P Flores
email: pflores@dem.uminho.pt

C S Koshy

H M Lankarani

J Ambrósio

J C P Claro

Numerical and experimental investigation on multibody systems with revolute clearance joints

Keywords: Multibody Dynamics, Clearance Joints, Contact Forces, Dry Contact, Experimental Test Rig

A comprehensive combined numerical and experimental study on the dynamic response of a slider-crank mechanism with revolute clearance joints is presented and discussed in this paper to provide an experimental verification and validation of the predictive capabilities of the multibody clearance joint models. This study is supported in an experimental work in a test rig, which consists of a slider-crank mechanism with an adjustable radial clearance at the revolute joint between the slider and the connecting rod. The motion of the slider is measured with a linear transducer and an accelerometer.

Dynamic tests at different operating crank speeds and with several clearance sizes are performed. The maximum slider acceleration, associated with the impact acceleration, is used as a measure of the impact severity. The obtained results demonstrate the dynamical behavior of a multibody mechanical system with a clearance joint. Finally, the correlation between the numerical and experimental results is presented and discussed leading to validated models of clearance revolute joints.

Introduction

A mechanical system is made of several components, which can be divided in two major groups, namely, links, that is, bodies with a convenient geometry, and joints, which introduce some restrictions on the relative motion of the various bodies of the system.

Usually, the bodies are modeled as rigid and/or deformable bodies, while the joints are modeled through a set of kinematic constraints. That is, the joints are not modeled as contact pairs in the strict sense of the word contact, but as algebraic constraints to which implicit forces are associated. The functionality of a joint relies upon the relative motion allowed between the connected components. In most cases, this implies the existence of a clearance between the mating parts, and thus surface contact, shock transmission and the development of different regimes of friction and wear. On the other hand, no matter how small that clearance is, it can lead to vibration and fatigue phenomena, lack of precision or, even random overall behavior.

Over the last few decades, a number of researchers have proposed various methodologies for modeling mechanisms with clearance joints [START_REF] Schwab | Dynamics of Flexible Multibody Systems, Small Vibrations Superimposed on a General Rigid Body Motion[END_REF][START_REF] Flores | Dynamic Analysis of Mechanical Systems with Imperfect Kinematic Joints[END_REF][START_REF] Koshy | Characterization of Mechanical Systems with Real Joints and Flexible Links[END_REF][START_REF] Flores | Kinematics and Dynamics of Multibody Systems with Imperfect Joints: Models and Case Studies[END_REF][START_REF] Flores | Lubricated revolute joints in rigid multibody systems[END_REF][START_REF] Tian | Simulation of planar flexible multibody systems with clearance and lubricated revolute joints[END_REF][START_REF] Erkaya | A neural-genetic (NN-GA) approach for optimising mechanisms having joints with clearance[END_REF][START_REF] Erkaya | Investigation on effect of joint clearance on dynamics of four-bar mechanism[END_REF][START_REF] Ambrósio | Improved bushing models for general multibody systems and vehicle dynamics[END_REF][START_REF] Flores | Spatial rigid-multi-body systems with lubricated spherical clearance joints: modeling and simulation[END_REF][START_REF] Mukras | Analysis of planar multibody systems with revolute joint wear[END_REF][START_REF] Mukras | Comparison between elastic foundation and contact force models in wear analysis of planar multibody system[END_REF][START_REF] Flores | Dynamics of Multibody Systems with Spherical Clearance Joints[END_REF][START_REF] Tian | Dynamics of spatial flexible multibody systems with clearance and lubricated spherical joints[END_REF][START_REF] Dong | Kinetic uncertainty analysis of the reheat-stop-valve mechanism with multiple factors[END_REF]. However, most of these studies only deal with numerical models. The literature reporting on experimental studies on mechanical systems with clearance joints is confined to a few publications.

Dubowsky and Moening [START_REF] Dubowsky | An Experimental and Analytical Study of Impact Forces in Elastic Mechanical Systems with Clearances[END_REF] studied experimentally the interactions between clearance joints and the system elasticity, using a Scotch-Yoke mechanism. Accelerometers were used in that study to measure the impact accelerations. A failure occurred at the Scotch-Yoke mechanism due to fatigue, caused by large impact forces developed at the clearance joint. Grant and Fawcett [START_REF] Grant | Effects of Clearance at the Coupler-Rocker Bearing of a 4-Bar Linkage[END_REF] investigated the effects of clearance size, lubrication and material properties on the contact loss in a four bar linkage with one clearance joint. Their experimental results confirmed the validity of the theoretical approach proposed, but only for a limited class of systems. Dubowsky et al. [START_REF] Dubowsky | An Analytical and Experimental Study of the Prediction of Impacts in Planar Mechanical Systems with Clearances[END_REF] studied analytically and experimentally a simple system, called the Impact Ring Model, to predict the impacts in planar mechanical systems with clearance joints. Haines [START_REF] Haines | An Experimental Investigation into the Dynamic Behaviour of Revolute Joints with Varying Degrees of Clearance[END_REF] carried out an experimental investigation on the dynamic behavior of a simple journal-bearing with varying degrees of freedom. In that study, the contact loss between the journal and bearing was predicted using proximity transducers. Bengisu et al. [START_REF] Bengisu | A Theoretical and Experimental Investigation of Contact Loss in the Clearances of a Four-Bar Mechanism[END_REF] studied the contact loss in revolute clearance joints of a four bar mechanism, being the relative motion between the journal and bearing measured by employing optical methods. Soong and Thompson [START_REF] Soong | A Theoretical and Experimental Investigation of the Dynamic Response of a Slider-Crank Mechanism with Radial Clearance in the Gudgeon-Pin Joint[END_REF] presented a theoretical and experimental investigation of the dynamic response of a slidercrank mechanism with a revolute clearance joint where the slider and the connecting rod accelerations were quantified by using accelerometers. More recently, Khemili and Romdhane [START_REF] Khemili | Dynamic analysis of a flexible slider-crank mechanism with clearance[END_REF] studied the dynamic behavior of a planar flexible slider-crank mechanism with clearance joints using simulation and experimental tests. Erkaya and Uzmay [START_REF] Erkaya | Experimental investigation of joint clearance effects on the dynamics of a slider-crank mechanism[END_REF] presented an interesting work, in which an extensive experimental data of a slider-crank mechanism with clearance joints were discussed. These two last works consider a similar test rig, where the system describes the motion in the vertical plane.

The main goal of this work is to provide an experimental verification and validation of the predictive capabilities of the clearance joint models revised in this work.

The experimental procedure complements the numerical studies in the literature and it provides a proposal for the coherent combination of numerical and experimental work which needs to be undertaken, in order to establish validated models and to identify directions for subsequent studies allowing for the improvement of the models. This study is supported in an experimental test rig that consists of a slider-crank mechanism with an adjustable radial clearance at the revolute joint between the slider and the connecting rod.

Dynamic tests at different operating crank speeds and with several clearance sizes are performed. The maximum slider acceleration, associated with the impact acceleration, is used as a measure of the impact severity. The obtained results demonstrate the dynamical behavior of a clearance joint and they provide qualitative measures that can be associated with fatigue and wear phenomena, when the system components have to operate with real joints. The correlation between the numerical and experimental results is presented and discussed allowing to define the range validity of the models. The motivation to develop a new experimental test rig comes from the necessity to have a flexible apparatus that allows for the study of the effect of the clearance in joints on the dynamic response of the system. In addition, the new experimental test rig allows studying of different test conditions such as, number and size of clearance joints, different contacting materials, lubrication actions, and several input conditions. Furthermore, the approach presented in this work can also be expanded to include spatial clearance joints with only minor modifications.

Modeling revolute joints with clearance

In the classical analysis of a revolute joint, the journal and bearing centers coincide, that is, the revolute joint is considered ideal or perfect, but the inclusion of the clearance allows for the separation of these two centers. Consequently, two extra degrees-of-freedom are added to the system. Figure 1(a) depicts a revolute clearance joint, the so-called journal-bearing, in which the radial clearance, c, is measured by the difference between the bearing radius and the journal radius, R B and R J , respectively. A revolute clearance joint does not impose any kinematic constraints on the system, but limits the journal orbit to stay inside the bearing's boundaries [START_REF] Flores | A parametric study on the dynamic response of planar multibody systems with multiple clearance joints[END_REF]. In a dry contact situation, the journal can move freely within the bearing until contact between the two bodies takes place. When the journal impacts the bearing wall, a normal contact force together with a friction force are evaluated to obtain the dynamics of the journal-bearing. These forces are of a complex nature, and their corresponding impulse is transmitted throughout the mechanical system [START_REF] Flores | Influence of the contact-impact force model on the dynamic response of multi-body systems[END_REF]. Figure 1(b) illustrates the normal and tangential force components due to the impact between the journal and the bearing. The impact, which has both normal and tangential relative velocities, is treated as an eccentric oblique collision between two bodies.

Bearing Journal R J C R B Plane of collision F T F N n t BJ e (a) (b) Bearing Journal R J C R B Bearing Journal R J C R B

Plane of collision

From the dynamic configuration of the system, the relative penetration depth between the journal and the bearing can be defined as

BJ δ e c = - (1)
where e BJ is the magnitude of the eccentricity vector defined between the bearing and journal center, as seen in Fig. 1b, and c is the radial clearance, which is a specified parameter. A more detailed description on the clearance joint modeling can be found in references [START_REF] Flores | Dynamic Analysis for Planar Multibody Mechanical Systems with Lubricated Joints[END_REF][START_REF] Flores | Revolute joints with clearance in multibody systems[END_REF][START_REF] Flores | Modeling and simulation of wear in revolute clearance joints in multibody systems[END_REF].

The dynamics of a dry journal-bearing is characterized by two different situations. First, when the journal and the bearing are not in contact, with each other, there is no contact force associated with the journal-bearing. Second, once the contact between the two bodies takes place, the contact-impact forces are evaluated based on a nonlinear Hertzian-type contact force law for normal force and on the Coulomb friction law for the tangential forces. The contact conditions can be expressed for penalty forces models as

0 N T F i f δ 0 F F F if δ 0 = < = + > (2)
in which, F N and F T are normal and tangential forces, respectively. In short, when the journal reaches the bearing wall an impact takes place. This impact is treated as a continuous event, that is, the local deformations and the contact forces are continuous functions of time. The impact analysis of the system is performed simply by including the normal and tangential contact forces into the system equations of motion [START_REF] Nikravesh | Computer Aided Analysis of Mechanical Systems[END_REF][START_REF] Nikravesh | Initial condition correction in multibody dynamics[END_REF][START_REF] Baumgarte | Stabilization of Constraints and Integrals of Motion in Dynamical Systems[END_REF][START_REF] Flores | Influence of the Baumgarte parameters on the dynamics response of multibody mechanical systems[END_REF][START_REF] Flores | A parametric study on the Baumgarte stabilization method for forward dynamics of constrained multibody systems[END_REF][START_REF] Shampine | Computer Solution of Ordinary Differential Equations: The Initial Value Problem[END_REF].

For any contact in multibody systems, such as the ones owing in revolute clearance joint, the contact between the journal and bearing can be modeled in the most fundamental form by using the Lankarani and Nikravesh force model given by [START_REF] Lankarani | A contact force model with hysteresis damping for impact analysis of multibody systems[END_REF] ( )

2 3(1 ) 1 4 n N e F K δ δ δ - ⎡ ⎤ - = + ⎢ ⎥ ⎣ ⎦ ( 3
)
where K is the contact stiffness, δ is the relative penetration depth, e is the restitution coefficient, δ is the relative penetration velocity and ) (δ is the initial impact velocity.

The exponent n is set to 1.5 for most metal contacts. The parameter K depends on the material and geometric properties of the contacting surfaces. For two spherical surfaces in contact, one inside the other, the generalized stiffness parameter is given by [START_REF] Goldsmith | Impact: The Theory and Physical Behaviour of Colliding Solids[END_REF] ( )

1 2 B J B J B J R R 4 K 3 h h R R ⎡ ⎤ = ⎢ ⎥ + - ⎣ ⎦ (4)
where the parameters h B and h J are given by

k 2 k k E ν 1 h - = , (k=B, J) (5)
R B and R J are the radius of the bearing and the journal, respectively, ν k is the Poisson's ratio and E k is the Young's modulus.

In a revolute clearance joint, three different modes of motion between the journal and the bearing can be considered, namely: continuous contact mode, free-flight mode, and impact mode. These three types of the journal motion are illustrated in Fig.

2 [START_REF] Flores | A parametric study on the dynamic response of planar multibody systems with multiple clearance joints[END_REF]. In the continuous contact mode, the journal and the bearing are in contact and a sliding motion related to each other is assumed to exist. In this mode, the penetration depth varies along the circumference of the journal. This mode is ended at the instant when the journal and bearing separate and the journal enters the free flight mode. In the free flight mode, the journal can move freely inside the bearing boundaries; i.e., the journal and the bearing joint are not in contact; and hence there is no reaction force between these two elements. After the end of the free flight mode, the journal enters the impact mode, in which impact forces are applied and removed. This mode is characterized by a discontinuity in the kinematic and dynamic characteristics, and a significant exchange of momentum occurs between the two impacting bodies. At the termination of the impact mode, the journal can enter either free flight or following mode. During the dynamic simulation of a revolute joint with clearance, if the path of the journal center is plotted for discrete instant of time, the different modes of motion of the journal inside the bearing can easily be observed.

Experimental test rig

In order to investigate the dynamic response of mechanical systems with clearance joints, an experimental test rig was designed and constructed. The A slider-crank mechanism, in which the revolute joint that connects the slider and the connecting rod has a variable and controlled radial clearance, was chosen due to its simplicity and importance within all possible candidate machines and mechanisms.

An overall view of the experimental apparatus built is shown in a photograph in Fig. The remaining kinematic joints were constructed as close to the ideal joints as possible, that is, with minimum clearance and friction in order to minimize any contamination of the data that are intended to be measured. Moreover, these joints are lightly oiled to minimize the friction in their connections. A standard sleeve element is press-fitted to the extremity of the connecting rod, working as bearing, with its diameter fixed to a very tight tolerance. The journal is rigidly connected to the sliding block and incorporates a standard pin with a variable diameter, as pictured in Figure 4(b). Thus, the clearance at the test journal-bearing can be altered by simply changing the pin. A particular journal-bearing set is also manufactured in order to simulate an ideal or zeroclearance joint, which is used to obtain the reference data associated with an ideal mechanism. Different sets of journal diameters are also built to allow the analysis of the influence of the clearance on the system's dynamic response. The crankshaft is keyed to the crank and it is supported by ball bearings. A needle bearing, with minimal radial clearance and high rigidity, connects the crank to the connecting rod. The sliding block component is screwed onto a linear translational bearing, which has a precision preloaded system with zero-clearances. Table 1 shows the type of joints used in the experimental slider-crank mechanism and their nominal or operating clearances. Thus, for numerical purposes, they can be considered as ideal or zero-clearance joints [START_REF] Flores | Dynamic Analysis of Mechanical Systems with Imperfect Kinematic Joints[END_REF]. Between the driven pulley and the crankshaft, an encoder and a torque sensor are incorporated. The encoder is used to measure the crank angular position and velocity, whereas the torque sensor allows the measurement of the reaction moment that acts on the crank. Furthermore, an accelerometer and a linear voltage differential transducer (LVDT) are used to monitor the slider acceleration and displacement, respectively. The slider velocity is obtained either by performing the numerical integration of the acceleration value, or the numerical differentiation of the displacement data. The impact force between the journal and bearing is measured indirectly, that is, the impact accelerations are directly related to the impact forces.

The slider-crank mechanism works on the horizontal plane and, due to its rigidity and alignment, the gravitational effects on the system's dynamic responses can be neglected. The mechanism components are built entirely from steel and, hence for practical purposes, are assumed to be perfectly rigid. The connecting rod is built with a hollow cross-section in order to reduce the mass, while maintaining a high stiffness and, thus, reducing the flexible effects.

Test scenarios and results

A comprehensive combined study on the dynamic response of the experimental slider-crank mechanism, with different clearances in one of the revolute joints, is used here to validate the numerical approach. The experimental test rig allows adjusting two parameters: the journal diameter, i.e., the radial clearance, and the driving frequency, i.e., the crank speed. Table 3 shows the test matrix performed with the experimental slider-crank mechanism. The radial clearance is obtained by changing the journal diameter and maintaining constant the diameter of the sleeve/bearing.

In the experimental setup, some difficulties have been experienced with the data acquisition by the torque sensor. These difficulties are related to a high residual torque that is always present in the output signal of the torquemeter, which contaminates the its outcome. Thus in what follows, only the dynamic characteristics of the slider motion are used. The dynamic behavior of the experimental slider-crank mechanism is monitored using the LVDT and accelerometer, and related to the crank angle. The crank speed is constant and equal to 200 rpm and the measured radial clearance is 0.25 mm, which is a large value for the journal-bearing used. Before carrying out the experimental tests, the joint elements that constitute the clearance joint are washed out using an alcohol to remove all traces of oil due to the manufacturing process, ensuring that the contact is dry. Figure 5 presents the experimental response of the slider-crank for ideal and clearance joints during one crank revolution. The dynamic response obtained with the experimental slider-crank setup is compared with that of the numerical models in which all the joints were considered to be ideal or perfect. Figures 5(a) and 5(c) show that experimental and numerical results for slider-crank mechanism with ideal joints are similar. In fact, the position data is almost identical and only a small deviation between the numerical and experimental acceleration response is observed. The position of the slider is not affected visibly by the clearance. This result is expected because the radial clearance is very small when compared to the maximum amplitude of the slider displacement. In sharp contrast to the slider position, the slider acceleration presents significant differences between the dynamic response histories of the system with and without clearance. 6 it is observed that the maximum slider impact acceleration increases with the radial clearance. Furthermore, when the radial clearance is increased, the number of impacts increases as well. For the lower radial clearance of 0.10 mm, the maximum slider acceleration is about 3.5 times larger than that the maximum slider acceleration for the ideal joint, whereas for a larger radial clearance of 1.00 mm, the maximum slider acceleration is about 20 times larger than for the ideal joint. When the radial clearance is reduced, the peaks on the slider acceleration curve are also lower; i.e., the response is smoother, and the slider-crank behavior tends to be closer to the ideal mechanism as the journal and bearing are in contact with each other for longer periods of time. Figure 7 presents a Fast Fourier Transformation analysis (FFT) of the slider acceleration data plotted in Figure 6. It can be observed that not only the number of dominant frequencies in the system increases with clearance on the joint, but also that the magnitude of the contribution of each frequency increases. Figure 8 shows the effect of varying the crank angular velocity on the slider acceleration response. The values of the crank velocity are 100, 150, 200 and 250 rpm, and the radial clearance is 0.25 mm. For the lower crank speeds the variation on the gross motion of the slider-crank mechanism is rather small, and the slider acceleration is similar to that observed for the ideal case. However, for higher crank speeds, the number of impacts and the size of the peaks in the slider acceleration increase, as observed in Figures 8(c)-(d) with particular emphasis. In Figure 9, the FFT analysis of the slider acceleration data for different crank speeds is presented. It is observed from this figure that the number and magnitude of the contribution of the dominant frequencies increase with the crank speed. With the actual construction of the experimental slider-crank setup, the impact force on the clearance joint can not be directly measured. Therefore, the maximum slider acceleration, i.e., the impact acceleration, is used to quantify the severity of the impact at the clearance joint. Figure 10 presents the maximum slider acceleration for the experimental dynamic response histories, as function of the crank speed and radial clearance. Figure 10 shows again that the maximum slider acceleration increases with crank speed and clearance. The three-dimensional design diagram for the maximum slider acceleration, which is distilled from Figures 6 and8, can be used for analysis and design purposes of mechanical systems with clearance joints. For instance, it is possible to estimate the maximum slider acceleration for a combination of clearance and crank speed by using the design diagram of Figure 10. For low crank speeds, the apparent gross motion characteristics of the slider-crank mechanism remains unchanged, but for high crank speeds and clearances, the dynamic response is significantly modified. This observation clearly demonstrates the severe variation of the dynamical behavior in the presence of a clearance joint and provides the required data to quantify the fatigue and wear phenomena.

Fixed parameters

Correlation between numerical and experimental results

In what follows, the experimental and numerical simulation results are correlated and discussed. The length and inertia properties of the experimental slider-crank mechanism components are listed in Table 4. Parameters used in the dynamic simulation of the slider-crank mechanism.

In the modeling of the impact phenomenon in multibody systems, the selection of the friction and restitution coefficients is of great importance and influences the outcome of the results. The selection of these parameters is made with the materials and the test conditions in mind, and based on the best published data. Wilson and Fawcett [START_REF] Wilson | Dynamics of Slider-Crank Mechanism with Clearance in the Sliding Bearing[END_REF][START_REF] Flores | Modeling and analysis of rigid multibody systems with translational clearance joints based on the nonsmooth dynamics approach[END_REF] used values of the friction coefficient in the range of 0.007-0.010 and of the restitution coefficient in the range of 0.4-0.6, to investigate the dynamics of the slidercrank mechanism with clearance in the sliding bearing. They showed that the values of friction and restitution coefficients equal to 0.01 and 0.40, respectively, agreed well with experimental data response. Soong and Thompson [START_REF] Soong | A Theoretical and Experimental Investigation of the Dynamic Response of a Slider-Crank Mechanism with Radial Clearance in the Gudgeon-Pin Joint[END_REF] developed an experimental device to quantify the restitution coefficient in the impact between a journal and a bearing. Based on the kinematic, or Newton restitution coefficient, Soong and

Thompson obtained a value of 0.46 for the restitution coefficient as the quotient between the impact velocity of 5.81 cm/s and the rebound velocity of 2.67 cm/s used in this test setup. In addition, Soong used the Coulomb's friction law to obtain the friction coefficient. Thus, in the present work, the values of 0.01 and 0.46 respectively for the friction and the restitution coefficients have been used to model the experimental slidercrank mechanism.

The differences between the experimental and numerical system response profiles, plotted in Figures 11 and12, must be considered in light of the assumptions made in the formulation for the mathematical model of clearance joints, namely in what concerns to the joints flexibility, which was ignored. The friction in the sliding bearing was also neglected.

Moreover, during the non-contact situation, between the journal and the bearing, in the slider velocity is not constant in the experimental model, but decreases due to the friction in the slider bearing, because a preloaded linear bearing type has been used. For the numerical simulation, when the journal and bearing do not contact each other, the slider velocity is obviously constant and, consequently, the slider acceleration is null.

This phenomenon is well visible as horizontal lines in the slider acceleration diagrams of Figures 11(f) and 11(h).

Furthermore, misalignments between journal and bearing elements, always present in actual mechanical systems, are not considered in the numerical simulations.

Another important feature in the numerical simulations is the choice of the restitution and friction coefficients, because small variations on these parameters can significantly change the system response. The friction and restitution coefficients depend on the materials and geometric properties of the contacting bodies and can vary during the contact. However, the detailed analysis of these phenomena is considered beyond the scope of the present work being the choice of the restitution and friction coefficients solely based on the best published data. The geometrical tolerances, such as the degree of roundness of the journal and bearing surfaces, also affects the results, whereas for the model, the cross-sections are treated as perfectly round. At this stage it is important to mention that the formulation of the equations of motion for constrained multibody system adopted in the present work follows closely the Nikravesh's work, in which the generalized absolute coordinates are used to describe the system configuration [START_REF] Nikravesh | Computer Aided Analysis of Mechanical Systems[END_REF]. The Newton-Euler approach was employed to obtain the equations of motion of constrained multibody systems that was augmented with the constraint equations, resulting in a set of differential and algebraic equations [START_REF] Nikravesh | Initial condition correction in multibody dynamics[END_REF]. A great merit of this formulation is that it is quite straightforward in terms of assemble of the equations of motion and providing all reaction forces. Moreover, the formulation is detailed for the type of coordinates adopted and expressions obtained for the contact-impact models used and presented throughout this work. The issues associated with the contact problem formulation, namely in what concerns the contact detection, the integration algorithm used, the selection of the appropriate time step, the lubrication problem, the influence of the number of clearance joints, and the computational efficiency of the methodology used in this study have been presented in detail in authors' previous works [START_REF] Flores | Lubricated revolute joints in rigid multibody systems[END_REF][START_REF] Flores | A parametric study on the dynamic response of planar multibody systems with multiple clearance joints[END_REF][START_REF] Flores | Influence of the contact-impact force model on the dynamic response of multi-body systems[END_REF][START_REF] Flores | Dynamic Analysis for Planar Multibody Mechanical Systems with Lubricated Joints[END_REF][START_REF] Flores | Revolute joints with clearance in multibody systems[END_REF][START_REF] Flores | Modeling and simulation of wear in revolute clearance joints in multibody systems[END_REF][START_REF] Flores | Modelling lubricated revolute joints in multibody mechanical systems[END_REF][START_REF] Flores | Dynamic behaviour of planar rigid multibody systems including revolute joints with clearance[END_REF][START_REF] Machado | Development of a planar multi-body model of the human knee joint[END_REF][START_REF] Flores | On the contact detection for contact-impact analysis in multibody systems[END_REF][START_REF] Lopes | A mathematical framework for contact detection between quadric and superquadric surfaces[END_REF]. Figure 13 presents the maximum slider acceleration, for both experimental and numerical dynamic response histories, as function of crank speed for various radial clearances. From Figure 13, it is clear that the maximum slider acceleration increases with crank speed and with clearance, for both experimental and numerical results.

Moreover, the maximum acceleration obtained from experimental tests agrees quite well with the numerical simulations, suggesting that the predictive capability of the proposed methodology is a reasonable approach to model multibody systems with impact at clearance joints. This observation confirms other data published on the field on dynamics of multibody systems with clearance joints [START_REF] Flores | Dynamic Analysis of Mechanical Systems with Imperfect Kinematic Joints[END_REF]. The fact that the existence of the clearance joint has an important effect on the slider acceleration supports the idea that the model of clearance joints must be considered in the analysis and design of the real mechanical systems.

Conclusions

In this paper, a comprehensive investigation was undertaken for a slider-crank mechanism in which the revolute joint between the slider and the connecting rod has an adjustable clearance. Dynamic response data for different operating frequencies; i.e., various crank speeds, with different radial clearance size were presented. Prior to present the dynamic response histories with clearance joints, the experimental model was used to obtain the global motion characteristics of the slider-crank mechanism with joints as close to ideal joints as possible, i.e., with a very low clearance. The experimental response data and the numerical results for these scenarios are in complete agreement.

The dynamic behavior of the experimental slider-crank setup was monitored with an LVDT and an accelerometer. The response histories were related to the crank angle using an encoder. For comparative purposes, the maximum amplitude of the slider acceleration was taken as a measure of the severity of the impact. This approach provides a reliable form of estimating the impact force that overcomes the deficiency of the actual test rig apparatus to acquire directly the impact force. It is observed that the maximum amplitude of the impact acceleration increases with clearance and crank speed. For low crank speeds, the gross motion characteristics of the slider-crank mechanism remain similar to those of the mechanism with perfect joints. However, for high angular frequencies and clearances, the slider dynamic response is significantly altered, being the maximum impact acceleration observed by more than 20 times. This information is important in the design of mechanical systems and allows estimating the tradeoff between journal-bearing life, cost and performance. In addition, the information relative to the maximum impact acceleration can be used to help the control and maintenance of industrial machines with clearance joints.

Some critical parameters and test uncertainties are also identified. The variety of material joint properties, mainly friction and restitution coefficients, should be considered since the selection of these parameters influences the outcome of the numerical results. In absence of better reference, values published in the literature were used. The correct alignment and geometrical precision of the impacting bodies is quite difficult to obtain experimentally, thus, the contact force experienced by these may not be distributed over the theoretical contact area. The methodology used in this work is based on some simplifications that include: the contact surfaces are considered as ellipsoidal; the center of the contact area does not change during the contact process; all the components of the system are rigid bodies. The basic fundamental form of Hertzian contact model was used in this study for the development of the numerical model.

Figure 1 .

 1 Figure 1. (a) Revolute joint with clearance; (b) Normal and tangential forces due to the impact between the journal and bearing surfaces.

Figure 2 .

 2 Figure 2. Different types of journal motion inside the bearing.

Figure 3 .

 3 Figure 3. (a) Photograph of the experimental test rig; (b) Schematic drawing of the test rig.

Figure 4 .

 4 Figure 4. This joint is designed as a dry journal-bearing, as illustrated in Figure 4(b).

Figure 4 .

 4 Figure 4. (a) Experimental slider-crank mechanism with an adjustable radial clearance at the revolute joint between the slider and the connecting rod; (b) Test revolute joint in which the clearance is altered by simply changing the journal diameter: 1bearing/sleeve, 2-journal/pin; 3-clearance. Clearance is exaggerated for illustration.

Figure 5 (Figure 5 .

 55 Figure 5. Dynamic response of the experimental slider-crank mechanism for a crank speed of 200 rpm with reference to the numerical ideal joint response: (a) slider position for ideal joint; (b) slider position for radial clearance equal to 0.25 mm; (c) slider acceleration for ideal joint; (d) slider acceleration for radial clearance equal to 0.25 mm.

Figure 6

 6 Figure 6 presents slider acceleration results of the experimental slider-crank model operating at 200 rpm, for radial clearances of 0.10, 0.25, 0.50 and 1.00 mm. In

Figure

  Figure 6 it is observed that the maximum slider impact acceleration increases with the

Figure 6 .

 6 Figure 6. Slider acceleration for crank speed of 200 rpm and for different clearance sizes: (a) c=0.10 mm; (b) c=0.25 mm; (c) c=0.50 mm; (d) c=1.00 mm.

Figure 7 .Figure 8 .

 78 Figure 7. FFT analysis of the slider acceleration for different clearance sizes and for 200 rpm: (a) c=0.10 mm; (b) c=0.25 mm; (c) c=0.50 mm; (d) c=1.00 mm.

Figure 9 .

 9 Figure 9. FFT analysis of the slider acceleration for different crank speeds and clearance of 0.25 mm: (a) 100 rpm; (b) 150 rpm; (c) 200 rpm; (d) 250 rpm.

Maximum slider acceleration [m/s 2 ]Figure 10 .

 210 Figure 10. Maximum slider acceleration as function of crank speed and radial clearance.

Figure 11 .Figure 12 .

 1112 Figure 11. Experimental and numerical slider acceleration for crank speed of 200 rpm and different radial clearance sizes: (a)-(b) c=0.10 mm; (c)-(d) c=0.25 mm; (e)-(f) c=0.50 mm; (g)-(h) c=1.00 mm.

Maximum slider acceleration [m/s 2 ]Figure 13 .

 213 Figure 13. Maximum slider acceleration as function of crank speed and radial clearance: numerical and experimental results.

Table 1 .

 1 Type of joints used in the slider-crank mechanism and the corresponding nominal clearances.

	Connection	Joint type	Diameter [mm] Clearance [mm]
	Ground -Crank	Ball bearing	17.0	0.009
	Crank -Connecting rod	Needle bearing	10.0	0.005
	Connecting rod -Slider	Journal-bearing	22.2	0.002
	Ground -Slider	Translational bearing	-	0.001

Table 2 .

 2 Similarly, the slider-block inertia properties take into account the linear bearing and the accelerometer characteristics. These values are used in the numerical simulations. The overall mass of the experimental equipment, including frame and moving parts, is about Physical properties of the experimental slider-crank mechanism.

	130 kg.
	Body	Length [m]	Mass [kg]	Moment of inertia [kgm 2 ]
	Crank	0.05	17.900	0.460327
	Connecting rod	0.30	1.130	0.015300
	Sliding block	-	1.013	0.000772

The slider-crank mechanism and all other mechanical components are mounted on a heavy stiff frame. The mechanical arrangement of the experimental test rig is schematically illustrated in Figure

3(b)

. A summary of the physical properties of the experimental slider-crank model is given in Table

2

, where the crank inertia properties include the shaft, encoder, torque sensor and flywheel.

Table 3 .

 3 Test matrix performed with the experimental slider-crank mechanism.

	Variable parameters

Table 2 .

 2 The parameters used in the dynamic simulations are given in Table4. Figures11 and 12present the numerical and experimental dynamic acceleration response histories of the slider-crank model with a clearance joint.

	Restitution coefficient	0.46	Baumgarte -α	5
	Friction coefficient	0.01	Baumgarte -β	5
	Young's modulus	207 GPa	Integration step size	10 -6 s
	Poisson's ratio	0.3	Integration tolerance	10 -7

Acknowledgement

This work is supported by the Portuguese Foundation for the Science and Technology under the research project BIOJOINTS (PTDC/EME-PME/099764/2008).


E De Bilbao

D Soulat

G Hivet

J Launay

A Gasser

Experimental study of bending behaviour of reinforcements

Keywords:

In composite reinforcement shaping, textile preform undergo biaxial tensile deformation, in plane shear deformation, transverse compaction and out-of-plane bending deformations. Up today, bending deformations are neglected in some simulation codes but taking into account them would give more accurate simulations of forming especially for sti and thick textiles. Bending behaviour is specic because the reinforcements are structural parts and out of plane properties cannot be directly deduced from in-plane properties, like continuous material. Because the standard tests are not adapted for sti reinforcements with non linear behaviour a new exometer using optical measurements has been developed to test such reinforcements. This new apparatus enables to carry out a set of cantilever tests with dierent histories of load.

A series of tests has been performed to validate the test method and to show the capacities of the new exometer to identify non linear non elastic behaviour.

Introduction

Composite parts contain resin and can be constituted with short or long bers [START_REF] Long | Design and manufacture of textile composites[END_REF].

For structural applications, long bers and continuous reinforcements are generally used which give to the piece the mechanical properties and the resin function is to restrain the motion of the yarns. Their use is increasing in automotive construction and above all in aeronautics for structural parts. Such reinforcements allow to manufacture composite structures with complex shapes for example by the RTM (Resin Transfer Molding) process. The rst stage of this process consists in a shaping of the dry woven preform before resin injection [START_REF] Rudd | Liquid Moulding Technologies[END_REF][START_REF] Potter | The early history of the resin transfer moulding process for aerospace applications[END_REF][START_REF] Parnas | Liquid composite molding[END_REF]. In prepreg draping [START_REF] Rudd | Liquid Moulding Technologies[END_REF] or in continuous bre reinforcements and thermo-plastic resin (CFRTP) forming [START_REF] Maison | Technical developments in thermoplastic composites fuselages[END_REF][START_REF] Soulat | Simulation of continuous bre reinforced thermoplastic forming using a shell nite element with transverse stress[END_REF], the matrix is present but is not hardened and the deformation of the structure is driven by those of the woven reinforcement. Textile reinforcements are especially ecient in case of double curve geometries because of the interlacing of warp and weft yarns. These geometries are dicult to obtain with unidirectional reinforcements. To reach double curve geometry, in-plane strains of the fabric are necessary. Because there are usually two directions of yarns (warp and weft) that are interwoven, the fabric can reach very large in-plane shear strain.

At the end of the rst stage of the RTM process the bers orientation and the local variations of ber volume fraction due to draping can have signicant eect on the mold lling process [START_REF] Long | A simulation of reinforcement deformation during the production of preforms for liquid moulding processes[END_REF][START_REF] Pillai | The role of dual permeability of ber preforms in resin transfer molding[END_REF][START_REF] Rudd | Processing and mechanical properties of bi-directional preforms for liquid composite moulding[END_REF]. Moreover, the orientation of reinforcement bers within the preform will be a major factor in determining the structural properties of the n-ished piece [START_REF] Advani | Flow and Rheology in Polymeric Composites Manufacturing, chap. 12 -Resin transfer molding[END_REF][START_REF] Rudd | Modelling the processing and performance of preforms for liquid moulding processes[END_REF][START_REF] Laroche | Forming of woven fabric composites[END_REF]. The prediction of the deformability of the fabric during forming, by simulation tool is essential for the understanding of the manufacture process and to reduce product development cycle times and cost. In order to determine the deformed shape of draped fabrics, several codes have been developed based on geometrical approaches so called shnet algorithms [START_REF] Long | A simulation of reinforcement deformation during the production of preforms for liquid moulding processes[END_REF][START_REF] Mark | The tting of woven cloth to surfaces[END_REF][START_REF] Van Der Weeën | Algorithms for draping fabrics on doubly-curved surfaces[END_REF][START_REF] Cherouat | Geometrical and mechanical draping of composite fabric[END_REF]. The alternative to these geometrical methods consists of a mechanical analysis of the fabric deformation under the boundary conditions prescribed by the forming process. This requires a model of the woven reinforcement and its mechanical behaviour, in order to compute the deformation through a numerical method, for instance, the Finite Element Method.

In one of these methods [START_REF] Boisse | A mesoscopic approach for the simulation of woven bre composite forming[END_REF][START_REF] Boisse | Importance of in-plane shear rigidity in nite element analyses of woven fabric composite preforming[END_REF][START_REF] Zouari | A woven reinforcement forming simulation method. inuence of the shear stiness[END_REF][START_REF] Boisse | Dierent approaches for woven composite reinforcement forming simulation[END_REF][START_REF] Hamila | Simulations of textile composite reinforcement draping using a new semi-discrete three node nite element[END_REF] specic nite elements are dened that are made of a discrete number of woven unit cells. The mechanical behaviour of these woven cells is obtained essentially by experimental analyses of the woven reinforcement.

Textile preforms undergo biaxial tensile deformations, in plane shear deformations, transverse compaction and out-of-plane bending deformations. If all these deformations can be signicant in some cases it is generally possible to use a simplied approach where only some strain energies are taken into account. The in plane shear strains are necessary for woven reinforcement forming on a double curvature surface. The shear angles can be very large (up to 50°) while the tensile strains remain small (1.5% for a carbon fabric) [START_REF] Launay | Experimental analysis of the inuence of tensions on in plane shear behaviour of woven composite reinforcements[END_REF]. In several approach bending behaviour is neglected but when the forming stage leads to the formation of defaults, like wrinkles, taking into account out of plane deformation, like the bending behaviour, could make more eective these simulation methods [START_REF] Hamila | Simulation de la mise en forme des renforts composites mono et multi plis[END_REF]. Wang et al. [START_REF] Wang | Energy analysis of reinforcement deformations during viscous textile composite forming[END_REF] demonstrated the relative importance of bending behaviour during composite forming by comparison between bending and shear energies in case of viscous composites.

Moreover, it has been shown that bending properties cannot be deduced from in-plane properties, like continuous material [START_REF] Yu | Analysis of exible bending bahavior of woven preform using non-orthogonal constitutive equation[END_REF]. Reinforcement bending behaviour is a specic structural out of plane behaviour. It depends, among other things, on the geometrical conguration of the yarns, their mechanical properties, and the contact behaviour. The determination of the specic bending behaviour of woven reinforcement, at macroscopic scale, is an objective of this study. This determination consists in dening a functional model. Macroscopic parameters of this model could be determined with bending tests using direct identication or inverse method. Thereby, the experimental bending test of composite reinforcements is necessary. The deformation behaviour of textiles has been studied since a long time, by experimental approaches and by development of models.

Even if these studies concerning deformation of woven fabrics have been realized for the clothing industry, especially in our case for the bending, we will apply in this paper the methodology for the studied composite reinforcements.

So and since Peirce's model [START_REF] Peirce | The geometry of cloth structure[END_REF], many studies have been made to present analytical model forecasting the bending behaviour of a fabric from the yarn properties and the weave geometry in the eld of clothing textiles. But these studies concern essentially plain-woven fabrics. At present, it is not possible to predict accurately the bending behaviour from only yarn properties for composite reinforcements with more complex structures such as multiplies, interlock and 3D reinforcements or non crimp fabrics.

For experimental aspects, two standard tests are used to determine the bending stiness of fabrics: the standard cantilever test [START_REF]ASTM: Standard Test Method for Stiness of Fabrics[END_REF][START_REF]ISO: Textiles glass -Woven fabrics -Determination of conventional exural stifness -Fixed angle exometer method[END_REF] and the Kawabata bending test (KES-FB) [START_REF] Kawabata | The standardization and analysis of hand evaluation[END_REF]. The rst is based on elastic linear behaviour and enables to determine only one parameter: the bending rigidity. However the bending behaviour is not linear elastic and the standard cantilever test is not adapted. The second test was designed by Kawabata and enables to record the moment versus the curvature during a bending cycle. Whereas it enables to show a non linear and hysteresis behaviour, it has been designed to test clothing textiles and is not very well adapted for composite reinforcements which are often thicker and stier and cannot be tested on this apparatus.

The purpose of this paper is to present a bending test for composite reinforcements with various thicknesses. In this aim, a new cantilever test using optical measurements has been designed. The sample can be either a yarn, a monoply or a multiply reinforcement which cannot be tested with standard apparatus. The bending test is performed for several overhang lengths which allow to identify non elastic behaviour. The test results are the shapes of the bent samples for the dierent lengths and the moment versus curvature curve. Associated to these experimental developments, we could deduce that macroscopic model for bending developed for clothing textiles are not very adapted for the composite reinforcement.

2 Reinforcements structure and properties

Structure and in-plane properties

As explained in the introduction, dry reinforcements studied here are composed with continuous yarns which contain thousands of bers [START_REF] Hivet | Consistent 3d geometrical model of fabric elementary cell. application to a meshing preprocessor for 3d nite element analysis[END_REF]. To allow the shaping into a non-developable shape, yarns have to be tied to one another. They can be woven, stitched or knitted. Four kinds of reinforcements have been tested in this study. All are used in aeronautical applications and enable to manufacture thick parts.

The rst considered example of dry reinforcement is an 2.5D carbon fabric (g. 1). Its area weight is 630 g/m 2 and the thickness is 1 mm. It will be denoted "fabric A".

The second example of dry reinforcement is an interlock carbon fabric G1151 (g. 2).

It is a laminate of four layers. Its area weight is 600 g/m 2 , the thickness is 0.6 mm and it has 7.4 yarns/cm in warp way and 7.4 yarns/cm in weft way. It will be denoted "fabric B".

The third example of dry reinforcement (g. 3) is a carbon non crimp fabric (NCF) [START_REF] Lomov | Carbon composites based on multiaxial multiply stitched preforms. part 1. geometry of the preform[END_REF].

It will be denoted "fabric C". It is a laminate of two unidirectional plies with yarn orientation at ± 45 °and plies tied by warp stitching. The area weight is 568 g/m 2 and the thickness is 1.1 mm. In general, such reinforcement will be characterized by the number of plies, orientation of plies, dimensions of the yarns, the pattern of the stitch (chain or tricot) and its dimensions.

To increase the number of plies, it is possible to assemble several NCF reinforcements and to stitch them with another stitching yarn (g. 4). A fourth sample has been tested. It will be denoted "fabric D". It is a multiply composed of two monoplies of the previous NCF "fabric C". Its nominal area weight is 1230 g/m 2 .

With such structures, dry reinforcements can support high geometrical transformations under low forces because the yarns are free to move. The mechanical behaviour which results from this freedom of movement is specic and gives to the structure low rigidities compared to those of tension in yarn directions. Because warp and weft yarns are joined by weaving or stitching, the tensile behaviour is biaxial and tensile tests results have been studied in several papers [START_REF] Boisse | Meso/macro-mechanical behaviour of textile reinforcements for thin composites[END_REF][START_REF] Boisse | Analyses of fabric tensile behaviour: determination of the biaxial tension-strain surfaces and their use in forming simulations[END_REF]. Moreover, reinforcement forming on a double curvature shape is possible by in-plane shear strains. Thereby, in-plane shear behaviour has been thoroughly studied and corresponding tests have been developed [START_REF] Launay | Experimental analysis of the inuence of tensions on in plane shear behaviour of woven composite reinforcements[END_REF].

Bending properties

Classically, bending behaviour of continuous shell is derived from the in-plane properties of the material. But Yu et al. [START_REF] Yu | Analysis of exible bending bahavior of woven preform using non-orthogonal constitutive equation[END_REF] investigated the bending behaviour of a woven preform through a cantilever experiment and simulation where the deection was only due to gravity. The authors showed the discrepancy between the experimental and numerical results and concluded that bending rigidity derived from in-plane properties gives unrealistically high value compared to the experimental bending rigidity of the woven preform. During deformations, a part of the yarn will have its curvature increasing while another part will have its curvature decreasing, involving interactions between laments and between yarns with high sliding. This multi-scale constitution gives to the reinforcement a specic bending behaviour independent of the tensile or the shear behaviour. For composite reinforcements, bending behaviour is a structural multi-scale problem which cannot be directly deduced from the in-plane material properties and it is then necessary to determine this specic behaviour. Up today, bending behaviour of reinforcement has not been the subject of many researches. On the contrary, there are a lot of studies in the eld of clothing. Because of the similarity between the geometries of textiles and reinforcement, the rst idea was to examine the bending behaviour of fabrics. The study of the bending behaviour of yarns or fabrics for the simulation means dening the relationship between the moment M and the curvature κ of a bent beam, plate or shell depending on the complexity of the model. Relationship between fabric behaviour, structural conguration and mechanical behaviour of yarns and their constituent bers is complex and a critical review has been proposed by Ghosh et al. [START_REF] Ghosh | The bending behaviour of plain-woven fabrics. part i : A critical review[END_REF].

Another way is to dene a macroscopic model based on rheological and experimental measurements. The simplest macroscopic model is the linear elastic (eq. 1) proposed by Peirce [START_REF] Peirce | The 'handle' of cloth as a measurable quantity[END_REF].

M = B • κ (1)
But Grosberg et al. proposed a more realistic model (eq. 2) [START_REF] Grosberg | The mechanical properties of woven fabrics part ii : the bending of woven fabrics[END_REF] taking into account the hysteretic behaviour of the fabric with the frictional restraint couple M 0 :

       κ = 0 if M < M 0 B × κ = M -sign(κ) • M 0 if M ≥ M 0 (2)
Recently, Ngo Ngoc et al. [START_REF] Ngo Ngoc | Modelling friction for yarn/fabric simulation application to bending hysteresis[END_REF] and Lahey [START_REF] Lahey | Mechanical modeling of fabrics in bending[END_REF] proposed models based on mechanical friction theory to t experimental data. In both ways, mesoscopic or macroscopic approach, it is necessary to dispose of a bending test to verify the model. Although there are many methods to test the bending behaviour of a material, reinforcement with low bending stiness needs specic equipment. Historically, it is again in the eld of clothing textiles that such bending tests have been designed to evaluate the quality of fabrics. The two more common tests are presented in the next section.

3 Standard bending tests

Cantilever test

Peirce was the rst to present both a macroscopic measurement of the bending behaviour [START_REF] Peirce | The 'handle' of cloth as a measurable quantity[END_REF] and a mesoscopic approach [START_REF] Peirce | The geometry of cloth structure[END_REF] to model the geometric conguration of the mesh. Assuming an elastic linear behaviour between the bending moment and the curvature of the strip, he proposed a cantilever test (g. 5) to determine the bending stiness. In this test the fabric is cantilevered under gravity.

In his model, bending moment M (N.m) is a linear function of the curvature κ (m -1 ):

M = G × b × κ (3)
where G (N.m) is the exural rigidity per unit width and b (m) is the width of the strip. Peirce dened the ratio S of the exural rigidity to the weight w per unit area:

S = G/w (4)
Assuming the fabric being an elastica and small strains but large deections, he dened the relation between the ratio S, the angle θ of the chord with the horizontal axis and the length l of the bent part of the sample (g. 5)

S = l 3 8 • cos θ/2 tan θ (5)
The cubic root of S allows to compare the fabrics. It has the unit of a length and is called by Peirce "the bending length". Today, the standard commercial apparatus are dened with a specic value equal to 41.5°for the angle of the tilted plane. With this value, the equation ( 5) becomes simpler:

S ≈ l 3 /8 ( 6
)
This conguration is described in standard tests [START_REF]ASTM: Standard Test Method for Stiness of Fabrics[END_REF][START_REF]ISO: Textiles glass -Woven fabrics -Determination of conventional exural stifness -Fixed angle exometer method[END_REF].

On this principle, Grosberg [START_REF] Grosberg | The mechanical properties of woven fabrics part ii : the bending of woven fabrics[END_REF][START_REF] Grosberg | The mechanical properties of woven fabrics part i : the initial load extension modulus of woven fabrics[END_REF] proposed a cantilever test to determine the both parameters of his model (eq. 2). Considering two specic values of θ (θ = 40 °and θ = 20 °) and assuming the load as a concentrated and a distributed load, the parameters are computed using functions derived from known solutions.

Lastly, Clapp et al. [START_REF] Clapp | Indirect measurement of the moment-curvature relationship for fabrics[END_REF] developed an indirect method of experimental measurement of the moment-curvature relationship for fabrics based on recording the coordinates of the deformed sample. Applying least square polynomial regression and numerical differentiation techniques, moment-curvature relationship is computed from coordinate data and weight per unit width. This method allows taking into account the non linear behaviour but assumes elastic behaviour.

KES -FB bending test

Kawabata's Evaluation System was originally designed to measure basic mechanical properties of fabrics [START_REF] Kawabata | The standardization and analysis of hand evaluation[END_REF]. It became a set of standard tests for tensile, shear, compression, surface roughness and bending behaviour. KES-FB tester (g. 6) is the test to quantify properties in pure bending deformation mode and enables then to record directly the evolution of the bending momentum per unit width versus the curvature during a load unload cycle.

The dimension of the sample in the bending direction is equal to 1 cm and its width with a constant rate of curvature equal to 0.5 cm -1 s -1 from -2.5 cm -1 to 2.5 cm -1 .

As indicated in the manual of the apparatus, the bending rigidity B and the bending hysteresis M 0 of Grosberg's model are computed as follow: the slopes are computed respectively between κ = 0.5 cm -1 and κ = 1.5 cm -1 for s 1 and between κ = -0.5 cm -1 and κ = -1.5 cm -1 for s 2 (see eq. 7).

               s 1 = ∆M =M (κ=1.5)-M (κ=0.5) ∆κ=1 s 2 = ∆M =M (κ=-0.5)-M (κ=-1.5) ∆κ=1 B = (s 1 + s 2 )/2 (7)
The bending hysteresis, that is the frictional restraint force M 0 , is the half average of the two hysteresis values h 1 and h 2 computed respectively at κ = 1 cm -1 and κ = -1 cm -1 (see eq. 8).

               h 1 = M l (κ = 1) -M ul (κ = 1) h 2 = M l (κ = -1) -M ul (κ = -1) 2 • M 0 = (h 1 + h 2 )/2 (8)
where M l and M ul are respectively the moment for load and unload curve. At the beginning of the test (g. 8(a)), the sample (S) is placed upon a xed board (F) and a special plane comprising laths (B). The length direction of the sample must be parallel to the bending direction and its free edge must be aligned with the lath (L1).

A translucent plate (C) is xed upon the both to ensure the embedding condition.

Thus the sample (S) will not slide. During the test, because of the translation of the drawer (T), the laths will successively retract, beginning with lath (L1), and the length of overhang will increase. The test is stopped for a chosen overhang length L and is continued for new lengths. Thus, the complete test is a succession of quasistatic tests with dierent loading cases. While single cantilever test provided only one conguration, the new exometer, with its set of loading cases associated with the dierent bent shapes, enables to identify a non elastic behaviour model.

Like in several studies of textile deformability during composite processing [START_REF] Lomov | Full-eld strain measurements in textile deformability studies[END_REF] full-eld strain measurements are applied to measure the deformed shape of the bent sample. A digital camera takes a picture for each length and the images are processed to extract the shapes of the bent sample (g. 8(b)). A previous step of pixel calibration [START_REF] Bailey | Pixel calibration techniques[END_REF] is required so that pixel measurements can be translated into real dimension by scaling.

Then, the image of the bent sample prole is captured, ltered [START_REF] Bailey | A rank based edge enhancement lter[END_REF], and binarized. The following step is to extract the borders of the binary object and to deduce the mean prole (see g. 9).

Post processing

At the end of the experimental test, a set of bent shapes is provided. Each bent shape is dened by the bending length at which the deformed shapes has been obtained and by the data points dened in a coordinate system. The subsequent post-processing of the proles aims to deduce from them the evolution of the moment with the curvature.

Each shape of the bent sample, dened by a set of data points, is smoothed by a series of exponential functions plus a rst order polynomial to ensure the boundary conditions at embedded point. For each length of bending test, bending moment and the curvature have to be computed along the prole and moment-curvature graph can be drawn.

Assuming the sample as a shell with its length L 0 = L in initial conguration, the new length is noted L b in bent conguration. The total strain energy U t is the summation of the bending energy U b , the membrane strain energy U m , and the transverse shear energy U T S . In this case, assuming that the bending moment M(s), the axial stress N(s) and the transverse shear T(s) are the only non zero stress components and that they depend only on the curvilinear abscissa s (see g. 10) along the prole:

κ = z (1 + z 2 ) 3/2 (9) L b = x F x E 1 + z 2 dt ( 10
)
s(P ) = x P x E 1 + z 2 du (11) M (s) = W L b s (u -s)cos(ϕ)du ( 12
)
P dened by the curvilinear abscissa s = s(P) is the point where the bending moment M(s) applied by the part PF and the curvature kappa are computed. W is the weight per unit length (N/m). u and ϕ are the Frenet's coordinates of the point Q moving along the shape from P to F.

       U t = U b + U m + U T S U t = L b 0 M (s) • κ(s)ds + L b 0 N (s) • ε s (s)ds + L b 0 T (s) • γ(s)]ds (13)
Assuming that we are in pure bending deformation the membrane strain energy is insignicant by comparison with the bending energy, it follows that membrane strains are negligible and that bent length L b is equal to initial length L. Finite element simulation of this bending test has been developed, in good agreement on the bending deection value and conrms this hypothesis that membrane strains are negligible [START_REF] De Bilbao | Bending test of composite reinforcements[END_REF]. Two points of the shape, P and Q are followed to illustrate the interpretation. During the test, points which are before the embedded point have a zero bending moment and a zero curvature: before L P for P and before L Q for Q. With increasing length of overhang, when a point becomes the embedded point (L =L P for P), its moment becomes equal to the resultant bending moment due to the bent part of the sample. Length of overhang continues to increase and this point has its bending moment decreasing because of the increasing inclination of the bent part.

U b = L 0 M (s) • κ(s)ds ( 14
)
L b = L ( 15
Thus, when Q becomes the embedded point (L = L Q ), moment has already decreased for point P. For each material point, its moment reaches maximum value when it becomes the embedded point and decreases after. Assuming an increasing relationship between moment and curvature, which seems to be realistic, the curvature reaches also its maximum value at the embedded point and decreases after (see curve κ(L)).

If the material has an elastic behaviour, load and unload curves of the momentcurvature graph are superposed (curve m(κ)). In this case P and Q go up and down along the elastic curve (continuous line). Thus, only one length of overhang, that is only one bent shape, will be sucient [START_REF] Clapp | Indirect measurement of the moment-curvature relationship for fabrics[END_REF].

For non elastic and nonlinear behaviour (g. 11) the locus of the points which have the moment and curvature at their maximum values (at the embedded point) gives the load part of moment-curvature curve (continuous line). On the other hand, following the moment and the curvature for a material point will give the unload curve (dash line).

It is then necessary to test the bending behaviour with several lengths of overhang.

This explanation points out the dierence between the new exometer with its complete test and the simple standard cantilever test. The standard test enables to provide only bending rigidity for linear elastic model if only one point of the shape is exploited.

It enables to provide the parameters of a non linear but elastic model if the complete shape is processed. But it does not enable to provide parameters for non elastic model contrary to the new exometer which takes into account the history of the deformations and enables then to identify non elastic models.

Moreover, each point has its maximum value of bending moment when it is the em-bedded point (E). From equations ( 12) and ( 15), it follows that this value depends on L:

M(E) = W L 0 s cos(ϕ)ds (16)
Each point undergoes a load at a maximum value increasing with the bending length and an unload. During the unload phase, the behaviour could be dierent in function of the level of load and in function of the material behaviour. For example, the behaviour of the sample can be quasi elastic for low bending length (low curvature), and can become strongly hysteretic with high bending length (high curvature). The new exometer test is then equivalent to a set of KES-FB tests with dierent ranges of curvature.

In practice, after having deduced moment-curvature curve for each bending length, we'll see if the curves are superposed. In this case we consider that the behaviour is elastic 1 and the moment-curvature graph for the greatest length enables to dene the bending model. If not, the moment and the curvature computed at embedded point for each length of bending test enable to plot a point on the moment-curvature load graph as explained above.

5 Bending tests Mulhouse.

For the exometer, the sample's width is equal to 100 mm and the test has been performed with the bent strip under its own weight only. The weft direction of the sample was parallel to the bending direction of the exometer. The usable bending length varied from 100 mm to 210 mm. The rst step within the results analysis is to verify if the behaviour is linear elastic. For each bent shape, the angle θ of the chord with the horizontal axis dened by Peirce's test has been determined (g.5). It follows the exural rigidity G according to equations ( 4) and ( 5). If the behaviour is linear elastic, this parameter should be constant. Figure 12 shows the evolution of this parameter with the bending length L. It turns out that G decreases while the length increases.

With a variation of about 42 % (tab. 2), it can be inferred that the behaviour is not linear2 for the fabric considered.

Figure 13 shows the evolution of the moment with the curvature computed along the prole for three lengths L = 100 mm, L = 150 mm and L = 210 mm. Moment is in N (moment per unit width). Other lengths have been tested but they are not plotted for more clarity. It can be observed inexions for low curvatures which don't represent physical reality but are due to wiggles of the smoothing function in quasi rectilinear parts of the proles. It ensues that the curve given for L = 100 mm may be not usable.

The curves seem to be superposed which could indicate a quasi-elastic behaviour. In this case, the bending behaviour is described by the moment-curvature curve given for the largest length as explained in section 4.3.

For each bending length, let curvature and moment computed at the embedded point be considered. Curvature increases with the length increasing but changes very slowly for length greater than 200 mm due to the low moment arm. For the largest length it is around 0.045 mm -1 (g. 14(a)). Concerning the maximum moment, it can be noticed a little inexion because of the decreasing moment arm and the softening of the structure (g. 14(b)). The combination of these both curves allows deducing the loading curve. In the case of elastic behaviour, loading curve is superposed with moment-curvature curve computed along the prole.

For the KES-FB bending test carried out on fabric A, the sample width has been reduced to 3 cm instead of 20 cm to complete the test. Figure 7 shows the momentcurvature graph recorded for a test performed on one of the samples. Figure 15 shows the moment-curvature curve recorded during the test KES-FB and with the exometer.

It can be noted that exometer measurements are in good agreement with KES-FB bending test. Due to the limitation on the KES system, it has been impossible to carry out a complete test with another composite reinforcements (like two plies of fabric A, nor with two plies of fabric C). We can conclude, as even, by these comparisons with the KES-FB test, that in our experimental methodology, the exometer test is validate.

Test with larger curvature

The second set of tests has been performed on fabric B (sec. 2.1). A rst series of ve tests has been performed under their own weights only and in weft direction. The samples width was equal to 100 mm and the usable bending length varied from 100 mm to 260 mm. Figure 16 show the repeatability of the deection of the samples for three lengths. The relative standard deviation of the maximal deection undergoes a variation by 1 % for larger length to 19 % for smaller length. Another test of intrinsic repeatability of the exometer gave a relative standard deviation of the maximal deection less than 0.2 % for large lengths and 5 % for small lengths. It follows that the observed repeatability is essentially due to the material scattering.

The Peirce's rigidity variation shows again that the behaviour is not linear elastic (tab. 3). Concerning the curvature (g. 17(a)), it increases with the length increasing with a more marked visible asymptotic behaviour. For the largest length it reaches around 0.036 mm -1 . For the moment (g. 17 Because the behaviour seems to be inelastic from the low curvatures and assuming an hysteretic behaviour, Dahl's model [START_REF] Dahl | Solid friction damping of mechanical vibrations[END_REF] can be chose to t with the moment-curvature curve using least square method. This model is used by Ngo Ngoc et al. to t on KES bending tests performed on clothing textiles [START_REF] Ngo Ngoc | Modelling friction for yarn/fabric simulation application to bending hysteresis[END_REF]. It is very ecient for clothing tex-tiles and it could be a good starting point for reinforcements. The bending curve is dened by a dierential equation where moment is only curvature dependent and rate independent:

dM dκ = B 0 1 - M M 0 sign( • κ) ( 17
)
The of the plies were free to slide completely, the bending rigidity would be near to the value of one ply. Because of the warp stitch, the plies are not completely free to slide and the bending behaviour of the multiply results of the bending behaviour of the structural monoply plus frictional interactions between the two plies and action of the stitch. However, again because of the diculty to compute the curvature, data points are scattered and the question arises as to whether Dahl's model is suitable for the reinforcement considered.

Conclusion

Bending behaviour of composite reinforcements is going to become a signicant behaviour to take into account in forming processes simulations, especially in case of the simulation of out-of-plane phenomenon during these processes, like the wrinkles. A rst test performed on the same kind of reinforcement both with new exometer and KES-FB tester allowed to validate the new test method. The second set of tests showed that the repeatability of the position of the shape ensued from natural repeatability of the material but it pointed out also the extensive uncertainties of curvature because of the numerical double derivative computation. This highlights the limit of the direct method. A set of tests with an added strip stuck to the free edge allowed to access to larger curvature and to identify the change from elastic to non elastic behaviour.

Finally tests performed on monoply and multiply allowed to compare their bending properties. These last tests have shown that the exometer enables to test thicker and stier reinforcements than KES-FB apparatus.

Associated to this experimental development this study permits us to show the limits of the bending models developed for clothing textile, when they are used for composite reinforcement. The loads are signicantly higher in composite materials applications than in the clothing industry, the constitution and the rigidity are dierent and consequently models for deformation of woven fabrics as developed for the clothing industry are often not applicable for composites [START_REF] Prodromou | On the relationship between shear angle and wrinkling of textile composite preforms[END_REF]. The denition of a specic model concerns ours futures works for the bending behaviour. This denition will be associated to an inverse method built on experimental results and results obtained by the simulation of the bending test by nite element method. The aim of inverse method is to optimize the parameters of the chosen model, by minimizing the gap between experimental shape and simulated shape.

  is 20 cm for exible fabrics. It is clamped between a xed (A) and a moving (B) clamps (g. 6). The xture setting of the sample in the clamps ensures pure bending deformation. During the test, the moving clamp (B) rotates round the xed one (A) ensuring a constant curvature through the sample length. The movement is made

Figure

  Figure 7 presents the results of the KES-FB test carried out on fabric A in weft

4 New exometer 4 . 1

 41 General descriptionWithin standard testers (cantilever, KES-FB and other less known ones) cantilever principle has been retained for a new exometer because of its simplicity and exibility to test dierent reinforcements. This new exometer is constituted by two modules: a mechanical module and an optical module. The mechanical module enables to place the sample in cantilever conguration under its own weight. It is also possible to add a mass at the free edge of the sample to reach larger curvatures. The optical module takes pictures the shape of the bent sample. The sample can be a yarn, a monoply or multiply reinforcement. It has a length about 300 mm and a width up to 150 mm. The thickness can reach several millimeters.

) 4 . 3

 43 Test interpretationThree curves are deduced of the experimental test (g. 11): M(L) (g. 11(a)): each curve presents the evolution of the bending moment applied at a material point with the bending length.

  κ(L) (g. 11(b)): each curve presents the evolution of the curvature of the shape at a material point with the bending length. M(κ) (g. 11(c)): this curve is obtained by combination of the both previous.It presents the evolution of the moment with the curvature as the actual behaviour of the material.

  (b)), a change of slope can be observed around L = 190 mm. To complete the repeatability of the test, a study of the scatterings of curvature indicated that the relative standard deviation is length independent and comprised between 5 and 21 %. These large variations are due still to the numerical double derivative of the smoothing exponential function (with wiggles) to compute the curvature (sec. 4.2). In opposite, the relative standard deviation of moment, between 1 and 5 %, is much less extensive because moment is computed by integration.A metal strip has been stuck on the free edge of the sample to reach larger curvatures in the second series of tests. This added mass increases the moment especially at the beginning of the test for small bending lengths. Consequently the deformation of the sample is accelerated. Test have been performed with strip which had the weight equal to the two third of the sample weight. This time, the bending length varied from 50 mm to 240 mm. Figure18presents the moment versus the curvature computed along the proles for all the lengths for one of the samples. Using the mass, the maximum curvature can reaches the signicant value of 0.15 mm -1 . It's an advantage of our experimental system, with the possibility to add a mass in fact to reach large value of the curvature, without changing the sample geometry. Moreover, it can be noted larger range of curvature. The curves are divided into two sets. The curves of the rst set are superposed which indicates an elastic behaviour. In opposite, the curves split o in the second set. The behaviour is non elastic. The behaviour changes for κ between 0.04 and 0.045 mm -1 .Finally, gure 19 shows the averages of the loading curves (e.g. moment and curvature computed at embedded point) for the two sets of tests (under own weight only and with added mass) performed on fabric B. For the tests carried out under own weight only, the curvature reaches 0.036 mm -1 and the moment 0.11 N. The behaviour is only elastic all over the range of curvature. For the tests carried out with added mass, the curvature reaches 0.10 mm -1 and the moment 0.12 N. The marked change of slope conrms that the material change from an elastic behaviour to an inelastic at curvature between 0.04 and 0.045 mm -1 . It can be observed a good continuity between the two series (without and with added mass).5.3 Non Crimped Fabric bending testAnother test has been performed on a fabric C (Non Crimp Fabric) presented in section 2.1. The sample had its width equal to 50 mm. The test has been performed with the bent strip under its own weight only. The results are presented for bending lengths between 100 mm and 170 mm. Because moment-curvature computed along the proles indicated a non elastic behaviour, bending moment and curvature have been computed at the embedding point. These results are directly presented in the gure 20. From low curvatures, points seem to be in an asymptotic zone and the moment increases very little. The non woven structure should enable the bers and the yarns to slide widely.

  optimization of Dahl's model gives B 0 = 6.11 N.mm and M 0 = 0.040 N. A second test has been performed on fabric D (2 stitched multiplies of fabric C). The sample's width is 100 mm. The test has been performed with the bent strip under its own weight only. The results are presented for bending length between 100 mm and 170 mm. Bending moment and curvature have been computed at the embedding point and reported on the gure 20. This time, rst points seem to be in an increasing zone while last points seem to have reached asymptotic zone. Assuming again an hysteretic behaviour, the Dahl's model has been chosen to t with the experimental curve and the optimization gives B 0 = 8.92 N.mm and M 0 = 0.104 N. With two plies, the initial bending rigidity B 0 increases by 45 % and the asymptotic momentum M 0 increases by 160 %. In the case of elastic linear shell hypothesis without in-plane shear strain, the bending rigidity should have increased by cubic variation of thickness. If both

Fig. Fabric A = 2 .Fig. 4 Fig. 5

 245 Fig. Fabric A = 2.5D carbon fabric

Fig. 6 Fig. 8 Fig. 9

 689 Fig. 6 Kawabata bending test -KES-FB2 / ENSISA Mulhouse

Fig. 10

 10 Fig. 10 Bending moment computing along the prole. E is the embedded point. F corresponds to the free edge. Curvature and moment are computed at general point P with curvilinear abscissa s

Fig. 13 FlexometerFig. 15 Fig. 17

 131517 Fig. 13 Flexometer test on fabric A. Moment-curvature computed along the proles for L = 100 mm, L = 150 mm and L = 210 mm.

  Fig. 17Flexometer test on fabric B. Curvature and moment computed at embedded point vs Bending length

Fig. 19 Fig. 20

 1920 Fig.19Bending curve for fabric B under gravity and with added mass.

  [START_REF] Long | A simulation of reinforcement deformation during the production of preforms for liquid moulding processes[END_REF] presents the results of the KES-FB test carried out on fabric A in weft direction. The Grosberg's curve is drawn on the gure and the parameters are presented in table 1. In this example, bending moment is in gf.cm/cm (1 gf 1 cN) and curvature is in cm -1 .

This experimental bending test was developed for exible textiles and testing sti or thick reinforcements requires to reduce length of the sample. Finally, it is neither possible to test stier multiply reinforcements nor to easily observe the behaviour in the thickness.

  This behaviour is a specic complex multi-scale mechanical problem. At macroscopic scale bending behaviour is described by the constitutive moment curvature relationship which is not linear and depends on the range of the curvature. Whatever the scale used to approach the problem, a macroscopic bending test is warranted to verify the model and identify experimentally the behaviour. Present standard bending tests designed for clothing textiles are not adequate for composite reinforcements. Thereby,

a new cantilever test has been designed to test various reinforcements with dierent thicknesses, dierent woven structures and with low or large bending rigidity. In the new exometer test, optical measure and image processing accurately provide cartesian coordinates of the deformed sample for each bending length. From these, a rst direct method enables to plot the moment-curvature graph. Contrary to the classical cantilever test, the exometer test is operated with several bending lengths which allows obtaining non linear non elastic bending behaviour because the test takes into account the history of the deformation. Moreover, a complete exometer test is equivalent to multiple KES-FB tests with dierent ranges of curvature because KES-FB tests only one point with only one history while the new exometer tests a set of points with dierent histories of load.

Table Grosberg '

 Grosberg s parameters computed on KES-FB test carried out on fabric A s 1 (gf.cm) s 2 (gf.cm) B (gf.cm)Table Flexometer test on fabric A. Variation of exural rigidity G with bending length L.

		3.95	3.40		3.68
		h 1 (gf)	h 2 (gf)	M 0 (gf)
		6.44	5.04		2.87
	G min	G max	G moy	∆G = Gmax-G min G moy	(%)
	4,49	6,78	5,42		42,3
		(N.mm)			(%)

Table

  Flexometer test on fabric B. Variation of exural rigidity G with bending length L.

	G min	G max	G moy	∆G = G max -G min Gmoy	(%)
	6,16	11,07	8,84	55,6
		(N.mm)		(%)

but it can be elastic nevertheless

Acknowledgements The authors thank Laurence Schacher of the Laboratoire de Physique et Mécanique Textiles de Mulhouse ( ENSISA) to have allowed us to perform the KES bending tests. The authors acknowledge also the support of the ITOOL European project.Integrated Tool for Simulation of Textile Composites, European Specic Targeted, Research Project, SIXTH FRAMEWORK PROGRAMME, Aeronautics and Space, http://www.itool.eu.