Potential Applications of Semiconductor Nanolasers in Particle Detection and

Speckle-Free Illumination T. Wang 1,2 , C. Jiang 1,2 , J. Zou 3 , J. Yang 1,2 , X. Lin 4 , H. Chen 4 , K. Xu 1,2 , C. Jin 5,6,7 , G. Wang 1,2 G. P. Puccioni 3 and G. L. Lippi 4

Implementation:

Scattering Emission is affected by feedback in both cases, to different extents, allowing for separate quantification ranges of particles and speed [START_REF] Wang | Second-order correlation function supported optical sensing for particle detection[END_REF].

Feedback coverage: With suitable sensitivity, all feedback values can be measured (covering diverse particle sizes, scattering coefficients and setup choices) from < 1% (microlaser) to ~50% (nanolaser).

Intermediate feedback (1% … 15%) is quantified with the time-delayed autocorrelation (g (2) (τ)) [START_REF] Wang | Second-order correlation function supported optical sensing for particle detection[END_REF].

Background:

Nanolaser response curve (black) consists of spontaneous (blue) and stimulated (red) events. Sharp photon bursts appear in the yellow region and can be exploited by adding an amount of external feedback (τ fb = 4 ns).

Predictions based on stochastic simulations [START_REF] Puccioni | Stochastic Simulator for modeling the transition to lasing[END_REF] require a careful analysis of the signal to avoid spurious interburst coherence [START_REF] Wang | Nanolasers with Feedback as Low-Coherence Illumination Sources for Speckle-Free Imaging: A Numerical Analysis of the Superthermal Emission Regime[END_REF]. A longer (fibered) feedback arm enables expanded sequences of independent pulses.

Speckle-Free Illumination

Background Photon bursts experimentally observed [START_REF] Wang | Photon bursts at lasing onset and modeling issues in micro-VCSELs[END_REF] in microlasers (orange background region).

• Predictions of photon bursts in stochastic simulations [START_REF] Puccioni | Stochastic Simulator for modeling the transition to lasing[END_REF] in micro-(e.g., β = 10 -4 ) and nanolasers (e.g., β = 10 -1 ). β: fraction of spontaneous emission coupled into the lasing mode.

• Result of the combination of large carrier fluctuations and poor excitation transfer from energy reservoir (carriers) to the field. Newly discovered regime offers opportunities for exploiting the unusual dynamics in different fields of application.

• Two applications are explored in which feedback modifies the spontaneous photon-burst regime.

Pulse power distribution

• The feedback fraction controls the distribution of pulse power.

Conclusions:

The peculiar emission in photon bursts offers so far unknown avenues for new kinds of applications based on the intrinsic properties of this low coherence regime.

  particles flow at variable speed -(0.01 < v < 10)m/s Medium: air or liquid Backscattering of incident beam introduces feedback into the laser The photon autocorrelation function is used to characterize the changes introduced by feedback (related to particle size and nature) Particle Sensing The zero-delay second order photon correlation function (g (2) (0)) characterizes the statistics of the emitted photons. Microlaser (right): superthermal statistics (0% feedback, black) signals strong photon bursts -progressively reduced to subthermal levels -in a very narrow range of pump values close to threshold. Nanolasers (left): weaker superthermal statistics persisting over a very broad pump range.

Independent pulse trains

• However, independence between neighbouring pulses (initiated by spontaneous seeds) allows for the generation of pulse trains which remain optically decorrelated even in the presence of partial correlation over a rountrip of the external cavity. • Increasing the feedback length (an easy task with an optical fiber) enables the generation of a larger number of pulse trains, which lower the mutual coherence.

• Partial coherence between pulses may emerge from photon reinjection by the feedback arm. The fraction of incoherent pulses [START_REF] Wang | Nanolasers with Feedback as Low-Coherence Illumination Sources for Speckle-Free Imaging: A Numerical Analysis of the Superthermal Emission Regime[END_REF] is analysed in the table.

• Increasing the feedback level leads to a larger degree of mutual coherence (second line).


M A Carroll

G D'alessandro

G L Lippi

G.-L Oppo

F Papoff

Quantum Modelling and Photon Squeezing in Nanolasers Quantum Modelling and Photon Squeezing in Nanolasers

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Methods and Main Results

• Dynamical systems methods identify laser threshold (bifurcation) depending on N, interaction and detuning • Universal transition: thermal, antibunched and lasing emission • Phase transition (incoherent-coherent emission) in the macroscopic limit

Conclusions

• We have derived a new quantum laser model that contains both coherent and incoherent variables. • Through linear stability analysis we find a critical number of QDs required for lasing to happen. • We show that there is a threshold for the so-called thresholdless laser.

• Our model predicts an antibunching regime that precedes the lasing regime even at mesoscopic emitter numbers. • We show that a second-order phase transition occurs in the thermodynamic or macroscopic limit.

System stability

Lasing (identified through Linear Stability Analysis) occurs if:

with -N; detuning, ∆; decay rates, g; light-matter coupling, g

Common condition for single and multi electron models [2]

• Red stars mark lasing bifurcation • Crosses and triangles mark threshold from the thermal to antibunching regime • The coherent variables reintroduce a lasing bifurcation (threshold) for the thresholdless laser • The bifurcation is followed by a non-zero coherent field (blue and green curves cf. Observable even for 10 4 QDs (mesoscopic laser)

Recovering the macroscopic threshold

• Comparing the single and multi electron models -the lasing bifurcation occurs at lower pump for the multi electron system • As the volume of the devices approaches the macroscopic limit (thermodynamic limit) the threshold moves from the upper branch towards the inflection point [3] • A bifurcation occurs in all cases however, it is only the curve that undergoes a second-order phase transition. • Below (above) the bifurcation the emission is fully incoherent (partially coherent).

N = 40

Quantum•

  Modelling and Photon Squeezing in Nanolasers Quantum Modelling and Photon Squeezing in Nanolasers M A Carroll, G d'Alessandro, G L Lippi, G.-L Oppo, F Papoff To cite this version: M A Carroll, G d'Alessandro, G L Lippi, G.-L Oppo, F Papoff. Quantum Modelling and Photon Squeezing in Nanolasers Quantum Modelling and Photon Squeezing in Nanolasers. 9ème édition du congrès OPTIQUE de la Société Française d'Optique, Jul 2022, Nice, France. hal-03723305 Fully quantised model for N Quantum Dots (Qds) • Two-level systems (with one electron) • Optical Cavity modes

  Photon NumberField Amplitude


Marouane Tliba

Aladine Chetouani

Giuseppe Valenzise

Fréderic Dufaux

REPRESENTATION LEARNING OPTIMIZATION FOR 3D POINT CLOUD QUALITY ASSESSMENT WITHOUT REFERENCE

Keywords: 3D Point Cloud, Point Cloud Compression, No-Reference Perceptual Quality Metric, Deep Learning, Unsupervised Learning

Recent information and communication systems have employed 3D Point Cloud (PC) as an advanced geometrical representation modality for immersive applications. Like most multimedia data, PCs are often compressed for transmission and viewing purposes, which can impact the perceived quality. Developing robust and efficient objective quality metrics for PCs is still an open problem. In this paper, we propose an end-to-end deep approach for evaluating the perceptual effects of point cloud compression solutions without reference. Our approach focuses on leveraging the intrinsic point cloud characteristics to quantify the coding impairments from few distant randomly selected patches using supervised and unsupervised training strategies. To evaluate the performance of our method, two well-known datasets have been used. The results demonstrate the effectiveness and reliability of the proposed method compared to to state-of-the-art methods.

INTRODUCTION

Following the advent of recent immersive technologies and the abundant availability of low-cost 3D sensors, several 3D representation media have come to birth allowing live interaction with the geometrical content from any point of view [START_REF] Chetouani | Classification of engraved pottery sherds mixing deep-learning features by compact bilinear pooling[END_REF] [START_REF] Tliba | 2d-based saliency prediction framework for omnidirectional-360[END_REF]. Among those representations, 3D Point Cloud (PC) has been adopted as one of the most preferable formats. PCs are sets of unordered points determined by their Cartesian's coordinates and potentially associated attributes such as color, curvatures, reflectance and normal vectors. It has emerged in many recent applications such as Extended Reality, Real-time Immersive Communications, Robotics, 3D gaming, and Cultural Heritage. In practice, the depiction of a realistic 3D PC scene with a high resolution needs up to millions of points. As this huge size of data requires further computational sensitive operations such as storing and transmission, the application of emerging compression schemes becomes inevitable, impacting the perceived quality. Achieving a balance between user satisfaction and data cost is thus important. Specifically, to evaluate the rate-distortion or performance of existing point cloud coding schemes, perceptual quality metrics need to be adopted in order to assess the perceptual fidelity of the decoded point cloud regarding a certain compression rate.

Generally, Point Cloud Quality Assessment (PCQA) could be obtained through subjective and objective experiments. The former requires a human intervention which is expensive and time-consuming, while the latter relies on computational methods that predict the perceptual quality. Depending on the availability of the reference image, objective quality metrics fall into three categories: Full-Reference [START_REF] Chetouani | Convolutional Neural Network for 3D Point Cloud Quality Assessment with Reference[END_REF], Reduced-Reference [START_REF] Chetouani | A reduced reference image quality metric based on feature fusion and neural networks[END_REF] and No-Reference [START_REF] Chetouani | On the use of a scanpath predictor and convolutional neural network for blind image quality assessment[END_REF][START_REF] Abouelaziz | 3D visual saliency and convolutional neural network for blind mesh quality assessment[END_REF][START_REF] Abouelaziz | No-reference mesh visual quality assessment via ensemble of convolutional neural networks and compact multi-linear pooling[END_REF]. Existing quality metrics for 3D PCs can be classified into three main groups: Point-based, Feature-based and Projection-based metrics. Point-based metrics such as Point-to-Point (Po2Po) [START_REF] Tulvan | Evaluation criteria for pcc (point cloud compression)[END_REF], Point-to-Plane (Po2Pl) [START_REF] Feng R. Cohen | Geometric distortion metrics for point cloud compression[END_REF], Planeto-Plane (Pl2Pl) [START_REF] Alexiou | Point cloud quality assessment metric based on angular similarity[END_REF] and Point-to-Mesh (Po2Mesh) [START_REF] Rochinni | Metro: measuring errors on simplified surfaces[END_REF], predict the quality through point-wise geometric and/or features distance between the reference PC and its distorted version. It is worth noting that MPEG is adopting Po2Po MSE and Po2Pl MSE with the associated PSNRs as the standard PC geometry quality metrics. Feature-based PC Quality Metrics, extract the geometry with the associated attributes form point-wise level in a global or local way. Among those metrics, we can cite, PC-MSDM [START_REF] Rochinni | Metro: measuring errors on simplified surfaces[END_REF] that extends the 2D SSIM metric [START_REF] Zhou Wang | Image quality assessment: from error visibility to structural similarity[END_REF] to PC by considering local curvature statistics, the Geotex [START_REF] Diniz | Towards a point cloud quality assessment model using local binary patterns[END_REF] metric that exploits the Local Binary Pattern (LBP) [START_REF] Pietikäinen | Two decades of local binary patterns: A survey[END_REF] descriptors, and PCQM [START_REF] Digne | Pcqm: A full-reference quality metric for colored 3d point clouds[END_REF] that combines the geometry and color features. In projection-based PC Quality Metrics, the 3D points are projected into 2D regular grids and 2D quality metrics are applied on these views. Lake of large 3D PC quality data-set, hinder the development of efficient deep based quality metrics, thus, finding a way to push representation learning from limited amount data is mandatory. Moreover, all the above-cited PCQA metrics need long time of pre-processing, which is computationally expensive and most of them require the reference PC. Therefore, deploying these metrics in edge devices for real-time quality assessment or adopting as loss function to optimize learnable compression models remains difficult.

To overcome the aforementioned shortage of PC quality metrics, and motivated by the nature of visual impairment resulting from emerging point cloud compression solutions [START_REF] Prazeres | Quality evaluation of static point clouds encoded using mpeg codecs[END_REF] that introduce somehow uniform effects distributed over all the geometrical content.In this paper we look to learn an efficient representation from local intrinsic characteristics of PC patches. More precisely, we extract PC patches by first selecting M centroid points using the farthest point sampling algorithm [START_REF] Qi | Point-net++: Deep hierarchical feature learning on point sets in a metric space[END_REF]. Then, we apply the K-nearest neighbor clustering method to form a patch around each centroid. Finally, we learn a lightweight permutation invariant feature descriptor function, followed by a shallow regressor that estimates the final quality score. To further optimize the model, beside training our shallow network using the supervised loss, we also use a self-supervised ranking loss. The goal is forcing the model to learn richer descriptive representations capturing local intrinsic characteristics of point clouds, and discard the rest of cues, that result in learning an inner representation related to the point cloud global shape. To the best of our knowledge, we are the first who consider learning representation directly from 3D PC without applying projection or transformation such us voxelization.

The main contributions of this paper are summarized in what follow:

• We propose a novel efficient end-to-end and shallow deep model for assessing the perceptual effects of PC compression methods relying on the local intrinsic characteristics of sub-sets of points. • We extend the self-supervised strategy of learning from rank as pseudo label to optimize the learning more on intrinsic point sub-set features, toward a potential descriptive representation related to our downstream task. • Two well-known datasets are used to demonstrate the effectiveness of the proposed approach comparing to state-of-the-art methods.

The rest of the paper is organized as follows: In section 2, we describe the proposed method and the learning strategy applied. In Section 3, we present the adopted experimental protocol and discuss the results. And finally, we give some conclusions in Section 4.

PROPOSED METHOD

The main goal of this work is to provide an efficient blind metric for estimating the quality of 3D compressed PCs without adding any projection step or applying changes on the specificity of the points. Motivated by the nature of compression schemes in producing uniform distributed effect over local features, we assume that the distribution of compressed PCs could be represented as a set of hidden features or descriptors related to the compression effects such as point consistency, sparsity, and density; plus a set of descriptors related to other characteristics such as position, global shape, and structure. In order to learn efficiently the representation of these hidden or intrinsic features and inspired by Point-Net [START_REF] Qi | Point-net++: Deep hierarchical feature learning on point sets in a metric space[END_REF], we use here a similar shallow permutation invariant encoder through series of shared 1D convolutions over all points. We then employ a shallow regressor to predict the quality. Inspired from previous works that learn representation from rank [START_REF] Severyn | Learning to rank short text pairs with convolutional deep neural networks[END_REF][19] [START_REF] Joachims | Optimizing search engines using clickthrough data[END_REF]. Our network is optimized to learn the inner representation from the labeled data, besides, the known implicit rank between data samples as pseudo labels, simultaneously. Fig. 1 demonstrates the training process and the architecture of the proposed approach.

Patch extraction and Pre-Processing

The PC is first divided into small patches by considering just the (x,y,z) coordinates and the RGB color information. For that, the farthest point sampling [START_REF] Qi | Point-net++: Deep hierarchical feature learning on point sets in a metric space[END_REF] and K nearest neighbor algorithms are used as follows:

• From a given PC X, we first apply the farthest point sampling algorithm to select M centroids (i.e. C = { P 1 , P 2 , P 3 , .., P m }) • In order to form a patch around each centroid, we then employ the K nearest neighboring clustering method.

The sub-set of points of the i th centroid, denoted here { P 1 i , P 2 i , P 3 i , .., P K i }, represents the i th patch. K was fixed here to 512.

• Finally, we compute the position and color differences between each sub-set of points and its corresponding centroid. We obtain the final sub-set of points, denoted here S i , that is fed as input to the model { P 1 i -P i , P 2 i -P i , P 3 i -P i , .., P K i -P i }. It is worth noting that the use of patches allows to augment the training data as the existing datasets for 3D PC quality are quite short. It also helps to generalize the learning from only local features, which leads to reduce the reliance of model to other global information and decrease the computational time.

Network Architecture

The design of our network is inspired from the efficient Point-Net backbone [START_REF] Qi | Point-net++: Deep hierarchical feature learning on point sets in a metric space[END_REF]. Since we are working on patches, we focus to learn useful representation only from local intrinsic features rather than the global shape. To sum up, the architecture of our model is composed of a symmetric invariant features extractor (i.e. series of 1D convolution followed by a max polling operation over point representation) and a shallow deep regressor followed by a sigmoid activation in order to produce a probability that represents the estimated quality score.

Self-supervised and supervised learning

As mentioned previously, we adopt two strategies to optimize our model: supervised learning from the subjective scores and self-supervised learning from the rank. The supervised learning stage aims to learn a mapping function to the Mean Opinion Score (MOS), while the goal of the self-supervised stage is to learn a better representation that maximizes the learning from intrinsic features. We look here to force the model to

S d = Q d + N d ( 1
)
where S d represents the distribution of a point sub-set. Q d is the distribution of a well representative intrinsic features related to the perceptual quality estimation, while N d is the distribution of other features that are not related to our downstream task.

Supervised Learning

The aim from the supervised step is to find a function

Self-Supervised Learning

The goal from the self-supervised step is to have a proxy supervised task with rich pseudo labels [START_REF] Tliba | Self supervised scanpath prediction framework for painting images[END_REF]. It allows to learn the separation from different degradations, and at the same time maximize the learning over the Q d distribution (i.e. intrinsic features).

As we know that higher compression level produces higher perceptual damage. So, for a given encoding module (i.e., MPEG G-PCC Octree/TriSoup [START_REF] Schwarz | Emerging mpeg standards for point cloud compression[END_REF] [23] [START_REF] De Queiroz | Compression of 3d point clouds using a region-adaptive hierarchical transform[END_REF], MPEG V-PCC), applying the compression levels l j , and l j+ on point cloud X results on point clouds X j and X j+ , respectively. Hence we could easily know the generated rank based on the introduced damage.

Considering the visual quality degradation of those encoding modules is uniform over the X geometry, we want our model to focus on intrinsic features and, ignore the learning from other features such as patch position (i.e., f (S d : θ) = f (Q d : θ). In other word, whatever the selected patches from X j and X j+ , the model should give outputs ordered according to the inverse order induced by the level of compression:

l j < l j+ =⇒ mean(f (S d j )) > mean(f (S d j+ )) (3)
To this end, we use a Siamese Network [START_REF] Hadsell | Learning a similarity metric discriminatively, with application to face verification[END_REF] to minimize the ranking loss between the activation of input pairs. Each branch of this Siamese Network is based on our model parameter.

Rank Loss

= max(0, ∆f + margin) (4)
The gradient of the Rank Loss in Eq (4) is:

∇ θ Rank Loss = 0, if ∆f + margin≤ 0 ∇ θ (∆f ), otherwise where ∆f = mean(f (S d j+ )) -mean(f (S d j ))
In much details, by optimizing the Siamese network through minimizing the Rank Loss, the gradient tends to zero when the network activation has the inverse order as induced by the compression. Whereas, the gradient of the higher activation decreases while the gradient of the lower activation increases when the activations have the same order as induced by the compression.

Training Protocol

At each training step, we update the weights by minimizing both the ranking self-supervised loss and the supervised loss as multi-task learning:

M ultiT ask loss = Rank Loss + 0.01 * Rg loss [START_REF] Chetouani | On the use of a scanpath predictor and convolutional neural network for blind image quality assessment[END_REF] It is worth noting that at each training step, the selected centroids for extracting patches change (i.e., (C j ) ̸ = (C j+ )). In other words, we implicitly urge the model to discard the position and global shape information, and maximize the learning on the quality related information which represent the mutual information across (S d ) patches. In addition, this strict protocol (i.e., the training data changes at each epoch) makes the model more robust and thus increase its generalization ability. The value of the margin changes based on the distance between compression levels, varying from 0.1 for the successive degradation level to 0.6 for the farthest degradation level. We set the initial model weights of randomly [START_REF] Glorot | Understanding the difficulty of training deep feedforward neural networks[END_REF], and train the whole parameters in end-to-end manner for 500 epochs using Adam optimizer [START_REF] Diederik | Adam: A method for stochastic optimization[END_REF].

EXPERIMENTAL RESULTS

We evaluate our model through two well-known datasets that employ different compression schemes with multiple encoding levels: ICIP20 [START_REF] Prazeres | Quality evaluation of static point clouds encoded using mpeg codecs[END_REF] and PointXR [START_REF] Alexiou | Pointxr: A toolbox for visualization and subjective evaluation of point clouds in virtual reality[END_REF]. ICIP20 is composed of 6 reference PCs from which 90 degraded versions were derived through three types of compression: V-PCC, G-PCC with triangle soup coding and G-PCC with octree coding. Each reference PC was compressed using five different levels. PointXR is composed of 5 PCs from which 45 degraded versions were derived through G-PCC with octree coding for geometry compression and, Lifting and RAHT for color compression. To fairly study the performance and for the sake of providing a solid validation baseline protocol to future works, we randomly select 64 centroids for validation as we do during the training. We adopt a 6 fold cross validation protocol on ICIP20 dataset through splitting the dataset into training validation 6 times, where 6 refers to the number of reference point cloud samples. More precisely, at each iteration 5 reference point cloud samples and their compressed versions are used for training, and 1 reference point cloud sample and its compressed versions are used for testing. Lastly, to emphasize the effectiveness, and validate the generalization ability of our model on unseen compression, we apply a crossdataset validation protocol by training our model on ICIP20 and testing it on PointXR as commonly used to rigorously benchmark deep learning methods. Pearson Correlation Coefficient (PCC) and Spearman Rank-Order Coefficient Correlation (SROCC) are computed to evaluate the quality predic-tion ability of our method. These correlations are computed regarding each fold independently during the validation, and finally the mean is reported.

Model

Table 1 shows the performance of our method on ICIP20 dataset using only supervised learning, denoted here as Our-(S) and supervised with self-supervised as multi-task learning, denoted here as Our-(S+SSL). The results are also compared to a set of state-of-the-art methods. As can be seen, using the self-supervised and the supervised losses allows to considerably improve the performance. Moreover, our method (S+SSL) is quite competitive with state-of-the-art models, especially in terms of ranking (i.e. SROCC) the compression distortion effects. It is also worth noting that unlike our method all of listed methods require the reference point cloud and take long time during the testing.

Table 2 shows the results obtained for the cross-dataset evaluation. As can be seen, high correlations are reached by our method, outperforming some of the compared ones. This results shows the generalization ability of our method to predict the quality of unseen PCs. It is remarkable that our method is the most consistent one comparing to the performance achieved on ICIP20.

CONCLUSION

In this paper, we proposed an efficient end-to-end shallow deep learning-based approach for interpolating quality scores induced by emerging compression effects. Unlike previous methods, the proposed approach works directly on point cloud patches without requiring prior computational expensive processing step. We urge our model to learn potential representation from point sub-set by focusing on the local intrinsic features related to our downstream task. To optimize the model, we also used self-supervised as rank learning strategy in order to learn a direct mapping to the accurate quality scores, while maximizing the distance between the representation of different degradations. The obtained results using a strict k-fold cross validation and cross dataset validation protocols demonstrated the effectiveness of our lightweight model. It also showed that our model is competitive comparing to state-of-the-art full reference methods. We look in our future works to extend our model by robust learnable aggregation mechanism in order to consider better the local and global effects on point distribution.

Fig. 1 .

 1 Fig. 1. : Training Process and model architecture. maximize the learning of local intrinsic features from point sub-set, in label unsupervised manner, considering:

  f (S d : θ) with parameters θ that captures the relationship between S d and quality scores Y ; Where (X, Y ) = {(S d0 , Y ), (S d1 , Y ), ..(S dm , Y )}. Since the output of our model is a probability value that represents the perceptual quality, we consider the binary cross entropy (BCE) as loss function to minimize the regression empirical risk over the S d set. Rg loss(P, Y ) = -Y log(P ) -(1 -Y ) log(1 -P ) (2) where P = f (S d : θ), and Y is the MOS of the point cloud sample.

Table 1 .

 1 Results obtained on ICIP20 dataset

		PLCC ↑	SROCC ↑
	po2pointMSE	0.945	0.950
	po2planeMSE	0.945	0.959
	PSNRpo2pointMSE	0.880	0.934
	PSNRpo2planeMSE	0.916	0.953
	Our-S	0.745	0.621
	Our-(S+SSL) P2	0.908	0.955
	Model	PLCC ↑	SROCC ↑
	po2pointMSE	0.887	0.978
	po2planeMSE	0.855	0.942
	PSNRpo2pointMSE	0.983	0.978
	PSNRpo2planeMSE	0.972	0.950
	Our	0.964	0.970

Table 2 .

 2 Results obtained on PointXR dataset


Mustapha Lahmar

Salah Ellagoune

Benyebka Bou-Sa Ïd

Elastohydrodynamic Lubrication Analysis of a Compliant Journal Bearing Considering Static and Dynamic Deformations of the Bearing Liner

Keywords: EHD Lubrication, Compliant Journal Bearings, Static Deformation, Dynamic Deformation, Stiffness And Damping Coefficients, Stability, Perturbation Technique, Compressible Thin Elastic Liner Model

In this article, the effect of both static and dynamic deformations of the bearing liner on the dynamic performance characteristics and stability of a water-lubricated, rubber-lined journal bearing operating under small harmonic vibrations is theoretically investigated. To take into account the dynamic deformations of the bearing liner, the first-order perturbation technique is used to determine the eight dynamic coefficients for a given excitation frequency value. The static and dynamic deformation of the fluid/bearing-liner interface is assumed to be proportional to the steady-state and dynamic fluid-film pressures. It was found that the dynamic properties and stability of the compliant finite-length journal bearing are affected by surface coatings from soft materials. It was also shown that when dynamic deformations are considered in the calculations, the dynamic coefficients depend on the excitation frequency, especially for higher values of this parameter. Moreover, the two cross-damping coefficients differ from each other, while the classical elastohydrodynamic (EHD) theory predicts them to be equal, when the dynamic deformations are ignored.

INTRODUCTION

The compliant plain journal bearings are the machine elements that consist of two components: a shaft in hard steel and a bearing with a liner. Technologically speaking, the coated bearings are used in order to reduce the friction coefficient in the boundary lubrication regime encountered mainly during the start and stop phases of machines.

Since 1965, steady-state performance characteristics of compliant journal bearings have theoretically and experimentally been investigated by many researchers (1-16). However, dynamic properties have mostly been calculated by taking into consideration only static-pressure-induced deformations of the bearing liner-even for low-elasticity modulus materials such as white metals and rubber materials. Indeed, the existing literature shows that studies on dynamic characteristics of compliant journal bearings considering dynamic-pressure-induced deformations of the bearing liner are scarce.

A good survey on EHD investigations in both static and dynamic operating conditions of steadily and dynamically loaded compliant journal bearings can be found in References (11,12,14).

An example of a compliant journal bearing is the waterlubricated rubber journal bearing that is used in many applications, including stern tube bearings on ships, submarines, and hydraulic pumps.

The rubber journal bearings reduce much noise and vibration (acoustic emissions) compared with equivalent composite bearings. These bearings are resistant to abrasion caused by the presence of solid particles in the water. The use of water as a lubricant provides efficient low-friction operation and a stable lubricant film due to the water's incompressibility.

Figure 1 shows a picture of a typical water-lubricated propeller shaft bearing system. To promote the formation of a hydrodynamic film between the propeller shaft and bearing, the lower loaded half of the bearing is smooth, while the upper one incorporates longitudinal grooves for flow of the water lubricant. The rigid outer shell is made of naval brass or nonmetallic composite, which acts as the compliant liner rigid backing.

The first designs of these bearings had a plain bore, but this was modified several times in order to improve the performance characteristics of the bearing.

There is another generation of marine bearings like straightfluted bearings widely used today. The design of this type of bearings consists of a number of rubber staves (load-carrying lands) bonded onto the outer rigid shell and separated by flutes. These flutes supply the bearing with water lubricant that enters at one end of the bearing and leaves at the other one.

1 NOMENCLATURE a εε , a εφ , a φε , a φφ = stiffness coefficients, N/m A εε , A εφ , A φε , A φφ = dimensionless stiffness coefficients, Certain marine bearings are made from molded rubber staves that, when assembled, form an uninterrupted cylindrical bore bounded by a hard backing metal. The steady-state performance characteristics of this type of bearings have been investigated extensively by several researchers, including Braun and Dougherty (11, 12). They have found that the liner compliance strongly affects the maximum steady-state pressure and the cavitation zone location, especially at high values of eccentricity and angular velocity.

A ij = a ij C 3 µωR 3 L ; (i, j ) = (ε,
In the present investigation, the governing equations relative to this situation are established, and a numerical simulation is performed. We analyzed the effects of both static and dynamic deformations of the bearing liner on the stiffness and damping coefficients, critical mass, and whirl frequency of a finite-length water-lubricated rubber-lined journal bearing operating under small harmonic vibrations.

The first-order perturbation technique was used to determine static and complex dynamic pressures developed in the fluid film. The eight dynamic coefficients can be obtained by means of numerical integrations for a given excitation frequency value. The dynamic coefficients were used as input data for studying the linear stability of the rotor-bearing system.

As a first approximation, the deformations of the thin elastic liner made from either compressible or almost incompressible materials were assumed to be proportional to the fluid-film pressures. The calculations were generally performed for the following cases:

1. rigid bearing liner; 2. only static deformation; 3. both static and dynamic deformations. the stationary X, Y, Z coordinate system is located at the center of the rigid bearing O b . The circumferential coordinate θ is measured from the negative ε axis (line of centers). The journal with radius R is assumed to be rigid and rotates with a constant angular velocity ω about the Z axis. The static position of its geometric center is defined by the steady-state eccentricity e 0 = Cε 0 and the steady-state attitude angle φ 0 . The radial clearance when the journal and bearing circles are concentric of the rigid bearing is given by C, and the steady-state film thickness including the static radial deformation of the bearing liner U 0 is measured by h 0 .The steady-state film thickness can be found using the cosine rule of triangle (O b -O j 0 -M):

THEORETICAL ANALYSIS

(R + h 0 ) 2 = (R + C + U 0 ) 2 + e 2 0 + 2 (R + C + U 0 ) e 0 cos θ [1]
Expanding this equation, dividing by R 2 , and neglecting second-order terms in h 0 /R, C/R, e 0 /R ,andU 0 /R yields

h 0 = C (1 + ε 0 cos θ) + U 0 , [2]
where

U 0 = (χ -1) (χ + 1) t h G p 0 . [3]
In Eq. [3], t h is the thickness of the bearing liner, p 0 is the steadystate hydrodynamic pressure,

G = E 2 (1 + σ)
is the shear modulus known as the Coulomb's elasticity modulus, and χ is the elasticity parameter depending on the Poisson's ratio σ (χ = 3 -4σ for the plane strain elasticity problem, and χ = 3-σ 1+σ for the plane stress problem).

Assuming plane strain (i.e., χ = 3 -4σ) and compressible materials case (σ< 1 /2), the radial deformation field of a thin elastic liner fixed on a rigid backing can be calculated with a sufficient accuracy by

U 0 = σ 0 t h E p 0 , [4a]
where

σ 0 = (1+σ)(1-2σ) 1-σ
, and E and σ are the Young's modulus and the Poisson's ratio of the bearing liner, respectively. Eq. [4a] can be rewritten as

U 0 = L 0 p 0 , [4b]
where Note that Eq. [4a] is obtained from Eq. [3] by replacing the elasticity parameter χ and the shear modulus G with their expressions.

L 0 = σ 0 th E is a scalar compliance operator. θ φ W Y φ h ε X Fluid-film C ε O j x Rigid journal . ω Fluid-liner interface S t a t i c f l u i d - f i l m p
For the rigid bearing liner (Fig. 3), Eq. [2] reduces to

h 0 = C(1 + ε 0 cos θ).
[5]

The simplified compressible thin elastic liner model (Eq. [4]) has been successfully applied by several investigators for EHD analyses of single-layered journal bearings (5, 6). Compared to the finite element method, the main advantage of this model is its low cost of space memory and CPU-time consumption as well as its easy implementation. However, the thin elastic liner model ceases to be valid when the bearing-liner Poisson's ratio becomes greater than 0.4, i.e., for almost incompressible materials, and in the case of thick liners. (See References (14, 16) for the detailed demonstrations about the derivation and the validation of thin elastic liner model.) For isoviscous and isovolume lubricants such as water, the pressure field under isothermal laminar flow condition in the clearance space of the journal bearing (Fig. 2) must satisfy the unsteady-state Newtonian Reynolds' equation (Eq. [START_REF] Lahmar | An Optimised Short Bearing Theory for Nonlinear Dynamic Analysis of Turbulent Journal Bearings[END_REF]):

1 R 2 ∂ ∂θ h 3 12 ∂p ∂θ + ∂ ∂z h 3 12 ∂p ∂z = 1 2 µω ∂h ∂θ + µ ∂h ∂t [6]
or its dimensionless form

∂ ∂θ h3 12 ∂ p ∂θ + (R/L) 2 ∂ ∂z h3 12 ∂ p ∂z = 1 2 ∂ h ∂θ + ∂ h ∂t , [7]
where h = h C ,p = p µω(R/C) 2 ,z = z L ,andt = ωt. The above assumptions used on the lubricating fluid and its flow in the thin gap between the journal and the bearing are appropriate for low-viscosity lubricants such as water, producing a small amount of dissipation energy and leading to negligible variations of the viscosity with respect to pressure and temperature. Indeed, the thermal and piezoviscous effects are more significant in the case of very viscous fluids such as oils used in rolling bearings and gears applications that operate at very high pressures.

As we suppose a pinching lubrication process, the hydrodynamic pressure p (θ, z, t) must satisfy the following boundary conditions:

(1) on the bearing sides, p(θ, z =-1/2, t) = p(θ, z = 1/2, t) = 0;

[8]

(2) the pressure is continuous and periodic in the circumferential direction, i. e., p(θ, z, t) = p(θ + 2π, z, t);

[9]

(3) at the leading edge of the cavitation zone θ = θ c (z), the pressure and its gradients vanish, i.e.,

p = ∂ p ∂θ = ∂ p ∂z = 0. [10]
These conditions, which were suggested in the early 1930s and known as the Reynolds or Swift-Stieber cavitation conditions [START_REF] Swift | The Stability of Lubricating Films in Journal Bearings[END_REF][START_REF] Stieber | Das Schwimmlajer, Hydrodynamische Theori des Glutlajers[END_REF], can be satisfied in a direct manner if the Christopherson's algorithm [START_REF] Christopherson | A New Mathematical Method for the Solution of Oil Film Lubrication Problems[END_REF] is employed. This algorithm has been successfully used for very long time (since the 1940s) in the simulation studies of lubrication problems. However, the use of Christopherson's algorithm leads to the determination of an erroneous boundary of film reformation (trailing edge of the cavitation zone). The mass flow continuity is not fulfilled. It should be noted that the Reynolds boundary conditions [10] are very frequently used, giving results comparable to those obtained experimentally or by using the conservation algorithms such as the JFO, Elrod-Adams, and Bayada cavitation models, with the exception of the value of the rate of flow [START_REF] Bayada | Sur Quelques Mod élisations de la Zone de Cavitation en Lubrification Hydrodynamique[END_REF][START_REF] Bayada | équations Variationnelles à Conditions aux Limites P ériodiques[END_REF](23).

In References (23), Dowson and Taylor presented an excellent review of cavitation phenomenon in the hydrodynamic journal bearings, where different cavitation models were largely discussed.

If the journal is excited into a simple harmonic motion of small amplitudes within the compliant bearing at frequency ν, the instantaneous eccentricity ratio and attitude angle may be expressed respectively as

ε = ε 0 + ε e iγ t and φ = φ 0 + φ e iγ t; | ε| ≪ ε 0 , | φ| ≪ φ 0 and i = -1, [11]
where γ = v/ω is the relative excitation frequency, and ε and φ are complex amplitudes of eccentricity ratio and attitude angle, respectively. The corresponding dynamic increment of dimensionless film thickness may be expressed as [START_REF] Jain | A Study on the Dynamic Response of Compliant Shell Journal Bearings[END_REF]

h = h0 + hd + Ũd , [ 12
]
where h0 = h0 C is the dimensionless steady-state film thickness including the steady-state deformation of the bearing-liner (Eq. [2]), hd = ( ε cos θ + ε 0 φ sin θ) e iγ t is the change in fluidfilm thickness due to dynamic movement of the journal, and Ũd = Ũe iγ t = ε Ũε + ε 0 φ Ũφ e iγ t is the dynamic deformation caused by the dynamic increment of film pressure where

i+1,j i-1,j i,j+1 i, -1 0 j i 2 1 2 1 - ∆ ∆ 2
Ũε = ∂ Ũ ∂ε , Ũφ = 1 ε 0 ∂ Ũ ∂φ ,
and Ũ is the complex amplitude.

Of course we suppose here that we are in the case of the application of the principle of superposition. As we assume that we are in small harmonic motion amplitudes and for the deformation in linear situation, we can consider this hypothesis reasonable if we add the additional terms to h0 , i.e., hd + Ũd is small compared to h0 . Of course with the help of nonlinear model- ing, one can more precisely give the significance of small and its limits.

Consequently, the instantaneous value of dimensionless film thickness may be expressed as

h = h0 + ( ε(cos θ + Ũε ) + ε 0 φ(sin θ + Ũφ )) e iγ t.
[13]

The first and second terms on the RHS of Eq. [13] correspond to the steady-state and dynamic components, respectively. The instantaneous value of the dimensionless resultant film pressure is then expressed as

p = p0 + Qe iγ t, [ 14
]
where p0 is the dimensionless steady-state pressure, and Q = ε Qε + ε 0 φ Qφ is the complex dynamic increment of the film pressure, where Qε

= ∂ Q ∂ε and Qφ = 1 ε0 ∂ Q ∂φ .
Note that adding the dynamic deformations Ũε and Ũφ in Eq.

[13] gives a more complete expression of the film thickness than if only static deformation is taken into consideration.

The dynamic deformations are also calculated using the thin elastic liner model, i.e., Ũε = L0 Qε and Ũφ = L0 Qφ , [START_REF] Lahmar | Elastohydrodynamic Analysis of Double-Layered Journal Bearings Lubricated With Couple-Stress Fluids[END_REF] where

L0 = σ 0 Cd th . [16]
In Eq. [START_REF] Boucherit | Misalignment Effects on Steady-State and Dynamic Behavior of Compliant Journal Bearings Lubricated With Couple Stress Fluids[END_REF], Cd = µω(R/C) 3 E is the dimensionless deformation coefficient varying from 0 to ∞,andt h = t h /R is the relative thickness of the bearing liner. Poisson's ratio of the bearing liner [START_REF] Hamrock | Fundamentals of Fluid Film Lubrication[END_REF] σ -0 . 3 5 Substituting Eq. [13] and [14] into Eq. [7] and collecting the zero and the first-order terms for ε and ε 0 φ,as e to fc o upled partial differential equations in p0 , Qε ,a n d Qφ can be obtained:

ℜ(p 0 ) = 1 2 ∂ h0 ∂θ , [17] ℜ( Qε ) =- 3 2 (cos θ + Ũε ) h0 ∂ h0 ∂θ + 1 2 -sin θ + ∂ Ũε ∂θ +iγ cos θ+ Ũε -3 h0 h0 -sin θ + ∂ Ũε ∂θ -∂ h0 ∂θ (cos θ + Ũε ) 12 ∂ p0 ∂θ + (R/L) 2 h0 ∂ Ũε ∂z -cos θ + Ũε ∂ h0 ∂z 12 ∂ p0 ∂z , [18]
ℜ( Qφ ) =- 3 2 (sin θ + Ũφ ) h0 ∂ h0 ∂θ + 1 2 cos θ + ∂ Ũφ ∂θ +iγ(sin θ + Ũφ ) -3 h0 h0 cos θ + ∂ Ũφ ∂θ -∂ h0 ∂θ sin θ + Ũφ 12 ∂ p0 ∂θ + (R/L) 2 h0 ∂ Ũφ ∂z -sin θ + Ũφ ∂ h0 ∂z 12 ∂ p0 ∂z , [ 19
]
where Note that the application of the first-order perturbation technique, which is considered as a superposition method, leads to a transformation of the transient Reynolds equation (Eq. [7]) into three partial differential equations. It is also noteworthy that Eq. [START_REF] Lahmar | An Optimised Short Bearing Theory for Nonlinear Dynamic Analysis of Turbulent Journal Bearings[END_REF] is a nonlinear equation because h0 depends on p0 , while complex Eq. [START_REF] Swift | The Stability of Lubricating Films in Journal Bearings[END_REF] and [START_REF] Stieber | Das Schwimmlajer, Hydrodynamische Theori des Glutlajers[END_REF] are linear in Qε and Qφ .

ℜ (•) = ∂ ∂θ h3 0 12 ∂ (•) ∂θ + (R/L) 2 ∂ ∂z
The perturbation technique also transforms the boundary conditions of Eq. [8], [9], and [10], i.e., p0 = 0atz =±1/2, [START_REF] Christopherson | A New Mathematical Method for the Solution of Oil Film Lubrication Problems[END_REF] p0 [START_REF] Bayada | équations Variationnelles à Conditions aux Limites P ériodiques[END_REF] where θ c0 is the stationary cavitation angle measured from the line of centers.

(θ = 0, z) = p0 (θ = 2π, z), [21] p0 = ∂ p0 ∂θ = ∂ p0 ∂z = 0atθ = θ c0 (z),
Qε = Qφ = 0atz =±1/2, [23] Qε (θ = 0, z) = Qφ (θ = 2π, z). [24]
Equation [START_REF] Bayada | équations Variationnelles à Conditions aux Limites P ériodiques[END_REF] expresses the steady-state Reynolds boundary conditions by assuming that the perturbations due to the journal vibrations do not affect the active zone extent. The first-order dynamic equations are therefore solved in the film domain iteratively defined by applying the Christopherson's method in the stationary case.

It is interesting to note that Eq. [START_REF] Bayada | équations Variationnelles à Conditions aux Limites P ériodiques[END_REF] is derived from Eq. [10] by expanding the normalized pressure p (θ, z) in a first-order Taylor series in the vicinity of (θ c0 , z0 ), i.e., p (θ, z) = p (θ c0 + θ, z0 + z) ≈ p (θ c0 , z0 )

+ ∂ p ∂θ 0 θ + ∂ p ∂z 0 z = 0. [25]
Equation [14] can be written as p = p0 + p1 , [START_REF] Fr Êne | Hydrodynamic Lubrication: Bearings and Thrust Bearings[END_REF] where | p1 |=| Qe iγ t| << p0 . Substituting Eq. [START_REF] Fr Êne | Hydrodynamic Lubrication: Bearings and Thrust Bearings[END_REF] into Eq. [START_REF] Lund | Review of the Concept of Dynamic Coefficients for Fluid Film Journal Bearings[END_REF], we obtain

p(θ, z) ≈ p0 (θ c0 , z0 ) + p1 (θ c0 , z0 ) + ∂ p0 ∂θ 0 θ + ∂ p0 ∂z 0 z = 0. [27]
This relation shows that the Reynolds boundary conditions read as [START_REF] Hamrock | Fundamentals of Fluid Film Lubrication[END_REF] p1 = 0, i.e., Qε = Qφ = 0.

p0 = ∂ p0 ∂θ = ∂ p0 ∂z = 0,
[29]

The complex distributions Qε and Qφ are obtained from Eq. [START_REF] Swift | The Stability of Lubricating Films in Journal Bearings[END_REF] and [START_REF] Stieber | Das Schwimmlajer, Hydrodynamische Theori des Glutlajers[END_REF], from which the eight dynamic coefficients in the ε, φ coordinate system can be calculated by integrations:

A εε + iγB εε =- 2π 0 1/2 -1/2 Qε cos θ dzdθ; A εφ + iγB εφ =- 2π 0 1/2 -1/2 Qφ cos θ dzdθ, [29a] A φε + iγB φε =- 2π 0 1/2 -1/2 Qε sin θ dzdθ; A φφ + iγB φφ =- 2π 0 1/2 -1/2
Qφ sin θ dzdθ,

where

A αβ = a αβ C 3 µωR 3 L and B αβ = b αβ C 3 µR 3 L ; (α, β) = (ε, φ) .
The first subscript of A αβ denotes the direction of the hydrodynamic force, and the second one denotes the direction of displacement. For B αβ , the first and second subscripts refer to the component of the hydrodynamic force and the direction of the velocity, respectively.

The dimensionless stiffness and damping coefficients in the X, Y coordinate system can be calculated from the stiffness and damping coefficients in the ε, φ coordinate system according to

A XX A XY A YX A YY = R T A εε A εφ A φε A φφ R, [30a] B XX B XY B YX B YY = R T B εε B εφ B φε B φφ R, [30b]
where the rotation matrix is defined by

R = cos φ 0 sin φ 0 -sin φ 0 cos φ 0 .
The dimensionless stability parameters (critical mass Mc and whirl frequency ratio γ c ) of the journal-bearing system can be determined at the threshold of instability by applying the Lund's stability criterion [START_REF] Lund | Review of the Concept of Dynamic Coefficients for Fluid Film Journal Bearings[END_REF]. The procedure of calculation of the stability parameters is outlined in the next section.

NUMERICAL ANALYSIS

In the present investigation, the partial differential equations (Eq. [START_REF] Lahmar | An Optimised Short Bearing Theory for Nonlinear Dynamic Analysis of Turbulent Journal Bearings[END_REF], Eq. [START_REF] Swift | The Stability of Lubricating Films in Journal Bearings[END_REF], Eq. [START_REF] Stieber | Das Schwimmlajer, Hydrodynamische Theori des Glutlajers[END_REF]) are solved satisfying the corresponding boundary conditions by finite difference method with successive over-relaxation scheme. Because of the axial symmetry of the bearing, only the half bearing is divided into N θ × N z equal rectangular cells with an area equal to θ × z, where

θ = 2π N θ and z = 0.5 N z .
are the mesh sizes in the circumferential and axial directions, respectively (Fig. 4).

Using the accurate central difference approximation for the pressures' partial derivatives, Eq. [START_REF] Lahmar | An Optimised Short Bearing Theory for Nonlinear Dynamic Analysis of Turbulent Journal Bearings[END_REF], Eq. [START_REF] Swift | The Stability of Lubricating Films in Journal Bearings[END_REF], and Eq. [START_REF] Stieber | Das Schwimmlajer, Hydrodynamische Theori des Glutlajers[END_REF] become the following: 1. Static governing equation:

p0ij = a (0) ij p0i+1,j + b (0) ij p0i-1,j + c (0) ij p0i,j+1 + d (0) ij p0i,j-1 -e (0) ij , [31]
where

a (0) ij = h 3 0 ij ( θ) 2 + 3 h2 0 ij ∂ h0 ∂θ ij 2 θ 2h 3 0 ij ( θ) 2 + 2 (R/L) 2 h 3 0 ij ( z) 2 ; b (0) ij = h 3 0 ij ( θ) 2 - 3 h2 0 ij ∂ h0 ∂θ ij 2 θ 2h 3 0 ij ( θ) 2 + 2 (R/L) 2 h 3 0 ij ( z) 2 ; c (0) ij = (R/L) 2 h 3 0 ij ( z) 2 + 3 h2 0 ij ∂ h0 ∂z ij 2 z 2h 3 0 ij ( θ) 2 + 2 (R/L) 2 h 3 0 ij ( z) 2 ; d (0) ij = (R/L) 2 h 3 0 ij ( z) 2 - 3 h2 0 ij ∂ h0 ∂z ij 2 z 2h 3 0 ij ( θ) 2 + 2 (R/L) 2 h 3 0 ij ( z) 2 ; e (0) ij = 6 ∂ h0 ∂θ ij 2h 3 0 ij ( θ) 2 + 2 (R/L) 2 h 3 0 ij ( z) 2 ;a n d ∂ h0 ∂θ ij ≈ h0i+1,j -h0i-1,j 2 θ ; ∂ h0 ∂z ij ≈ h0i,j+1 -h0i,j-1 2 z ;
2. Dynamic governing equations:

Qεij = a (ε) ij Qεi+1,j + b (ε) ij Qεi-1,j + c (ε) ij Qεi,j+1 + d (ε) ij Qεi , j -1 -e (ε) ij , [32] Qφij = a (φ) ij Qφi+1,j + b (φ) ij Qφi-1,j + c (φ) ij Qφi,j+1 + d (φ) ij Qφi,j-1 -e (φ) ij , [33]
where

a (ε) ij = h 3 0 ij ( θ) 2 + 3 h2 0 ij ∂ h0 ∂θ ij 2 θ + L0 2 θ 3 h2 0ij ∂ p0 ∂θ ij -6 2h 3 0 ij ( θ) 2 + 2 (R/L) 2 h 3 0 ij ( z) 2 + L0 -18 ∂ h0 ∂θ ij h0 ij + 12iγ + 3 h0ij ∂ h0 ∂θ ij ∂ p0 ∂θ ij + 3 (R/L) 2 h0ij ∂ h0 ∂z ij ∂ p0 ∂z ij b (ε) ij = h 3 0 ij ( θ) 2 - 3 h2 0 ij ∂ h0 ∂θ ij 2 θ + L0 2 θ -3 h2 0ij ∂ p0 ∂θ ij + 6 2h 3 0 ij ( θ) 2 + 2 (R/L) 2 h 3 0 ij ( z) 2 + L0 -18 ∂ h0 ∂θ ij h0 ij + 12iγ + 3 h0ij ∂ h0 ∂θ ij ∂ p0 ∂θ ij + 3 (R/L) 2 h0ij ∂ h0 ∂z ij ∂ p0 ∂z ij c (ε) ij = (R/L) 2 h 3 0 ij ( z) 2 + 3 h2 0 ij ∂ h0 ∂z ij 2 z + 3 L0 h2 0 ij ∂ p0 ∂z ij 2 z 2h 3 0 ij ( θ) 2 + 2 (R/L) 2 h 3 0 ij ( z) 2 + L0 -18 ∂ h0 ∂θ ij h0 ij + 12iγ + 3 h0ij ∂ h0 ∂θ ij ∂ p0 ∂θ ij + 3 (R/L) 2 h0ij ∂ h0 ∂z ij ∂ p0 ∂z ij d (ε) ij = (R/L) 2 h 3 0 ij ( z) 2 - 3 h2 0 ij ∂ h0 ∂z ij 2 z - 3 L0 h2 0 ij ∂ p0 ∂z ij 2 z 2h 3 0 ij ( θ) 2 + 2 (R/L) 2 h 3 0 ij ( z) 2 + L0 -18 ∂ h0 ∂θ ij h0 ij + 12iγ + 3 h0ij ∂ h0 ∂θ ij ∂ p0 ∂θ ij + 3 (R/L) 2 h0ij ∂ h0 ∂z ij ∂ p0 ∂z ij e (ε) ij = -18 cos θ i ∂ h0 ∂θ ij h0 ij -6sinθ i + 12iγ cos θ i + 3 h2 0ij sin θ i ∂ p0 ∂θ ij + 3 h0ij ∂ h0 ∂θ ij ∂ p0 ∂θ ij cos θ i + 3 (R/L) 2 h0ij ∂ h0 ∂z ij ∂ p0 ∂z ij cos θ i 2h 3 0 ij ( θ) 2 + 2 (R/L) 2 h 3 0 ij ( z) 2 + L0 -18 ∂ h0 ∂θ ij h0 ij + 12iγ + 3 h0ij ∂ h0 ∂θ ij ∂ p0 ∂θ ij + 3 (R/L) 2 h0ij ∂ h0 ∂z ij ∂ p0 ∂z ij a (φ) ij = a (ε) ij ; b (φ) ij = b (ε) ij ; c (φ) ij = c (ε) ij ; d (φ) ij = d (ε) ij ; e (φ) ij = -18 sin θ i ∂ h0 ∂θ ij h0 ij + 6cosθ i + 12iγ sin θ i -3 h2 0ij cos θ i ∂ p0 ∂θ ij + 3 h0ij ∂ h0 ∂θ ij ∂ p0 ∂θ ij sin θ i + 3 (R/L) 2 h0ij ∂ h0 ∂z ij ∂ p0 ∂z ij sin θ i 2h 3 0 ij ( θ) 2 + 2 (R/L) 2 h 3 0 ij ( z) 2 + L0 -18 ∂ h0 ∂θ ij h0 ij + 12iγ + 3 h0ij ∂ h0 ∂θ ij ∂ p0 ∂θ ij + 3 (R/L) 2 h0ij ∂ h0 ∂z ij ∂ p0 ∂z ij and ( ∂ p0 ∂θ ) ij ≈ p0i+1,j -p0i-1,j 2 θ ;( ∂ p0 ∂z ) ij ≈ p0i,j+1 -p0i,j-1 2 z ; i = -1.
Applying the successive over-relaxation algorithm, Eq. [31], Eq. [32], and Eq. [33] take the following form:

p(k+1) 0ij = (1 -)p (k) 0ij + a (0) ij p(k) 0i+1,j + b (0) ij p(k+1) 0i-1,j + c (0) ij p(k) 0i,j +1 + d (0) ij p(k+1) 0i,j -1 -e (0) ij [34a] Q(k+1) εij = (1 -) Q(k) εij + a (0) ij Q(k) εi+1,j + b (0) ij Q(k+1) εi-1,j + c (0) ij Q(k) εi,j +1 + d (0) ij Q(k+1) εi,j -1 -e (ε) ij [34b] Q(k+1) φij = (1 -) Q(k) φij + a (0) ij Q(k) φi+1,j + b (0) ij Q(k+1) φi-1,j + c (0) ij Q(k) φi,j +1 + d (0) ij Q(k+1) φi,j -1 -e (φ) ij [34c]
where k and are the number of iterations and the relaxation factor, respectively. Different mesh sizes were tried, and a mesh size with 91 × 21 grid was adopted. This size gives more accurate results with a rapid rate of convergence when the value of is equal to 1.80. This optimal value, which is predicted by numerical experimentation, will always lie between 1.0 and 2.0 as shown in Fig. 5.

The solution of elastohydrodynamic problem under static and dynamic conditions is obtained by an iterative numerical procedure. The following steps are performed: 5. Solve the zeroth-order equation (Eq. [START_REF] Lahmar | An Optimised Short Bearing Theory for Nonlinear Dynamic Analysis of Turbulent Journal Bearings[END_REF]) for the static pressure field p(n+1) 0ij using the finite difference method with successive over-relaxation scheme. The iterative process is repeated until either the pressure convergence criterion is achieved or the maximum number of iterations is exceeded. 6. Update the dimensionless film thickness profile according to h(n+1

) ij = h+ 0ij L0 (1 -α) p(n) 0ij + α p(n+1) 0ij ,
where α is an under-relaxation factor that ranges from 0 to 1.

7. Check the static film pressure convergence criterion

1 N i,j p(n+1) 0ij -p(n) 0ij p(n+1) 0ij
≤ 10 -5 ,w h e r eN is the total number of nodes for which the pressure is positive. If convergence is not achieved, then increment the iteration number n by1andreturn to step 5. 8. Code the nodes for which the static pressure is positive and calculate the static pressure and film thickness derivatives at each node. This step is necessary to solve the first-order equations Eq. [START_REF] Swift | The Stability of Lubricating Films in Journal Bearings[END_REF] and Eq. [START_REF] Stieber | Das Schwimmlajer, Hydrodynamische Theori des Glutlajers[END_REF]. 12. Calculate the steady-state hydrodynamic force components and the attitude angle by means of numerical integrations:

F ε0 F φ0 = 2 1/2 0 θc 0 0 p0 cos θ sin θ dθdz, andφ 0 = Ta n -1 -Fφ0 Fε0 .
[35] 13. Solve the first-order equations Eq. [START_REF] Swift | The Stability of Lubricating Films in Journal Bearings[END_REF] and Eq. [START_REF] Stieber | Das Schwimmlajer, Hydrodynamische Theori des Glutlajers[END_REF] by the finite difference method with successive over-relaxation scheme to obtain the complex dynamic pressures Qεij and Qφij . It should be noted that the calculations are performed for each coded node belonging to the stationary active region without vanishing the computed negative pressure terms. The pressure convergence criterion is similar to that used for the calculation of static pressure. 14. Calculate the eight fluid-film dynamic coefficients (Eq. [START_REF] Piteau | Banc d'Essais de Paliers: Mesure des Coefficients Dynamiques d'un palier circulaire[END_REF] and Eq. [30]), from which the equivalent stiffness coefficient A eq and whirl frequency ratio γ c at stability threshold are obtained:

A eq = A XX B YY + A YY B XX -A XY B YX -A YX B XY B XX + B YY , γ 2 c = (A XX -A eq )(A YY -A eq ) -A XY A YX B XX B YY -B XY B YX .
Note that there is dependence between dynamic coefficients and frequency ratio when the deformations due to the dynamic pressures at the fluid-film-bearing-liner interface are considered. 15. Test the sign of γ 2 c :I fγ 2 c < 0, then the equilibrium position is stable [START_REF] Fr Êne | Hydrodynamic Lubrication: Bearings and Thrust Bearings[END_REF]; otherwise, γ 2 c > 0. The dimensionless critical mass is then calculated:

Mc = A eq γ 2 c .
16. Test the calculated value of γ c with γ:Ifγ c does not coincide with the estimatedγ,t h e ns e tγ ← γ c , and return to step 13 until convergence is reached with a sufficient accuracy, i.e., γγ c γ c ≤ 10 -4 .

RESULTS AND DISCUSSION

Based on the analysis described in the present article, a computer code was developed to study the effects of both static and dynamic deformations on the dynamic behavior of a compliant journal bearing using the linearized theory.

Validation

To validate the static and dynamic results obtained from the computer program, we compared the Sommerfeld number, attitude angle, stiffness and damping coefficients, and the dimensionless stability parameters (critical mass and whirl frequency ratio) for a rigid finite-length journal bearing (R/L = 0.5a n d Cd = 0.) with those published by Constantinescu, et al. [START_REF] Constantinescu | Sliding Bearings[END_REF]. The results were obtained for three values of static eccentricity ratio (ε = 0 0.0962, 0.5374, and 0.8349). As can be seen in Table 1, the results are in good agreement even at high values of static eccentricity ratio. Furthermore, all dynamic results presented in [START_REF] Constantinescu | Sliding Bearings[END_REF] for R/L = 0.25 and 1 were recovered.

Effects of Elastic Deformations on the Steady-State and Dynamic Fluid-Film Pressures

Table 2 shows the details of bearing geometry and operating conditions of a sample problem investigated in the present study. The dimensionless parameters calculated from numerical values given in Table 2 are as follows: Cd = 0.04, th = 0.40, R/L = 0.5, C R = 2×10 -3 , which are the deformation coefficient, the relative thickness of the bearing liner, and the aspect and clearance ratios of the journal bearing, respectively.

Figure 6 depicts the steady-state pressure profiles and contours calculated in the half bearing for a highly loaded journal bearing operating at ε 0 = 0.9. It is observed that the effect of the bearing-liner elasticity leads to a spreading of the pressure distribution in the circumferential direction of the journal bearing and to an important reduction of the peak pressure inducing a reduction of the journal-bearing carrying capacity.

Figure 7 shows the fluid film thickness distributions obtained for rigid and compliant bearings. In the rigid case, the film thickness has a sinusoidal shape whose minimum value occurs at θ = 180 degrees regardless of the cross-section of bearing. The film thickness profile calculated by taking into account the compliance of the bearing liner is different from that obtained in the rigid case and presents maximum and minimum values at the midplane section and the free edges of bearing, respectively. The increasing of the film thickness at the midplane section of the journal bearing explains the pressure drop.

Figure 8 compares the real and imaginary parts of complex dynamic pressures Qε and Qφ calculated in the half bearing for rigid and compliant bearing liners and ε 0 = 0.9. As found under steady-state conditions, the elastic deformations also affect the maximum value of both dynamic pressures, and this effect is more pronounced when the dynamic deformations are considered in addition to the static ones.

Influence of Elastic Deformations on the Sommerfeld Number

Evolutions of the dimensionless Sommerfeld number S (also commonly known as the load number) versus the steady-state eccentricity ratio for rigid and elastic bearing liners are depicted in Fig. 9. This number, which represents the inverse of the applied load (S → 0asε 0 → 1), depends on several design parame-ters, namely the absolute viscosity of lubricant µ, the geometric characteristics of the journal bearing (R, L and C) and the operating conditions ω and W 0 . It is clearly shown that the elasticity effect on the Sommerfeld number becomes significant when the journal bearing operates at eccentricities greater than 0.80, i.e., when the journal bearing system is moderately or highly loaded.

Influence of Dynamic Deformations on the Dynamic Properties

The graphical results were obtained for three cases:

-rigid bearing-liner case (doted lines), -compliant bearing liner when only static deformation is considered (dashed lines), -compliant bearing liner when both static and dynamic deformations are considered (solid lines).

Figure 10 shows the variations of the four synchronous dimensionless fluid-film stiffness coefficients with the steady-state eccentricity ratio obtained for both rigid and compliant bearing liners. It is observed that when the dynamic deformation is ignored, the effect of static deformation on stiffness coefficients is significant for eccentricities greater than 0.7.

Consideration of dynamic distortions changes the evolution of the direct stiffness coefficients regardless of the running eccentricity.

The stiffening effect shown in that same figure comes, in our opinion, from the additional pressure (fig. 8) coming from the dynamic deformation. The dynamic movement of the shaft creates additional dynamic pressure leading to an increase of stiffness. At high eccentricity, this phenomenon is counterbalanced by the extent of the positive pressure zone, and thus the maximum pressure magnitude decreases, and the pressure appearance is more flat.

Note that the change of sign from negative to positive for the cross-coupling stiffness coefficient A YX plays a significant role in the stability of system. The figure clearly shows that the A YX change of sign, calculated when both static and dynamic deformations are considered, occurs at an eccentricity ratio around 0.60.

Figure 11 shows the four synchronous dimensionless fluid-film damping coefficients versus the steady-state eccentricity ratio calculated for γ = 1. It can be observed that the four damping coefficients generally decrease with the journal eccentricity, i.e., when the system is heavily loaded.

Consideration of dynamic distortions changes the evolution of the cross-coupling damping coefficients whatever the running eccentricity is.

Especially, the two cross-coupling damping coefficients' values B XY and B YX differ from each other significantly when dynamic deformation is taken into consideration instead of being equal as predicted by the classical hydrodynamic lubrication theory (see the detailed demonstration in Appendix A) or by the EHD approach when the dynamic deformations are neglected. It should be specified that the dynamic deformation might be one of the main causes of the significant discrepancy between the values of the two cross-coupling damping coefficients determined experimentally by other researchers [START_REF] Piteau | Banc d'Essais de Paliers: Mesure des Coefficients Dynamiques d'un palier circulaire[END_REF].

Influence of Dynamic Deformations on the Stability Parameters

According to the obtained stiffness and damping coefficients, the critical mass and the whirl ratio of the rigid rotor-bearing system are determined and plotted in Fig. 12. These figures show variations of the aforementioned parameters against the steadystate eccentricity ratio. It is observed that significant reduction of the instability boundary can occur if the effects of both static and dynamic deformations are considered in the calculations because of the enhancement of the fluid-film stiffness coefficients. Figure 13 depicts the variations of fluid-film dynamic coefficients obtained for γ = γ c versus Sommerfeld number. Exciting the journal into a harmonic motion of small amplitude at frequency ν gives steady state pressure and complex dynamic pressures Q ε and Q. Taking into account the dynamic deformation leads to an increase of the real part of Q ε and Q (see Eq. [START_REF] Swift | The Stability of Lubricating Films in Journal Bearings[END_REF] and Eq. [START_REF] Stieber | Das Schwimmlajer, Hydrodynamische Theori des Glutlajers[END_REF]) and the magnitude of the imaginary part. This additional pressure increases the stiffness coefficients by increasing the attitude angle for a given eccentricity. For direct stiffness coefficients, no effects occur until ε 0 = 0.6 ≈ 0.7. For crossed coefficients, taking into account the imaginary part plays an immediate role on attitude angle.

Influence of Excitation Frequency on the Stiffness and Dynamic Coefficients

Figures 14 and 15 present respectively dimensionless fluid-film stiffness and damping coefficients as functions of the relative excitation frequency for ε 0 = 0.5. The results show that the dynamic coefficients calculated for rigid and compliant bearing liners considering solely static deformations are independent of the excitation frequency. When both static and dynamic deformations are considered, these coefficients stay mainly constant for low magnitudes of the frequency excitation. However, we observe a nonlinear evolution of these coefficients when considering highfrequency values, which can happen in some machines with gear elements.

CONCLUSIONS

According to the results obtained, the following conclusions can be drawn:

1. The elastic deformations affect the maximum value of both dynamic pressures, and this effect is more pronounced when the dynamic deformations are considered in addition to the static ones. 2. Consideration of dynamic distortions changes the evolution of the direct stiffness coefficients and the cross-coupling damping coefficients, regardless of the running eccentricity. 3. The effects of dynamic deformations of the bearing liner on dynamic performance characteristics and stability parameters are nonnegligible, especially for high values of the operating eccentricity ratio and/or low-elasticity modulus bearing liners. 4. The cross-coupling damping coefficients differ from each other significantly when dynamic deformation is included. 5. It is observed that significant reduction of the instability boundary can occur if the effects of both static and dynamic deformations are considered. 6. At high values of excitation frequency, the fluid-film dynamic coefficients become nonlinear as functions of this frequency.

APPENDIX A SYMMETRY OF THE FLUID-FILM DAMPING MATRIX

To demonstrate the property of symmetry of the rigid hydrodynamic journal bearings damping matrix, it is convenient to write the Reynolds equation (Eq.

[6]) in the form 1 R 2 ∂ ∂θ h 3 12µ ∂p ∂θ + ∂ ∂z h 3 12µ ∂p ∂z = 1 2 (ω -2 φ) ∂h ∂θ + 2ė cos θ , [A1]
where

h = C (1 + ε cos θ) , ∂h ∂θ =-Cε sin θ, and ( • ) = d () dt ,
or in its dimensionless form,

ℜ ( p) = 1 2 ∂ h ∂θ + ε ′ cos θ 1 -2φ ′ , [A2] where h = h C , p = p µω (R/C) 2 (1 -2φ ′ ) , z = z L , t = ωt, () ′ = d () dt = 1 ω ( • ), and
ℜ (•) = ∂ ∂θ h3 12 ∂ ∂θ (•) + (R/L) 2 ∂ ∂z h3 12 ∂ ∂z (•) .
The radial and tangential fluid film forces are expressed as

F ε F φ = L/2 -L/2 θ2 θ1 p cos θ sin θ Rdθdz = ηω (1 -2φ ′ ) 1/2 -1/2 θ2 θ1 p cos θ sin θ dθdz, [A3]
where η = µRL(R/C) 2 ,andθ 1 and θ 2 are the film reformation and film rupture boundaries, respectively. The positive (uncavitated) pressure region lies between θ 1 and θ 2 . These angular coordinates depend on the position of the journal center within the bearing. Thus, the fluid-film reaction forces are general functions of the journal center displacements and velocities, i.e.,

F ε e,φ,ė, φ F φ e,φ,ė, φ = ηω (1 -2φ ′ ) Fε (ε, φ, ε ′ ,φ ′ ) Fφ (ε, φ, ε ′ ,φ ′ ) , [A4]
where e = Cε,and

Fε Fφ = 1/2 -1/2 θ2 θ1 p cos θ sin θ dθdz.
The assumption of small amplitude motions about an equilibrium position allows expressing the bearing reaction forces as a Taylor series expansion around the static journal position (ε 0 ,φ 0 ), i.e., Note that (C ε, Cε 0 φ) are the radial and tangential displacements of journal center in the (ε, φ) coordinate system, and Cε, Cε 0 φ are the radial and tangential velocities, respectively. According to Eq. [A4], we can write

f ε f φ = F ε -F ε0 F φ -F φ0 = e ∂ ∂e + φ ∂ ∂φ + ė ∂ ∂ė + φ ∂ ∂ φ F ε F φ (e=e0,φ=φ0,ė= φ=0) = ε ∂ ∂ε + φ ∂ ∂φ + ε ∂ ∂ε + φ ∂ ∂ φ F ε F φ (ε=ε0,
f ε f φ = F ε -F ε0 F φ -F φ0 = ηω (1 -2φ ′ ) ε ∂ ∂ε + φ ∂ ∂φ + ε ′ ∂ ∂ε ′ + φ ′ ∂ ∂φ ′ Fε Fφ (ε=ε0,φ=φ0,ε ′ =φ ′ =0) -2ηωφ ′ Fε Fφ (ε=ε0,φ=φ0,ε ′ =φ ′ =0) [A6]
It is assumed that the perturbed pressure field due to small amplitude journal motions about the equilibrium position (ε 0 ,φ 0 ) does not affect θ 1 andθ 2 delimiting the active film region. So, the normalized Reynolds equation [A2] is quite linear. In this situation, the hydrodynamic forces do not depend on the angular velocity φ ′ , i.e.,

∂ Fε ∂φ ′ ε ′ =0 = ∂ Fφ ∂φ ′ ε ′ =0
= 0, and Eq. [A6] becomes:

f ε f φ = ηω ε ∂ ∂ε + φ ∂ ∂φ + ε ′ ∂ ∂ε ′ Fε Fφ (ε=ε0,φ=φ0,ε ′ =φ ′ =0) -2ηωφ ′ Fε Fφ (ε=ε0,φ=φ0,ε ′ =φ ′ =0) [A7]
Note that Fε Fφ (ε=ε0,φ=φ0,ε ′ =φ ′ =0) = Fε0 Fφ0 = -W 0 cos φ 0 W 0 sin φ 0 , which is the equilibrium equation of the journal, W 0 being the appliedstaticloadintheX direction (e.g., the weight of rotor per bearing). According to Eq. [A5] and Eq. [A7], the stiffness and dynamic coefficients can be expressed as The dimensionless dynamic coefficients can be expressed according to the definition

A ij = a ij C W 0 , and
B ij = b ij Cω W 0 ,
where (i, j ) = (ε, φ), or

A εε A εφ A φε A φφ =-πS ⎡ ⎢ ⎢ ⎢ ⎣ ∂ Fε ∂ε 0 ∂ Fε ε 0 ∂φ 0 ∂ Fφ ∂ε 0 ∂ Fφ ε 0 ∂φ 0 ⎤ ⎥ ⎥ ⎥ ⎦ ; B εε B εφ B φε B φφ =- ⎡ ⎢ ⎢ ⎢ ⎣ πS ∂ Fε ∂ε ′ 0 2cosφ 0 ε 0 πS ∂ Fφ ∂ε ′ 0 -2sinφ 0 ε 0 ⎤ ⎥ ⎥ ⎥ ⎦ , [A10]
where

S = µωRL R C 2 πW 0 = 1 π W0
is the dimensionless Sommerfeld number.

In the following, we will demonstrate that the cross-damping coefficients of hydrodynamic journal bearings with rigid-liner are always identical. According to Eq. [A10], we have [A13]

Taking into account the linearity of the Reynolds equation Eq.

[A2] with respect to ε ′ ,wecanwrite p = p0 + ε ′ p1 ,where p0 represents the nondimensional steady-state pressure field, and p1 represents the dynamic pressure field added to p0 for ε ′ = 1. Equation [A13] becomes

∂ Fφ ∂ε ′ = 1/2 -1/2 θ2 θ1 p1 sin θdθdz [A14]
Note that p1 and p0 are the solutions of the following equations:

ℜ ( p1 ) = cos θ, [A15] ℜ ( p0 ) = 1 2 ∂ h ∂θ =- ε 0 sin θ 2 . [A16]
From Eq. [A16], we can obtain sin θ =-2 ε 0 ℜ ( p0 ) . we get

B φε =- 2cosφ 0 ε 0 .
Thus, the cross-damping coefficients are quite identical: (B φε = B εφ ).

For the plain journal bearings (without axial groove), the expression and the calculation of dynamic coefficients are simplified because the pressure field does not depend on the attitude angle due to the circumferential symmetry. For this particular case, and according to the relations stated above, we have A εφ = sin φ 0 ε 0 ; A φφ = cos φ 0 ε 0 ; B φφ = 2A εφ ;a n dB εφ =-2A φφ .

[A21]

It should be noted that the pressure boundary conditions related to the environment and to the flow of lubricant in the clearance space of the journal bearing appreciably modify the values of the dynamic coefficients; some of them vanish or become negative.

=

  AbbreviationsEHD= Elasto-Hydro-Dynamic JFO = Jakobsson Floberg Olsson Im( ) = Imaginary part of ( ) Real( ) = Real part of ( )

Fig. 1 -

 1 Fig. 1-View of a typical water-lubricated propeller shaft bearing system.

Figure 2 Fig. 2 -

 22 Figure 2 schematically shows the aligned compliant plain journal bearing at its static equilibrium position with an elastically deformed bearing liner where the undeformed (rigid) configuration is represented by the circle of radius (R + C). The origin of

Fig. 3 -

 3 Fig. 3-Geometry of a rigid plain journal bearing.

Fig. 4 -

 4 Fig. 4-Finite difference grid for the solution of the static and dynamicReynolds' equations.

Fig. 5 -

 5 Fig. 5-Determination of the optimum value of the over-relaxation factor.

Fig. 6 -

 6 Fig.6-Steady-state pressure distributions for rigid and compliant journal bearings, ε 0 = 0.9.

Fig. 8

 8 Fig. 8-3-D representations of real and imaginary parts of complex dynamic pressures calculated for rigid and compliant journal bearings, ε 0 = 0.9: (a) Rigid bearing liner, (b) Static deformation, (c) Static and dynamic deformations (γ = 1).

Fig. 9 -

 9 Fig. 9-Variation of Sommerfeld number as a function of steady-state eccentricity ratio.

  ε 0 , R L ,t h , Cd ,σ ,N θ , N z ,γ, relaxation factors, convergence criteria, and maximum number of iterations. 2. Compute the undeformed film thickness h0ij from Eq. [5]. 3. Initialize the iteration number n to 0. 4. Initialize the dimensionless static pressure field p(n) 0ij = 0a n d the dimensionless static film thickness profile h(n) ij = h0ij .

Fig. 10 -

 10 Fig. 10-Fluid-film dimensionless synchronous stiffness coefficients as a function of steady-state eccentricity ratio, γ = 1.

Fig. 11 -

 11 Fig. 11-Fluid-film dimensionless synchronous damping coefficients as a function of steady-state eccentricity ratio, γ = 1.

Fig. 13 -Fig. 14 -

 1314 Fig. 12-Stability parameters as functions of steady-state eccentricity ratio.

Fig. 15 -

 15 Fig. 15-Dimensionless damping coefficients as a function of relative excitation frequency, ε 0 = 0, 5.

- a εε a εφ a φε a φφ C ε Cε 0 φ - b εε b εφ b φε b φφ

  φ=φ0,ε= φ=0) =-film bearing stiffness and damping, respectively.

  . [A15], Eq. [A20] takes the following form: W0 cos φ 0 , andS = 1 π W0 .

TABLE 1 -

 1 COMPARISON OF STEADY-STATE AND DYNAMIC PERFOR-

	ε 0	0.0962	0.5374	0.8349	Note
	S	1.03540	0.1549	0.0345	(1)
		1.03610	0.1559	0.0350	(2)
	φ 0 (deg)	84.03	56.07	33.03	(1)
		84.07	56.22	33.59	(2)
	A XX	1.004	2.154	6.585	(1)
		1.100	2.140	6.467	(2)
	A XY	10.630	3.181	4.302	(1)
		10.613	3.182	4.237	(2)
	A YX	-10.250	-0.886	0.837	(1)
		-10.233	-0.887	0.757	(2)
	A YY	2.157	2.002	1.961	(1)
		2.144	1.994	1.901	(2)
	B XX	21.149	5.867	7.383	(1)
		21.133	5.874	7.434	(2)
	B YX	2.112	2.093	2.020	(1)
		2.147	2.085	2.061	(2)
	B YY	20.703	3.110	1.314	(1)
		20.694	3.117	1.372	(2)
	Mc	6.416	6.757	Stable position	(1)
		6.422	6.757	"	(2)
	γ c	0.501	0.474	Stable position	(1)
		0.500	0.473	"	(2)

MANCE CHARACTERISTICS FOR A RIGID FINITE-LENGTH PLAIN JOUR-

NAL BEARING ( R L = 0, 5)

TABLE 2 -

 2 GEOMETRIC CHARACTERISTICS AND OPERATING CONDI-

	TIONS OF THE COMPLIANT JOURNAL BEARING ( R L = 0, 5)
	Parameter	Symbol Unit	Value
	Bearing length	Lm	0.050
	Journal radius	Rm	0.025
	Radial clearance	Cm	50×10 -6
	Thickness of the bearing liner	t h	m	0.010
	Dynamic viscosity of lubricant (water at	µ	Pa.s	0.001
	20 • C)
	Angular velocity of the journal	ω	rad.s -1 100 × π
	Young's modulus of the bearing liner	EP a	0.9×10 9
	(Polyethylene high density at 20 • C
	[28])

Published results from Reference[START_REF] Constantinescu | Sliding Bearings[END_REF].

Present work.


Anurag Goel

Angshul Majumdar

Emilie Chouzenoux

Giovanni Chierchia

DEEP CONVOLUTIONAL K-MEANS CLUSTERING

Keywords: Convolutional Neural Network, K-means Clustering, Convolutional Transform Learning

Conventional Convolutional Neural Network (CNN) based clustering formulations are based on the encoder-decoder based framework, where the clustering loss is incorporated after the encoder network. The problem with this approach is that it requires training an additional decoder network; this, in turn, means learning additional weights which can lead to over-fitting in data constrained scenarios. This work introduces a Deep Convolutional Transform Learning (DCTL) based clustering framework. The advantage of our proposed formulation is that we do not require learning the additional decoder network. Therefore our formulation is less prone to over-fitting. Comparison with state-of-the-art deep learning based clustering solutions on benchmark image datasets shows that our proposed method improves over the rest in challenging scenarios where there are many clusters with limited samples.

INTRODUCTION

In a regular feedforward neural network, the data are projected by the network to form the representation. Usually, such a neural network is used for supervised tasks and the generated features are projected through another network to the label / output space. While training, the network weights are learnt by gradient descent / backpropagation. When such a neural network does not have any output, backpropagation leads to a trivial solution -network weights are zeroes and representations are also zeroes. This solution is reached irrespective of the cost function; changing it from Euclidean to Manhattan or KL divergence does not change the trivial solution. This is the reason conventional feedforward neural networks without any output cannot be used for unsupervised representation learning. The situation does not change when an unsupervised clustering loss is incorporated after the unsupervised representation layer. Irrespective of the clustering loss (K-means, Sparse Subspace, Spectral, etc.) [START_REF] Bauckhage | K-means clustering is matrix factorization[END_REF], the same trivial solution (network weights are zeroes and representations are zero) is reached. This is the reason clustering loss cannot be embedded in a standard feedforward neural network. To avoid this issue, prior studies on deep learning based clustering (to be discussed in detail in section II) incorporated clustering losses into the autoencoder framework. Since autoencoder is a self-supervised neural network, backpropagation does not lead to the trivial solution and the framework can be used for unsupervised feature extraction. Embedding a clustering loss after the representation layer of an autoencoder leads to a meaningful non-trivial solution. However, an autoencoder requires learning twice the number of network weights compared to a standard feedforward neural network. This is because they need to learn an encoder and decoder network; a regular feedforward neural network only requires learning the encoder portion. The decoder portion of the autoencoder is not useful for analysis; its sole purpose is to prevent the trivial solution. The requirement of learning twice the number of network weights may lead to over-fitting in data constrained scenarios. Conversely, one could have used the unsupervised framework of the restricted Boltzmann machine (RBM) for embedding clustering losses. These are undirected graphs and do not suffer from the trivial solution. In a sense, RBM is more optimal than autoencoders since the former needs to learn only half the number of parameters compared to the latter. However, the RBM cost function is not mathematically amenable; its training via contrastive divergence does not have the same flexibility as that of backpropagation. Strictly speaking, contrastive divergence can only optimize the RBM cost function approximately. This perhaps had been the main deterrent behind the use of RBM based clustering solutions. The shortcomings of existing solutions are pertinent to the Convolutional Neural Network (CNN) as well. Unless there is an output, CNN ends up in the trivial solution. The only way to prevent the trivial solution is to have deconvolution layers, but then the formulation suffers from the possibility of over-fitting. In the recent past, we have developed the framework of con-volutional transform learning (CTL) [START_REF] Maggu | Convolutional Transform Learning[END_REF] and deep CTL [START_REF] Maggu | Deep Convolutional Transform Learning[END_REF]. This framework is able to learn convolutional filters in an unsupervised fashion without leading to the trivial solution. Furthermore, CTL guarantees the uniqueness of the learnt filters. These advantages of CTL and its deep version makes them ideal candidates for embedding clustering losses. Here we have incorporated K-means clustering since it is the most popular. However, it is possible to incorporate other clustering losses as well. The contributions of this work can be summarized as -• We propose a new formulation for deep convolutional transform learning based K-means clustering that is less prone to over-fitting compared to existing solutions since we do not need to learn the deconvolution/decoder layers.

• Our proposed method shows superior clustering performance in the challenging scenarios where data comprise of high number of clusters and low volume of samples.

• The execution speed of our proposed method is of the same order as that of K-means clustering and is much faster than the deep learning based clustering approaches compared against. The rest of the paper is organised into several sections. A brief review of various deep learning based clustering formulations is discussed in the next section. Furthermore, as CTL is a relatively new framework, we will discuss it in depth in Section 2. This is necessary for understanding our formulation discussed in Section 3. The experimental evaluation is detailed in Section 4. The conclusion is discussed in Section 5.

BACKGROUND

Deep Learning Based Clustering

In [START_REF] Tian | Learning deep representations for graph clustering[END_REF], a deep clustering model based on stacked autoencoder is proposed where the clustering layer is embedded after the encoder network. This deep embedded clustering model is trained in piecemeal fashion. Later, [START_REF] Peng | Deep Sub-space Clustering with Sparsity Prior[END_REF] proposed the jointly learnt formulation of the stacked autoencoder embedded with the sparse subspace clustering. The performance of jointly learnt formulation [START_REF] Peng | Deep Sub-space Clustering with Sparsity Prior[END_REF] is better than the piecemeal technique [START_REF] Tian | Learning deep representations for graph clustering[END_REF]. In [START_REF] Xie | Unsupervised deep embedding for clustering analysis[END_REF][START_REF] Yang | Towards k-means-friendly spaces: Simultaneous deep learning and clustering[END_REF][START_REF] Fard | Deep k-means: Jointly clustering with k-means and learning representations[END_REF], K-means clustering is embedded in the stacked autoencoder and trained in joint end-to-end fashion. The distance metric used in K-means clustering is the Student's tdistribution kernel in [START_REF] Xie | Unsupervised deep embedding for clustering analysis[END_REF] while the standard Euclidean distance in [START_REF] Yang | Towards k-means-friendly spaces: Simultaneous deep learning and clustering[END_REF][START_REF] Fard | Deep k-means: Jointly clustering with k-means and learning representations[END_REF]. In [START_REF] Guo | Deep clustering with convolutional autoencoders[END_REF], the stacked autoencoder is replaced with the convolutional autoencoder. In [START_REF] Yang | Deep Spectral Clustering Using Dual Autoencoder Network[END_REF], the autoencoder is embedded with the spectral clustering loss. In [START_REF] Trigeorgis | A Deep Matrix Factorization Method for Learning Attribute Representations[END_REF], deep matrix factorization, which is deep dictionary learning with ReLU activation, is proposed and argued that different layers correspond to different concepts in the data. A recent work [START_REF] Huang | Robust Deep K-Means: An Effective and Simple Method for Data Clustering[END_REF] claims improvement over [START_REF] Fard | Deep k-means: Jointly clustering with k-means and learning representations[END_REF] by using hierarchical K-means; the authors of [START_REF] Huang | Robust Deep K-Means: An Effective and Simple Method for Data Clustering[END_REF] claim that such a scheme improves robustness. Another work that proposed a minor variation to the previous autoencoder based schemes is [START_REF] Diallo | Deep embedding clustering based on contractive autoencod-er[END_REF]; instead of a simple autoencoder, they proposed using a contractive autoencoder. We have mentioned before that existing studies that proposed embedding clustering losses in representation learning, were based on autoencoders. The only study that overcomes this issue (over-fitting in autoencoders) is [START_REF] Peng | Deep Subspace Clustering[END_REF]; they only need to learn the encoder network. The said study manages to bypass the trivial solution by imposing constraints on the learnt encoder network. We have focused on deep learning based clustering papers that are algorithmic in nature.

Deep Convolutional Transform Learning

In convolutional transform learning (CTL) [START_REF] Maggu | Convolutional Transform Learning[END_REF] a set of filters are learnt such that when operated on the data they produce the corresponding representations. Formally this is expressed as

t m * x (k) = z (k) m , ∀m ∈ {1, .., M }and∀k ∈ {1, .., K} (1)
Here x (k) denotes the k th input sample, t m the m th convolutional filter and z m (k) the representation of k th sample after applying the m th convolutional filter. There are a total of K samples and M filters. The symbol '*' denotes a convolutional operation with zero padding. During training, the filters and the representations are learnt by solving the following optimization problem. min (tm)m,(z

(k) m ) m,k K k=1 M m=1 (∥t m * x (k) -z (k) m ∥ 2 2 + ψ(z (k) m )) +λ{∥T ∥ 2 F -logdet(T )} (2) Here λ is a positive constant. T is defined as T = [t 1 |...|t m ].
The log determinant term prevents the trivial solution t m =0, z m (k) =0 and also promotes linear independence among the learnt filters; the other penalty ∥T ∥ 2 F prevents degenerate solutions where t m → ∞, z

(k) m → ∞.
The penalty on T is borrowed from transform learning [START_REF] Ravishankar | Online Sparsifying Transform Learning-Part I: Algorithms[END_REF]. One can see how it is possible to learn convolutional filters from training data in an unsupervised fashion. CNNs do not have the said penalty on the learnt filters and hence can end up at the trivial solution. Furthermore, without the penalty on T, there is no guarantee that CNNs will learn unique filters. In matrix-vector form, (2) can be expressed as follows

min T,Z ∥T • X -Z∥ 2 F + ψ(X) + λ{∥T ∥ 2 F -logdet(T )} (3) where X = [x 1 |...|x K ], Z = [z 1 (k) |...|z M (k) ] 1≤k≤K , T•X =   t 1 * x (1) ... t M * z (1) ... ... ... t 1 * x (K) ... t M * z (K)
  and ψ amounts to applying the penalty term column-wise on matrix X and summing.

In [START_REF] Maggu | Deep Convolutional Transform Learning[END_REF], a deeper extension of the unsupervised formulation was proposed leading to deep convolutional transform learning (DCTL). As the name suggests, in the said formulation, multiple convolutional filters were being applied on the samples one after the other to generate the representation. This is expressed as follows, min T1,T2,T3,Z

∥T 3 • (T 2 • (T 1 • X)) -Z∥ 2 F + ψ(Z) +λ 3 i=1 {∥T i ∥ 2 F -logdet(T i )} (4)
Here, T i refers to the i th layer of convolutional filters. In each layer, the T i is formed by stacking the filters as columns of a matrix. The formulation ( 4) is shown for three layers but can be extended further.

PROPOSED FORMULATION

The schematic diagram of the proposed architecture is shown in Fig. 1. We embed a K-means clustering loss into the DCTL formulation. The image is convolved by a set of convolutional filters (in T1). The resulting feature map is max-pooled followed by a scaled exponential linear unit (SELU) activation function. This constitutes the first stage of the convolutional layer. The thus obtained feature map is convolved through the second set of convolutional filters (T2). The resulting feature map undergoes max-pooling before being input to K-means clustering.

A piecemeal solution where the features were learnt separately via CTL / DCTL and passed onto K-means clustering separately was shown in [START_REF] Maggu | Convolutional Transform Learning[END_REF][START_REF] Maggu | Deep Convolutional Transform Learning[END_REF]. In this work, we propose to train the clustering embedded DCTL as a single optimization problem. Mathematically the expression is shown as:

min T1,T2,Z,H ∥T 2 • (T 1 • X) -Z∥ 2 F + ψ(Z) +λ 3 i=1 {∥T i ∥ 2 F -logdet(T i )} +µ∥Z -ZH T (HH T ) -1 H∥ 2 F (5)
Here, H is is the matrix of binary indicator variables h ij ; h ij =1 if x j ∈ cluster i and 0 otherwise. Problem ( 5) is solved iteratively in two parts. In the first part, the H is assumed to be constant and T1, T2, Z are updated -

P 1 : min T1,T2,Z ∥T 2 • (T 1 • X) -Z∥ 2 F + ψ(Z) +λ 3 i=1 {∥T i ∥ 2 F -logdet(T i )} +µ∥Z -ZH T (HH T ) -1 H∥ 2 F ( 6
)
In the second part, T1, T2, Z are assumed to be fixed and H is updated -

P 2 : min H ∥Z -ZH T (HH T ) -1 H∥ 2 F ( 7
)
The first part (P1) is solved via ADAM optimizer and the second part (P2) is solved via K-means clustering. Solutions to P1 and P2 are carried out alternately till convergence. By convergence, we mean a condition when the cluster centers do not vary significantly in subsequent iterations.

EXPERIMENTAL RESULTS

We evaluate our algorithm on three well known image databases. They are YaleB [20], Extended YaleB [21] and AR Faces [22]. The YaleB consists of 5760 images of 10 different subjects; each under 576 different lighting conditions. The Extended YaleB contains 16128 images of 28 subjects under different poses and illumination conditions. The AR Faces contains over 4000 images of 126 different subjects. For all the datasets the dense shift invariant feature transform (DSIFT) features were first extracted; then principal component analysis (PCA) was used to further reduce the dimensions to 300. This protocol was followed by several prior clustering studies [START_REF] Chen | Active Orthogonal Matching Pur-suit for Sparse Subspace Clustering[END_REF][START_REF] Peng | Deep Subspace Clustering with Sparsity Prior[END_REF][START_REF] Maggu | Transformed Subspace Clustering[END_REF][START_REF] Maggu | Deeply Transformed Subspace Clustering[END_REF]. We compared our proposed formulation with several benchmarks including Deep Learning friendly Clustering (DLC) [START_REF] Yang | Towards k-means-friendly spaces: Simultaneous deep learning and clustering[END_REF], Deep K-Means (DKM) [START_REF] Fard | Deep k-means: Jointly clustering with k-means and learning representations[END_REF], Deep Clustering with Convolutional Autoencoder (DCEC) [START_REF] Guo | Deep clustering with convolutional autoencoders[END_REF] and AutoEncoded K-Means (AEKM) [START_REF] Tian | Learning deep representations for graph clustering[END_REF]. We have also used the K-means algorithm with standard Euclidean distance as distance metric for comparison. Normalized Mutual Information (NMI), Adjusted Rand Index (ARI) and Accuracy are used as metrics [START_REF] Peng | Deep Sub-space Clustering with Sparsity Prior[END_REF][START_REF] Peng | Deep Subspace Clustering[END_REF]. We have used µ=1 and λ=.001 in all the experiments. For all the experiments, 3 filters of sizes 9x9 have been used in both the first and second layer of convolutions. The max-pooling kernel size is 2x2. The results are shown in Table 1. For YaleB and AR Faces our method yields the best results; for Extended YaleB our results are a close second. The YaleB and the AR Face datasets are more challenging compared to Extended YaleB. This is because the first two have a larger number of clusters (an order of magnitude higher than Extended YaleB) and fewer images (an order of magnitude lower than Extended YaleB). On these two challenging datasets, we do better than the existing benchmarks. In the relatively simpler case (Extended YaleB) we are doing slightly worse than DCEC in terms of ARI and NMI but is better in terms of accuracy. It is interesting to note that, existing deep learning algorithms are doing worse than the simple K-means for the more challenging datasets. The experiments were run on a 64 bit Intel i5 clocked at 1.6 GHz with 16GB of RAM. The operating system is Ubuntu.

All the algorithms were run on Python. The runtime (in seconds) for various techniques are shown in Table 1 under Time column. Unsurprisingly K-means is the fastest, our proposed method is in the same order as that of K-means and is much faster than the rest of the deep learning based clustering techniques.

Ablation Studies

In this section, we show how the results vary with the number of layers. We also show how the results vary when the problem is solved jointly (as proposed) versus the piecemeal solution; by piecemeal we mean features are generated from (deep) convolutional transform learning and the features are separately fed into K-means clustering. The results are shown in Table 2. The joint solution, be it one layer or two-layer, yields better results than the piecemeal solution. This is expected; even in the past, jointly formulated solutions yielded better results than piecemeal ones. For both the piecemeal and joint solutions, going deeper helps, that is, the results obtained from two layers are always better than the ones from one layer. Finally we show the empirical convergence plot in Fig. 2. We find that the proposed algorithm converges within 50 iterations. The convergence plot for other depths show a similar trend.

CONCLUSION

We have proposed a convolutional representation learning based clustering formulation that does not require learning additional decoder/deconvolutional filters as is required by all existing approaches. This results in the requirement of learning fewer parameters, which in turn is less prone to over-fitting and hence can cluster from fewer data. The im- In the future, we would like to embed popular clustering techniques like subspace clustering [START_REF] Vidal | Subspace Clustering[END_REF], fuzzy C-means clustering [START_REF] Bezdek | FCM: The fuzzy c-means clustering algorithm[END_REF] and hierarchical clustering [START_REF] Murtagh | Algorithms for hierarchical clustering: an overview[END_REF] into the framework.

Fig. 1 .

 1 Fig. 1. Schematic diagram of the proposed approach

Fig. 2 .

 2 Fig. 2. Empirical Convergence Plot

  . and G.C. acknowledge support from the Agence Nationale de la Recherche of France under MAJIC (ANR-17-CE40-0004-01) project.

Table 1 .

 1 Clustering

	Results

Table 2 .

 2 Results From Ablation Studies

				YaleB			Extended YaleB			ARFaces
	Metric	Prop1L Piece1L Prop2L Piece2L Prop1L Piece1L Prop2L Piece2L Prop1L Piece1L Prop2L Piece2L
	Accuracy 0.620	0.588	0.649	0.636	0.146	0.135	0.349	0.320	0.142	0.122	0.159	0.151
	NMI	0.686	0.648	0.708	0.691	0.204	0.193	0.448	0.432	0.453	0.434	0.463	0.456
	ARI	0.447	0.412	0.510	0.462	0.033	0.031	0.132	0.123	0.044	0.040	0.051	0.048


Ana Cristina Mafla

Ramiro José Benavides

Pierre Meyer

Nicolas Giraudeau

Falk Schwendicke

Association of children's toothbrushing and fine motor skills: a cross-sectional study

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