in a generic way and cannot completely be canceled. The excitations taken into account are due to the residual mass unbalance and to the imposed movements of the support. The transverse deflections of the elastic line of the rotor shaft are studied.

Let the assumptions be: the disk is rigid, the shaft is deformable and modeled with Timoshenko beam elements of constant cross-section solicited in bending along two orthogonal directions, the bearing supporting the shaft is rigid, the disk and/or the shaft can be asymmetric, the mass unbalance is modeled by concentrated masses, the support is rigid and mobile, the rotor runs at a constant speed of rotation Ω .

The consideration of the base motion modifies the form of the equations of motion of a rotor in bending compared to those obtained in the case of a fixed base. In order to undertake a modeling as simple as possible, the approach presented by [START_REF] Duchemin | Dynamic behavior and stability of a rotor under base excitation[END_REF] is adopted. Three frames of reference are used to take into account the movement of the rotor with respect to the base and that of the base with respect to the ground.

The derivation of the equations of motion is inspired from the theory described in [START_REF] Lalanne | Rotordynamics Prediction in Engineering[END_REF]. First, the coordinate systems needed to describe the motion of an on-board rotor are defined and the vectors expressing the rotations between them are calculated. Then, the kinetic energy of a disk, the kinetic and strain energies of a shaft element as well as the kinetic energy of a concentrated mass unbalance are calculated. Moreover, the rotating flexible rotor is discretized using the Timoshenko beam FEs. Finally, the differential equations of motion of an on-board rotor in bending are deduced from the Lagrange's equations being applied with respect to the generalized coordinates i q by the following form

, i q i i i d T T U F dt q q q   ∂ ∂ ∂ - + =   ∂ ∂ ∂   ɺ (1) with ( ) 1 df df n i n ≤ ≤
the number of degrees of freedom, T and U the kinetic and strain energies respectively and i q F the external forces corresponding to i q . The symbol " i " refers to the differentiation with respect to time t .

Preliminary calculations

The utilization of more than one frame of reference needs to establish the relationships between the different defined systems. Three principal Cartesian frames of reference shown in Figure 1 are introduced to take into account the mobility of the rotor support:

( ) , , The elastic line of the undeformed shaft is along the Oy axis on which the disk mass center is located. Moreover, xOy and yOz are both planes of symmetry for the disk and the undeformed shaft. Thus Ox , Oy and Oz are principal axes of inertia. )

1 1 1 1 , , i i i i R x y z and ( ) 2 2 2 2 , , i i i i
R x y z . The relationship between the components of any vector V expressed in R and those expressed in l R is described by

, l l R R R R V V = ℜ = ℜ = ℜ = ℜ (2)
l R R ψ φ ψ θ φ ψ φ ψ θ φ θ φ ψ θ ψ θ θ ψ φ ψ θ φ ψ φ ψ θ φ θ φ - + -     = -     + -   ℜ ℜ ℜ ℜ (3)
is the rotation matrix of the rotor l R with respect to its base R . Using the intermediate coordinate systems

( ) 4, the relationship between the components of any vector V projected in g R and those projected in R is expressed as follows

3 3 3 3 , , i i i i R x
, g g R R R R V V = ℜ = ℜ = ℜ = ℜ (4)
R R α γ α β γ α γ α β γ β γ α β α β β α γ α β γ α γ α β γ β γ - + -     = -     + -   ℜ ℜ ℜ ℜ (5)
is the rotation matrix of the rotor base R with respect to the ground g R . Computing the energies of the rotor components requires establishing the angular velocity vectors of the coordinate systems and the position vectors of their origins. The instantaneous angular velocity vector

x R y R z R R ω ψ θ φ θ φ ω ψ θ φ ψ θ φ θ φ ω     - +       = = +         +       ω ɺ ɺ ɺ ɺ ɺ ɺ (7)
As before, the instantaneous angular velocity vector g R R ω

which achieves the orientation of the frame attached to the rotor base R with respect to the Galilean frame g R is expressed in the frame R by cos sin cos sin , cos cos sin

g x R y R z R R ω α β γ β γ ω α β γ ω α β γ β γ     - +     = = +         +     ω ɺ ɺ ɺ ɺ ɺ ɺ (8)
where x ω , y ω and z ω are the components of the angular velocity vector ω presented in equations ( 7) and ( 9), then

x x R y R y R R z z R R ω ω ω ω ω ω           = =               ω ℜ ℜ ℜ ℜ (9)
1 2 1 2 1 2 . l l l g g l l l l l l l l l l l x x x R R R y y y R R R z z z R R R ω ω ω ω ω ω ω ω ω                   = + = + =                         ω ω ω (10)
The absolute position vector g O O of the origin O of the frame attached to the base R expressed in the Galilean frame of reference g R is given by .

g g O g O g O R x y z g O O           =             (11)
Using equations ( 4) and ( 5) expressing the relationship between the frames g R and R , the position vector

g O O is expressed in the frame R as . g g g O O R g O R O g O O R R x x y y z z g O O ℜ ℜ ℜ ℜ                     = =                         (12)
In what follows, the calculations are performed using the rotational ( )

Energy calculations of the rotor components

This section includes formulations for the disk, the shaft and the rotating mass unbalance. The kinetic and strain energies are measured by an observer located at the ground and their terms are written with respect to the frame linked with the rotor rigid base R . The rotatory inertia, the gyroscopic inertia, the shear deformation of shaft as well as the geometric asymmetry of disk and/or shaft are taken into consideration.

Disk

According to the assumptions presented in Section 2, the disk is assumed to be rigid. Therefore only its kinetic energy d T is calculated. This energy is composed of the scalar sum of the absolute translational kinetic energy and the absolute rotational kinetic energy about the disk mass center located at the generic abscissa d y along the Oy axis

( ) ( ) 1 , 2 2 g g g g l l l l d d d T T R R R R d d m O O R R m T = + v v ω I ω (
    +     = =         -         I (14)
where the superscript T is the matrix transposition symbol, d m is the mass of the disk,

O d d O O d d O O d d O R R R x u u x y y y y z w w z +             = + = + = +             +       g l g l d d O O O O OO (15)
Using the transport theorem relating the time derivative of a vector measured from a fixed frame to that of the same vector but measured from a moving frame, the derivation of the position vector

g l d O O yields , g g g l d R R R R R O d d dt dt = + ∧ = g l g l g l d d d O O O O ω O O v (16) then ( ) ( ) ( ) ( ) ( ) ( )
, ω for the disk, the expression of its kinetic energy becomes

l d g l l d d l d y z O d O d O d O R x z O d O d O O O x y d O d O d O R O R u u x w z y y y w z u x v w z y y u x w ω ω ω ω ω ω     + + + - +       = - + + + =         + + + - +       v ɺ ɺ ɺ ɺ ɺ ɺ ɺ ɺ (17)
( ) ( ) ( ) ( ) 2

Shaft

The kinetic energy of the shaft element can be obtained by taking a shaft elementary volume which can be considered as a disk of very small thickness dy . ω , the expression for the kinetic energy sh T of the shaft element, whose end-points have the abscissas 1 y and 2 y along the Oy axis, is written in the following form The rigid base motion relative to the ground has no influence on the strain energy of the shaft because the latter depends only on the stresses and therefore on the transverse deflection of the shaft with respect to the rotor base R . In addition to the bending deformation, the shear effects highlighted by Timoshenko and the secondorder (non-linear) terms of the strain field are accounted. The non-linear strains are linked with the centrifugal stiffening which is proportional to the square of the base angular velocity around the transverse axes and which corresponds to the geometric potential energy. Thus the strain energy sh U of the shaft element is defined by

( ) ( ) ( ) ( ) 2 2 2 2 1 1 1 2 2 2 2 2 2 2 2 1 1 2 ,
2 2 2 1 1 1 2 2 2 2 2 2
κ ψ θ ψ θ ψ θ κ ψ θ                    ∂ ∂ ∂ ∂ ∂ ∂        = + + + + - - -                    ∂ ∂ ∂ ∂ ∂ ∂                        ∂ ∂ - + - -     ∂ ∂     ∫ ∫ ∫ ( ) ( ) ( ) 2 2 1 1 2 2 1 1 2 2 2 2 2 2
ψ θ ρ κ ψ θ ω ω     ∂ ∂    Ω -    ∂ ∂               ∂ ∂ ∂ ∂   - + - Ω + + - + +           ∂ ∂ ∂ ∂           ∫ ∫ ∫ ∫ (22) with ( ) ; ; , 2 1 2 2 x z x z mo di sh sh sh sh sh sh sh sh sh E G κ κ κ κ κ κ ν + - = = = + (23)
where sh E , sh G and sh ν are respectively the Young's modulus, the shear modulus and the Poisson's ratio of the isotropic shaft material. mo sh κ and di sh κ are respectively the mean shear correction factor of the cross-sectional area and that relative to the section asymmetry of the shaft.

Mass unbalance

The mass unbalance is modeled by a concentrated mass mu m placed at a point mu P with a distance mu r from the geometric center of the cross-sectional area of the shaft. Its initial angle with the Oz axis of the frame R at rest is mu η . The mass unbalance remains in a plane perpendicular to the Oy axis with a constant abscissa mu y along the Oy axis relative to the base R . Its kinetic energy mu T is expressed as follows

( ) . 2 g g mu mu T R R mu mu P P m T = v v (24)
The translational velocity vector

 + + Ω +    = + + = +     + + Ω +   g g l l mu mu mu mu O P O O OO O P (25)
Applying the transport theorem, the time derivative of the position vector g mu O P is of the following form ,

g g g mu R R R R R P d d dt dt = + ∧ = g g g mu mu mu O P O P ω O P v (26) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (
η η ω ω η ω η ω η ω η ω   + + Ω Ω + + + + Ω + - +     = - + + Ω + + + + Ω +     + -Ω Ω + + + - + + Ω +     v ɺ ɺ ɺ ɺ ɺ (27)
The external force vectors obtained by the application of the Lagrange's equations to the kinetic energy of the mass unbalance contain the so-called "centripetal and centrifugal force vectors", harmonic tangential force vectors and harmonic force vectors fixed in space with respect to the base R . Thus the excitation due to the mass unbalance combined with the base rotations is sinusoidal and has a frequency equal to the speed of rotation Ω of the rotor.

Equations of motion of an on-board rotor

Finite element procedure

The FE analysis is selected to discretize the rotor in bending. Therefore each node of nodal interpolation contains four degrees of freedom (two lateral translations and two rotations). The disk and the mass unbalance are modeled by one-node FEs situated at the geometric center of the cross-sectional area of the shaft. On the other hand, the shaft is modeled by esh n two-node beam FEs based on the Timoshenko beam theory and thus has κ where ( )

1 esh i i n ≤ ≤
is the shaft FE number. Its nodal displacement vector i n esh δ expressed in the coordinate system attached to the rotor rigid base R is given by

1 1 1 1 , , , , , , , . i
i i i i i i i i i i i i i i i i T n n n n n n n n n esh esh esh esh esh esh esh esh esh R u w u w θ ψ θ ψ + + + + = δ (28)
The above vector is separated into two vectors ,

i i i i i i i i i i i i i i i i i i T T n n n n n n n n n u n w esh esh esh esh esh esh esh esh esh esh R R u u w w ψ ψ θ θ + + + + = = δ δ , , , ; , , , .
The lateral displacements

( ) ( ) ( ) ( ) , , , , ; , ;
, The Lagrange's equations are applied to the energies for the disk, the shaft FEs as well as the mass unbalance (see equations ( 18), ( 20), ( 22) and ( 24)) and the resulting elementary vectors and matrices are appropriately assembled. In order to deal with the boundary conditions bc n , the corresponding row and column entries of the assembled vectors and matrices are set to zero and eliminated. Thus the obtained global vectors and matrices are respectively of dimension 1 g n × and g g n n × where ( )

i i i i i i i i i i i i u n u n u esh esh esh esh esh esh w n w n w esh esh esh esh esh esh u y y w y y ψ θ ψ θ = = = = F δ F δ F δ F δ (
4 +1 g esh bc n n n = -
is the total number of degrees of freedom after canceling those relative to the boundary conditions bc n . The obtained linear second-order differential equations describing the behavior of the FE on-board rotor in bending are written with respect to the non-inertial frame connected to the rigid base R as follows

( ) ( ) ( ) ( ), r r r r r r r t t t t + + = M δ C δ K δ F ɺɺ ɺ (31)
where r δ , r δ ɺ and r δ ɺɺ are the global displacement, velocity and acceleration vectors conforming to the connectivity of the FEs.

( )

r t M , ( ) r t C and ( ) r t K
are the global mass, damping and stiffness matrices with time-varying parameters due to the geometric asymmetry of the rotating rotor and to the rotations of its moving base. These matrices are not affected by the translational motions of the rotor base. Lastly, ( )

r t F
is the global external force vector including the excitations due to the influence of the mass unbalance as well as to that of the rotational and translational base motions. In what follows, two different rotor configurations (symmetric and asymmetric) are to be treated and their governing differential equations for the transverse vibrations are presented. The following abbreviations and terms used in the equations are detailed hereafter. The subscripts " d ", " sh " and " b " refer to the disk, the shaft as well as the base respectively and express the contribution to the phenomena represented by the corresponding matrix. The superscript " id " stands for the rotor internal damping introduced by estimating the Rayleigh damping coefficients, " g " for the rotor gyroscopic effect, " e " for the shaft elasticity corresponding to the bending and shear deformations, " re " for the rotational effects due to the base rotations (these effects come from the kinetic energies of the disk and the shaft) and " gse " for the geometric stiffening effects associated with the centrifugal stressing due to the base rotations (these effects come from the strain energy of the shaft). In addition, the superscripts " c " and " s " denote the coefficients of the timevarying trigonometric (cosine and sine) functions. The load vectors mu V ,

F

are associated respectively with the mass unbalance, the inertia force due to base motions and that due to coupling between both phenomena.

Symmetric rotor

The disk and the shaft are symmetric. Moreover, the rotor is excited by a mass unbalance and base motions which consist of a constant angular velocity x ω (or y ω ) around the Ox axis (or the Oy axis) and a harmonic translation along the Oz axis given by sin

+ = Ω + Ω + Ω + Ω + Ω + Ω + M δ C δ K K K δ F F F F F F F ɺɺ ɺ (32)
where ( )

2 2 2 2 , 2 , 2 , 1 , , 1 , 2 2 0 , , 1 , , 1 , , , , , , ; ; ; , ;
; .

ω = Ω = Ω = = = = Ω + = - Ω F V F V F V F V F 0 F V F V (33) ( ) ( ) 2 , , , 2 , , , , , , , , , 2 , 2 ,
+ Ω + + + Ω + = Ω + Ω + Ω + Ω + Ω + Ω M δ C C δ K K K δ F F F F F F ɺɺ ɺ (34)
where

( ) 2 2 , , 2 , , 2 , 2 , , , 2 , , 2 2 , , 2 , , , , 2 , , ; , 2 ;
.

Z Z ω ω ω ω ω ω ω ω ω ω Ω Ω = Ω + = Ω + = - Ω = Ω + F V V F V V F V F V (35)
In equations ( 33) and ( 35

F

is a constant force vector.

As x ω and z ω have similar influences on rotordynamics, only x ω is considered in this study.

Asymmetric rotor

The disk is assumed to be asymmetric, while the shaft is symmetric. The rigid base and thereby the rotor are subjected to the same excitations as indicated in Section 5.2. However, the left-hand side of the linear equations of motion contains periodic parametric coefficients and their right-hand side contains load vectors multiplied by trigonometric functions due to the geometric asymmetry of the rotor. Thus the equations containing the Rayleigh damping are solved by a numerical time integration method. The equations of motion (31) are rewritten in equation ( 36) due to the motion x ω and equation ( 38) due to the motion y ω (

+ Ω + Ω + + Ω + Ω Ω + Ω Ω + + + + Ω + Ω = Ω + Ω + M M M δ C C C C δ K K K K K δ F F F ɺɺ ɺ 0 , 1 , , 1 , , 1 , , , , 3 ,
ψ θ ω ω = Ω = Ω F V F V (37) ( ) ( ) ( ) ( 2 2 , , , , , , , , , , , , 2 , , , , 2 , , , , , , , , , , , , ,
+ Ω + Ω + + Ω + Ω Ω + Ω Ω + + + Ω + + Ω + Ω + M M M δ C C C C C δ K K K K K K ɺɺ ɺ ( ) ) 2 , , 2 , , , 2 , 2 ,
+ Ω = Ω + Ω + Ω + Ω + Ω + Ω K δ F F F F F F (38)
The elementary vectors and matrices involved in equations ( 32) to (38) are presented in Appendix A.

Dynamic analysis

Deterministic rotational and translational excitations of the rotor base have been stated in the previous section for two rotor configurations. The first one is a symmetric rotor and treated through analytical methods to seek its natural frequencies and to compute its steady-state responses and its orbits. The second one is an asymmetric rotor and examined through the Floquet theory to evaluate its dynamic stability as well as the Newmark method to predict its steady-state orbits in the time domain.

Equations with constant coefficients

In this case, the system of linear differential equations (31) contains constant matrices (according to equations ( 32) and ( 34)). It is transformed into a first-order differential equation by introducing the state-space vector r D , i.e.,

) -1 -1 -1 ; ; . r r r r r r r r r r r t       = = =       - -       0 δ 0 I D A B M F δ M K M C ɺ (40)

Natural frequencies

The rotor is studied in free motion and the homogeneous system is thus considered, i.e., equation (39) with no external forces. The eigenvalues of the matrix r A are calculated for seeking the natural frequencies r f of the rotor. Therefore the Campbell diagram can be plotted to represent the frequencies r f as a function of the speed of rotation Ω of the rotor.

Steady-state responses and orbits

Since the steady-state responses are sought, only the particular solution of the equations of motion is considered. Due to the form of the external force vectors in the right-hand side, this solution adopts the following

( ) ( ) ( ) ( ) 0 cos sin cos sin , c s c z s z r mu mu b b b t t t t = Ω + Ω + Ω + Ω + δ δ δ δ δ δ (
  +     -Ω Ω     =       -Ω -Ω +           F F δ K M C C K M δ F F (42 ) 2 , , 2 , ,
,

c c z z d sh b b r r r z z s s r r r b d sh b       -Ω Ω     =       -Ω -Ω           F δ K M C C K M δ F (43) 0 -1 0 , , . b r d sh b = δ Κ F (44)
The constant vector 0 b δ is nil in the case of the angular velocity y ω , i.e., of the base motion around the Oy axial axis (see equation 34) but not in the case of x ω , i.e., of the base motion around the Ox transverse axis (see equation 32). It is concluded that equation ( 42) is used to investigate the mass unbalance response of the rotor, equation ( 43) is employed to observe the frequency response of the rotor to the translational base motion and equation ( 41) helps to illustrate the steady-state orbits described by the points situated along the shaft elastic line.

Equations with time-varying coefficients

In this case, the system of linear differential equations (31) has periodic parametric coefficients (according to equations ( 36) and ( 38)) of period τ π = Ω .

Dynamic stability

The stability of the periodic solution is predicted by employing the Floquet theory. In other words, by determining the so-called "Floquet transition matrix" (or monodromy matrix) during one period of the parametric excitation and by finding its eigenvalues, the zones of instability are identified, see [START_REF]Resonance Oscillations in Mechanical Systems[END_REF], [START_REF] Dufour | Parametric instability of a beam due to axial excitations and to boundary conditions[END_REF] as well as [START_REF] Berlioz | Bifurcation in a nonlinear autoparametric system using experimental and numerical investigations[END_REF]. First, the application of the Floquet theory requires a homogeneous differential system represented in the state-space form. Namely, the linear second-order differential equations (31) of the rotor become homogeneous and have to be transformed into a first-order system . r r r

= D A D ɺ (45)
Then, the system of equation ( 45) of the rotor is integrated over one period τ π = Ω using the Runge-Kutta time-step integration scheme with the following initial conditions at time 0 t = ( )

0 0,0, ,1, ,0 , T r = D … … (46)
where only the -th i component is not nil and equal to 1. For each integration, the vector 45) is unstable if the modulus of at least one of the characteristic multipliers is greater than 1.

Time history responses and orbits

In the presence of time-varying parametric coefficients, the linear equations of motion (31) are solved using the Newmark time-step integration algorithm based on the average acceleration to predict the rotor orbits when the steady-state responses are reached, i.e., the integration algorithm requires the presence of the damping to overcome the transient regime.

Simulation results and discussion

Symmetric rotor

Configuration and data

Figure 5 and Table 1 present the components and the characteristics of the symmetric on-board rotor. The origin O of the reference frame R coincides with the shaft end-point positioned at the center of the bearing # 1. The symmetric shaft is modeled with 12 identical two-node Timoshenko beam FEs. The symmetric disk is located at node 5. The rigid short bearings # 1 and # 2 generate a pinned-pinned rotor and are located at nodes 1 and 13. Thus the corresponding degrees of freedom are canceled. The mass unbalance is placed on the disk. The investigated rotor runs at a constant speed of rotation Ω and is subjected to mass unbalance excitation as well as to combined constant rotation and harmonic translation of the rigid base: the constant rotation around the Ox axis (or the Oy axis) is given by x ω (or y ω ), while the harmonic translation along the Oz axis is expressed as

sin z O O z Z t =
Ω . The steady-state dynamic analysis is performed by means of Campbell diagrams, mass unbalance responses, responses to the harmonic translational base excitations and rotor orbits. In addition, the equations of motion of the on-board rotor are parametrically affected (i.e., the left-hand side of equations ( 32) and ( 34)) by the base rotations, while the base translations have only an effect on the external force vector (i.e., on the right-hand side). Therefore the overall behavior is analyzed as a function of the base rotation.

Campbell diagrams

In the case of a symmetric rotor, the linear equations of motion always display constant coefficients, see equations ( 32) and (34). It should be mentioned that a fixed base or a rotating base around the Oy axis keep the system isotropic, while a base rotation around the Ox axis makes it anisotropic. Moreover, the sign of the components of the damping and stiffness matrices depends on the direction of the base rotation around Oy but not on that of the base rotation around Ox . In the case of a positive direction (counterclockwise) of the base rotation around Oy and any direction around Ox , the symmetric stiffness matrices can lose their positive definiteness because of the presence of negative diagonal terms containing the angular velocity x ω or y ω . Thus a possible dynamic instability of the rotor can occur. In the presented application, since the eigenvalues of the state-space system (39) with no external forces are purely imaginary quantities whatever the constant angular velocity of the base rotations in the zone of interest . The natural frequencies r f being deduced from these eigenvalues depend on the speed of rotation Ω of the rotor and on the angular velocity of the base rotations either around Ox or Oy . They are represented by the Campbell diagrams plotted in Figure 6 for the rotor subjected to the angular velocity either x ω or y ω . When the rotor base is fixed ( 0x y ω ω = = ), the system is isotropic and therefore the dynamic behavior of the rotor is symmetric, i.e., the natural frequencies of each pair of backward and forward whirl modes are equal at rest ( 0 Ω = ). Even if the natural frequencies strongly do not change, the base rotation x ω breaks the symmetry of the rotor behavior, see Figure 6(a). Since the corresponding stiffness matrices are symmetric with anisotropic terms and there is no damping matrix associated with x ω , the zoom shows that the natural frequencies of the first backward and forward whirl modes at rest are non-equal, this justifies that the behavior of the rotor is non-symmetric. The rotational effects due to

x ω decrease the natural frequencies, while the geometric stiffening effects associated with the centrifugal stressing relative to x ω increase them. Unlike the previous case, the solicitation due to the base rotation around Oy does not bring any asymmetry to the dynamic behavior of the rotor (because the corresponding matrices are symmetric and skew-symmetric with isotropic terms) but shifts and changes the natural frequencies (due to additional terms in the damping and stiffness matrices), see Figure 6

Responses to the mass unbalance or to the translational base excitation

The rotor response due to the mass unbalance effect can be related to the constant rotational base motions, see equations ( 32) to (35). Figure 7 ω increases because the backward and forward whirl modes at rest ( 0 Ω = ) are reversed. It is also observed that even if the rotor is at rest, the amplitudes of the mass unbalance response are not nil due to the presence of the base rotation either around Ox or Oy . In addition, these amplitudes become larger when the angular velocity of the base rotation increases, see Table 3. At the highest speeds of rotation of the rotor, the corresponding response amplitudes reach the limiting values which are roughly equal to 12.431×10 -6 m for x ω and 11.512×10 -6 m for y ω .

Under combined motions of the base, the rotor response due to the harmonic translational base motions depends on the constant rotations of the base, see equations (32) to (35). Figure 8

Influence of the base excitations on the orbits

In this section, the orbits of the symmetric rotor subjected successively to a translation, a rotation and combined rotation and translation of the base are calculated with the finite element (FE) method. For validation purposes, a comparison is performed with results obtained by the Rayleigh-Ritz (RR) method which is based on the work of [START_REF] Duchemin | Contribution à l'étude du comportement dynamique d'un rotor embarqué[END_REF] and which has been extended in the present paper to account for a pure rotation and combined motions of the rotor base. The FE model is based on 48 degrees of freedom, while the RR method uses only one classical trigonometric function as a displacement function, i.e., two generalized coordinates ( ) 1 q t and ( )

2 q t 1 2 sin ; sin . sh sh y y u q w q l l π π     = =        
(47)

Pure translation base and validation against the Rayleigh-Ritz method

The orbits presented in Figure 9 . Figure 9(a) compares the orbits obtained with the RR approaches and no deviation can be observable. (=25 Hz=the mass unbalance frequency) and a fixed base. Since the matrices of the rotor system are symmetric and skew-symmetric with isotropic diagonal and cross-coupling components, the dynamic behavior is symmetric and the orbit is circular and periodic of period equal to the inverse of the mass unbalance frequency Ω . The orbit center coincides with the point O (bearing center) because the bearings are rigid and the motion amplitude of the rotor in both Ox and Oz directions is 3.628×10 -6 m at its middle. Figures 10(b , there is no such a static position and the orbit center remains at the bearing center O . The periodic orbit keeps a circular shape because the symmetric dynamic behavior of the rotor is never broken, while its radius representing the x and z vibration amplitudes becomes greater than that obtained for a fixed base because the mass unbalance excitation increases in a way more important than the stiffness matrix of the rotor. Lastly, this radius is 6.183×10 -6 m at the middle of the shaft, i.e., the amplitude increases by a factor of about 1.7 in both transverse directions. The RR and FE mass unbalance orbits are qualitatively similar and show quantitatively very low x and z magnitude deviations:

5.75% 32) and ( 34)), the rotor orbits contain the response to the mass unbalance excitation superimposed with the response to the rotational and translational base motions. Thus the orbit characteristics (shape and size) depend on both excitations. In addition, the steady-state orbits are periodic because the external excitations due to both the mass unbalance and the base translation are sinusoidal and commensurable, see the solution in equation ( 41). The period of the orbits is equal to the inverse of the greatest common factor for the mass unbalance frequency Ω and the base translation frequency ). The frequency of the harmonic translational base excitation changes extremely the shape of the orbits, while it modifies slightly their magnitudes (when compared to those shown in Figure 11). As stated previously, the orbit period is equal to the inverse of the greatest common factor of the mass unbalance frequency Ω and the base translation frequency z Ω . Namely, one period of the above orbit corresponds to one period of the mass unbalance excitation and to four periods of the base translation for 100 Hz z Ω = , while one period of the below orbit corresponds to five periods of the mass unbalance excitation and to 24 periods of the base translation in the case where 120 Hz z Ω = . It should be noted that for several base translation frequencies different from the mass unbalance frequency, the orbit shapes become more complicated with respect to those obtained when the base is subjected to the pure rotation, see Figures 10(b) and 10(c). In Figure 13

O C sp O

unbalance orbits for a base excited by a pure rotation, see Figure 10. The base translation has a large influence on the z orbit magnitudes but a slight influence on the x magnitudes.

The RR and FE orbits shown in Figure 14

Configuration and data

The description of the asymmetric rotating rotor is similar to that in Section 7.1 but the disk is asymmetric and its geometric properties change. The shape of the disk presented in Figure 15 is defined as a function of the angle d θ with the 0 C x axis of the frame R . The rotor data are the same as given in Table 1. The rotor is subjected to the same excitations as indicated in the previous section. The shaft is discretized into 12 identical two-node Timoshenko beam FEs. The disk as well as the rigid short bearings # 1 and # 2 are located at nodes 5, 1 and 13 respectively. Thus the degrees of freedom of the bearing nodes are eliminated. The linear differential equations of motion have time-varying periodic parametric terms. In order to treat them and predict the steady-state behavior, the Runge-Kutta and Newmark time-step integration methods are used. The proportional Rayleigh damping . The dynamic analysis is performed by means of stability charts, rotor orbits and fast Fourier transforms (FFTs). In addition, the equations of motion of the on-board rotor (see equations ( 36) and ( 38)) show that the base rotations have a parametric influence on these equations, while the base translations have only an influence on the external force vector. Therefore the overall behavior is evaluated as a function of the base rotation.

Stability charts

The instability coming from the parametric excitation due to the geometric asymmetry of the rotor can be related to the base rotations, see equations ( 36) and (38). For a certain angular velocity of the rotor base, two factors affect the parametric excitations generated in the asymmetric rotor: the speed of rotation Ω of the rotor and the moment of inertia The stability chart plotted in Figure 18(a) is predicted with the RR method. Figure 18(b) compares the bounds of the primary instability regions calculated by the RR and FE methods. As can be seen in this figure, the bounds predicted by the RR method are 4-7% greater than those predicted by the FE method and the FE instability region is narrower.

Orbits

Pure rotation ω

x (or ω y ) of the base Since the equations of motion are linear with periodic parametric coefficients (see equations ( 36) and ( 38)), the steady-state response to the mass unbalance is superimposed with the steady-state response to the base excitations. Therefore the orbit characteristics are related to both excitations. Figure 20 presents the orbits and their FFTs at the middle of the asymmetric rotor (node 7) for 45 , the frequency of the harmonic base translation has a limited influence on the magnitudes of the orbits. In addition, the stabilized orbits have shapes more complex than those obtained for the symmetric rotor (see Figure 12) since they involve frequency components due to the mass unbalance excitation (25 Hz), to the harmonic base translations (100 Hz, 120 Hz and 200 Hz) and to the combination of frequencies of both excitations as shown in FFTs of Figure 20.

The RR and FE orbits seen in Figure 21 have the same shape but quantitatively are different. The x and z magnitude deviations are 36.74% . The observed deviations are due to the fact that the RR expansion is based only on two generalized coordinates, while the FE model is built with 48 degrees of freedom, i.e., the FE model is more flexible because it uses more degrees of freedom than the RR model.

Conclusions

The dynamics of two different rotor configurations (symmetric and asymmetric), whose rigid base is subjected to combined constant rotation and sinusoidal translation, are analytically and numerically investigated. A new finite element model based on the Timoshenko beam theory is presented to analyze the steady-state dynamic behavior of the on-board rotor mounted on rigid bearings. It contains the effects associated with the rotary inertia, the gyroscopic inertia, the shear deformation of shaft, the geometric asymmetry of disk and/or shaft as well as six types of deterministic motions (rotations and translations) of the rotor rigid base. Moreover, the rotational effects and the geometric stiffening effects relative to the centrifugal stressing caused by the base rotations are taken into account. The derivation of the equations of motion has shown that the base rotation brings not only a parametric contribution to them but also additional external force vectors, while the base translation only adds external force vectors.

In the case of the symmetric rotor, the equations of motion always are ordinary differential equations with constant coefficients. A dynamic instability can occur in the rotor due to the presence of negative diagonal terms containing the angular velocities of the base in the symmetric stiffness matrices which can lose their positive definiteness. It is shown that the base rotations change not only the natural frequencies of the rotor but also can break the symmetry of the dynamic behavior in the case of the rotation around a transverse axis. The sinusoidal base translation does not change the natural frequencies of the rotor. It has only an influence on the corresponding response amplitudes as well as on the magnitude and the shape of the rotor orbits.

The asymmetric rotor introduces time-varying periodic parametric coefficients in the linear equations of motion. Thus possible regions of instability of the rotor can occur and their sizes cannot be related to the base rotations. On the other hand, the base rotations shift the instability regions along the speed of rotation of the rotor. The sinusoidal base translation has no influence on the instability regions but only an influence on the magnitude and the shape of the rotor orbits.

The finite element calculations have been validated against the Rayleigh-Ritz formulation. Using the finite element method is useful because it is more accurate and can be used for industrial rotating machinery modeling.

i i i i ω ω ω = = = = = + = + = + = + ∑ ∑ ∑ ∑ C C C C C 0 C C 0 C C C (49) 2 2 2 2 2 2 2 2 , , , , , , , , , , , 1 1 1 , , , , , , , , , , , , , , 1 1 ;
i i ω ω ω ω ω ω ω ω ω ω ω = = = Ω Ω Ω = = = = + = = + = + ∑ ∑ ∑ ∑ ∑ K K K K K K K K K K K K K 2 2 2 2 2 2 , , , , , , , , , , , , , , , , , , , , , 1 1 1 , , , , , , , , , , , , , , 1 1 ;
= + = + ∑ ∑ ∑ ∑ ∑ K K 0 K K 0 K K 0 K K 0 K K 0 2 2 , , , , , , , 1
.

ω ω = = + ∑ K K 0 (50) 2 2 2 2 2 2 , , , , , , , , 1 1 1 1 , , , , , , , , , , , , 1 1 =
+ = + = + ∑ ∑ ∑ ∑ ∑ ∑ V V 0 V V 0 V V 0 V V 0 V V 0 V V 0 V V 0 2 2 , , , , 1 1 , , , , , , , , , , , , , , , , , , , ,
+ + + = + ∑ ∑ ∑ ∑ ∑ ∑ V V 0 V V V V V V V V V V V 0 V V 0. sh ∑ (51)
The symbol ∑ stands for the assembly operator after following an appropriate globalization way. The elementary matrices and vectors for the disk are defined by 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ; ; ; . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

                        = = = =       -           - -               M M M M (
I ω                   = = =       - - -       -               -  =  - - -  C C C C .          (53) ( ) 2 2 2 , , , , , , , ,
I ω ω ω ω Ω -             - -       = = =       -       - - -             = K K K K 2 2 , , , , , , , , , , , , 0 0
Ω                   = =             - -                   = =         K K K K 2 , , , 0 0 0 0 0 0 0 0 0 0 0 ; . 0 0 0 2 0 0 0 0 0 2 0 0 0 0 y d d d d re s di di ed b m m di di m m I I I I ω             =     - -     - -         K (54) , , , , , , , 0 0
ψ ψ θ                               = = = = =                                         V V V V V (55)
The elementary matrices and vectors for the shaft are given by 2 ,1 ,

= = = = + M M M M C M C M K M K M M K ( ) ( ) 2 2 2 2 , , ,8 ,9 ,10 , , , ,2 , ,1 ,2 , =
ρ ρ ρ ρ ρ ρ ρ Ω - - - = - - = M M M K M K M M V V ,1 , , 2 , ,3 ;
; =2 . i i i i i i i esh i i w y mo h esh b esh esh esh esh b esh S esh S I ψ ρ ρ = V V V V (56)
( )

2 0 1 2 ,1 ,1 ,1 ,1 2 2 2 2 2 2 2 2 2 2 0 ,
  = + +     +   - - - = M M M M M 2 2 2 2 2 2 2 2 2 2 1 ,
l M l l l l l υ υ υ ϑ ϑ ϑ ϑ υ υ υ             - -       -           - - = M 2 2 2 2 2 2 2 2 2 2 2 2 2 ,1 0 2 0 0 9 2 0 2 9 0 0 2 , 84 0 0 11 84 11 0 2 0 2 i i i i i i i i i i i i i i i i i i i i i i i i i sh esh
esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh

esh l l l l l l l S l Y l M l υ υ υ υ ϑ ϑ ϑ ϑ ϑ υ υ υ ϑ         -     - -       -           M 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
i i i i i i i i i i i i i i i i i i i i i i i i i i i
esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh l l

l l l l l l l l l S ϑ ϑ ϑ ϑ υ υ υ υ υ υ υ ϑ ϑ ϑ ϑ ϑ υ - - - - - = 2 2 2 2 2 2 . 5 0 0 i i i i i i i esh esh esh esh esh esh l Y l M l υ υ ϑ                  -           (57) ( ) ( ) 0 1 2 2 2 ,2 , 2 ,2 , 2 2 2 2 2 2 2 2 2 2 2 0 , 2 1 5 5 , 30 1 36 0 0 3 36 0 0 3 36 3 0 0 36 3 0 4 0 0 3 0 4 i i i i i i i i i i i i i i i i i i i i i i i i i z z esh
esh esh esh esh esh esh esh x esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh

esh ea l l l l l l l l l l l ϕ ϕ ϕ υ υ υ υ υ υ υ = + + + - - - - - - = M M M M M 2 2 2 2 2 2 2 2 2 2 2 2 1 , 2 3 0 0 , 36 0 0 3 36 3 0 4 0 4 0 0 0 3 0 0 0 3 0 3 0 0 0 3 0 0 0 3 0 3 0 i i i i i i i i i i i i i i i i i i i i i i i
esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh

ea l l l S l Y l M l l l l υ υ υ ϑ ϑ υ υ υ υ υ ϑ ϑ             -       -           - - - - = M 2 2 2 2 2 ,2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 ; . 0 0 0 3 0 0 0 0 0 0 0 0 3 0 2 0 0 2 i i i i i i i i i i i i i i i esh esh esh esh esh esh ea esh esh esh esh esh esh esh esh l S S Y Y l M M l υ υ ϑ ϑ ϑ ϑ υ υ υ ϑ ϑ                       -     =     -                           M (58) ( )( ) 0 1 2 2 ,3 ,3 ,3 ,3 2 2 2 0 ,3 5 1 5 , 2 30 1 1 0 36 3 0 0 36 3 0 0 0 3 36 0 0 3 0 4 3 0 0 0 0 3 0 0 3 i i i i i i i i i i i i i i i i i i i i i i z esh esh
x z esh esh esh esh esh esh esh x z esh esh esh esh esh esh esh esh esh esh esh esh esh

l l l l l S l l K l l l E l l W ϕ ϕ ϕ ϕ ϕ   = + +     + +   - - - - - - - - - - = - M M M M M ( ) ( ) ( ) ( ) 2
l M S K l l E l W S Y l M ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ                       -               - + +    - +  =  -   -   - +     M 2 ,3
, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 1 0 0 0 1 0 . 0 0 0 0 0 0 0 0 2 0

i esh S K E W S Y M                  - -     =         -       M (59) ( )( ) 0
l l l S l l K l l l E l l W l ϕ ϕ ϕ ϕ ϕ   = +     + +   - - - - - - = - M M M M M ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2
i i i i i i i i i i i i i i i i i i i i i esh
S l Y l M l l S l l K l l l E ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ ϑ                   -           + + + - + + + - + + - + - + = M ( ) ( ) ( ) ( ) ( ) ( ) 2
i i i i i i i i i i i i i i i i i i esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh ea esh l l W l S l Y l M l l S l l K l l l E ϑ ϑ ϑ ϑ ϑ ϑ               - + +     + - +     - +     +         - - - - - - = M 2 2 0 . 0 40 5 0 0 0 5 0 0 i i i i i esh esh esh esh esh l l W l S l Y l M                 -   -           (60) ( ) 2
 = + +     +   - - - - - = M M M M M 2 2 2 2 2 2 2 2 2 2 2 2 2 1 ,5 6 0 0 2 , 12 0 0 6 12 6 0 4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 2 0 2 0 0 0 2 0 0 0 0 0 0 0 2 i i i i i i i i i i i i i i i i i i
i sh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh

esh es l l l S l Y l M l l l l l S Y l υ υ υ υ υ ϑ ϑ                   -           - - = M ( ) ( ) 2
M l l l l l l l l υ ϑ ϑ ϑ ϑ ϑ υ υ υ υ ϕ υ υ ϕ                             - - - - + - - = M ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 3 6 0 0 3 . 12 0 0 6 12 6 0 3 0 3 i i i i i i i i i i i i i i i i i i i i i i
h esh mo mo esh esh esh esh esh esh esh esh esh esh esh esh esh esh mo esh esh esh mo esh esh esh

l l l l S l Y l M l υ ϕ ϑ ϑ ϕ ϑ ϑ ϑ υ υ ϕ υ ϕ ϑ               + -       -     +     +     (61) ( ) 2 0 1 2 ,6 ,6 ,6 , 6
  = + +     +       - - -     - -     =    -   -     M M M M M 2 2 1 ,6 2
           - - -     - -     =       -     -      
                  -     -       (62)
= + + + -           - -   =   - -           -     M M M M M 1 , 7
-           -   =             -             - -   =             -   M M (
                            - - - - = M
                            (
            -     -             - - = M 2 2
  - -   - -   -   -   .                   (
            -     -             - - - - = M 2 2 2 2
        - -     -     -             M 2 2 2 2 2 2 2 2 2 2 2 2 2
i i i i i i i i i i i i i i i i i i i i i i i i i i i
esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh esh

                            (
                - -         = = = =                                 = V V V V V 2 2 , ,
                - -         = = =                                 V V V (69)

  z is the Galilean frame of reference fixed to the ground,( ), , R x y z is the non-inertial frame of reference attached to the moving rigid support, z is the noninertial local frame of reference (moving with the rotor during its operation) attached to the center of mass of the rigid disk or the geometric center of a section of the deformable shaft. The origins of the frames of reference g R , R and l R are g O , O and l O respectively.

Figure 1 .

 1 Figure 1. Mechanical components and frames of reference of the on-board rotor model. Let us consider an arbitrary point 0 C along the elastic line, i.e., it represents the geometric center of the undeformed shaft. Its coordinates in the frame R are ( ) 0, , 0 y . Let its lateral displacements along the Ox and

(

Figure 2 .

 2 Figure 2. Lateral displacements of the rotor in bending.

Figure 3 .

 3 Figure 3. Passage from the frame linked with the rotor base R to the local frame R l .

  base R with respect to the ground g R projected in the local frame l R . Their expression is obtained by equations (2) and (3) defining the relationship between the frames R and l R

Figure 4 .Finallyω

 4 Figure 4. Passage from the Galilean frame R g to the frame linked with the rotor base R.Finally, the instantaneous angular velocity vector

  due to the mean moment of inertia of the disk mass and those due to the moment of inertia modeling the geometric asymmetry of the disk. Thus in the presence of an asymmetry, the moments of inertia the arbitrary abscissa d y relative to the frame R . The latter is observed from the ground and expressed in the frame attached to the rotor base R by .

  are respectively the density, the cross-sectional area and the length of the shaft element. The mean moment of inertia of the cross-sectional area and the inertia characterizing the asymmetry of the shaft are

  latter is measured from the ground and written relative to the frame linked with the base R as

1 eshn

 1 + nodes. Let us consider an undeformed shaft FE i esh which consists of two nodes i n + along the Oy axis. This FE is of density i esh ρ , cross-sectional area

  the planes xOy and yOz , i.e.,

  point 0 C due to bending respectively along the Ox and Oz axes of the frame R as well as the corresponding rotations Ox and Oz directions are described as a function of the nodal displacement vectors ,

F

  are the shape function vectors of a Timoshenko beam FE. They are third-degree and second-degree polynomial equations, functions of the dimensionless variable y ( axis Oy and contain the shear effect (see chen and Ku, 1992).

(

δ

  are unknown displacement amplitude vectors. Using r δ , the velocity as well as acceleration vectors r δ ɺ , r δ ɺɺ are obtained. Substituting them into equation (39) yields the steady-state solutions

.

  of equation (45) at the end of one period τ and the -To provide the Floquet transition matrix at time t τ = , 2 g n integrations are necessary where g n is the total number of degrees of freedom of equation (31). The eigenvalues or characteristic multipliers of the Floquet (45) are calculated. Finally, the system (

Figure 5 .

 5 Figure 5. Symmetric on-board rotor configuration.Table 1. Main characteristics of the investigated rotor Disk material density Disk radius Disk thickness Disk location Shaft material density Shaft radius Shaft length Young's modulus of the shaft Poisson's ratio of the shaft Shear correction factor of the shaft Mass unbalance and its phase

,

  the on-board rotor is stable for all the considered mode shapes included in its operating range [ ] 0; 6000 rpm Ω ∈

Figure 6 .

 6 Figure 6. Campbell diagrams for the first three natural frequencies of the rotor subjected to the base angular velocity: (a) ω x , or (c) ω y , with the values: 0 (solid line), 5 Hz (dashed line), 10 Hz (dotted line), (b) Zoom on the intersection of the first forward whirl frequency with the speed of rotation of the rotor. Table2. Natural frequencies as a function of the speed of rotation of the rotor and the constant rotations of the base (B and F stand for backward and forward whirl modes)

  presents the mass unbalance responses at the middle of the shaft (i.e., node 7) of the rotor excited by the angular velocity either x ω or y ω . In the operating range critical speed of rotation (one peak II ) when the base is fixed. It exhibits only one peak in the presence of the base rotation y ω (due to the isotropic system), see Figure7(b), but three peaks ( I , II , III ) in the presence of x ω (due to the anisotropic system), see Figure 7(a). These peaks correspond to the intersection of the straight line 60 f = Ω ( Ω in rpm) with the natural frequencies of the rotor as shown in Figure 6, i.e., the intersection with the frequencies of the first and second backward whirl modes ( I , III ) and of the first forward whirl mode ( II ) for x ω and the intersection with the frequencies of the first forward whirl mode for y ω . The responses confirm that the rotor keeps a symmetric dynamic behavior in the case of the base rotation around Oy but has a non-symmetric dynamic behavior in the case of the base rotation around Ox as stated in Section 7.1.2 and presented in Figure 6. For increasing values of x ω , the principal critical speed ( II ) increases slightly as shown by the points ( A , B , C ) in Figure 6(b) and the amplitudes of the additional peaks ( I , III ) become larger and larger. As shown by the points ( A , B , C ) in Figure 6(c), the critical speed position is shifted down when the angular velocity y

Figure 7 .

 7 Figure6and presented in Table 2. In the case of increasing values of y ω , the peak relative to the third natural frequency and produced by the harmonic base translation along the Oz axis becomes smaller and smaller, see Figure8(b). This is because equation (34) presents a negative cosine term containing the frequency z Ω of the base translation. Lastly, it is noted that the operating zones of the frequency responses along the z Ω axis do not change mostly for the angular velocity x ω , while they can change significantly for y ω .

  are produced at the middle of the rotor presented in Section 7

.Figure 8 .Figure 9 .

 89 Figure 8. Frequency responses of the rotor to the sinusoidal base translation for a speed of rotation Ω=1500 rpm and the constant rotation of the base around: (a) the Ox axis, or (b) the Oy axis.

For

  Figure10(a) displays the classical orbit at the middle of the rotor (node 7) due to the mass unbalance for a constant speed of rotation 1500 rpm Ω = (=25 Hz=the mass unbalance frequency) and a fixed base. Since the matrices of the rotor system are symmetric and skew-symmetric with isotropic diagonal and cross-coupling components, the dynamic behavior is symmetric and the orbit is circular and periodic of period equal to the inverse of the mass unbalance frequency Ω . The orbit center coincides with the point O (bearing center) because the bearings are rigid and the motion amplitude of the rotor in both Ox and Oz directions is 3.628×10 -6 m at its middle. Figures10(b) and 10(c) compare the steady-state mass unbalance orbits in the case of a pure rotation of the base either around Ox or Oy , see the solution in equation (41).For the base angular velocity 5 Hzx ω =

Figure 10 .

 10 Figure 10. Mass unbalance orbits at the middle of the rotor running at Ω=1500 rpm and subjected to three values of base rotation: (a) ω x =ω y =0, i.e., fixed base, (b) ω x =5 Hz, ω y =0, (c) ω x =0, ω y =5 Hz.

.

  z Ω . Figures 11 to 13 give the orbits at the middle of the shaft (node 7) for a constant speed of rotation 1500 rpm Ω = (=25 Hz) and different cases of combined rotational and translational motions of the rotor base. In Figure 11, the base translation is kept constant ( to the sinusoidal base translation, complicated orbits appear instead of the classical circle or ellipse. As shown previously when the rotor is subjected to the angular velocity x ω , the rotor orbits oscillate about a static position sp C . The coordinates along Ox of the displacements yielding the static position are calculated using equation (44) and increase with x ω . They are roughly 3.2×10 -The plot adopts the static position sp C as an origin. The magnitudes of the orbits remain almost the same for increasing values of x ω . When the rotor is excited by the angular velocity y ω , the plot adopts the bearing center O as an origin. The magnitudes of the orbits increase for increasing values of y ω because the mass unbalance excitation increases in a way more important than the stiffness matrix of the rotor. In Figure 12, the base rotation and the amplitude of the harmonic base translation are kept constant (

  , the base rotation and the frequency of the base translation are kept constant ( x the amplitudes makes the orbits much larger and more complicated especially when compared with the mass

Figure 11 .Figure 12 .Figure 13 .Figure 14 .

 11121314 Figure 11. Influence of the angular velocity ω x and ω y of the base. Orbits at the middle of the rotor running at Ω=1500 rpm (=25 Hz) and subjected to a base translation: zO=ZOsin(Ω z t) with ZO=1×10 -6 m and Ω z =200 Hz.

  rotating parts (disk and shaft) is taken into account (see[START_REF] Spears | Approach for selection of Rayleigh damping parameters used for time history analysis[END_REF] and its corresponding coefficients estimated for a modal damping factor 0

Figure 15 .

 15 Figure 15. Geometry of the asymmetric disk.

.

  latter depends on the angle d θ describing the geometric asymmetry of the disk. Therefore the stability analysis covers the two parameters Ω as well as d θ and the stability charts of the rotor shown in Figures 16 and 17 are established for the primary resonance with 5 are bounded by transition curves. The increase of the base angular velocity x ω roughly doesnot change the size of the instability zones and slightly shifts them down (see Figure16). For the considered values of the angle d θ , the on-board rotor can become unstable for In the same way, the increase of y ω does not change the size of the instability zones but this time, their limits are remarkably shifted down (see Figure17). For the considered values of d θ , the rotor can be unstable for

Figure 19 Figure 16 .Figure 17 .Figure 18 .Figure 19 .

 1916171819 Figure 19(a) presents the mass unbalance orbit at the middle of the asymmetric rotor (node 7) for an angle of disk asymmetry 45 d θ =

  ) and combined rotational and translational excitations of the rotor base. In this figure, the base rotation and the amplitude of the harmonic base translation are kept constant ( orbits at the middle of the shaft are elliptical and the displacement amplitudes are mostly the same as shown in Section 7.2.3.1 whatever the value of the base translation frequency z Ω . FFTs exhibit only one frequency component equal to 50 Hz 2 = Ω because of the presence of x ω in the cos 2 t Ω and sin 2 t Ω terms of the external excitation produced by the geometric asymmetry of the rotor, which are prevalent regarding the other forcing terms, see equation (36). When the rotor is excited by the base angular velocity 5 Hz y ω =

Figure 20 .

 20 Figure 20. Influence of the frequency Ω z of the base translation: zO=ZOsin(Ω z t) with ZO=1×10 -6 m. Orbits of the asymmetric rotor running at Ω=1500 rpm and subjected to two angular velocities of the base: ω x =5 Hz or ω y =5 Hz.

Figure 21 .

 21 Figure 21. Comparison of RR and FE orbits at the middle of the asymmetric rotor running at Ω=1500 rpm and subjected to a base translation: zO=ZOsin(Ω z t) with ZO=1×10 -6 m and Ω z =100 Hz, combined with two angular velocities of the base: ω x =5 Hz or ω y =5 Hz.

  )

	u	mu	O x r mu	cos	t	mu	mu w	O z r mu	cos	t	mu	y	y	mu	O y	z
	g mu R P		O y	mu w	z	O	mu r	cos	t	mu	x	u	mu	O x r mu	sin	t		mu	z	.
	mu w		O z r mu	sin	t	mu	y	mu	O y	x	u	mu	O x r mu	sin	t	mu	y	R

  in rotordynamics with base motions. The equations of motion (31) are rewritten in equation (32) due to the rotation x

	, d sh r	+	ω and equation (34) due to the rotation y ω ( ) ( ) 2 2 , , 2 , , , , cos x x g e re gse x c d sh r sh d sh b sh b r mu ω ω ω Ω + +	t	s mu	sin	t
	, 1 mu b c	cos	t	, 1 mu b s	sin	t	, , 1 d sh b c	cos	z	t	, , 1 d sh b s	sin	z	t	0 , , d sh b	,

z O O z Z t =

Ω . The resulting linear second-order differential equations of the dynamic system include constant terms and are solved by analytical methods in order to display the major basic phenomena occurring

Table 2 .

 2 Natural frequencies as a function of the speed of rotation of the rotor and the constant rotations of the base (B and F stand for backward and forward whirl modes)

	50
				III
	0 40	I	1500 II	f=Ω/60					C	B
										A
				f=Ω/60
					A
				B
				C
	Ω rpm		fr Hz	Fixed base	5	ω x Hz	10	5	ω y Hz	10
			f1 B	44.93	44.96		45.04	49.42		53.88
	0		f1 F f2 B	44.93 123.79	45.19 123.80		45.93 123.84	40.42 124.49		35.88 125.42
			f2 F	123.79	123.88		124.15	123.32		123.09
	1500		f1 B	42.28	42.40		42.73	46.65		50.99

O O C sp Stable Unstable Stable Unstable Stable Stable Stable Stable Unstable Stable Stable Unstable Unstable Stable Stable

	7.2.3.2 Combination of translation z O =Z O sinΩ

z t and rotation ω

x (or ω y ) of the base