C Gaudillière

N Ranc

A Larue

A Maillard

P L O Ro N G

C Gaudillieère

P Lorong

High Speed Blanking: An Experimental Method to Measure Induced Cutting Forces

Keywords: High speed blanking, Blanking force measurement, Hopkinson device

A new blanking process that involves punch speed up to 10 ms -1 has obvious advantages in increased productivity. However, the inherent dynamics of such a process makes it difficult to develop a practical high speed punch press. The fracture phenomenon governing the blanking process has to be well understood to correctly design the machine support and the tooling. To observe this phenomenon at various controlled blanking speeds a specific experimental device has been developed. The goal is to measure accurately the shear blanking forces imposed on the specimen during blanking. In this paper a new method allowing the blanking forces to be measured and taking into account the proposed test configuration is explained. This technique has been used to determine the blanking forces experienced when forming C40 steel and quantifies the effect of process parameters such as punch die clearance, punch speed, and sheet metal thickness on the blanking force evolution.

Introduction

Blanking is a shearing process which is commonly used in the metal forming industry [START_REF] Lascoe | Handbook of fabrication processes[END_REF][START_REF] Nee | Fundamentals of tool design[END_REF][START_REF] Smith | Die design handbook[END_REF]. Using this process, large and very large production runs of thin parts can be manufactured from sheet metal stock. Currently, two main blanking technologies exist: traditional blanking and fine blanking. In traditional blanking the sheet metal is located on the die and the punch blanks it to obtain the desired workpiece geometry. The punch speed and the clearances between the punch and the die are about 0.1 ms -1 and 10 % of the metal sheet thickness. In the fine blanking process the metal sheet is compressed between the die and a guide plate. An upper and a lower punch extract the workpiece. In this technique, the punch speed and the clearances between punch and die are lower: about 0.01 ms -1 for the punch speed and 1 % for the clearances. The main advantages of fine blanking are the tight tolerances and also the elimination of finishing machining operations . However, the tool costs are higher than the conventional blanking technique.

The constant improvement of the technology of the presses and particularly of hydraulic actuators allows an increase in punch speed. It is thus possible to develop high speed blanking with a punch speed of about 10 ms -1 . Research and industrial experiments show that high speed blanking has many potential advantages, especially the possibility to obtain a high quality blanking edge geometry, comparable to those obtained from the fine blanking technique. The new process also reduces the zone affected by plastic strain. This allows a reduction in the distance between each workpiece and a consequent saving in materials. With high speed blanking, unlike fine blanking, heat treatments such as spheroidizing for carbon steel are not necessary. Furthermore, the technique does not require the use of lubricant.

Nevertheless, a general lack of experience with the process tend to lead to short tool life and poor prediction of the geometrical and dimensional quality of the product. The tooling has to be better designed to produce parts in a way that satisfies industrial requirements. Tooling design is a crucial part of the development of such a process, and necessitates a reliable estimation of the cutting forces. This study aims to measure these forces.

The first attempt to increase punch speed to reduce the blanking energy was undertaken by Zener and Hollomon in 1944 [START_REF] Zener | Effect of strain rate upon plastic flow of steel[END_REF]. Many studies were carried out thereafter to determine the effect of the punch speed and punch die clearance on blanking energy [START_REF] Johnson | A comparison of the energy required for slow speed and dynamic blanking using an improved linear motor[END_REF][START_REF] Johnson | Further experiments in quasi-static and dynamic blanking of circular discs from various metals[END_REF][START_REF] Johnson | High-speed blanking of copper[END_REF][START_REF] Slater | The effects of temperature, speed and strain-rate on the force and energy required in blanking[END_REF][START_REF] Stock | The energy required for highspeed shearing of steel[END_REF]. These different works established the link between the reduction in blanking energy when the punch speed is increased and the occurrence of adiabatic shear bands [START_REF] Rogers | Adiabatic plastic deformation[END_REF] . Some of the studies focused on the determination of the evolution of blanking forces during the process [START_REF] Dowling | Dynamic punching of metals[END_REF][START_REF] Zurek | The study of adiabatic shear band instability in a pearlitic 4340 steel using a dynamic punch test[END_REF][START_REF] Roessig | Adiabatic shear localization in the dynamic punch test, part i: Experimental investigation[END_REF]. Measurement of blanking forces was carried out with the Hopkinson bar or tube system. However, the effect of process parameters, materials choice and the geometry of the tools on the history of the blanking force are still not well described and the experimental methods used are difficult. Knowledge of these blanking forces is necessary to design tools and to identify and predict the nature of the fracture mechanisms in a precise and useful way.

The objective of this work is to propose and test an experimental technique which enables the estimation of the blanking forces by taking into account the dynamic contribution of the three-dimensional geometry of the die. These blanking forces data will allow the characterization of the effect of parameters such as punch speed, blanking sheet metal thickness and the clearance between punch and die.

The second part of the paper describes the proposed experimental device. The results of a typical test are presented. The third part of the paper discusses in detail the method used to determine the blanking induced forces. The last part shows and discusses the effects of punch speed variation on blanking force.

High Speed Blanking Tests

The Loading Device and Measurement To reach very high speeds during blanking process, a specific experimental device, which allows the measurement of the cutting forces and observation the shear zone was designed and built. This device which is shown in Fig. 1(a), can be divided into three component parts. The "punch impulse device" is a punch propelled by an air gun. This device contains a hydraulic control system allowing the user to adapt the pressure in the air receiver of the air gun to produce the desired launching speed. A linear guide system enables precise control of the gap between the cutting punch flanks and the workpiece cut faces. The general geometry of the punch is detailed in Fig. 1(d). The second part is the "active zone" and is composed of a d i ew i t ham a s so f1 .4 kg (Fig. 1(b)) which supports the specimen before and during the blanking action. The test specimens are rectangular milled sheets of metal with a thickness of 2 mm,3mm or 4 mm with an accuracy to within 0.02 mm. The general shape of the specimen is shown in Fig. 1(c). The width is 17 mm andlengthis50mm. By changing the die, the clearance can be set to 0.05 mm or 0.2 mm. A stop-system has been designed to stop the punch just after blanking to avoid any direct interference between the punch and the die. In the last part of the rig, the "the measurement device", the cutting forces are estimated using a Hopkinson tube device. The section of the Hopkinson tube noted S tube is about 6.28 × 10 -4 m 2 and length is 3.5 m. The tube is made from 100C6 steel with a Young's modulus E tube = 206 MPa and a density ρ tube = 7800 MPa.The impedance of the tube defined as Z tube = S tube √ ρ tube E tube is about 2.6 × 10 4 P a.s.m.

The strain wave generated in the Hopkinson tube by the cutting process is measured using strain gages. The location of the gages on the tube and the remaining length on the tube allows measurement duration of 1.2 ms without overlapping of the signal on the gages. The bandwidth of the gage amplifier is about 500 kHz. A laser extensometer with a bandwidth of 250kHz (manufacturer: Polytec; model: HSV-2000 series) is located at the end of the tube targets the front face of the punch through a hole drilled in the center of the sample and measures the speed. The radius of the hole is about 1 mm and the distance between the hole and the shear zone is quite large (8 mm). Moreover the plastic strain due to the punching process remains very close to the shear zone. The punch speed and the Hopkinson tube strain signals are recorded by a data acquisition system with a sampling frequency of 1 MHz.

A high speed blanking test can be separated into different steps. First, the pressure increases in the air gun compressor tank (step 1). Then, when the test is activated, the air tank is opened and propels the piston and the punch (step 2). In the third step, the piston is stopped and the punch flies freely along the punch guides. In the step 4, the punch cuts the specimen. Finally, the punch is stopped and the cut part of the specimen is ejected into the Hopkinson tube.

Blanking Test Results

Tests were carried out with a fixed clearance between die and punch equal to 0.05 mm and an initial punch speed between 7 ms -1 and 18 ms -1 . The punch mass is equal to 1.5 kg. The specimen has a parallelepipedal geometry with dimensions: 17 mm width, 50 mm length and 2 mm,3mm or 4 mm thickness (Fig. 1(c)). The specimen material is a C40 steel with the composition given in Table 1. Figure 2(a) shows the change in punch speed with time during a blanking test on a specimen with a 2 mm thickness. The first phase corresponds to the punch propulsion phase where the punch speed increases quasi linearly. Then, when the piston is stopped, the punch slips freely along the guide rails. The speed decreases slightly because of friction. At the beginning of the cut, the punch strikes the specimen front face and the speed decreases very quickly. Just after the cut the punch moves again before finally stopping. Then the punch speed oscillates around zero with possibly a small negative mean speed due to punch rebound.

Figure 2(b) is an enlargement of Fig. 2(a) of the cutting duration shows the decrease and the oscillation of the punch front face during blanking. The oscillations of the punch speed are strongly characterized by a dominating harmonic frequency of about 20 kHz. This frequency corresponds to the first longitudinal eigen frequency of the punch. For the test shown in this Fig. 2(b), the punch speed before the cut is equal to 11.5 ms -1 and at the end of the cut, the punch speed oscillates around a constant value which is estimated at 9.4 ms -1 . It is also possible to calculate the cutting energy which is assumed to be equal to the variation of the kinetic energy of the punch. In the case of the test conducted at a punch speed of 11.5 ms -1 the calculated blanking energy is 33 J . It is possible from Fig. 2(b) to estimate the cutting time which corresponds to the duration of the speed decreasing phase, and the mean blanking force can also be determined. The estimated blanking time and the mean blanking force are equal to 220 μs and 14 kN.

This method becomes difficult to apply when the blanking duration is small i.e. when the punch speed is high or when the thickness of the specimen is small. Moreover this method only gives an estimate of the mean blanking forces during the cut. The blanking forces are also determined from the Hopkinson tube device. Figure 3 shows the strain measurements from the Hopkinson tube gages during the same test. This signal shows a compressive stress in the tube resulting from the blanking forces on the die. The origin of the timescale corresponds to the beginning of the cut. The delay of 135 μs between the beginning of the cut and the decrease of the strain is due to the wave propagation between the die and the gages on the Hopkinson tube. The loading duration at the half amplitude is about 79 μs and the maximum longitudinal strain in the Hopkinson tube is about 240 × 10 -6 . For a tube section of S tube = 6.28 × 10 -4 m 2 and a Young's modulus for the tube material of E tube = 206 GP a (mild steel), this strain value corresponds to a maximum force in the tube of 31 kN. This value can be different from the maximum cutting force because the complex die geometry and the die tube interface perturbs elastic wave propagation. This problem will be discussed in the next section. In our experimental device, the strain gage enables the estimation of the forces in the Hopkinson tube induced by the blanking process. The one-dimension elastic wave theory which is often used in the Hopkinson bar technique could not be used in this study because of the complex 3D geometry of the die. The die has been designed to be as much as possible like an industrial die. To estimate the blanking forces, our device was calibrated to make a link between blanking forces on the die and the strains measured on the Hopkinson tube.

The main difficulty in calibrating the device is how to apply a known force with a controlled amplitude and a duration similar to that of the blanking forces: the loading duration of a blanking test can vary between 35 and 400 μs. As a consequence, another Hopkinson apparatus was chosen (Fig. 4). Two projectiles of 68 mm and 132 mm allow a controlled pulse duration of respectively 30 and 52 μs. The loading amplitude is controlled by the projectile speed. To limit bending in the incident bar, the projectiles have a rounded end. Using this experimental device an incident cylindrical bar in 100C6 steel with a diameter of 32 mm is inclined against the die in a similar fashion to the specimen during the cut. The impedance of the incident bar is 3.2 × 10 4 P a.s.m. A gage placed on the incident bar is used to measure the longitudinal strain in this bar. When the projectile impacts against the incident bar, a compressive wave is created in the calibration bar which propagates up to the die. At the interface the wave is split into a reflected and a transmitted wave which are respectively measured with the gages J i and J tube . The applied force noted F calib on the die is expressed using the one dimensional elastic wave theory:

F calib (t) = E i S i ε Incident (t -t 0 ) + E i S i ε Ref lected (t + t 0 ), (1)
with E i = 210 GP a the Young's modulus, S i = 8.04 × 10 -4 m 2 the section of the incident bar, t 0 = 230 μs the propagation time of a compressive elastic wave between the gage and the die and ε Incident and ε Ref lected the strains in the incident bar associated respectively with the incident and the reflected wave. Figure 6 depicts the evolution of the strain measured on the tube and the bar versus time while Fig. 5 shows the calculated force on the die, applied by the incident bar. A classical procedure of post processing Hopkinson bar measurements [START_REF] Kolsky | Stress waves in solids[END_REF][START_REF] Zhao | A three dimensional analytical solution of longitudinal wave propagation in an infinite linear viscoelastic cylindrical bar. Application to experimental techniques[END_REF] is applied to deduce these applied forces. On Fig. 5 the signal duration at half amplitude is 30 μs and the maximum of the force is around 64 kN, which is the order of magnitude of the measured blanking forces. The rise time of the force signal which is about 30 × 10 -3 ms (Fig. 6) is due to the rounded geometry of the front face of the projectile.

To calibrate our experimental device, the force signal must be related to the measured strain on the Hopkinson tube. blueThe black curve on Fig. 7 shows the strain measured on the Hopkinson tube during. This curve shows that the duration of the transmitted wave in the Hopkinson tube is similar to the force signal on the incident bar. The maximum compressive strain on the Hopkinson tube noted ε max is about 320 × 10 -6 . The known maximum force applied to the die and the maximum strain measured on the tube enables the calculation of a calibration ratio K max = F max ε max and this value is about 196 × 10 6 N. Using the second projectile, the ratio becomes about 187 × 10 6 N. This result shows that the coefficient K max depends slightly on the loading time. These values can be compared with the calculation of K = E tube S tube = 130 × 10 6 N which corresponds to the case of a tube without the die perturbation.

The red dashed curve on Fig. 7 represents the strain obtained from the applied force divided by K max . A difference can be observed between the black and the red curves at the end of the signals. This difference can be explained by the presence of the die and the dynamics of the apparatus which induce 3D effects such as wave reflection inside the die.

History of the Blanking Forces

The calibration gives the elementary forces signals, F calib (t), and the related strain, ε calib (t) measured on the Hopkinson tube. The blanking force identification method approximates the measured strain ε m using a combination of different ε calib (t). The measured strain signal is decomposed in several elementary calibration strains and using the property of linearity inherent in the device, it is possible to estimate the related blanking force. To have a good approximation of the strain signal ε m (t) and therefore a good approximation of the blanking force, the calibration curve ε calib (t) must be improved by combination with other calibration signals. This family of calibration curves is noted ε calib,i (t) and the associated forces F calib,i (t).

A temporal lag in the force generates the same temporal lag on the strain measured on the tube. A calibration curve family ε calib,i (t) = ε calib (t + i t c ) is thus obtained by time shifting the calibration curve ε calib (t). The time-gap t c between each curve is chosen equal to 1 μs, the sampling time of the data acquisition. The associated forces are also F calib,i (t) = F calib (t + i t). With this calibration curve family, strains measured on the Hopkinson tube, ε m (t), can be written as a linear combination of ε calib,i (t)

ε m (t) = n i α i ε calib,i (t) + r(t) with ε calib,i (t) = ε calib (t + i t c ) (2)
with n the number of curve and r(t) the residual strain. This approach can be used because the relation between measured strain and the blanking forces is quite linear. This hypothesis can be justified experimentally during the calibration of the system. Therefore, a sum of calibration forces F calib,i(t) which are nearly the same as the force obtained during the blanking process is applied and if the

Micro-strain

Strain measured on the Hopkinson tube during the calibration Strain calculated from the applied force during calibration and Kmax corresponding strains ε i (t) are known, an estimation of blanking forces F m (t) noted F recons can be reconstructed:

F recons (t) = n i α i F calib,i (t) with F calib,i (t) = F calib (t + i t c ) (3)
Ag a pG between the measurements and the strains obtained with the calibration is defined as follows:

G = 1 t t f t i ε m (t) - n i α i ε calib,i (t) 2 dt (4)
with t = t f -t i the measurement duration. To get the best approximation of ε m (t) and also the best approximation of F m (t) the residual r(t) and the gap G must be minimized. The derivative expression of equation ( 4) is therefore calculated and optimized α i which minimizes the gap between ε m (t) and i α i ε i (t) are obtained. The Fig. 8 shows the result of this construction for a blanking test with a 2 mm the sheet metal thickness and a punching speed of 11.5 ms -1 . In this case, the time-gap between each calibration curve is 12 μs. Only the 10 first curves are shown on the Fig. 8, multiplied by the coefficient α i calculated with the method. The red curve represents the measurements made during the blanking test and the black one represents the sum of all curves used on the calibration. After the optimization step, it is also possible to estimate the blanking force F c (t) using the optimized α i and the forces signal family F i (t) using equation [START_REF] Smith | Die design handbook[END_REF]. Figure 9 also represents in red the estimation of the blanking forces with the reconstruction method for a punching speed around 11.5 ms -1 , a thickness about 2 mm and a gap equal to 0.05 mm. To obtain this curve, the time-gap between each calibration curve is equal to 1 μs. The black curve in this figure corresponds to the forces calculated directly by multiplying the strain in the Hopkinson tube by the factor K = E tube S tube (called the "K method"). For this test, the maximum force calculated with the two methods is 34 kN with the reconstruction method and 31 kN with the "K method". Figure 10 shows the results of the two methods for a test using the same thickness and the same clearance but using a higher punch speed of 16.7 ms -1 . The maximal blanking forces are respectively 26 kN and 35 kN obtained from the "K method" and the reconstruction method. From these two tests it can be concluded that the correction of the proposed reconstruction method becomes more significant for the shorter loading duration induced by a higher blanking speed. The shape of the blanking force evolution is dependent on the punch speed: for the blanking speed of 11.5 ms -1 the blanking forces show two maxima and for the blanking speed of 16.7 ms -1 the blanking force shows only one maximum. This result can be more precisely explained by the effect of the punch geometry on the blanking forces evolution and by the complex wave propagation in the punch.

From each test, a loading duration defined as the time during which the blanking force is higher than 50 % of the maximum blanking force can be estimated from the Figs. 9 and 10. For a punch speed of 11.5 ms -1 and 16.7 ms -1 ,the loading durations are respectively 69 μs and 44 μs.

Results and Discussion

Different thicknesses of 2 mm,3mm and 4 mm and different punch speeds between 7 ms -1 and 18 ms -1 have been tested. For each test the maximum blanking force obtained with the reconstruction method is shown in Fig. 11. The evolution of the maximum forceversus the initial punch speed for the various thicknesses is quasi-linear. For a punch speed of 14 ms -1 the maximum force is equal to 34 kN,5 4kN and 76 kN respectively for the thickness of 2 mm,3mm and 4 mm. These blanking force values divided by two times the cross section of the specimen give blanking stresses equal to 500 MPa, 529 MPa and 558 MPa: these values are very close. The evolution of the blanking forces according to the punch speed seems to be constant for a thickness of 2 mm and decreases with increasing thickness. This thickness effect has not previously been described in the literature. The reduction of the blanking forces with the punch speed is because for high punch speeds, the thermal softening of the materials plays a predominant role compared with the strain and strain rate hardening [START_REF] Stock | The energy required for highspeed shearing of steel[END_REF]. The greater the thickness of the specimen, the greater the decrease in the blanking forces with speed. For thicknesses of 3 mm and 4 mm, the maximum blanking forces decrease by 0.6 kN and 1.0 kN when the blanking speed increases of 1 ms -1 .

For each test, a characteristic time denoted t c can be defined as the ratio of the thickness and the initial punch speed. The Fig. 12 represents the evolution of the ratio of the loading duration and the characteristic time and shows that this ratio is very close for the different thicknesses and it slightly decreases when the blanking speed increases. So when the blanking speed increases, the drop of the blanking forces occurs earlier and therefore the fracture takes place for a lower penetration depth of the punch. These results are in agreement with work of Dowling et al. [START_REF] Dowling | Dynamic punching of metals[END_REF].

Conclusion

To better understand the high speed blanking process, an experimental device has been designed and built to match certain industrial characteristics such as punch-die clearance or tool geometry. This device works at high punch speeds, up to 25 ms -1 and allows the measurement of the punch speed evolution and the blanking forces during the process. In this study, a new method for measuring the blanking force evolution has been developed. This method is based on the Hopkinson tube method and takes into account the dynamic effects and the complex geometry of the device and in particular, of the die.

Dynamic punching tests were carried out on C40 steel specimens. Thicknesses of 2 mm,3mm and 4 mm and different punch speeds were used to examine the punching speed effect. For an initial punch speed of 14 ms -1 and a thickness of 3 mm, a maximum blanking force was estimated at about 54 kN. For the C40 steel, the maximum force calculated with the method decreases with increasing punch speed. The loading duration evolution from the punch speed shows that the specimen fracture takes place earlier for a lower penetration depth of the punch when the punch speed increases.

Fig. 1

 1 Fig. 1 Scheme of the experimental device (a) general view, (b) enlargement on the active zone, (c) specimen, (d) geometry of the punch

Fig. 2

 2 Fig. 2 Punch speed variation during a cutting test (C40 Steel; initial speed: 11.5 ms -1 ; thickness: 2 mm; clearance: 0.05 mm)(a) general view (b) enlargement during the cut

Fig. 3

 3 Fig. 3 Evolution of the measured strain on the Hopkinson tube ε m versus time

Fig. 4

 4 Fig. 4 Diagram of the experimental calibration device

Fig. 5

 5 Fig. 5 Calibration force F calib applied on the die

Fig. 6

 6 Fig.[START_REF] Johnson | Further experiments in quasi-static and dynamic blanking of circular discs from various metals[END_REF] Evolution of the measured strain in the tube and the bar versus time

Fig. 7

 7 Fig. 7 Calibration strains on the Hopkinson tube ε calib

Fig. 8 Fig. 9

 89 Fig. 8 Representation of the reconstruction of the measured strain with the calibration curve family

Fig. 10

 10 Fig. 10 Estimation of the blanking forces (C40 Steel; initial speed: 16.7 ms -1 ; thickness: 2 mm; clearance: 0.05 mm)

Fig. 12 1

 12 Fig.[START_REF] Zurek | The study of adiabatic shear band instability in a pearlitic 4340 steel using a dynamic punch test[END_REF] Ratio of the loading time to the characteristic time evolution versus punch speed

Table 1

 1 Metallurgical composition of the C40 steel Weight % 0.35 ± 0.40 .65 ± 0.15 0.4 0.035 0.035 0.10

	CM nS i	P	S	M o

Acknowledgment

This study was carried out with the financial support of CETIM.


Hugues Thouin

Tiffanie Lescure

Pascale Gautret

Catherine Joulian

Claude Le Milbeau

Brunet Fabienne Battaglia

Influence of organic matter and microbial activities on the mobility of arsenic and metals in polluted soils

Oral

Session: Biogeochemical processes in anthropogenic ecosystems

Abstract text;

Processes of remediation on anthropogenic polluted sites often involve the addition of organic amendments to restore soils biological functions and physic-chemical properties. However, interactions between organic matter and bacterial activities with inorganic pollutantsdrive many biogeochemical reactions influencing metal(loids) mobility. Incubation experiments were performed with four polluted soils sampled on industrial and mining sites all,containing high concentration of arsenic, iron, together with lead and antimony on one site. Soils were incubated in aerobic slurries, with or without addition of a complex mixture of organic substances. Abiotic controls were prepared with autoclaved soils. Arsenic speciation and concentrations of total arsenic and metals were determined at the beginning and at the end of incubation together with bacterial biomass and diversity. Results showed that without organic matter addition, micro-organisms contribute to reduce As and metal concentrations in the aqueous phase. In contrast, when organic matter was added, microorganisms enhanced As(III), total arsenic and lead concentrations in the liquid phase. Only iron concentration was not increased by microbial activity when organic matter was added, suggesting that As and lead mobilization were not linked to bioreduction of iron oxides, but involved other biogeochemical mechanisms. The addition of organic matter may (i) enhance the solubilizationby chelation of metal(loids),and/or (ii) influence bacterial activitiesdirectly influencing metal(loids) speciation and mobility. These processes need to be better understood in order to avoid mobilization of toxic elements from soils that may be enriched in organic matter, through amendment or growth of vegetal cover.This work was performed in the frame of Labex VOLTAIRE ANR,10,LABX,100,01.


MD Pierre Alexis Geoffroy
email: pierre.a.geoffroy@gmail.com

Carole Boudebesse

Frank Bellivier

Mohamed Lajnef

Chantal Henry

Marion Leboyer

Jan Scott

Bruno Etain

Sleep in remitted bipolar disorder: a naturalistic case-control study

Keywords: bipolar disorder, actigraphy, euthymia, remission, sleep, circadian rhythms

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

Bipolar disorder (BD) is a severe mental disorder and the worldwide prevalence of the BD spectrum is 1-4%. Although researchers agree that BD is multifactorial with inherited, genetic and environmental risk factors, the patho-physiological determinants are unknown [START_REF] Geoffroy | Reconsideration of bipolar disorder as a developmental disorder: Importance of the time of onset[END_REF].

Recent studies of circadian genes and phenotypes indicate that sleep/activity dysregulation is a plausible model for BD episode recurrence [START_REF] Mcclung | How Might Circadian Rhythms Control Mood? Let Me Count the Ways[END_REF]. Abnormalities in sleep and circadian biomarkers during the euthymic phase have been identified, revealing that this dysregulation is not purely an epiphenomenon of acute illness episodes; indeed, increased disruption of the sleep homeostasis and circadian system are frequently associated with relapse [START_REF] Etain | Genetics of circadian rhythms and mood spectrum disorders[END_REF][START_REF] Harvey | Sleep and circadian rhythms in bipolar disorder: seeking synchrony, harmony, and regulation[END_REF]. Consequently, the validation of sleep and circadian biomarkers of BD may help in the development both of methods for earlier and more accurate detection of unstable mental states, and of novel strategies to prevent relapse [START_REF] Frey | Biomarkers in bipolar disorder: a positional paper from the International Society for Bipolar Disorders Biomarkers Task Force[END_REF].

Subjective reporting of sleep profiles is useful in clinical practice, although objective measures are increasingly utilized for research. Actigraphy (recording the subject's sleep/wake cycles generally with a mobile portable device) is considered to be the most relevant non-invasive technique for the measurement of sleep/wake irregularities [START_REF] Morgenthaler | Practice parameters for the use of actigraphy in the assessment of sleep and sleep disorders: an update for 2007[END_REF]. Actigraphy has been used extensively in the field of chronobiology, and has more recently been applied to research in mood disorders, and in particular the characterization of sleep homeostasis and circadian rhythm abnormalities in BD [START_REF] Kaplan | Evaluating sleep in bipolar disorder: comparison between actigraphy, polysomnography, and sleep diary[END_REF]. We identified nine published actigraphic studies of BD: although most suggest that, even in remission, BD cases were more likely than 'controls' to display a range of sleep abnormalities, there was significant heterogeneity in the methodologies employed [START_REF] Gershon | Restless pillow, ruffled mind: sleep and affect coupling in interepisode bipolar disorder[END_REF][START_REF] Harvey | Sleep-related functioning in euthymic patients with bipolar disorder, patients with insomnia, and subjects without sleep problems[END_REF][START_REF] Jones | Actigraphic assessment of circadian activity and sleep patterns in bipolar disorder[END_REF][START_REF] Kaplan | Evaluating sleep in bipolar disorder: comparison between actigraphy, polysomnography, and sleep diary[END_REF][START_REF] Millar | The sleep of remitted bipolar outpatients: a controlled naturalistic study using actigraphy[END_REF][START_REF] Mullin | A preliminary study of sleep in adolescents with bipolar disorder, ADHD, and non-patient controls[END_REF][START_REF] Ritter | The characteristics of sleep in patients with manifest bipolar disorder, subjects at high risk of developing the disease and healthy controls[END_REF][START_REF] Salvatore | Circadian activity rhythm abnormalities in ill and recovered bipolar I disorder patients[END_REF][START_REF] St-Amand | Sleep disturbances in bipolar disorder during remission[END_REF]. The five major between-study differences were: 1) the mean age of sample (e.g. ranging from adolescents with early onset BD to older adults with chronic, persistent BD); 2) the overall sample size (the total number of participants including cases and controls ranging from 27 to 68); 3) the duration of actigraphic recording (some studies recorded data for only 3 days with a median of 6 days for the nine studies); 4) different approaches to 'matching' cases and controls, with a number of studies failing to match on demographic characteristics; and/or 5) failure to control for confounding factors such as body mass index (BMI) and/or sleep apnoea syndrome (SAS).

Most of these previous studies examined 'quantitative' differences (mean scores of each sleep measure for each group), and mainly focused on four basic sleep variables (sleep duration, latency and efficiency, and waking after sleep onset). However, it is increasingly argued that differences in terms of quality or variability (which can be represented by the standard deviations [SD] in the mean scores for a particular measure extracted from actigraphic recordings) may be more sensitive markers for distinguishing BD sleep patterns from those of controls [START_REF] Scott | Clinical parameters of circadian rhythms in affective disorders[END_REF]. Only three studies have published data relevant to this approach [START_REF] Gershon | Restless pillow, ruffled mind: sleep and affect coupling in interepisode bipolar disorder[END_REF][START_REF] Millar | The sleep of remitted bipolar outpatients: a controlled naturalistic study using actigraphy[END_REF][START_REF] Mullin | A preliminary study of sleep in adolescents with bipolar disorder, ADHD, and non-patient controls[END_REF], but comparing the findings is difficult. Indeed, the study among adolescents [START_REF] Mullin | A preliminary study of sleep in adolescents with bipolar disorder, ADHD, and non-patient controls[END_REF], and the study involving three days of actigraphic recording [START_REF] Salvatore | Circadian activity rhythm abnormalities in ill and recovered bipolar I disorder patients[END_REF] differed from the study in older adults over a longer period (28 days) of actigraphic monitoring [START_REF] Gershon | Restless pillow, ruffled mind: sleep and affect coupling in interepisode bipolar disorder[END_REF]. Also, none of these studies controlled for the possible effects of BMI or level of risk for SAS on sleep profiles. Thus, published findings for the quality and quantity of sleep in remitted BD may have been confounded by other individual characteristics, including those known to affect sleep directly, and by heterogeneity in sampling and/or in study design.

To overcome the limitations noted in previous studies, we report an analysis of mean scores of subjective and objective measures, and variability in objective measures, using methodological approaches designed to avoid potential confounders. Previous studies examined each reported sleep variable in independent analyses and none of them adequately addressed potential confounders in their statistical analytic strategy. We therefore tried to overcome some of these previous weaknesses in design by examining which combination of sleep measures best distinguishes cases from controls. Our hypothesis was to identify and/or confirm reliable and robust circadian/sleep markers differentiating euthymic BD cases from HC when potential confounders of sleep and circadian differences are taken into account (most notably, age, gender, daytime sleepiness, mood symptoms, BMI and risk of sleep apnoea).

MATERIALS AND METHODS

Sample

With ethical approval from the Institutional review board, written informed consent was obtained from 55 adults (BD=26; HC=29). The included cases were recruited from our universityaffiliated psychiatric clinic (University Paris Est) and the HC were recruited from individuals attending the blood donor service at the adjacent general hospital (Henri Mondor Hospital, Creteil).

The groups (BD and HC) were matched as closely as possible for age and gender: perfect matching was not feasible, so we increased the size of the HC group to ensure that we had a control of the same age and/or a control of the same gender for each case recruited. Individuals eligible for the study were included if, during the preceding three months, they had 1) not experienced any periods of severe sleep disruption due to somatic conditions (eg organic insomnia/ hypersomnia, or sleep-wake disorders) and/or any life event that may have altered their sleep patterns (e.g. shift work, jet-lag, child birth, trauma, or somatic disease known to be associated with sleep disturbances); 2) not been hospitalised or received a treatment that may disrupt sleep (e.g. for cases: electro-convulsive therapy);

3) not been prescribed medication or taken drugs that may disrupt sleep (e.g. sympathomimetic stimulants, corticosteroids, thyroid hormone, antiarrhythmics, beta-blockers, clonidine, diuretics, theophylline, and medications containing alcohol or caffeine) and not changed either the dose or type of psychotropic treatment.

Bipolar cases also had to fulfil the following inclusion criteria: the DSM-IV criteria for BD according to the French version of the Diagnostic Interview for Genetic Studies (DIGS) [START_REF] Nurnberger | Supersensitivity to melatonin suppression by light in young people at high risk for affective disorder. A preliminary report[END_REF]; currently euthymic i.e. they scored <8 on both the Montgomery-Asberg Depression Rating Scale (MADRS; [START_REF] Montgomery | A new depression scale designed to be sensitive to change[END_REF] and the Young Manic Rating Scale (YMRS; [START_REF] Young | A rating scale for mania: reliability, validity and sensitivity[END_REF]; and the International Society of Bipolar Disorder task force criteria [START_REF] Tohen | The International Society for Bipolar Disorders (ISBD) Task Force report on the nomenclature of course and outcome in bipolar disorders[END_REF] for remission (i.e. they had not experienced a BD episode in the prior three months). Patients experiencing a BD relapse whilst participating in the study were excluded.

Controls were assessed with the DIGS and the Family Interview for Genetic Studies [START_REF] Maxwell | Family Interview for Genetic Studies[END_REF]; individuals were excluded if they had any personal history of DSM-IV disorders and no family history of schizophrenia, affective disorders and/or suicide attempts.

Assessment Procedures

Cases and controls were first assessed for affective symptoms using the MADRS and YMRS, and their BMI was calculated. Participants then completed the Berlin Questionnaire (a measure of the risk of SAS) [START_REF] Netzer | Using the Berlin Questionnaire to identify patients at risk for the sleep apnea syndrome[END_REF], and the Epworth Sleepiness Scale (a measure daytime sleepiness) [START_REF] Johns | A new method for measuring daytime sleepiness: the Epworth sleepiness scale[END_REF], to allow these potential confounding factors to be taken into account.

Self-Ratings of Sleep Profile

All participants recorded their subjective sleep quality using the French version of the 19-item Pittsburgh Sleep Quality Index (PSQI) [START_REF] Blais | Evaluation of insomnia: validity of 3 questionnaires[END_REF]. Here, we report the mean total score over 21 consecutive days and seven other specific measures of sleep (with higher scores indicating lower quality or worse outcome): sleep quality, latency, duration and efficiency, disturbances, use of sleeping medication and daytime dysfunction (e.g. sleepiness and enthusiasm).

Actigraphy

All participants were asked to wear an actigraph (AW-7 CamNtech) on the wrist of their nondominant hand for the 21 consecutive days, the days for which they completed the PSQI self-ratings.

The AW-7 actigraph is an accelerometer that detects the intensity and the amount of movement as a function of time. For this study, data were sampled in one-minute epochs and participants were instructed to press the event-marker when they went to bed to go to sleep and when they got out of bed at the start of the next day. They also completed a sleep diary for these 21 days. Other requirements were kept to a minimum to try to minimise intrusion into sleep/activity routines. Recording was made during periods when the participants were not on vacation and not involved in unusual activities.

Two experienced psychiatrists (PAG and CB) analysed the actigraphic records of cases and controls using the sleep detection algorithm provided by Actiwatch software (Actiwatch Activity & Sleep Analysis Ltd CamNtech® 7.28) and any incongruences between the sleep diary and actigraphy rest periods were clarified prior to statistical analyses (see reference Boudebesse et al., 2012 for details).

We selected the ten clinically interpretable actigraphic measures that have been most frequently reported in previous studies [START_REF] Gershon | Restless pillow, ruffled mind: sleep and affect coupling in interepisode bipolar disorder[END_REF][START_REF] Harvey | Sleep-related functioning in euthymic patients with bipolar disorder, patients with insomnia, and subjects without sleep problems[END_REF][START_REF] Jones | Actigraphic assessment of circadian activity and sleep patterns in bipolar disorder[END_REF]Kaplan et al., 2012;[START_REF] Millar | The sleep of remitted bipolar outpatients: a controlled naturalistic study using actigraphy[END_REF][START_REF] Mullin | A preliminary study of sleep in adolescents with bipolar disorder, ADHD, and non-patient controls[END_REF][START_REF] Ritter | The characteristics of sleep in patients with manifest bipolar disorder, subjects at high risk of developing the disease and healthy controls[END_REF][START_REF] Salvatore | Circadian activity rhythm abnormalities in ill and recovered bipolar I disorder patients[END_REF][START_REF] St-Amand | Sleep disturbances in bipolar disorder during remission[END_REF]. There were six measures of basic sleep profile: time in bed, sleep duration, sleep latency, wake after sleep onset (WASO), sleep efficiency, fragmentation index (a measure of sleep continuity); two measures of activity during sleep: mean activity in active periods of sleep, mean sleep activity; and two measures of variability: inter-daily stability and intra-daily variability. To facilitate comparison with other studies that used different measures of variability, we also estimated the variability of the basic sleep measures over 21 days (using standard deviations (SD) of the mean scores) (see references of [START_REF] Millar | The sleep of remitted bipolar outpatients: a controlled naturalistic study using actigraphy[END_REF] and [START_REF] Mullin | A preliminary study of sleep in adolescents with bipolar disorder, ADHD, and non-patient controls[END_REF] for the rationale and details of this approach).

Statistical Analysis

All analyses were performed using SPSS 19.0. We included several covariates in the analysis: age and/or gender because matching was incomplete, and also BMI, SAS, daytime sleepiness (Epworth) and the presence of residual or low-grade mood symptoms (as measured using the MADRS or YMRS) that can all be associated with differences in sleep profile independently of case-control group status (i.e. BD or HC). We undertook a power calculation to assess the sample size required to detect a between-group Effect Size (ES) of ≥0.35 (classified as a medium ES according to [START_REF] Cohen | Statistical power analysis for the behavioral sciences[END_REF][START_REF] Cohen | Statistical power analysis for the behavioral sciences[END_REF]) for self-rated and objective measures of sleep whilst controlling for these potential confounders. We thereby estimated that the sample size of 55 individuals had >80% power to detect differences at a significance level of 0.05 (and an ES≥0.35) in a multivariate analysis.

The analysis involved three stages. First, we checked the normality of the distributions of variables and undertook log transformation if required (none were needed). Second, we used multivariate Generalized Linear Modelling (GLM) for the analysis of sleep variables (self-ratings; actigraphy measures). GLM is a flexible statistical model that incorporates normally distributed dependent variables and categorical or continuous independent variables (we also report Wilk's Lambda, to indicate the proportion of generalized variance in the dependent variables explained by the model and also to give the best estimate of overall power of the analysis to differentiate between groups). We used bootstrapping (i.e. resampling) to reduce biases introduced by any 'outliers' and to obtain the best estimates of summary statistics. Any variables that were significantly different between groups (even allowing for interactions between group and other categorical or continuous variables) were then entered into a Backward Stepwise Logistic Regression (BSLR) to assess the combination of these selected variables that best classified individuals as cases or controls. The proportion of cases correctly classified is reported along with the odds ratios (OR) and 95% confidence intervals (95% CI) for each variable included in the final optimal model.

RESULTS

Socio-demographic and clinical characteristics

The sample comprised of 55 individuals (26 BD cases; 29 HC), the mean age was about 54 years and more than half (30/55) were female (Table 1). Cases fulfilled the criteria for BD subtype I (n=16) or BD subtype II (n=10). One patient did not receive medication, and nine patients received only one psychotropic medication and 16 received ≥2 medications. The psychotropic treatments used were: lithium (n=12), anti-convulsants (n=15), atypical antipsychotics (n=7), typical antipsychotics (n=4), benzodiazepines (n=4), antidepressants (n=5).

There were no statistically significant differences between the groups for age, gender, BMI, level of risk of SAS, or daytime sleepiness. Levels of mood symptoms were very low in both groups, but the median score on the MADRS was slightly higher for the BD group than controls (0.50 vs. 0.00); there were no between-group differences on the YMRS.

-Insert Table 1 about here -Mean scores for a number of subjective and objective ratings of sleep differed between cases and controls. There were statistically significant differences between the groups for several PSQI and actigraphy measures, even when taking into account gender, risk of SAS, age, BMI, and scores on the Epworth, MADRS, and YMRS. When considering the multivariate values for the model as a whole, there was a significant difference between the patients with BD and HC (Wilks' lambda, F=90.89; df 26,16; p=0.0001; statistical power 0.93).

Several interactions between group and other variables were identified; in particular, five PSQI items (Total score and scores for PSQI subscales 1, 4, 5 and 7), four mean actigraphy measures (sleep latency, duration, efficacy, and fragmentation index, inter-daily stability) and five actigraphy variability measures (variability in sleep duration, sleep efficiency, fragmentation index and time in bed) differed between the groups (Table 2). As measured by the PSQI, patients with BD were more likely to report poorer sleep quality (p=0.001), poorer sleep efficiency (p=0.001), more frequent sleep disturbances (p=0.001), and more daytime dysfunction (p=0.001); consequently they had poorer global sleep quality as measured by the PSQI total score (p=0.0001). As determined from mean actigraphy scores, patients with BD were more likely to have longer sleep latency (p=0.007), longer sleep duration (p=0.02), poorer sleep efficiency (p=0.04), higher fragmentation index (p=0.05) and poorer inter-daily stability (p=0.03). For actigraphy variability over 21 days, patients with BD were more likely to show more variable time in bed (p=0.005), sleep duration (p=0.04), sleep efficiency (p=0.03) and fragmentation index (p=0.03).

-Insert Table 2 about here -

We next used BSLR to determine the optimal combination of variables that best classified participants as cases or controls. When the variables identified by the GLM were entered into a BSLR; a combination of four variables (mean sleep duration, mean sleep latency, variability in fragmentation index over 21 days, and mean score on PSQI sub-scale 7: daytime dysfunction) correctly classified 89% of the study participants as cases or controls (Chi-square=39.81; df=6; p=0.001). The OR for these variables ranged from 1.03 (sleep duration) to 8.35 (PSQI 'daytime dysfunction' sub-scale) (Table 3). However, whilst daytime dysfunction is the variable with the highest OR, on its own it classified about 75% cases, so the other measures significantly contributed to the improved classification obtained by the final model.

-Insert Table 3 about here -

DISCUSSION

We report a study of sleep abnormalities in remitted bipolar (BD) patients compared with HC that also controlled for major potential confounders, and replicate and extend previous findings. In particular, objective and subjective assessments of sleep indicate that, compared to controls, BD patients show: longer sleep duration, longer sleep latency, poorer sleep efficiency, higher fragmentation index (with actigraphy) [START_REF] Gershon | Restless pillow, ruffled mind: sleep and affect coupling in interepisode bipolar disorder[END_REF][START_REF] Harvey | Sleep-related functioning in euthymic patients with bipolar disorder, patients with insomnia, and subjects without sleep problems[END_REF][START_REF] Jones | Actigraphic assessment of circadian activity and sleep patterns in bipolar disorder[END_REF][START_REF] Kaplan | Evaluating sleep in bipolar disorder: comparison between actigraphy, polysomnography, and sleep diary[END_REF][START_REF] Millar | The sleep of remitted bipolar outpatients: a controlled naturalistic study using actigraphy[END_REF][START_REF] Mullin | A preliminary study of sleep in adolescents with bipolar disorder, ADHD, and non-patient controls[END_REF][START_REF] Ritter | The characteristics of sleep in patients with manifest bipolar disorder, subjects at high risk of developing the disease and healthy controls[END_REF][START_REF] Salvatore | Circadian activity rhythm abnormalities in ill and recovered bipolar I disorder patients[END_REF][START_REF] St-Amand | Sleep disturbances in bipolar disorder during remission[END_REF], lower sleep quality, weaker sleep efficiency, more sleep disturbances, and more daytime dysfunction (with PSQI) [START_REF] Rocha | Significant sleep disturbances in euthymic bipolar patients[END_REF]. The partial discrepancy between PSQI (subjective measure) and actigraphy (objective measure) findings has also been observed previously: patients with BD underestimated their sleep latency and duration on questionnaire assessments compared to actigraphy examination. This may arise as a consequence of misperceptions in patients with BD regarding their sleep quality [START_REF] Gershon | Restless pillow, ruffled mind: sleep and affect coupling in interepisode bipolar disorder[END_REF][START_REF] Harvey | Sleep-related functioning in euthymic patients with bipolar disorder, patients with insomnia, and subjects without sleep problems[END_REF][START_REF] Millar | The sleep of remitted bipolar outpatients: a controlled naturalistic study using actigraphy[END_REF].

Nevertheless, actigraphy demonstrated a high correlation with polysomnography (PSG) measures in patients with BD regarding sleep latency, sleep duration, fragmentation index and sleep efficiency, which further validate our findings [START_REF] Kaplan | Evaluating sleep in bipolar disorder: comparison between actigraphy, polysomnography, and sleep diary[END_REF].

We report several new findings for issues that have been under-explored: patients with BD presented with more variability for time in bed, sleep duration, sleep efficiency and fragmentation index. Indeed, sleep duration, sleep efficiency and fragmentation index are altered not only in quantity (mean) but also in variability (SD) in BD patients. This indicates that analyses of actigraphic markers should not be restricted to classical sleep markers (latency, duration, efficiency and WASO), but should also be extended to the variability of markers. Indeed, Millar et al. were the first to explore the variability of markers in different groups and showed that the means of none of the four actigraphic measure differed between cases and controls, but variability of two of the four measure did (sleep duration and night waking time) with an effect size around 0.35 [START_REF] Millar | The sleep of remitted bipolar outpatients: a controlled naturalistic study using actigraphy[END_REF]. Similarly, Gershon et al. confirmed more variability of both time in bed and sleep duration for BD patients than controls (although the latter did not survive their Bonferroni's correction) [START_REF] Gershon | Restless pillow, ruffled mind: sleep and affect coupling in interepisode bipolar disorder[END_REF].

Millar et al. also found more variability of night wake time among remitted cases [START_REF] Millar | The sleep of remitted bipolar outpatients: a controlled naturalistic study using actigraphy[END_REF].

Both [START_REF] Gershon | Restless pillow, ruffled mind: sleep and affect coupling in interepisode bipolar disorder[END_REF] and our study (0.40±0.3 vs. 0.36±0.3) replicate this finding. Further, we observed significant variability of sleep efficiency and fragmentation index, which confirmed the greater variability of sleep patterns and circadian rhythms in patients with remitted BD than controls. Mullin et al., despite also assessing the variability of actigraphic measures, did not find any significant differences [START_REF] Mullin | A preliminary study of sleep in adolescents with bipolar disorder, ADHD, and non-patient controls[END_REF]; however, that study involved recordings for only four nights, which may have been too short to detect variability. Studies involving sufficient duration of recording would be useful to explore further the variability of actigraphic measures in BD patients, and to examine trait-markers of BD.

The results of our regression analysis deserve comment. The fact that 89.3% of BD and 88.5% of HC could be correctly classified indicates that a combination of different methods of assessment (i.e PSQI, mean and SD actigraphy measures) may serve as a useful marker of BD. Daytime dysfunction, sleep duration, sleep latency and fragmentation index variability appeared, when combined, to identify clear circadian biomarkers of remitted BD. Indeed, it may be possible to define a biosignature of BD using these four features. We partially replicate the findings of Millar et al. whose best multivariate model involved a combination of one actigraphic (variability of sleep duration) and two subjective sleep variables (average sleep duration, and average onset latency) [START_REF] Millar | The sleep of remitted bipolar outpatients: a controlled naturalistic study using actigraphy[END_REF].

These two studies indicate that individuals are best classified by a combination of subjective and objective measures (quantity and variability). The other comment is that it is noteworthy that-although daytime dysfunction-was a subscale score of the PSQI, it is debatable whether this is a specific measure of sleep pattern/dysfunction. Indeed in BD cases it may be a measure of overall effects of illness or possible effects of medication, not only of daytime consequences of disturbed sleep.

Our study has some limitations. Controlling for psychotropic medications is always difficult in bipolar disorder research. In this study, the size of the case sub-group did not allow us to include treatment data as a covariate in any 'within group' sub-analyses. However, the type and dose of psychotropic medications have been found to be unrelated to actigraphic measures in patients with remitted BD [START_REF] Salvatore | Circadian activity rhythm abnormalities in ill and recovered bipolar I disorder patients[END_REF]. Moreover, we believe that on balance, it can be hypothesized that medication would be as likely or more likely to improve the patients' quality of sleep as to have a detrimental effect. We also attempted to control for daytime sedation by using scores on the Epworth scale as a covariate in the preliminary between group comparisons. As such, the observed differences between cases and controls may have been attenuated rather than exaggerated by such treatments. Indeed, antipsychotics, valproate and lithium have been shown to regulate sleep and circadian rhythms in patients with bipolar disorder [START_REF] Geddes | Treatment of bipolar disorder[END_REF]. Therefore, we believe that it is unlikely that psychotropic drugs had a major influence. Our sample size is small; however, it is of reasonable size for an actigraphic study and larger than the median for the previous studies cited (median = 40) [START_REF] Gershon | Restless pillow, ruffled mind: sleep and affect coupling in interepisode bipolar disorder[END_REF][START_REF] Harvey | Sleep-related functioning in euthymic patients with bipolar disorder, patients with insomnia, and subjects without sleep problems[END_REF][START_REF] Jones | Actigraphic assessment of circadian activity and sleep patterns in bipolar disorder[END_REF][START_REF] Kaplan | Evaluating sleep in bipolar disorder: comparison between actigraphy, polysomnography, and sleep diary[END_REF][START_REF] Millar | The sleep of remitted bipolar outpatients: a controlled naturalistic study using actigraphy[END_REF][START_REF] Mullin | A preliminary study of sleep in adolescents with bipolar disorder, ADHD, and non-patient controls[END_REF][START_REF] Ritter | The characteristics of sleep in patients with manifest bipolar disorder, subjects at high risk of developing the disease and healthy controls[END_REF][START_REF] Salvatore | Circadian activity rhythm abnormalities in ill and recovered bipolar I disorder patients[END_REF][START_REF] St-Amand | Sleep disturbances in bipolar disorder during remission[END_REF]. We studied several markers of subjective and objective sleep. Although GLM is an appropriate method for assessing multiple, potentially inter-dependent measures, and the power calculation (and estimated effect sizes) suggest we have reduced the risk of type II error, it is clear that it would be beneficial to use a larger sample for further studies. Nevertheless, our study allowed controlling for confounding factors that were insufficiently taken into account in previous studies; our greater attention to potential confounding factors was an important methodological strength relative to most previous studies.

We were thus able to confirm that patients with BD in remission experienced substantial sleep disturbances that can be measured both objectively and subjectively. This provides further evidence of the involvement of sleep and circadian rhythm abnormalities in the pathophysiology of BD, as proposed by, e.g., [START_REF] Harvey | Sleep and circadian rhythms in bipolar disorder: seeking synchrony, harmony, and regulation[END_REF]. Indeed, genetic variants of candidate genes (mainly circadian genes) may predispose individuals to be relatively less able to adapt their circadian rhythms appropriately to their environment, and therefore to greater vulnerability to sleep disturbances. Since circadian and neurotransmission systems are highly connected, circadian and/or sleep-related abnormalities may affect the functioning of the dopamine and serotonin circuitry, which in turn may affect mood regulation. These chronic sleep and circadian rhythm disturbances (observable in acute episodes and in remission) may thereby contribute to the more general issues persistently encountered bipolar patients such as poor quality of cognitive functioning, residual emotional hyper-reactivity or vulnerability to metabolic disturbances [START_REF] Boland | Life events and social rhythms in bipolar spectrum disorders: An examination of social rhythm sensitivity[END_REF][START_REF] Boudebesse | Emotional hyper-reactivity and sleep disturbances in remitted patients with bipolar disorders[END_REF][START_REF] Mcclung | How Might Circadian Rhythms Control Mood? Let Me Count the Ways[END_REF][START_REF] Soreca | Sleep duration is associated with dyslipidemia in patients with bipolar disorder in clinical remission[END_REF].

The findings we report have several important clinical implications. Sleep and circadian rhythm disturbances in remitted BD might be treatable by manipulating the circadian system using chronobiological medication (e.g. melatonin or melatonin analogues) or sleep focused psychological interventions [START_REF] Kaplan | Behavioral treatment of insomnia in bipolar disorder[END_REF][START_REF] Livianos | Is melatonin an adjunctive stabilizer?[END_REF][START_REF] Pacchierotti | Melatonin in psychiatric disorders: a review on the melatonin involvement in psychiatry[END_REF][START_REF] St-Amand | Sleep disturbances in bipolar disorder during remission[END_REF]. Attention to rhythm abnormalities is of particular interest because biological rhythm dysfunctions have been found to be associated to a wide range of sources of poor functioning and are the earliest markers of impending mood relapses [START_REF] Giglio | Functional impact of biological rhythm disturbance in bipolar disorder[END_REF]. Thus, better identification of these abnormalities in clinical practice might facilitate the prevention of the evolution of prodromes of mood relapses, and also help improve functioning in inter-episodic BD. It may allow a more complete approach including combinations of treatments acting on several dimensions including sleep or circadian disturbances [START_REF] Geoffroy | Combination Therapy for Manic Phases: A Critical Review of a Common Practice[END_REF]. These findings more generally indicate a new area for research both into possible therapeutic targets and to improve our understanding of the circadian pathophysiology of BD.

CONCLUSION

This 21-day case-control study demonstrates the presence of subjective and objective disturbances of sleep and activity markers in patients with BD during remission. Disturbances identified by actigraphy are both quantitative (mean scores) and qualitative (variability). We further confirm the greater variability in sleep markers in remitted patients with BD. These findings indicate that future studies should examine both the mean scores and the variability over extended periods of time. In addition, we show that a combination of subjective and objective measures (quantity and variability) may be a better circadian biosignature of BD than any single measure on its own.

Table 1 :

 1 Baseline characteristics of bipolar disorder (BD) cases and healthy controls (HC) Risk of Sleep Apnoea Syndrome assessed using the Berlin Questionnaire **Daytime sleepiness assessed using the Epworth Sleepiness Scale

		BD (n=26) Mean (±SD)	HC (n=29) Mean (±SD)	p
	Age (years)	53.50 (±11.49)	54.10 (±9.11)	0.78
	Montgomery Asberg	1.85 (±2.80)	0.48 (±1.35)	0.02
	Depression Rating Scale
	Median score	0.50	0.00
	Young Mania Rating	0.65 (±1.38)	0.14 (±0.44)	0.16
	Scale	0.00	0.00
	Median score
	Body Mass Index	26.74 (±5.72)	26.64 (±4.17)	0.68
	Daytime Sleepiness**	8.85 (±3.90)	6.86 (±3.72)	0.12
		n (%)	n (%)
	Sleep Apnoea			0.09
	Syndrome*: High Risk	6 (23%)	2 (7%)
	Category	20 (77%)	27 (93%)
	Low Risk Category
	Gender Distribution:			0.13
	Female	17 (65%)	13 (45%)
	Male	9 (35%)	16 (55%)

*

Table 2 :

 2 Differences between sleep measures for cases (BD) and controls (HC)

				Multivariate Generalized Linear
		BD (n=26)	HC (n=29)		Model*
		Mean (±SD)	Mean (±SD)
				F	p	Observed Power
			PSQI measures
	Total score	(±3.49) 7.38	(±2.13) 4.11	4.45	0.0001	0.99
	Sub-scale 1	1.19	0.71	3.64	0.001	0.99
	Subjective sleep quality	(±0.69)	(±0.66)
	Sub-scale 2	1.19	0.82	0.70	0.74	0.33
	Sleep latency	(±0.80)	(±0.91)
	Sub-scale 3	0.62	0.50	1.53	0.15	0.70
	Sleep duration	(±0.94)	(±0.75)
	Sub-scale 4	0.88	0.43	3.86	0.001	0.99
	Habitual sleep efficiency	(±0.95)	(±0.69)
	Sub-scale 5	1.58	1.18	3.67	0.001	0.99
	Sleep disturbances	(±0.58)	(±0.48)
	Sub-scale 6	0.69	0.14	1.08	0.18	0.68
	Use of night sedation	(±1.09)	(±0.59)
	Sub-scale 7	1.23	0.32	3.53	0.001	0.98
	Daytime dysfunction	(±0.99)	(±0.48)
		Mean actigraphy scores
	Time in bed (mins)	511.81	482.31	1.79	0.08	0.79
		(±50.28)	(±49.36)
	Sleep duration (mins)	475.42	455.83	2.38	0.019	0.91
		(±64.5)	(±53.98)
	Sleep latency (mins)	25.23	11.59	2.81	0.007	0.95
		(±33.65)	(±7.98)
	Wake after sleep onset	57.88	52.62	0.57	0.85	0.27
	(mins)	(±23.17)	(±27.33)
	Sleep efficiency (%)	81.54	84.90	2.13	0.036	0.86
		(±9.87)	(±6.32)
	Fragmentation index	32.04	28.69	2.01	0.05	0.84
		(±10.51)	(±10.88)
	Mean activity in active	115.54	91.10	1.21	0.31	0.58
	sleep periods**	(±37.67)	(±24.39)
	Mean overall sleep activity	19.35	12.69	1.69	0.11	0.76
		(±13.10)	(±7.07)
	Inter-daily stability	0.503	0.512	2.17	0.033	0.87
		(±0.14)	(±0.12)
	Intra-daily variability	0.82	0.80	0.90	0.56	0.43
		(±0.17)	(±0.18)
		Actigraphy measure variability over 21 days
	Time in bed (mins)	1.44	1.33	2.97	0.005	0.96
		(±0.7)	(±0.5)
	Sleep duration (mins)	1.3	1.13	2.08	0.04	0.86
		(±0.6)	(±0.4)
	Sleep latency (mins)	0.67	0.43	1.40	0.21	0.66
		(±1.2)	(±0.7)
	Wake after sleep onset	0.49	0.36	1.80	0.08	0.79
	(mins)	(±0.3)	(±0.3)
	Sleep efficiency (%)	8.77	6.13	2.26	0.026	0.89
		(±7.5)	(±3.8)
	Fragmentation index	12.52	9.28	2.16	0.033	0.96
		(±7.3)	(±2.7)

*Model controlled for age, gender, daytime sleepiness (Epworth), current mood symptoms (MADRS and YMRS), BMI and risk of sleep apnoea (Berlin); **Intensity of movement during periods of sleep when activity occurs

Table 3 : Backward Stepwise Logistic Regression (BSLR) showing (a) the best combination of variables for correctly classifying study participants as cases or controls and (b) the overall classification rate by group 3a

 3 ) Variables included in the final BSLR model

	Variable		p	Odds Ratio	Lower	95% C.I. Upper
	Sleep Duration		0.015	1.03		1.01	1.05
	Sleep Latency		0.012	1.24		1.05	1.47
	Fragmentation Index: Mean Variability	0.01	1.63		1.12	2.31
	Daytime Dysfunction		0.005	8.35		2.50	18.11
	(PSQI Subscale-7)
	3b) Classification table and summary statistics
			Predicted
	Observed			% Correctly
		Controls	Cases	Classified
	Controls*	25	3		89.3%
	Cases	3	23		88.5%
	% of all participants correctly classified 89%
	Chi-square 39.81 (df 6) p=0.001
	*1 control participant was excluded from the analysis due to inadequate data


B França De Paula

G Rejdych

T Chancelier

G Vermot Des Roches

E Balmes

Vibrations, Shocks and Noise On the influence of geometry updating on modal correlation of brake components

Keywords: Correlation, finite-elements models, model updating, morphing, modal assurance criterion, reduction, test-analysis correlation

In most industries dealing with vibration, test/analysis correlation of modal properties is considered a key aspect of the design process. The success of test/analysis methods however often show mixed results. The aim of this paper is to assess and answer some classical correlation problems in structural dynamics. First an investigation of correlation problems from tests is proposed. Tools based on the modal assurance criterion are presented to provide a deeper analysis of correlation and results improvement. In a second part, the need of FEM topology correlation and update is demonstrated, using an efficient morphing technique. Tolerances in the manufacturing process that are well accepted in design and production stages are shown to lead to significant degradation of the test/analysis correlation. An application to an industrial brake part is eventually presented, in an approach of correlation procedure automatization for recurrent use.

Introduction

NVH (Noise Vibration Harshness

) is a common source of problems that need to be addressed in the design of vehicles. To improve the design cycle in quality and resources needed, finite element models (FEM) are used to make predictions, discriminate between designs and analyze quantities that are not accessible to test. At various steps, test/analysis correlation is then used to validate the predictions made of components and assemblies.

While the quality of FEM models is very well established, test analysis correlation very often shows mixed results. Understanding the origin of problems in correlation is thus a critical aspect of a proper use of such procedures.

The first objective of this paper is an analysis of correlation problems coming from tests. When using rich scanning laser vibrometer measurements, analyzing measurement quality is a problem. Intermediate levels of correlation are thus shown in section 2.1 to be possibly due to measurement errors. It is first shown how the MACCo [START_REF] Balmes | Vermot des Roches, Structural Dynamics Toolbox User's Guide[END_REF] is appropriate for outlier removal thus leading to improved correlations. The second common problem is associated with closely spaced modes. It is then shown how a subspace based Modal Assurance Criterion (MAC) [START_REF] Allemang | The modal assurance criterion (MAC): twenty years of use and abuse[END_REF] is a more relevant indication when correlating very close modes.

The second part of the paper addresses errors coming from FEM. Classical updating procedures consider that model properties are not well known. These are material properties for components and junction properties for assemblies. The assumption generally made is that since objects and FEM come from the same CAD geometry errors there are minimal. Using morphing tools to adjust geometries for sensitivity and geometry updating studies, section 3 illustrates the fact that this assumption is far from being true. Section 4 eventually presents the integration of these correlation tools into a full updating process applied to a real component.

Experimental sources of poor correlation

When performing test/simulation correlation it is a common procedure to consider test data as exact. This is of course not the case, and a good correlation methodology should consider that error sources can come from test data and from simulation data. This section focuses on the exploitation of the MAC and illustrates how poor results can arise although an objectively good correlation exists.

Efficient use of the MAC for test correlation with rich data

All experimental results contain errors -this is usually improved by a pass of so-called outliers' removal. The detection of outliers however becomes difficult when the number of measurement points increases. Laser vibrometry performed on meshes of over 1,000 points in the three directions lead to large amounts of data. Some points are noisy, due to poor local reflection of the laser beam (orientation, surface roughness, partially masked edge …) and should be removed. Performing Operational Modal Analysis (OMA) adds a level of difficulty and error source, so that it is tempting to base the correlation on Operational Deflection Shapes (ODS) at peaks of the measured response.

The MACCo is a pure what if algorithm, looping on the sensors set and targeting the one that if removed would most improve MAC results. The sensor removal is then chained sequentially up to a desired number of sensors. This process is an efficient procedure for outlier detection assuming that the correlation is not too poor. Figure 2 illustrates this pattern by showing the MAC results between the experimental and the FEM modes (without additional updating) of a test bench component presented in figure 1. shows that with 1% of sensors removed the results can already be improved. Pushing the algorithm further highlights the fact that an optimum exists where only relevant information can be kept, without significant degradation of the result. This optimum was here found around 40% of sensor removal (200 over a total of 504 sensors), as illustrated in figure 2(c), where 12 modes can be identified. Removing more sensors is then becomes difficult as the measure autoMAC degrades. Figure 3 eventually shows the modes identified by MAC after the MACCo loop.

∑ ∑ = = + ⋅ × = EXP FEM N k N l MAC l i MAC i k MAC i i MAC i i MAC i Qual 1 1 ) , ( ) , ( , 2 , ) ( (1)
where N EXP is the number of experimental modes, and N FEM is the number of numerical modes. It can thus be seen that a clear and global MAC improvement is obtained during the removal of the first percents, for all frequency bands. Improvements over this threshold depend on the frequency band. Indeed, the optimum for the low frequency band (figure 4(c)) occurs for a removal of 30%, 50% for middle band and over 60% for higher frequencies.

Since lower frequency modes are more global in terms of deformation, the presence of very poor sensors disturbing the results are quickly found by MACCo. This is less true for higher frequency modes, where local aspects become important. Modes with low observability can thus be sensitive to specific poor sensors with a local effect. Such type of sensor is not efficiently detected by the MACCo, hence the difficult optimization of higher frequency modes with a raw MACCo. Improvement of this work would consider a MACCo for mode subsets with equivalent observabilities, thus focusing on the removal of a minimal set of sensors with efficient effect on global and local scales.

Subspace correlation for closely spaced modes

When measuring components with closely spaced modes, the order between experimental result and numerical modes can change. A fact that tends to be ignored and will be illustrated here is that the shapes themselves can then be different, although the subspaces generated by the shapes of both models are the same.

The realization of subspace correlation based on the shapes compared can be realized by considering a cross projection. Given two sets of shapes,φ from the numerical model andψ from the experiment, expressed on the sensor sets, one projects the experimental shapes on the subspace generated byφ such that

ψ φ ψ = (2)
whereψ ˆis the projection of ψ onφ .

A singular value decomposition onψ ˆ can then be performed to obtain a second displacement basis of the subspace spanned byψ ˆ, in the space spanned byφ

T v uΣ = ψˆ (3)
T uv is thus a normed and reordered basis of the subspace spanned byψ ˆ in the space spanned byφ . If both subspaces are well correlated, the restitution of this reordered basis in the space spanned byψ andψ itself are identical, else, a poor result will be obtained.

The proposed application is a symmetry breaking effect on a test bench component. Several configurations are tested, with different heights of the large left bracket height, shown in figure 5. Two modes are identified involving flexions of the component flenches, highly impacted by the brackets. When studying the MAC between the modes computed with the reference, taken with identical bracket heights, and the modes with asymmetrical brackets, a cross pattern occurs as the shapes frequencies come closer, illustrated in figures 6 and 7(a). The shapes evolve as function of the braket height. It can be seen in particular that the MAC between modes 14 and 15 decreases while modes 17 and 18 cross. Using the subspace correlation method, used to produce the MAC results of figure 7(b), it is understood that in fact the space spanned by these 5 five shapes is strictly identical in all cases.

Effect of geometry variation on modal properties of a simple component

The common practice is to update material properties of component parts. This methodology however assumes that the FEM geometry is identical to the real component. For parts with local geometrical refinement, and whose production method shows significant variability (in the mm range for die cast brake parts), this assumption is often false (see [START_REF] Blain | Etudes experimentales et numériques de la dispersion vibratoire d'assemblages soudés par points[END_REF] for example). This section aims at showing that errors on geometry can yield errors on modes which may be larger than those due to errors on material properties thus implying the need to update geometries.

An efficient morphing technique for geometry updating

Geometry updating is set on the same basis than usual shape optimization methods. Here modifications are deemed to be slight from a given FEM, so that only morphing is considered. Morphing defines a methodology of topology deformation by moving certain nodes, edges and/or surfaces of the component. The additional constraint placed here is that the component FEM already exists and remeshing is excluded thus allowing easy comparisons of models with a constant number of nodes and degrees of freedom.

Classical morphing techniques often fail as the surface mesh deformation used to adapt the component shape is not properly transmitted to the volume mesh, thus leading to mesh geometries that are not suitable for FEM computations. To succeed, morphing must continuously and smoothly transmit the component surface deformation to the underlying volume mesh. This effect is obtained here by successively morphing edges, surfaces and volumes. Applying such morphing methodology thus allows much higher amplitude of topology deformation without having to remesh the component.

Figure 8 shows the application of this morphing methodology on the component left bracket. Looking at figure 5, the three top points of the bracket are commanded in height. For a nominal height of 50mm, a range of 17mm is available (-7 to +10mm) without need of remeshing. Figure 8 shows in particular that a large portion of the flench surface is impacted by this height modification, although the topology outside the bracket is unchanged. A first application of morphing is the study of component sensitivity to geometric modification. Areas of interest can for example be found by looking at the localization of high strain energy density areas. Figure 9(a) indeed shows that modes 14 and 15, sensitive to the bracket height have high strain energies in this area, and show high frequency variation as function of this parameter in figure 9(b).

Improving correlation using measured geometries

To update the component model, a measure must be provided, based on an accurate topology measurement, using for example a COMET bench measure. Figure 10 shows the real component measurement on the COMET bench and the component FEM along with their orientation basis. The comparison between both meshes is not straightforward. The measure output is a very fine surface mesh, corners are not straight, and some faces are missing. The FEM is much coarser, and has straight edges. Once both meshes are overlayed, the geometry update aims at sticking the FEM surface onto the measured one. Figure 11 shows the initial mesh superposition, with clear differences. A manual pass has to be done to correct outstanding errors. Control points on the FEM model are thus defined, here on the surface edges of the FEM, as illustrated in figure 11 and 12. This allows a manual input of macroscopic surface displacements, as illustrated in figure 12 Fig. 12. Manual input corrections based on control points of the FEM mesh, before/after positions.

Full updating process applied to real brake parts

The results presented in section 2 and 3 are here integrated to a full updating process of a real brake component. The focus is also set on the procedures automatization for recurrent use in production.

Geometry updating of a brake anchor and correlation

Tolerances between a nominal model and an industrially produced part can show some variations. A brake anchor is tested here, presented in figure 15(a). A first pass of manual correction on macroscopic errors is first performed to correct outstanding errors, like illustrated in figure 15(b). A pass of automatic correction can then be performed to optimize shape corrections over multiple faces for fine tuning. It is indeed unpractical to manually control surface movement under 0.5mm. The methodology applied to the automatic correction computes for each FEM element of the surface the distance to paired elements of the measured surface, along the FEM normal. A correction field can then be computed by applying these differences along their normals.

Figure 16 shows two passes of automatic correction and the visualization of the faces. It can be seen a reduction of topology errors, from an initial mean error of distances of 0.5mm to a maximum of 0.03mm. The impact on the model frequencies is presented in the conclusive figure 19. Experimental data are obtained performing ODS on the component, using a laser vibrometer with 656 positions in the three directions, yielding 1968 sensors. The view positioning on the FEM model for MAC computation must be handled with care. Indeed, the laser measure has to be matched on the solid surface, although the mesh position output by the laser is not always very precise. Figure 17 The MAC results of figure 17(c) can be further improved by performing the MACCo pass presented in section 2. Figure 18 shows the results of the MAC improvement by sensor removal, using the MACCo and the MAC quality criterium ( 1). An optimum arises around a removal of 40% of the total set of sensors. A significant improvement of the MAC results is obtained for the higher frequency modes.

Property updating

Once modeshapes and geometry have been correlated in section 4.1, material properties can be updated correlating frequencies. Indeed, one is now ascertained that material updating will not be biased by geometry imprecision. An automatic updating procedure has been developed to update simple material formulations in mass and Young modulus.

The total mass of the component can be computed by projection of the mass matrix on the system geometrical rigid body modes normalized to one. Indeed, the mechanical energy of the component following a rigid body trajectory is

rbm T rbm m M E φ φ 2 1 = (4)
If the rigid body trajectory is normalized to one, the mass is directly integrated over the structure, thus yielding a very quick mass estimation. A proportionality coefficient can then be applied to conform to the measured mass of the real component.

Updating the Young modulus of an isotropic material is also straightforward. Brutal force is chosen here, computing the system frequencies over a large range of interest, with a refined Young modulus discretization. That is to say 2,000 computation points between 213 and 225 GPa.

Model reduction is here employed (see [START_REF] Balmes | Parametric families of reduced finite element models[END_REF] for example) to optimize computation time. Current modeshapes are used as Rayleigh-Ritz vectors since shapes are well correlated. The Young modulus is a factor of the reduced stiffness matrix, so that its parameterization is trivial. The total system size thus decreases from over 200,000 DOF to 30 DOF. Several optimum detection strategies can be explored, trying either to minimize mean frequency errors, or standard deviation, or maximum error. Since the parameter is very macroscopic to the system, it was chosen to minimize the average frequency error, i.e. finding the Young modulus opt

E verifying         ∆ = ∆ ∑ ∑ es i i i E es i i opt i f E f f E f mod mod ) ( min ) ( (5)
The final update results of the component can thus be summarized in figure 19. It can be seen that an average error of 0.96% is obtained on frequencies. A reminder of the effect on frequencies of previous updating steps is also presented.

Conclusion

The aim of this paper was to show how an eventually good FEM/test correlation was improved. Test data improvement was performed using MAC based tools outlier removal and subspace correlation. The key aspect of FEM topology updating regarding the real tested component has also been illustrated, through the use of an efficient morphing technique. This appears to be as sensitive and influent as the classical material parameters used in updating procedures. This kind of error, if not handled, usually ends-up as a bias into the materials property update. Although some applications can bear it, some where material identification is sought, or where topology has a great influence (shells) should avoid this bias.

The idea that the measure and the FEM have inherent and unavoidable defaults must be assessed, to only exploit the better of both. This becomes a clear stake in an industrial environment where this work is classically performed by two separate teams, one mastering experiments, and the other mastering FEM computations.

A natural step further in this work is to master assemblies. Control of mechanical assemblies' modal properties is sought at the design stage, which implies a fine test/correlation analysis validation at each level. The role of geometry in the way contacting solid interfaces behave indeed requires a robust model at the component level.

Fig. 1 .

 1 Fig. 1. Presentation of test test bench component studied. (a) FEM and denomination of areas of interest. (b) Illustration of a vibrometry measurement. (c) FEM and test-mesh wireframe positioning.

Figure 2 (

 2 Figure 2(a) thus shows at first glance a poor correlation with only two to three modes properly identified.

Fig. 2 .

 2 Fig. 2. MAC between FEM and experimental results for the test bench component (504 sensors) (a) Raw MAC result; (b) MAC result with 1% of removed sensors with MACCO, (c) MAC result with 40% of removed sensors (d) MAC result with 80% of removed sensors.

Figure 2 (

 2 Figure 2(b)shows that with 1% of sensors removed the results can already be improved. Pushing the algorithm further highlights the fact that an optimum exists where only relevant information can be kept, without significant degradation of the result. This optimum was here found around 40% of sensor removal (200 over a total of 504 sensors), as illustrated in figure2(c), where 12 modes can be identified. Removing more sensors is then becomes difficult as the measure autoMAC degrades. Figure3eventually shows the modes identified by MAC after the MACCo loop.

Fig. 3 .

 3 Fig. 3. Modes identified after the MACCo pass

Fig. 4 .

 4 Fig. 4. Evolution of the MAC quality criterion as function of the number of removed sensors. (a) Modes up to 2.8kHz. (b) Modes between 2.8 and 5.7kHz. (c) Modes between 5.7 and 10kHz.

Fig. 5 .

 5 Fig. 5. Identification of the test bench component brackets linking the base and the flench parts When both brackets have the same height, both flexions are at the same frequencies, yielding an in-phase, and out-of-phase mode. When different heights are applied, the frequencies are separated and two modes featuring a single side flexion are found.

Fig. 6 .

 6 Fig. 6. A mode crossing due to a symmetry braking pattern

Fig. 7 .

 7 Fig. 7. Mode crossing patterns as function of the component bracket height. (a) Upper figures: Raw MAC. (b) Lower figures: subspace correlation.

Fig. 8 .

 8 Fig. 8.(a) Morphing applied to a height reduction of 7mm of the 50mm height bracket (b)Morphing applied to a height increase of 10mm of the 50mm height bracket.

Fig. 9 .

 9 Fig. 9.(a)Localization of strain energy density for modes 14 and 15 (b)Frequency variation of modes 14 and 15 as function of the bracket heights (symmetrical modification).

Fig. 10 .

 10 Fig. 10.(a) Measurement component mesh output by the COMET test bench. (b)Component FEM. Blue lines are orientation bases of both models for initial positionning

Fig. 11 .

 11 Fig. 11. Initial superposition of the COMET bench measure and the FEM, over two 2D views. The FEM is represented by its control points forming the component surface edges (in red).

Figure 15

 15 Figure 15. (a) Industrial brake part studied. (b) Manual correction of macroscopic errors

Figure 16 .

 16 Figure 16. (a)Mean topology error over two passes of automatic corrections. (b) Visualization of the first three faces..

  (a) shows the FEM and an initial test frame positioning, output by laser. The first MAC obtained in figure 17(b) is not fully satisfying. Performing a proper surface match with projection on the FME surface yields the MAC of figure 17(c), which is clearly improved.

Figure 17 .

 17 Figure 17. (a) Positioning tolerances between the FEM and the wire frame imported. (b) first MAC using a direct match between both meshes. (b) Improved MAC by performing a proper test wire frame projection.

Figure 18 .

 18 Figure 18. (a) MAC quality criterion evolution as function of the sensor removal. (b) Final MAC after 40% sensor removal

Figure 19 .

 19 Figure 19. (a) Evolution of the FEM frequencies during the updating sequence. (a) after topology update, (b) after density update, and (c) after Young modulus update.

The effect of the topology update on mode frequencies is provided in Figure 13. It can be seen that errors up to 9% was introduced by the geometry imprecision. The most impacted modes also shown in figure 13 are the modes with high strain energy in the outstanding areas. Such errors are very high regarding usual errors on Young modulii and density for isotropic steels.

The impact on MAC is also of interest. Here the MAC between initial and updated FEM is computed for identical geometrical points. The aim is not to measure mesh points having moved during the geometry update. Figure 14 shows that some modeshapes significantly evolves as function of the topology update.