T Deloze

Y Hoarau

M Braza

F Deliancourt

Turbulent simulation of the flow around two cylinders in tandem configuration

The turbulent flow around a generic configuration of a landing gear (the tandem cylinders) is simulated and analysed physically at Re = 1.66 10 5 , by means of hybrid RANS-LES turbulence modelling approaches. In the present study, the Delayed Detached Eddy Simulation (DDES) approach has been employed for two cylinders spaced of 3.7D and with a height of 4D. The DDES-OES modelling has been considered, especially involving turbulence length scale reconsiderations in the statistical part, by means of the Organised Eddy Simulation, (OES), to take into account non-equilibrium turbulence effects. The results are compared with experiments carried out at the NASA Langley Research Centre in the context of ATAAC EU-program in which the tandem cylinders is one of the stepping stones. In the present study, the benefits of these hybrid approaches have been discussed for capturing the vortex dynamics and frequency modes responsible for aerodynamic noise production in the context of landing gear configurations.

Introduction

Within the ATAAC (Advanced Turbulence Simulations for Aerodynamic Application Challenges) EU program, it has been shown through its main test-cases, that DDES (Delayed Detached Eddy Simulation) is a promising approach to capture the unsteady dynamics and the turbulence statistical content in strongly detached unsteady flows, by using reasonably fine grids, comparing to the grid sizes that would be needed by LES, for the present high-Reynolds number range flows around bod-T. Deloze, M. Braza and F. Deliancourt IMFT-UMR CNRS-INPT-UPS-N 5502, Alle du Prof. Camille Soula, F-31400 Toulouse, e-mail: tdeloze@imft.fr Y. Hoarau IMFS, Institut de Mcanique des Fluides et des Solides, 2 rue Boussingault, F-67000 Strasbourg e-mail: hoarau@unistra.fr ies, involving strong detachment. The present configuration highly interests aeroacoustics and aerodynamic noise control, generated by the two supports (tandem cylinders) of a generic configuration of a landing gear. The present test case has been the object of detailed experimental and numerical studies leaded by the NASA Langley Research Center, (Jenkins et al [START_REF] Jenkins | AIAA 2005-2812[END_REF], Lockard [START_REF] Lockard | Tandem Cylinder Noise Predictions, 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference) AIAA[END_REF][START_REF] Lockard | Summary of the Tandem Cylinders solution from the benchmark problems for Airframe noise computations I -Workshop[END_REF], among other). The experiments were carried out by standard 2D PIV, at Reynolds number 166 000. In order to avoid natural transition development, the experiments applied two transition strips at two specific upstream locations. A first synthesis of numerical simulations carried out for this test case was carried out by Lockhard, regrouping an order of 13 contributions involving different modelling approaches, as well as previous simulations by Khorrami et al. [START_REF] Khorrami | Simulations of bluff body flow interaction for noise source modelling[END_REF] using URANS-SST. These simulations indicated that the majority of the approaches captured quite well the Strouhal number of the vortex shedding frequency around the first cylinder, (St = 0.24) . Furthermore, as is seen in the ATAAC European program, the DES approaches better capture the complex vortex dynamics of the present flow, especially the formation of Kelvin-Helmholz vortices in the separated shear layers. In the experimental context, it was found that the shear layers formed downstream of the first cylinder wrap around the second cylinder and interact non-linearly with the complex turbulence background, producing predominant frequencies in the energy spectrum, in the range of acoustic noise.

The aim of the present paper is to evaluate the ability of the DDES methodology for an accurate prediction of the pressure fluctuations and frequencies responsible for the acoustic noise. In the present study, the DDES approach is considered, involving OES modelling (Braza et al. [START_REF] Braza | Turbulence Properties in the cylinder wake at high Reynolds number[END_REF]), in the URANS part, accounting for improvement of the nearregion flow physics in respect of non-equilibrium turbulence. In Bourguet et al. [START_REF] Bourguet | [END_REF], the DES-OES modelling was successfully applied for the simulation of the strongly detached flow around an airfoil beyond stall. The present DDES-OES, as the previous one, aims at providing a smooth passage from the URANS towards the LES region of flow detachment and at keeping the statistical region extent quite significant around the body.

Description of the test case

The test case is composed of two cylinders in in-line tandem configuration (see figure 1). The diameter (D) is identical for the both cylinders and the distance between the cylindres (center to center) is equal to L = 3.7D. The Reynolds number (Re = U D/ν) associated with the diameter D is Re = 166000. The height of the cylinders is H = 4D. The Mach number is Ma = 0.128, corresponding to an incompressible flow. The Reynolds number is subcritical but it is closed to the drag crisis.

The experiments with the same parameters were conducted at the NASA Langley Basic Aerodynamic Research Tunnel (BART), which is a subsonic, atmospheric wind tunnel. The free stream velocity was set to 44m/s. The velocitie correspond to a Reynolds number based on the cylinder diameter of Re = 166000 and the distance between the cylinders is L = 3.7D. The difference with the present numerical study is the height of cylinder equal in the experiments to H = 16D. The free stream turbulence level at these conditions is about 0.09%. Figure 1 shows schematically the configuration with the tandem cylinders placed in the wind tunnel, as well as a sketch of the computational used. The simulations have been performed with the NSMB solver, (Navier-Stokes Multi Block), based on structured grids architecture. IMFT is part of the NSMB consortium (Vos et al. [START_REF] Vos | Recent advances in aerodynamics inside the NSMB (Navier-Stokes Multi-Block) consortium[END_REF]), contributing with turbulence modelling development for strongly detached unsteady aerodynamic flows. For the present simulations, central difference scheme has been applied, among the variety of spacial discretisation schemes available in the solver, after performing detailed tests (Barbut [START_REF] Barbut | Analyse physique par simulation numrique dcoulements turbulents instationnaires autour de surfaces portantes fixes ou en mouvement, nombres de Reynolds et de Mach levs[END_REF], Gual et al. [START_REF] Gual-Skopek | Hybrid RANS-LES Modelling on a strongly detached turbulent flow[END_REF]). Dual time stepping is used for the temporal discretisation. The computational grid is of 16 millions of finite volumes. The closest point to the wall at the second cylinder is at a distance of y/D = 3.0 10 -5 . The computations, using 512 processors in the present study, have been carried out at the SGI Altix supercomputer JADE of CINES, (Centre Interuniversitaire National de lEnseignement Suprieur) and at the supercomputer CURIE of the CEA (Commisariat dEtudes Atomiques) of France.

The time step is equal to ∆t = 0.001 in order to be able to allow resolution of frequencies corresponding to a Strouhal number of order St = 0.25 and of higher frequencies predominant peaks due to Kelvin-Helmholtz eddies, responsible for the acoustic noise.

The turbulence modelling is the DDES approach, in which the RANS turbulence length scale is improved by means of the Organised Eddy Simulation (Braza et al. [START_REF] Braza | Turbulence Properties in the cylinder wake at high Reynolds number[END_REF]). It is recalled that distinction of coherent and chaotic turbulent structures is performed in OES by dual spectrum splitting, solving the ensemble-averaged Navier-Stokes equations, where the turbulence stresses are modelled by modified turbulence scales modelling (as presented schematically in the figure 2). The fact that part of the spectrum to be modeled in OES extend from the low to the high frequencies allows the use of statistical turbulence modeling with appropriate modification due the non-equilibrium. In the time-domain, the equations are the phase-averaged Navier-Stokes equations. Due to the nonlinear interaction between the coherent part and the incoherent one, there is a slope modification of the inertial part in the spectrum, in the vicinity of the peak. This yields to a reconsideration of the Eddy-diffusion coefficient for the class of the two-equation modeling, as well as improved damping functions to attenuate turbulence towards the wall. Figure 4 shows the averaged pressure coefficient corresponding, where the pressure peaks values are quite closer to the experimental ones. Concerning the first cylinder, the averaged maximum pressure values are lower than in the experiment, where a better agreement is obtained for the second cylinder. In the numerical results, the detachment provides earlier than experiments and can explain the lower value of pressure. The 2D turbulence kinetic energy

(2DT KE = 1 2 (< u ′ >< u ′ > + < v ′ v ′ >)/U 2 ∞
) is performed and plotted along the x-line (in the gap region and rear the cylinders) and along the z-line (cross-flow) (figure 4). On the same x-line, the time averaged x-velocity is plotted (figure 4). The results are in good agreement and especially on the time averaged x-velocity. The 2D turbulence kinetic energy shows the same behaviour but it quite over-estimated in the gap region and it is quite underestimated in the rear of cylinder.

The power spectral density are performed with the time-evolution of the pressure at four points (two points in the upstream cylinder at 135°and two points in the downstream cylinder at 45°) and represented at the figure 6. The turbulence spectra show the principal instability modes, among which the Von Krmn mode, as well as formation of a spectral bump corresponding to the Kelvin-Helmholtz eddies. Of course, the chaotic turbulence effect provides the fact that these eddies are governed by a random vortex smearing around a predominant K-H mode. The main frequency is identical for the four points and the value is 159Hz (178Hz for experiment). This frequency corresponds to the Von-Kármán vortex shedding. The seond pic corresponds to the second instability of Kelvin-Helmotz with a frequency of 316Hz always see by the four points (362 for experiment). A third pic is visible on the downstream cylinder (476Hz). We find similar frequencies to those obtained by experiment with a slight underestimation.

Conclusion

A numerical study has been carried out to analyse the complex vortex pattern around two cylinders in tandem at Reynolds number 166,000, by means of Hybrid Turbulence modelling approaches. The Delayed Detached Eddy Simulation DDES-SST has been applied and comparison of results with the experiment is carried out. The results have shown that the flow structure between the two cylinders and past the second cylinder are well produced by the present methods. The statistical content of the DDES-OES is detailed and the shear layer interfaces are fine and contain a number of Kelvin-Helmholtz vortices. Moreover, the spectral analysis shows that the flow is mainly governed by the von Krmn mode. A reasonably good agreement with the experiments carried out in the NASA-Langley Research Center is shown, especially concerning the mean streamlines structure, velocities and predominant frequencies. The interaction between the Von Krmn mode and the shear layer frequencies bump is quantified by means of advanced signal processing. The present study shows the ability of the DDES to simulate the present complex flow dynamics and predominant modes, associated with landing gear aerodynamic noise.

Fig. 1

 1 Fig. 1 Schematic representation of the two cylinders in in-line tandem arrangement (left) and picture of NASA experiment (right) for the dimensionless parameters L = 3.7D, Re = 166000

Fig. 2 Figure 3

 23 Fig.2Dual spectrum splitting : the distinction between the structures to be resolved and those to be modeled is based upon their organised or random character. Part (2) of the non-equilibrium energy spectrum has to be modeled by reconsidering advanced statistical turbulence modeling efficient in high-Re wall flows, due to the inertial-range modulation from equilibrium turbulence, schematically shown on the right.

Y

  Vorticity: -20 -10 -5 -3 -2 -1 1 2 3 5 10 20

Fig. 3

 3 Fig. 3 Instantaneous spanwise vorticity contours (left) and averaged streamlines field for present study and NASA experiments (right)

Fig. 4

 4 Fig. 4 Wall pressure coefficient (C p ) on the first cylinder (left) and the second one (right) copared with the experiment.

Fig. 5 Fig. 6

 56 Fig. 5 Time-averaged velocity profile in the gap region and at the rear flow (left), the 2D turbulence kinetic energy following the same line (middle) and following the cross-line x = 4.45 (right)

Frequency

Acknowledgements This work is part of the ATAAC research program, Advanced Turbulence Simulations for Aerodynamic Application Challenges, coordinated by DLR-Germany (D. Schwamborn). The authors are grateful for the use of the French supercomputing facilities of CINES, IDRIS, CEA, this last in the context of the PRACE EU initiative.


Thomas Denoual

Jean-François Petiot

Franck Mars

Andras Kemeny

Predicting the subjective evaluation of vehicle behaviour in a driving simulator

This paper investigates the possibility of using objective indicators to predict the subjective evaluation of a driver in a simulator. Situations of loss of adherence (LOA) were controlled and modulated in intensity and duration on a static and on a dynamic simulator (with and without a motion base). Multiple regression analyses were performed using the subjective evaluation of participants as the dependent variable, and the objective physical variables of the interaction driver/vehicle as the independent variables. The results assigned the most contributive variables to the accuracy of the model's prediction for each subjective item, lateral acceleration and yaw rate for "control feeling" for instance. They also underlined the consistency of our approach and the influence of motion rendering on the perception of LOA intensity. A similar method could be used to evaluate the perception of various configurations of electronic stability control (ESC) systems.

INTRODUCTION

Electronic Stability Control (ESC) is an advanced driving assistance system (ADAS) that dynamically corrects vehicle trajectory according to the driver's intentions in emergency situations [START_REF] Liebemann | Safety and Performance Enhancement: The Bosch Electronic Stability Control (ESP)[END_REF] [START_REF] Erke | Effects of electronic stability control (ESC) on accidents: A review of empirical evidence[END_REF]. It is particularly useful in cases of loss of adherence (LOA) in bends that are caused by excessive speed or a change in road grip, and can help to maintain vehicle control. The calibration and validation processes of ESC systems are time consuming. They require the development of physical prototypes based on expert drivers and specific terrain, especially for low adherence situations. Understanding drivers' perceptions of and reactions to LOA situations would be useful for the engineering specification of ESC. Driving simulators are useful tools in vehicle design and perceptual studies. They allow the safe exploration of critical situations with naive drivers and without environmental bias [START_REF] Kemeny | Driving Simulation for Virtual testing and perception studies[END_REF]. Consequently, driving simulators are being used to study LOA episodes and ESC performance [START_REF] Watson | Design of Simulator Scenarios to Study Effectiveness of Electronic Stability Control Systems[END_REF] [START_REF] Papelis | An empirical study of the effectiveness of electronic stability control system in reducing loss of vehicle control[END_REF]. This study aimed at evaluating the possibility of using objective indicators to predict the subjective evaluation of vehicle behaviour in a simulator.

Measuring human perception has been a major challenge since the middle of the 19 th century. In recent decades, with the emergence of human-centred design approach [START_REF] Rossi | Measurement involving human perception and interpretation[END_REF],there has been a clear need to provide measurement standards and methods. Numerous sensitivity analysis methods have been proposed and used in various fields [START_REF] Christopher Frey | Identification and Review of Sensitivity Analysis Methods[END_REF]. These methods, together with the relationship between stimuli and sensory response, have been widely investigated in areas such as textile [START_REF] Giboreau | Sensory evaluation of automotive fabrics: the contribution of categorization tasks and non verbal information to set-up a descriptive method of tactile properties[END_REF] and food quality [START_REF] Williams | Flavour quality -Understanding the relationship between sensory responses and chemical stimuli. What are we trying to do? The data, approaches and problems[END_REF] [START_REF] Skovgaard | Modelling relations between instrumental and sensory measurements in factorial experiments[END_REF]. Multiple linear regression models based on the Weber-Fechner law seem to give good results for predicting textile quality [START_REF] Mazzuchetti | Objective Measurement of Tactile Sensitivity Related to a Feeling of Softness and Warmth[END_REF] or food preferences [START_REF] Cliff | Development and evaluation of multiple regression models for prediction of sweet cherry liking[END_REF]. However little has been done in the field of vehicle dynamics. We thus propose to explore the use of this approach to explain and predict drivers' perceptions of LOA situations.

This paper presents two driving simulator experiments in which episodes of LOA were triggered to produce significant modifications of the vehicle trajectory without loss of control and road departure. Intensity and duration of the LOA were manipulated. Previous studies [START_REF] Denoual | Drivers' perception of simulated loss of adherence in bends[END_REF] [START_REF] Denoual | Drivers' Perception of Loss of Adherence in Bends: Influence of Motion Rendering[END_REF] have focused on developing an evaluation method to describe LOA episodes by means of subjective indicators using a non-structured-scaled questionnaire [START_REF] Strigler | Les épreuves[END_REF] and determining to what extent objective and subjective indicators were related [START_REF] Petiot | Measuring consumer perceptions for a better comprehension, specification and assessment of product semantics[END_REF].

In this study, the first objective was to develop models to explain each subjective item from the questionnaire and to determine which objective variables contributed the most to the accuracy of the predictions. Another objective was to determine the influence of motion rendering on the selected models and their predictions.

MATERIALS AND METHODS

PARTICIPANTS

STATIC SIMULATOR

A total of 20 participants (4 females, 16 males) aged between 20 and 24 (mean age: 21.4) participated in the first experiment on the static simulator. They had held a driving licence for 3.4 years on average and drove between 1000 and 25000 km per year (mean: 6325 km).

DYNAMIC SIMULATOR

A total of 20 participants (5 females, 15 males) aged between 19 and 58 years old (mean age: 36.8) participated in the second experiment. They had held a driving licence for 16.7 years on average and drove between 1000 and 40000 km per year (mean: 17538 km).

PROCEDURE

The same graphics database was used in both experiments. It reproduced an open countryside driving environment. Behavioural measures (lateral position, steering angle, lateral acceleration, etc.) were recorded during the trials at 20 Hz. All trials were performed on a short section of the driving environment, which comprised a straight line followed by a bend (total distance: 700 m; mean radius in the bend: 111 m) without traffic.

Modifying the adherence under the wheels when the vehicle reached a defined point in the bend yielded LOA. The intensity (adherence coefficient) and duration of the simulated LOA in the bend were manipulated as independent variables (IV). An adherence coefficient decrease corresponds to an increase in the intensity of LOA. These values of intensity and duration values were chosen to induce perceptible but controllable LOA simulated on the four wheels. The LOA situation induced a skid towards the outside of the bend. The environment did not give clues about a potential LOA (such as snow, rain or a mark on the road).

Participants were asked to keep to their lane without cutting the corner, even if there was no oncoming traffic. After a 10-minute practice session, they drove around the test bend at a predefined speed. An automatic gearbox and a speed regulator helped subjects in the first experiment. For the second experiment, the subjects received verbal assistance from the person conducting the experiment, in order to maintain a constant speed and stay focused on steering control. Four trials without any LOA were performed first in order to offer the participants the time to familiarize themselves with the task. For the LOA, a 3*3 factorial design was used (3 levels of intensity: 0.1, 0.3 & 0.5; 3 levels of duration: 250 ms, 500 ms & 750 ms). Nine experimental trials followed four preliminary trials representing mild and strong LOA episodes. The preliminary trials were conducted in order to familiarize the participants with the range of steering perturbations they would encounter during the experiment, but they were not analysed. A Williams Latin Squares design [START_REF] Williams | Experimental Designs Balanced for the Estimation of Residual Effects of Treatments[END_REF] was adopted to control rank and carry-over effects. After each trial, a questionnaire about the subjects' perception of the event was displayed. As a first approach, the items were chosen to evaluate LOA perception using "actual" drivers rather than professional expert drivers. The latter are commonly used to evaluate the dynamic handling of a car during the design process.

Subjective item Question

Perceived intensity

The LOA appeared to be weak/strong

Perceived duration

The LOA appeared to be short/long Danger I perceived a danger during the bend Control feeling I easily kept my vehicle in the lane Thus, a voluntary choice was to made to use nontechnical vocabulary. Two items were about the perception of the LOA itself (intensity and duration) and two others about its consequences in terms of perceived danger and feeling of control (Table 1). The aim here was not to use an extensive vocabulary to evaluate the participants' perceptions in detail, but to evaluate a method that relies on subjective and objective indicators.

Answers to the questions were given by means of continuous horizontal scroll bars representing two ends of a continuous scale (0: totally disagree to 10: totally agree).

OBJECTIVE INDICATORS

Three types of objective indicators can describe a driving situation: vehicle dynamics, driver's inputs and environmental indicators. The latter are even more easily recordable or computable in a driving simulator where the environment is managed. The specific case of LOA induces significant variations in vehicle dynamics and requires an appropriate response from the driver to keep the vehicle on the road.

For each trial, a time to stability (TTS) was computed. It corresponds to the time taken by the driver after the onset of LOA to bring the vehicle speed drift back into a stability envelope. The stability envelope is defined as the average standard deviation of the speed drift measured in the control condition for all the participants. The angular drift speed 2).

APPARATUS

STATIC SIMULATOR

The first experiment was conducted on a fixed-base simulator at the IRCCyN laboratory (Fig. 1a). It consists of a compact size passenger car with actual instrument panel, clutch, brake and accelerator pedals, handbrake, ignition key and an adjustable seat with seat belt. It is equipped with a TRW© active steering system for realistic "scale one" forcefeedback. Transmission was carried out using an automatic gearbox. Vibrators are installed underneath the driver seat and upper position of the steering column to render engine noise and vibrations. The audio system reproduces the audio environment for an interactive vehicle. It comprises an amplifier, four speakers and a subwoofer. The SCANeR©II software package was used with the CALLAS© dynamic vehicle model [START_REF] Lechner | Méthodologie de validation du logiciel de dynamique automobile CALLAS[END_REF]. The visual environment was displayed on three 32-inch LCD monitors, each with a resolution of 1280 x 720. One monitor was positioned in front of the driver, with two laterals inclined at 45° from the front one, viewed from a distance of about 1 m and covering 115° of visual angle. A simple generic speed regulator was used, consisting of a PID corrector with a nominal speed of 75 km/h, using the automatic gearbox mode in order to reject inter-subject velocity bias. This also allowed the subject to concentrate on the steering task.

DYNAMIC SIMULATOR

The second experiment was conducted on the highperformance dynamic Ultimate simulator [START_REF] Dagdelen | Validation process of the ULTIMATE high-performance driving simulator[END_REF] at the Renault Technical Center for Simulation (Fig. 1b). It consists of a compact size passenger car based on a real Laguna interior design. The cab is mounted on a large X-Y table and a hexapod motion system to render physical accelerations and rotations. Transmission is carried out using a manual gearbox. A system of sound synthesis is used to reproduce engine noise and the audio environment for an interactive vehicle. Active steering force feedback is computed by a proprietary model and reproduced by a TRW electric power steering system.

The SCANeR© Studio software package was used with a real-time version of the MADA (Advanced Modelling of Vehicle Dynamic) vehicle dynamic software, developed by RENAULT. The visual environment was displayed on a cylindrical screen (radius 1.9 m) by three single-chip DLP projectors, each with a resolution of 1024 x 768. The system covers a visual angle of 150°. Speed regulation was unavailable for this experiment.

DATA ANALYSIS

The objective indicators mentioned above are composed of a set of basic properties i, for which a measured value Xi has been obtained. Each basic property can be considered as a stimulus that may be perceived by the driver, and may create an associated sensation A i . It is said that the Weber-Fechner law can be applied to represent the relation between a stimulus and its sensory response in a fairly wide area of this stimulus. We hypothesize that the Weber-Fechner law is also valid for the study of such a complicated stimulus as basic property i. Thus, we used a logarithmic transformation on the measured values X i to obtain A i , objective variables used in the next modelling step.

The purpose of this study was not to find all the parameters for each basic property. We concentrated on "high level" subjective descriptors, which enable us to describe the situation from a non-expert driver point of view. In order to simplify the method, we assume that the perception of these descriptors is a weighted sum of the basic properties. Let j be a subjective descriptor and Y j the magnitude of its sensation, we propose a multivariable linear model, where Y j is the dependent variable and A i are the independent variables:

(Equation 2)
Multiple linear regression models were determined to predict each subjective descriptor from both static and dynamic simulator data. The selection of the best model was made with a step-by-step procedure according to the AIC criterion (Akaike information criterion, a measure of the relative goodness of fit of models -best model selection procedure; STATISTICA, 2004).

Parameters estimates for all models were tested using ttest. To evaluate the quality of the models, we used a two-fold validation procedure. For each subjective descriptor and each simulator, we have a dataset of 180 observations (20*9). We split it randomly into two blocks of N=90 observations, the first one to determine the multiple regression models (training) and the second one to evaluate the predictive capabilities of the models (holdout validation).

Three indicators were used to estimate the quality of the linear adjustment:

• The determination coefficient of the regression R 2 . It represents the percentage of variance taken into account by the linear model, • The p-value of the Fisher's test (variance analysis -significance of the model),

• The Mean Absolute Percentage Error (MAPE). It represents the forecast accuracy of the model (equation 3): the smaller the MAPE, the better the forecast accuracy. MAPE is a measure of how high or low are the differences between the predictions and actual data. For e.g., a 15% MAPE means on average that the predictions from a model will be 15% higher or lower than in actuality. The MAPE was computed on the validation set.

(Equation 3)

The MAPE can be compared to the MAPEn of a « naive » model, which would simply predict the average value of all the data, for each observation.

(Equation 4)

To assess the quality of the model, we propose to compute the predictive confidence, a measure of the improvement gained by the model over chance. The predictive confidence is defined by: (Equation 5)

We also compute the Pearson coefficient correlation (r) between the predicted values by the models and the actual observed values Y j on the validation sample, to evaluate the quality of the predictions.

RESULTS

For each subjective variable Y j , a two variables model was chosen according to the AIC criterion (Eq.6).

(Equation 6)

In the followings tables, all parameters are given with their standard deviation for each multiple regression linear model on the static (table 3) and the dynamic simulator (table 4). For each subjective variable A i , the standardized coefficient (β i *) is given to show the more contributive variable of each model. Residuals analyses have been performed to check basic hypotheses of the generalized analysis. The followings tables give the results of the MAPE criteria (equation 3), the predictive confidence (equation 5) and the coefficient correlation between predicted and actual values for each model on static (table 5) and dynamic simulator (table 6). A significant between-subject effect was observed (Table 7) on the subjective items for both experiments, except on "perceived intensity".

Yj

DISCUSSION

The first objective of the study was to evaluate how the chosen models could explain the variability of the subjective items and which objective variable was the most influential in each model. The results of the model selection method applied on the subjective items showed similar trends with both the static and dynamic simulators in terms of explanation capabilities. The models were able to explain more than 50 % of variability for perceived intensity and feeling of control. However, poorest results were observed for perceived duration (static: R² = 0.25 and dynamic: R² = 0.35) and danger (static: R² = 0.39 and dynamic: R² = 0.38). This could be explained by the large between-subjects variability observed on these items.

Heading speed was the second contributive variable for explaining perceived intensity in both dynamic and static simulators. This suggests that heading speed variation had a primary influence in detecting and assessing variation of LOA intensity. Interestingly, the most contributive variable in the models differed in both experiments. On the static simulator, the slip angle was the most contributive variable, while the lateral acceleration dominated on the dynamic simulator. This suggests that the perception of LOA intensity was mostly influenced by vestibular and other inertial cues. However, in the absence of motion rendering, drivers principally relied on visual cues to estimate the strength of the LOA .

The feeling of control was mainly influenced by heading speed variation, in both the static and dynamic simulators. In both cases, the second contributive variable has a very weak normalized coefficient. This suggests that heading speed variation was the most influent variable in explaining the subjective assessment of feeling of control.

As suggested by a previous study [START_REF] Denoual | Drivers' Perception of Loss of Adherence in Bends: Influence of Motion Rendering[END_REF], the duration of the LOA was poorly perceived in the simulator, which could explain the poor fit of the model. This also does not allow us to discuss the selected contributive objective variables, which differed in both experiments. The models obtained for perceived danger did not show a much larger explanatory power, but the same objective variables contributed the most for both simulators. The level of lateral acceleration and quantity of steering wheel angle needed to correct the trajectory had a primary influence on perceived danger. This supports the idea that drivers control a variable safety margin of perceived lateral acceleration according to their anticipated steering deviations when negotiating bends [START_REF] Reymond | Role of lateral acceleration in curve driving: driver model and experiments on a real vehicle and a driving simulator[END_REF].

From a global point of view, the models obtained with the static simulator explained more variability than those obtained in the dynamic condition. A plausible explanation is that the static simulator gave fewer cues to subjectively assess LOA events and that a simple multiple linear regression model is sufficient to fit the data. Through motion rendering, the dynamic simulator provided more complex stimuli to the driver, closer to a real driving situation. Models with interactions should be preferred, especially to take into account visuo-vestibular interactions that come into play in the perception of self-motion [START_REF] Kemeny | Evaluating perception in driving simulation experiments[END_REF]. In this case a linear model may not be sufficient.

It remains now to examine the capabilities of the selected models to predict new scores. Even if the selected models for both static and dynamic simulators were not able to capture all the variability of subjective items assessment, they gave good correlation scores (around 0.75) for predicting values of perceived intensity and feeling of control. For feeling of control, errors in prediction (as measured by the MAPE) were quite similar in the static and dynamic conditions. This suggests that motion rendering had little influence on this subjective item. Given that the variation of heading speed was the most contributive variable, the fidelity of the heading motion restitution should be investigated. For perceived intensity, errors in prediction are markedly better in the dynamic condition (MAPE = 0.43) than in the static condition (MAPE = 0.72); again, this suggests a determining influence of motion rendering on the perception of LOA intensity. The selected models for predicting perceived duration and danger have coefficients of correlation that are too small; thus, further investigations will be necessary to improve them.

The predictive confidence values are the highest for "perceived intensity" and "feeling of control ". For both simulators, these subjective items are predicted with the highest accuracy. For all subjective items, we noticed that the predictive confidence values are in the same range for both static and dynamic simulator. This underlines the consistency of our approach.

CONCLUSION

Multiple regression analyses were used to evaluate the relations between the subjective judgements and the vehicle dynamics for loss of adherence episodes. Mathematical models based on established psychophysical principles (the Weber-Fechner law) were tested. They were able to explain more than 50% of the variance for two subjective items out of four. This study highlights the objective indicators of vehicle behaviour that influence the variation of each subjective indicator the most. This study is a preliminary approach to objectivize drivers' perceptions and to help the tuning of ADAS by means of a simulator. Further research is planned to optimize the model prediction in various situations. A similar method might be used to objectivize how drivers perceive various configurations of ESC.

BIBLIOGRAPHY

(Table 2 :

 2 ) was calculated from the longitudinal speed (Vx) and the lateral speed (Vy): Definition of the objective indicators (variables X i ) According to the consequences of an LOA event on vehicle behaviour, pertinent indicators were computed in the TTS interval. The maximum lateral acceleration (LA, in m.s -2 ), maximum slip angle (SA, in deg), maximum heading speed (HS, in deg.s -1 ) and TTS were chosen for the vehicle dynamics; the maximum steering wheel angle (SW, in deg) for drivers' inputs; and the maximum lateral deviation (LD, in m) and root mean square of the lateral deviation during TTS (LDrms, in m) for the environmental indicators (table

Figure 1 :

 1 Figure 1: (a) IRCCyN driving simulator. (b) Ultimate Renault driving simulator

Figure 2

 2 Figure 2 represents the correlation between observed and predicted scores for all subjective indicators on the static and dynamic simulators for the evaluation data sample.

Figure 2 :

 2 Figure 2: Correlation between actual and predicted scores for feeling of control , perceived intensity and danger (n=90) for static (a,b,c) and dynamic (d,e,f) simulator. Solid line: actual correlation, dotted line: perfect correlation (actual=predicted)

Table 1 : Questionnaire and definition of the rating scales for the LOA (Variable Yj)

 1

Table 3 : Models parameters, selected objective variables and statistics of multiple linear regressions for subjective variables on static simulator (without motion base)

 3

	Yj	β 0	SD	β 1	SD	β 1 *	A 1	β 2		SD	β 2 *	A 2	F		p	R²	Total variance
	Perceived	-8.49	3.34 2.41 0.57 0.53	SA	3.45	1.36 0.32	HS	91.38	<.05	0.68	10.39
	Intensity
	Perceived	-12.69 4.34 3.97 0.98 0.62	SW	-0.49 0.48 -0.16 LDrms 14.81	<.05	0.25	9.08
	duration
	Danger	-11.94 2.60 3.80 2.13 0.32	LA	2.42	1.37 0.32	SW	2.30	<.05	0.39	12.56
	Control	28.38	2.94 -0.49 0.41 -0.13 TTS	-7.55 1.13 -0.72 HS	99.88	<.05	0.70	9.47
	feeling
	Yj	β 0	SD	β 1		SD	β 1 *	A 1	β 2	SD	β 2 *		A 2	F	p	R²	Total variance
	Perceived	-17.90	2.77 6.62		1.18 0.53	LA 3.57	1.28 0.26	HS	46.36	<.05 0.52 11.39
	intensity
	Perceived	8.68	3.58 -6.73	2.72 -0.62	LA 7.74	0.48 0.13	SA	23.02	<.05 0.35 10.00
	duration
	Danger	-13.57	3.98 4.71		1.19 0.42	LA 2.50	1.20 0.22	SW 22.20	<.05 0.38 9.19
	Control	26.64	2.55 -0.86	0.64 -0.13	SA -6.90	1.11 -0.62	HS	44.20	<.05 0.50 9.11
	feeling

Table 4 : Models parameters, selected objective variables and statistics of multiple linear regressions for subjective variables for dynamic simulator (with motion base)

 4

Table 5 : MAPE, Pc and correlation coefficient indicators for model forecast on static simulator

 5

					Yj	MAPE	Pc	r
					Perceived	0.43	42.70	0.74
					intensity
					Perceived	0.48	36.52	0.58
					duration
		MAPE	Pc	r	Danger	0.63	12.88	0.62
	Perceived intensity	0.72	45.57	0.77	feeling Control	0.38	50.55	0.77
	Perceived	0.93	23.76	0.58
	duration
	Danger	0.82	30.92	0.55
	Control	0.30	55.38	0.77
	feeling

Table 6 : MAPE, Pc and correlation coefficient indicators for model forecast on dynamic simulator

 6

Table 7 : Kruskal-Walis test results for between-subject effect on subjective items; df=19 and n=90 for each experiment

 7

			Without		With
			motion base	motion base
	Yj	H	Sig.	H	Sig.
	Perceived intensity	26.63	n.s	20.63	n.s
	Perceived duration	34.84	p<.05	34.84	p<.05
	Danger	83.8	p<.05	48.19	p<.05
	Control feeling	34.58	p<.05	41.42	p<.05


Thomas Denoual
email: thomas.denoual@irccyn.ec-nantes.fr

Franck Mars
email: franck.mars@irccyn.ec-nantes.fr

Jean-François Petiot
email: jean-francois.petiot@irccyn.ec-nantes.fr

Andras Kemeny
email: andras.kemeny@renault.com

Toward predicting the subjective assessment of ESC in a driving simulator

Keywords: Sensory evaluation, ESC, Driving simulators, Model prediction, Virtual engineering

  L'archive ouverte pluridisciplinaire

Introduction

Loss of adherence (LOA) can lead to loss of vehicle control, a major factor in many accidents. Electronic Stability Control (ESC) is an advanced driving assistance system (ADAS) that dynamically corrects the vehicle trajectory according to the driver's intentions in emergency situations. It is useful in particular in case of loss of adherence (LOA) in bends due to excessive speed or alteration of road grip, which can lead to loss of vehicle control. Using electronic stability control (ESC) can limit the consequences of these situations [START_REF] Liebemann | Safety and Performance Enhancement: The Bosch Electronic Stability Control (ESP)[END_REF] [START_REF] Erke | Effects of electronic stability control (ESC) on accidents: A review of empirical evidence[END_REF]. The calibration and validation processes of ESC are time consuming and require the use of physical prototypes and expert drivers at specific test sites, especially for very low-adherence situations. Consequently, driving simulators are being used to study LOA episodes and ESC performance [START_REF] Watson | Design of Simulator Scenarios to Study Effectiveness of Electronic Stability Control Systems[END_REF] [START_REF] Papelis | An empirical study of the effectiveness of electronic stability control system in reducing loss of vehicle control[END_REF]. Driving simulators are useful tools in vehicle design and perception studies. They allow the safe exploration of critical situations with naive drivers and without environmental bias [START_REF] Kemeny | Driving Simulation for Virtual testing and perception studies[END_REF]. The present study is a step of a research program that is aimed at understanding how drivers perceive and react to trajectory perturbations and to the intervention of an ESC system. This could be useful for using driving simulators to develop the engineering specifications of ESC and to evaluate how actual drivers perceive different system configurations. Previous works [Den1] [Den2] have sought to develop an evaluation method to describe LOA episodes by means of subjective indicators using a non-structured-scaled questionnaire [START_REF] Strigler | Les épreuves » dans évaluation sensorielle : manuelle méthodologique[END_REF] and to determine to what extent objective and subjective indicators were related. In this paper, an explicative and predictive model of driver subjective assessment in LOA situations has been proposed. It is based on objective measurement of vehicle behaviour such as lateral acceleration, heading speed or slip angle [START_REF] Denoual | Predicting the subjective evaluation of vehicle behaviour in a driving simulator[END_REF]. The present study evaluated the influence of ESC activation on subjective ratings and the predictiveness of the model for ESC triggering cases in LOA situations. An experiment was conducted on the high fidelity dynamic ULTIMATE simulator using the SCANeR © Studio software package with a real-time version of the MADA (Advanced Modelling of Vehicle Dynamic) vehicle dynamic software, developed by RENAULT. The intensity (0.1 to 0.3 adherence coefficient) and duration (250ms to 750ms) of the simulated LOA in the bend were manipulated as independent variables on the four wheels. Situations of LOA inducing a significant modification on the vehicle without involving a brutal road departure were chosen. ESC availability (on or off) was the third independent variable. All subjects experienced the same LOA situations with and without ESC assistance. They were not aware of the presence of the assistance. The objective of the study is to evaluate the accuracy of the developed model on a new data set, with and without the presence of an ESC system. The first results show a good accuracy of the predictive model on the data set without ESC. ESC triggering seems to slightly perturb the model prediction. Although further experiment will be necessary to improve the robustness of the model and enlarge to various applications, the methodology proposed seems to have a strong potential to evaluate ESC strategy in the early steps of engineering design.

Method

Participants Four female and ten male drivers between 27 and 59 years old (mean age of 43.6) participated in the experiment. They had held a driving licence for 20.6 years on average and drove between 1000 and 20000 km per year (mean = 13400). All of them had normal or corrected-to-normal vision.

Apparatus

The experiment was conducted on the high-performance dynamic Ultimate simulator [START_REF] Dagdelen | Validation process of the ULTIMATE high-performance driving simulator[END_REF] at Renault Technical Center for Simulation (Fig. 1). It consists of a compact size passenger car based on a real Laguna interior design. The cab is mounted on a large X-Y table and a hexapod motion system to render physical accelerations and rotations. Transmission is carried out using a manual gearbox. A system of sound synthesis is used to reproduce engine noise and the audio environment for an interactive vehicle. Active steering force feedback is computed by a proprietary model and reproduced by a TRW electric power steering system. The SCANeR© Studio software package was used with a real-time version of the MADA (Advanced Modelling of Vehicle Dynamic) vehicle dynamic software, developed by RENAULT. The visual environment was displayed on a cylindrical screen (radius 1.9 m) by three single-chip DLP projectors, each with a resolution of 1024 x 768. The system covers a visual angle of 150°. The graphics database reproduced an open countryside driving environment (Fig. 3). Behavioural measures (lateral position, steering angle, lateral acceleration, etc.) were recorded during the trials at 20 Hz. All trials were performed on a short section of the driving environment, which comprised a straight line followed by a bend (total distance: 700 m; mean radius in the bend: 111 m) without traffic (Fig. 2). A software-in-the-loop generic ESC system was used (Fig. 4). The system was set up with the characteristics of the simulated vehicle, but the tuning did not reflect the supplier tuning mounted on the real car. The complex proprietary control laws to handle the vehicle stability will not be detailed.

Procedure

LOA was simulated by modifying the adherence under the wheels when the vehicle reached a defined point in the bend. The intensity (adherence coefficient) and duration of the simulated LOA in the bend were manipulated as independent variables (IV). An adherence coefficient decrease corresponds to an increase in the intensity of LOA. ESC availability (on or off) was the third independent variable. These values of intensity and duration were chosen to induce perceptible but controllable LOA simulated on four wheels (Table 1). The LOA situation induced a skid towards the outside of the bend. The environment did not give clues about a potential LOA (such as snow, rain or a mark on the road). Subjects faced each of the four conditions two times, with and without ESC. A Williams Latin Squares design [START_REF] Williams | Experimental Designs Balanced for the Estimation of Residual Effects of Treatments[END_REF] was adopted to control rank and carry-over effects. After each experimental condition, the subject was asked to assess the LOA on a continuous unstructured scale according to 3 descriptors: perceived intensity, control feeling, danger.

Participants were asked to keep to their lane without cutting the corner, even if there was no oncoming traffic. After a 10-minute practice session, they drove around the test bend at a predefined speed (75 km/h). Subjects received verbal assistance from the person conducting the experiment in order to maintain a constant speed and stay focused on steering control. Four trials without any LOA were performed in order to allow the subjects to familiarize themselves with the task. • The mean absolute error (MAE). It represents the forecast accuracy of the model (equation 4): the smaller the MAE, the better the forecast accuracy.

MAE = 1 N Ŷkj -Y kj k=1 N ∑ (4)
where N is the data sample size

• The root mean square error (RMSE)

In order to evaluate the quality of the predictions, we also computed the Pearson coefficient correlation (r) between the predicted values Ŷj by the models and the actual observed values Y j .

Paper Number

-5-

DSC'12

Results

The followings tables give the results of the MAE, the RMSE and the correlation coefficients between predicted and actual values for each descriptor, without (table 3) and with ESC triggering (table 4). Situations where ESC has not triggered were excluded for the analysis. In order to have same data sample size for both group (ESC vs. no ESC), we also excluded the paired data in the "without ESC" group. Finally, we had 18 situations in each group.

Discussion

The present study shows the use of multiple linear models to predict driver subjective assessment in LOA situations with and without ESC triggering. For the data set without ESC, we observed good predictions of the models in the same range as seen as previously [START_REF] Denoual | Predicting the subjective evaluation of vehicle behaviour in a driving simulator[END_REF]. This confirms the capabilities of the model with a different panel of participants, which validates the methodology we used.

The same model was applied to characterize different situations in which ESC triggered. Results show that the prediction errors were more important with ESC activated. In this case, the predictive evaluations were also less correlated with the actual subjective evaluations. This probably means that the model parameters need adjustments to fit with the action of ESC. The correlation plots seem to show that the model overestimated the subjective evaluations compared to the observations. Unfortunately, results do not show a significant effect of ESC on subjective evaluation. This could be explained by the tuning of our ESC that provided limited trajectory corrections. Further investigations will be made on this question with different tuning of ESC and different LOA situations.

A traditional problematic with subjective evaluation on simulator is the simulation rendering fidelity compared to real perception. In our case, for obvious safety reasons, it would be difficult to reproduce the same experiment on a real track with non-expert drivers and compare the results. However, the global fidelity of a simulator can be estimated indirectly. On the Ultimate simulator, subjective evaluation campaign has been performed by Renault expert drivers. They have notably rated steering wheel, heading and pitch as "Satisfying" for a 0.2Hz / 0.2G slalom manoeuvres on Laguna simulation [START_REF] Dagdelen | Comportement Dynamique du Simulateur de Conduite ULTIMATE mi-2007[END_REF]. We can also identify and compensate for the simulator motion cueing delays [START_REF] Fang | Performance identification and compensation of simulator motion cueing delays. Trends in driving simulation design and experiments[END_REF] and adjust scale factor for lateral and yaw motion rendering [START_REF] Filliard | Combined scale factors for lateral and yaw motion rendering[END_REF]. Moreover, the fidelity of the simulator has been evaluated within the framework of the Eureka Moves European Project [START_REF] Berthoz | High-performance motion cueing for driving simulators[END_REF]. Thus, we are confident on the level of fidelity of Ultimate for this application.

This study shows the strong potential of this methodology to evaluate and characterized ESC with the help of a driving simulator, in particular in the early stage of the development process when no physical prototypes are available. It also adds the possibility to evaluate ESC system from a non-expert point of view, which may complement traditional methods. However, further studies are necessary to improve the quality and robustness of the model. A similar method could be applied on new projects like electrical vehicle [START_REF] Fang | Multi objective analysis on a driving simulator applied to an electric vehicle: vehicle stability, handling and drivability Proceedings SIA[END_REF] for which drivers' reactions are not well known.

Conclusions

A model presented in this paper allows predicting subjective evaluations of loss of adherence in driving simulator experiments. Further studies will allow improving the robustness of the model and its application in various situations. This study shows the strong potential of this methodology to evaluate and characterized ESC on a driving simulator in the early stages of the engineering design.

Figure 1 .

 1 Figure 1. Ultimate Renault driving simulator

Figure 2 .

 2 Figure 2.Layout if the country track

Figure 3 .

 3 Figure 3. Visual environment in the bend

Figure 4 .

 4 Figure 4. ESC software model integrated in the MADA vehicle dynamics model

Figure 6

 6 Figure 6 and 7 represent the correlation between observed and predicted scores for subjective indicators with and without ESC activation for the data sample.

Figure 6 .Figure 7

 67 Figure 6. Correlation between actual and predicted scores for control feeling, perceived intensity and danger without ESC triggering (n=18). Solid line: actual correlation, dotted line: perfect correlation

Table 1 . LOA conditions

 1 Ŷj in order to assess its accuracy as a function of ESC activation.

	Paper Number

Figure 5. Input and Output data

Table 2 : Input and output data for the predictive model Input (objective indicators) Output (subjective descriptors)

 2

	Lateral acceleration (m.s	-2 )	Perceived intensity
	Heading speed (deg.s	-1 )		Control feeling
	Slip angle (deg)			Danger
	Steering wheel angle (deg)
	Two indicators were used to evaluate the predictions errors:

Table 3 . RMSE, MAE and correlation coefficients for model forecast without ESC triggering

 3

	Yj	RMSE	MAE	r
	Intensity	1.08	0.90	0.81
	perceived
	Control	1.9	1.61	0.65
	feeling
	Danger	1.47	1.19	0.65

Table 4 . RMSE, MAE and correlation coefficients for model forecast with ESC triggering

 4

		RMSE	MAE	r
	Intensity	1.39	1.07	0.60
	perceived
	Control	2.39	1.64	0.52
	feeling
	Danger	1.58	1.24	0.60


Guilherme Cesar Augusto Fonseca

Rodrigues Sampaio

Geraldo F De

S Rebouças

Marcelo Pereira
email: marcelocpereira@globo.comamerico.cunha@uerj.br

Americo Cunha

Guilherme Rodrigues Sampaio

Modelling and analysis of vibrations on an aerial cable car system with moving mass

Keywords: cable car system, nonlinear dynamics, moving boundary problem, time-scale separation

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.

Introduction

Transporting persons and items by aerial cable cars, as seen in Figure 1, has been in use over the last hundred years for many different applications. This type of transport is present in thousands of installations, Switzerland has alone more than 130 in operation. The study of cable car systems is an interesting subject in modern literature, with many different aspects of the system approached, but published works on this topic are still only a few years old. Since it is a complex mechanical system with many bodies, flexibility, and subject to environmental perturbations, nonlinear phenomena may appear such as excitation of sub and super harmonics on the cable causing vibration to the traction mechanism, which can damage the machinery and endanger the operation.

The mechanical system studied consists of two cables, a supporting cable fixed in both ends, and a variable-length traction cable. The car is modeled as a concentrated mass pulled by the traction cable and connected to the supporting cable through a spring. In this work, we analyze the effect of vibrations in the traction cable according to the speed of operation and position of the car. The coupling of the two cables and a mass-spring system creates a very complex dynamic behavior that causes a nonlinear movement and can even show some chaotic characteristics. Some references found in the literature such as the modeling of elevators show resemblance to the present study.

The work of Brownjohn [START_REF] Brownjohn | Dynamics of an Aerial Cableway System[END_REF] investigates the cable car that connects the island of Sentosa and Singapore's main island trough a vertical plane motion perspective. The article presents mathematical simulations representing the normal operation of the system, as well as free vibration, and the situation when the system is halted. Accelerometers measurements in the real system are used to obtain real vibrations for each operational condition. Investigation also includes a finite element analysis of the system followed by the results analysis.

Terumichi et. al. [START_REF] Terumichi | Nonstationary Vibrations of a String with Time-Varying Length and a Mass-Spring System Attached at the Lower End[END_REF] analyzes the non-stationary lateral vibration of a timevarying length string with a mass-spring system guided by rail, attached to the inferior end, a system similar to an elevator. A sinusoidal horizontal displacement at its upper end excites the string. An analysis of the influence of the string's axial velocity on the vibrations is performed as well as an experimental setup investigation. Data generated by experiment and simulation are compared.

Bao et. al. [START_REF] Bao | Dynamic Analysis of Flexible Hoisting Rope with Time-Varying Length[END_REF] analyzes as well, a vertical motion system similar to an elevator, focusing on lateral vibrations of the string. In this article, the lateral motion of the mass, coming from imperfections on the rail guide are the source of excitation. The Hamilton principle is used to obtain the string's equations of motion. In the end, experimental tests results are shown and the results used validate the numerical simulations.

Lastly, Kaczmarczyk and Iwankiewicz [START_REF] Kaczmarczyk | Dynamic response of an elevator car due to stochastic rail excitation[END_REF] present a stochastic approach to the excitation of an elevator model. The stochastic parameter are the guide rail imperfections that excite the elevator car laterally. In order to do so, the imperfections are considered as a zero-mean stationary Gaussian process. Although the model proposed in this work uses a strict deterministic approach, the knowl-edge on time-scale separation shown is essential for solving the problem, as seen in another work of the same author [START_REF] Kaczmarczyk | The passage through resonance in a catenary-vertical hoisting cable system with slowly varying length[END_REF].

Those works show that despite the presented cable vibration theories, there are still very few applications to aerial cable car systems, a common system around the world, where a failure can endanger many lives. Cable car systems differ significantly from elevators as the gravity acts transversely to the cable.

The main originality of this work is to develop a mechanical model of a cable car system, to investigate the influence of its parameters using numerical methods, and to explore some of its nonlinearities. The manuscript is organized as follows: the mechanical-mathematical model for the system of interest is presented in Section 2, the numerical results, and a brief discussion in Section 3, and finally the conclusions are made in Section 4.

Mechanical-mathematical model

The proposed mechanical system is illustrated in Figure 2, where a pair of overhung cables has a constant value of tension and is fixed on both ends with density, circular cross-section, and constant length. The purpose of this cable is to provide support to the cable car and to serve as the rails, where the cable car slides on. Later, the rail cables are simplified as a single spring K attached to the cable car represented by the concentrated mass M . The car is pulled by another cable with a linear density ρ, cross-section A, and tension T . However, the length of the traction cable varies in time with constant velocity v, pulling the mass M to the origin, where the pulley is located. To find the equivalent stiffness of the pair of supporting cables, a simple analytical approach is used. The car passing through the extent of the supporting cables to be simplified as a concentrated load F traveling with constant velocity, then the solution of this problem is analytically solved and demonstrated in Hagedorn [START_REF] Hagendorn | Nonlinear Oscillations[END_REF]. In this case the cable deflection is given by

w(x, t) = - 2F L0 ρAπ 2 (c 2 -v 2 ) ∞ j=1 1 j 2 sin jπvt L0 - v c sin jπct L0 sin jπx L0 , (1)
where c = T /ρA is a constant having the dimension of speed and the product ρA is the linear density of the cable. Then stiffness can be determined by taking the inverse of the deflection of the supporting cables.

The analytical solution of Eq.( 1) is illustrated in Figure 4 for different time instants using the mentioned parameters. It shows the deflection of the cable at different instants while the load moves with velocity v. The parameters used to generate these results are shown in Table 1. The deflection of the rope reaches its maximum value when the load is close to half of the length. Therefore, the rail cable equivalent stiffness value is estimated as

K = F w(L 0 /2, L 0 /2 v) . (2)
The following analysis does only concern with the mass being pulled by the cable. We begin by stating the kinetic T and the potential energy of the rope and the mass. The total kinetic energy is

T = 1 2 L(t) 0 ρ[ v2 + ( ẏ + v y ) 2 ] dx + 1 2 M v2 + ( ẏ + v y ) 2 x=L(t) , (3)
and the total potential energy U is

U = 1 2 L(t) 0 T y 2 dx + 1 2 K y 2 x=L(t) . (4)
Here, y is the vertical displacement of the moving mass M attached to the pulling cable with linear density ρ and constant traction T . The dot represents the derivation in time and the ( ) the spatial derivation.

For the traction cable, using the Hamilton principle with the adequate definition of the boundaries conditions and considering the variation of the length gives the following initial-boundary value problem

ρ (ÿ + 2 v ẏ + v 2 y ) = T y , (5)
M (ÿ + 2 v ẏ + v 2 y ) + K y = T y at x = L(t), (6)
y = 0 at x = 0. ( 7
)
The first boundary condition, given by Eq.( 6), is not homogeneous. This term can be transferred to the differential equation of motion, Eq.( 5), by considering the car's dynamic forces as a concentrated load at x = L(t), rendering a simpler and more intuitive approach [START_REF] Meirovitch | Fundamentals of Vibrations[END_REF], given by:

ρ (ÿ + 2 v ẏ + v 2 y ) + 2 [M (ÿ + 2 v ẏ + v 2 y + g) + K y] δ(x -L(t)) = T y , (8)
y = 0 at x = 0, (9)
T y = 0 at x = L(t), [START_REF] Blevins | Flow-induced Vibration[END_REF] where ẏ is the time-derivative of y, and y is the corresponding spatial-derivative with respect to x.

To promote a separation of time scales in the problem, a slow time variation τ = ε t is introduced, in which ε = v/ω 0 L 0 , being ω 0 is the natural frequency of the cable without mass and spring, and L 0 the maximum extent of the cable, see [START_REF] Cunha | Analysis of Vibrations on an Aerial Cable Car System With Moving Mass[END_REF][START_REF] Colón | On dynamic analysis and control of an elevator system using polynomial chaos and Karhunen-Loève approaches[END_REF] for details.

Knowing that L(t) = L 0 + v t, so, v = L = ∂τ ∂t ∂L ∂τ = εν, where ν = ∂L ∂τ . Therefore, we seek solutions of the form

y(x, t) ≈ N i=1 φ i (x, τ ) q i (t), (11)
where φ i (x, τ ) = sin(λ i τ x) and q i (t) are mode shapes and modal coordinates associated to the underlying eigenvalue problem, respectively. The term λ i τ is a slow variation in time of the natural frequency of the cable

λ i τ = 1 2 (2 i -1) π L τ . (12)
Applying Eq.(11) into Eq.( 8) multiplying by φ j (x, τ ) and integrating between 0 and L τ , it is possible to write the discrete dynamical equation of motion

N i=1 [M ij qi (t) + G ij qi (t) + K ij q i (t)] = 0, ( 13
)
where mass, gyroscopic and stiffness operators components are, respectively, defined by the following expressions

M ij = Lτ 0 ρ φ i φ j dx + M φ i (L τ ) φ j (L τ ) G ij = Lτ 0 2 ρ ν φ i φ j dx + 2 M ν φ i (L τ ) φ j (L τ ) K ij = Lτ 0 ρ ν 2 2 -T φ i φ j dx + M ν 2 2 φ i (L τ ) + K φ i (L τ ) φ j (L τ ). ( 14
)
To construct the mass, gyroscopic, and stiffness matrices, the following orthogonality relationships are taken into account

Lτ 0 φ j φ i dx = 1 2 L τ δ ij , Lτ 0 φ j φ i dx = - 1 2 λ 2 i L τ δ ij (15)
and that Lτ 0

φ j φ i dx = -2 (i -1/2)(j -1/2) (i -1/2) 2 -(j -1/2) 2 , ( 16
)
where δ ij is the Kronecker delta. As seen in Blevins [START_REF] Blevins | Flow-induced Vibration[END_REF], the result from Eq.( 16) exists only when the sum i+j is odd, otherwise it is zero. Also, one has φ i = sin ( 1 2 (2i-1)π Lτ

x), and φ j = sin ( 1 2 (2j-1)π Lτ

x). Thus, for a two-mode approximation (i = 1..2 and j = 1..2) one gets

1 2 ρ L τ + M q1 (t) -M q2 (t) + 3 2 ρ ν q2 (t) + 1 8 L 2 τ -2 π 2 2 ν 2 1 2 ρ L τ + M + L τ π 2 T + 8 K L 2 τ q 1 (t) + 1 4 L 2 τ M ν 2 π 2 2 -4 K L 2 τ q 2 (t) = 0, ( 17
)
1 2 ρ L τ + M q2 (t) -M q1 (t) - 3 2 ρ ν q1 (t) + 1 4 L 2 τ 9 M ν 2 π 2 2 -4 K L 2 τ q 1 (t) + 1 8 L 2 τ -18 π 2 2 ν 2 1 2 ρ L τ + M + 9 L τ π 2 T + 8 K L 2 τ q 2 (t) = 0, (18)
that can be rewritten as

q1 (t) -µ τ q2 (t) + α τ q2 (t) + (-β τ -η τ + σ τ ) + ω 2 τ q 1 (t) + η τ -ω 2 τ q 2 (t) = 0 (19) q2 (t) -µ τ q1 (t) -α τ q1 (t) + 9 η τ -ω 2 τ q 1 (t) + 9 (-β τ -η τ + σ τ ) + ω 2 τ q 2 (t) = 0, ( 20
)
where

m τ = 1 2 ρ L τ + M, µ τ = M m τ , α τ = 3 2 ρ ν m τ , β τ = π 2 2 ν 2 ρ 8 L τ m τ , η τ = 1 4 π 2 2 ν 2 L 2 τ µ τ , σ τ = π 2 T 8 L τ m τ , and
ω 2 τ = K m τ . ( 21
)
The first term, m τ is the modal mass of the cable along with the cable car mass, which also brings the ratio µ τ of the masses. The latter couples the system with its acceleration terms. The parameter α τ represents the Coriolis, and β τ and η τ are the centripetal forces of the cable and the mass, respectively. The traction can now be written as σ τ and the natural frequency from the attached spring as ω τ . Each of these terms varies in the slow time τ as the pulling cable varies its length.

Numerical results and discussion

By inspecting the terms in (21), as well as the Eqs.( 19) and (20), it is possible to observe the overall dependency of the cable variation in the slow time τ . Moreover, some of them are proportional to O(1/L τ ) like ω 2 τ and η τ ; others are proportional to O(1/L 2 τ ) like β τ and σ τ . It means that β τ , and σ τ can become extremely high when the cable length is getting closer to zero. Although, the effect is rather small for η and β τ , it is usually not for σ τ . It is worth pointing out that there is a set of values in which σ τ surpasses the value of ω 2 τ at a specific length of L τ . A numerical investigation is conducted and a set of values are given for each of parameters from Eq.( 21). Then a simulation of equations ( 19) and (20) using a 4 th and 5 th Runge-Kutta solver is done from initial condition of a fully extended length of cable to its almost entire retraction. The parameters used in the simulations match the ones from a real cable car installation, meaning that their values completely feasible and the results may represent the behavior of an actual situation. Figure 4 shows that the parameters η τ , β τ , α τ have little influence on the overall dynamics, albeit the increase of the travelling speed.

When comparing the parameters σ τ and ω 2 τ , one observes in Figure 4a that there is a specific length of the cable that their values are equal, and as the cable car approaches the end, σ τ becomes even higher that may be relevant to the dynamics. This results in the Fig. 4b where it shows the variation of the natural frequencies as a function of the cable length. To further investigate the proposed mechanical model a harmonic force is added to the equation presented in section 2 applied to the concentrated mass M . The nature of this force can be seen as an early approximation of the wind mean effect acting directly on the cable-car. A Rayleigh-model damping is also included to emulate the energy dissipation, which also prevents numerical instabilities.

A constant value for the external force frequency is chosen to be ω f = 0.26 Hz, because it has long period time compatible to the natural behaviour of the wind. Note in Fig. 4b that this becomes the resonance of the system twice as the cable retracts. The result can be seen in Fig. 5, which shows the cable-car displacement time-series y(t). As expected the cable car develops high amplitudes of oscillations twice as the cable gets shorter, and the maximum value of the amplitude gets smaller as the travelling velocity v is higher, due to the shorter period of time that the forcing frequency matches with one of the natural frequencies.

The presence of high amplitude oscillations shown by the numerical simulation is in accordance with the real functioning cable-car seen by the maintenance crew as it approaches the terminal station. These vibrations are mitigated by the breaking system changing the value of the travelling speed v. Also, the model has its own limitations, since by shortening the cable means that the mass M is reaching its boundary condition, which would lead to numerical instabilities or misrepresentation of the real phenomena.

Conclusions and future perspectives

This work presented a mechanical-mathematical model for a cable car system. First, the equations of motion are developed, and numerical analysis is conducted with parameters inspired by a real case. This analysis shows the changes of each parameter as a function of the cable length, among them the increase of the stiffness parameter related to the cable. This paper paves the way for a series of different analysis, which may include the variation of the stiffness parameter related to the pair of supporting cables, the investigation of the influence of a harmonic forcing at the origin, which would represent the machinery responsible for actuating the whole system.

Results show that some points of the trajectory and some speeds present higher amplitude vibrations than others, this is an important non-linear phenomenon that poses challenges to the operation of systems similar to the one modeled. The understanding of those phenomena, as well as its numerical modeling, is of high importance to the safe operation of aerial cable cars.

Another possibility is to include the presence of a random force concentrated at the mass, representing the wind or passengers moving inside the car. Lastly, these models can be compared to other numerical analyses with finite elements models or even with experimental data. This research can also be adapted to other mechanical equivalent found in installations such as in ski lifts.

Fig. 1 .

 1 Fig. 1. Line art drawing of a typical aerial cable car system.

Fig. 2 .

 2 Fig.2. Schematic representation of the mechanical system used to emulate the aerial cable car system: a variable-length cable coupled to a mass-spring oscillator.

Fig. 3 .

 3 Fig. 3. Parameter variation of the first mode in relation to the length Lτ .

Figure 4

 4 Figure4illustrates the variation of the parameters as the traction cable is being retracted by a constant velocity. Three different velocities are tested. Figure4shows that the parameters η τ , β τ , α τ have little influence on the overall dynamics, albeit the increase of the travelling speed.

Fig. 4 .

 4 Fig. 4. In (a): The specific length where the natural frequencies depend more of the cable (blue and red) than the rail stiffness (yellow). Travelling speed v = 6m/s. In (b): Variation of the natural frequencies as function of the cable length.

Fig. 5 .

 5 Fig.5. Displacement for different velocities v of the cable-car when the system is excited by a harmonic force at F (t) = 300 sin (ω f t) [N] at x = Lτ .

Table 1 .

 1 Parameters used to model the rail cable and the traction cable

	Supporting Cables	Traction Cable
	Parameter	Value Unit Parameter Symbol Value Unit
	Linear density 39.20 kg/m Linear density	ρ	3.83 kg/m
	Length	600	m Initial length	L0	600	m
	Load	32.8	kN mass		M	6550 kg
	Mean Tension	10.0	kN Mean Tension	T	250 kN
	Velocity	6.0	m/s Stiffness		K	13.4 kN/m

Acknowledgments

The fourth author would like to thank the financial support given to this research by the Brazilian agencies Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001, and the Carlos Chagas Filho Research Foundation of Rio de Janeiro State (FAPERJ) under the following grants: 211.304/2015, 210.021/2018, 210.167/2019, and 211.037/2019.