Ulf R Kristiansen

Pierre-Olivier Mattei

Cedric Pinhde

Muriel Amielh

Measurements on tones generated in a corrugated flow pipe with special attention to the influence of a low frequency oscillation

Keywords: no, 43, 20, Ks,43, 28, Ra

It is well known that an air flow in a corrugated pipe might excite the longitudinal acoustic modes of the pipe. In this letter is reported experiments where a low frequency, oscillating flow with velocity magnitudes of the same order as the air flow has been added. Depending on the oscillation strength, it might silence the pipe or move the resonances to higher harmonics. It is also shown that a low frequency oscillation by itself might excite a higher frequency acoustic resonance of the pipe.

Introduction

Sound production in corrugated pipes has been a topic for scientific investigation for almost a century. Short pipes has been used in physics demonstrations and served as musical toys, but also been a topic for scientific research. The flow acoustic interaction is complicated as it involves a close interaction between the flow and the acoustics. A review of the literature on the topic up to about 2006 is given by Kristiansen and Wiik [4]. The studies up to that time were largely experimental. Simplified models of sound sources interacting with the pipe flow has later been presented by Goyder [START_REF] Goyder | On the modelling of noise generation in corrugated pipes[END_REF], Debut et al. [START_REF] Debut | Flow-acoustic interaction in corrugated pipes: time domain simulation of experimental phenomena[END_REF], and Tonon et al. [START_REF] Tonon | Whistling of a Pipe System With Multiple Side Branches: Comparison With Corrugated Pipes[END_REF]. In a more direct numerical approach, Popescu and Johansen recently showed, by solving the compressible Navier Stokes equation numerically for a short corrugated pipe, that strong cavity vortices would interact with the boundary layer in parts of the pipe [START_REF] Popescu | Modelling of aero-acoustic wave propagation in low Mach number corrugated pipe flow[END_REF]. No full explanation of the sound generation by the flow and its feedback effect has, to the knowledge of the authors, been published.

The topic has received renewed interest as the so called "singing riser" problem has become apparent in the natural gas industry. The long flexible pipes used for conveying gas are corrugated on the inside and are known to exhibit strong sound levels at pure tones on some off shore installations. Some recent publications directly treating the singing riser problem are published by Reinen [START_REF] Reinen | Singing risers experimental work. Summary of results and conclusions[END_REF], Belfroid et al. [START_REF] Belfroid | Flow induced pulsations due to flexible risers[END_REF]. In the study by Reinen, a 20m long plastic tube, embedded in concrete, was used. Resonances were established between an open entry end, and a downstream side branch resonator. With air flowing through the pipe, it would sound at the side branch resonances. In a special test it was found that injecting sound from an attached loudspeaker, at a frequency one order of magnitude lower, and at a level of the same order as the flow induced resonance, would silence the pipe. In the present communication we report the influence of low frequency sound on corrugated pipe resonances in a simple experimental arrangement. A hard walled box has been constructed with two openings, one connected to a small diameter corrugated pipe, and the other to a vacuum cleaner able to draw air through the system and thereby excite the corrugated pipe's longitudinal acoustic resonances. A loudspeaker is attached to one side of the box. For a low frequency pure tone sound from the loudspeaker, the arrangement effectively makes up an excited Helmholtz resonator with the air in the corrugated pipe oscillating as a single fluid body. For frequencies close to the Helmholtz resonance, high sound pressure levels and particle velocities (pipe fluid volume oscillations) are expected in the pipe. The Helmholtz resonance made up by the box and the corrugated pipe is in the present arrangement around 8.8Hz. The vacuum cleaner connecting pipe and different ways of attaching the vacuum cleaner in order to get the right velocity, would slightly change this value. At 10Hz the loudspeaker would give an adequate signal to observe the phenomenon and was therefore chosen as the test frequency. For a Helmholtz resonator of this type the oscillation velocity is expected nearly constant over the length of the pipe, while the pressure will decrease from the box towards the open end.

Experimental set up

A sketch of the experimental set up is shown in figure 1. The inner box volume measures 0.29x0.29x0.55 m 3 . The corrugated pipe has an inner diameter of 25.4mm, and is 0.64m long. One end is mounted flush with the box's inner surface. The cavity pitch is 5.5 mm. The pipe geometry is similar to the one discussed reference 4. A 35mm inner diameter pipe connects the box to a vacuum cleaner. A 240 mm diameter loudspeaker is positioned at one of the box walls. The loudspeaker was fed 10Hz signals between 0 and 12V. To minimize static pressure differences on both side of the loudspeaker in a flow situation, it is backed by a 0.07 m 3 volume communicating with the box through small holes. The velocity is measured with a Pitot tube at the box side opening of the pipe. The Pitot tube is connected to an electronic manometer of type Alnor AXD. In a special test, a standard hot wire anemometer probe (type 55P11 operated with a DANTEC Streamline CTA system) was used to measure the fluctuating velocity at the pipe's entry section. The sound levels were recorded with a 160mm long probe microphone (type G.R.A.S. 40SC), positioned so that it measured the sound levels 30mm into the pipe. A second probe microphone was used as a control microphone in the middle of the tube. The signals were transferred to a computer and analyzed by the dBFA suite 4.9 (01dB Metravib). The microphone was calibrated using standard calibration equipment. The probe corrections are supplied by the manufacturer and were accounted for.

Results

Flow generated resonances

In figure 2 are shown the resonances generated by letting the vacuum cleaner draw air through the pipe. The sound pressure level of the dominant peak (measured 30mm into the pipe from the air entry end) is plotted against the velocity measured at the box end of the pipe by the Pitot tube, U pt . The fundamental longitudinal resonance could not be excited. By increasing the velocity, harmonics were found at 506 (square), 751(circle), 1012(star), 1256(diamond), and 1517(plus) Hz.

Resonances generated by 10Hz oscillation alone

It was also observed that with no air flow drawn through the system, the 10Hz oscillation could by itself excite resonances in the corrugated tube. Figure 3 shows the different dominant peaks as function of L 10 (30), the 10Hz sound pressure level measured 30mm from the flow entry opening. Figure 4 shows the spectrum at a L 10 (30) level of 107.6dB. We see that the peak is rather broad and modulated by the 10Hz tone.

In what follows, we chose to represent the oscillating field by a sound pressure level value at a given position, namely the L 10 (30) value.

Hot wire measurements

Some hot wire velocity measurements were done at the pipe's inflow end to record oscillation velocities directly. As no simultaneous velocity and pressure measurements were done, we have not tried to relate velocity values to pressure values at specific positions. The purpose of the velocity measurements were to record order of magnitude values. The measurements were done in the entry plane of the pipe on the axis position. Figure 5 shows an example of a fluctuating flow driven by the 10Hz tone alone. The difference in heights was interpreted as caused by the difference in exit and entry flows at this position. The exit flow would be more jet like than the entry flow, where air is sucked from all directions. In figure 6 is plotted the velocity when air is drawn through the system by the vacuum cleaner. The flow is here always entering the pipe which makes the signal more symmetric. The velocity measured with no loudspeaker signal is also plotted. As this signal also contains the flow excited resonance, we can observe the orders of magnitude difference in 10Hz oscillation velocities and the high frequency particle velocities. The difference in the mean level measured by the hot wire and the Pitot value is attributed to the difference in exit and entry flows discussed above.

Influence of 10Hz tone on flow generated resonances

In figure 7 are plotted spectra showing the influence of the 10Hz oscillation on the 2nd. longitudinal pipe resonance. With no 10Hz tone present, this resonance was easily excited with an air flow measured at 6.5m/s with the Pitot tube. Comparing the upper spectra with the one below, it is seen that the resonant peak is reduced at this position by about 40dB with an added L 10 (30) value of 89.4dB. Increasing the tone's level even more, higher order resonances become apparent. It is also seen that these are much broader. A zoom of the lower panel of figure 7 around the 5'th resonance peak shows it to be strongly influenced by the 10Hz signal, see figure 8. The sound levels are measured In figures 9, 10, and 11 the 2nd, 3rd and 4th longitudinal resonances were generated by flow velocities U pt = 6.3, 9.9 and 12.0m/s respectively. It is seen also here that by adding and increasing the 10Hz signal, the resonance levels are first lowered before the dominant peaks are shifted to the higher harmonics.

Summary

The experimental results showed that a constant air flow above about 5m/s would excite the longitudinal acoustic resonances in a 0.64m long, 25mm inner diameter corrugated pipe, and that a superposed oscillating flow, with velocity amplitudes of the same order, would alter the acoustic field in a systematic manner. The oscillating (10Hz) flow would also by itself excite the acoustic resonances. The oscillating flow was for all measurements represented by the 10Hz sound pressure level measured at a position 30mm from the end of the pipe. For a given steady flow velocity exciting one of the acoustic resonances, the addition of a low frequency flow oscillation of a certain velocity amplitude would reduce the resonant pressure considerably. A further increase in the oscillation velocity would then shift the dominant acoustic peak to the higher resonant frequencies. Some hot wire measurements are presented to show that in the present experiment, the oscillation velocities were of the same order as the constant flow velocities. No detailed measurements have however yet been done on the details of the boundary layer in order to better understand the sound generating mechanism, or features like for instance the presence of acoustic streaming. We hope to give further information on this in future communications.

Figure 1 :

 1 Figure 1: A sketch of the experimental set up.

Figure 2 :

 2 Figure 2: Sound pressure levels of dominant frequencies measured at 30mm position as function of flow velocity.

Figure 3 :

 3 Figure 3: Resonances generated by 10Hz tone alone.

Figure 4 :

 4 Figure 4: Spectrum measured for L 10 (30) = 107.6dB.

Figure 5 :

 5 Figure 5: Hot wire measurement at center of the pipe's entry section. U pt = 0m/s; Vacuum cleaner end closed. Loudspeaker voltage 7.5V, L 10 (30)=101.1dB.

Figure 6 :

 6 Figure 6: Hot wire measurement at center of entry section, U pt = 10.9m/s. Loudspeaker voltage 9.0V. L 10 (30) = 110.5dB

Figure 7 :

 7 Figure 7: Influence of 10Hz tone on the second longitudinal pipe resonance. Upper figure: resonance generated with flow velocity U pt = 6.5m/s (no 10Hz tone). Lower panels show the effect of an added 10Hz tone at increasing levels: (L 10 (30) = 0, 89.4, 96.6, and 109.4dB)

Figure 9 :

 9 Figure 9: Effect of increasing 10Hz tone level on dominant spectrum peak. Start point is 2nd. harmonic excited with U pt = 6.3m/s .

Figure 10 :

 10 Figure 10: Effect of increasing 10Hz tone level on dominant spectrum peak. Start point is 3rd. harmonic excited with U pt = 9.9m/s.

Figure 11 :

 11 Figure 11: Effect of increasing 10Hz tone level on dominant spectrum peak. Start point is 4th harmonic excited with U pt = 12.0m/s.


Abdel Kader Zaiter
email: abdel-kader.zaiter@irccyn.ec-nantes.fr

Philippe Wenger
email: philippe.wenger@irccyn.ec-nantes.fr

Damien Chablat
email: damien.chablat@irccyn.ec-nantes.fr

Keywords: parallel manipulator, assembling mode, characteristic surfaces, singularity, cusp. 1

teaching and research institutions in France or abroad, or from public or private research centers.

A study of the singularity locus in the joint space of planar parallel manipulators: special focus on cusps and nodes

INTRODUCTION

Most parallel manipulators have singularities that limit the motion of the moving platform. The most dangerous ones are the singularities associated with the direct kinematics, where two direct kinematic solutions (DKSs) or assembly modes (AM) coalesce. Indeed, approaching such a singularity results in large actuator torques or forces, and in a loss of stiffness. Planar parallel manipulators have received a lot of attention [1-4, 6, 7, 9, 10, 15-16] because of their relative simplicity with respect to their spatial counterparts. Moreover, studying the former may help understand the latter.

Planar manipulators with three extensible leg rods, referred to as 3-RPR manipulators, have often been studied. Such manipulators may have up to six assembly modes (AM) [START_REF] Hunt | Structural Kinematics of In Parallel-Actuated Robot-Arms[END_REF]. The direct kinematics can be written in a polynomial of degree six. Moreover, the singularities coincide with the set of configurations where two direct kinematic solutions coincide. It was first pointed out that to move from one assembly mode to another, the manipulator should cross a singularity [START_REF] Hunt | Assembly Configurations of some In-Parallel-Actuated Manipulators[END_REF]. However, [START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators[END_REF] showed, using numerical ex-periments, that this statement is not true in general. More precisely, this statement is only true under some special geometric conditions, such as similar base and mobile platforms [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF]. Relying on geometric arguments, [START_REF] Wenger Ph | Definition Sets for the Direct Kinematics of Parallel Manipulators[END_REF] conjectured that the workspace of 3-RPR parallel manipulator is divided into two singularity-free regions called aspect and that there should be 3 solutions in each aspect. Recently, [START_REF] Husty | Non-singular assembly mode change in 3-RPRparallel manipulators[END_REF] provided a mathematical proof of the decomposition of the workspace into two aspects using geometric properties of the singularity surfaces. Also, non-singular AM changing motions were described in a 3-D representation. McAree and Daniel [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF] pointed out that a 3-RPR planar parallel manipulator can execute a non-singular change of assembly-mode if a point with triple direct kinematic solutions exists in the joint space. The authors established a condition for three direct kinematic solutions to coincide and showed that the encirclement of a cusp point is a sufficient condition for a non-singular AM change. This condition was exploited by [START_REF] Zein | An algorithm for computing cusp points in the joint space of 3-RPR parallel manipulators[END_REF], where authors provided an algorithm to detect the cusp points of a 3-RPR parallel manipulator in a section of the joint space, which corresponds to an input joint variable set to a constant value. Wenger and Chablat [START_REF] Wenger Ph | Workspace and Assembly modes in Fully-Parallel Manipulators: A Descriptive Study[END_REF] investigated the question of whether a change of assembly-mode must occur or not when moving between two prescribed poses in the workspace. They defined the uniqueness domains in the workspace as the maximal regions associated with a unique assembly-mode and proposed a calculation scheme for 3-RPR planar parallel manipulators using octrees. They showed that up to three uniqueness domains exist in each singularity-free region. When the starting and goal poses are in the same singularity-free region but in two distinct uniqueness domains, a non-singular change of assemblymode is necessary. However they did not investigate the kind of motion that arises when executing a non-singular change of assembly-mode. [START_REF] Zein | Singular curves and cusp points in the joint space of 3-RPR parallel manipulators[END_REF] analyzed the variation of the topology of singularity curves and the distribution of the cusp points from one section of the joint space to another. By calculating the image in the workspace of a trajectory Abdel Kader Zaiter, Philippe Wenger and Damien Chablat A study of the singularity locus in the joint space of planar parallel manipulators: special focus on cusps and nodes that encircles a cusp point, [START_REF] Zein | Non-singular assembly-mode changing motions for 3-RPR parallel manipulators[END_REF] described all corresponding trajectories. Also, the authors introduced the notion of reduced configuration space for an aspect and used this notion to show which cusp points must be encircled to achieve a non singular AM change in the associated aspect. [START_REF] Macho | Singularity free change of assembly mode in parallel manipulators: Application to the 3RPR planar platform[END_REF] showed the singularity surface in the 3D workspace for the 3-RPR parallel manipulator, with the six solutions corresponding to the same point in the joint space. Moreover, they studied the coincidence of DKSs on the singular curves in the reduced configuration space. Another type of AM change was reported by [START_REF] Bamberger | Assembly mode changing in parallel Mechanisms[END_REF], which corresponds to the encirclement of an α-curve in the joint space. Moreover, the authors showed the possibility to produce an AM change when approaching a singularity for micro mechanisms that present relatively large joint clearances. Later, [START_REF] Macho | Transitions between multiple solutions of the Direct Kinematic Problem[END_REF] related the encirclement of a α-curve to the encirclement of a loop characterized by a node, which corresponds to the simultaneous coalescence of two couples of DKSs. [START_REF] Hernandez | Defining conditions for non singular transitions of assembly mode[END_REF] provided a mathematical condition of the existence of cusps and nodes.

[15] provided a tool to calculate the practical workspace and, therefore, the reduced configuration space, keeping one input variable constant. This work was extended in [START_REF] Urizar | Analysis of the Direct Kinematic Problem in 3-DOF Parallel Manipulators[END_REF] with more explanations and more examples. In [START_REF] Urizar | Researching into nonsingular transitions in the joint space[END_REF], the authors showed, using numerical experiments, that not any cusp point may be encircled to perform a nonsingular AM changing motion. However, their work did not make it possible to identify definite rules.

In this work, a detailed analysis is provided to explain the role of the singular curves in the joint space and in the workspace as pertained to the loss of solutions. More attention is drawn to the study of cusps and nodes. It is shown that a cusp (resp. a node) defines a point of tangency (resp. a crossing point) in the workspace between the singular curves and the curves associated with the so-called characteristics surfaces. This study is illustrated with a 3-RPR planar parallel manipulator. It is a first step towards a rigorous and complete analysis of nonsingular changing motions in parallel manipulators.

ILLUSTRATIVE MANIPULATOR

A 3-RPR planar parallel manipulator is used in this paper to illustrate the analysis. As showed in figure 1, this manipulator has triangular base and platform denoted The base and the platform are related by three similar legs.

Each leg consists of three joints; the first and third ones are passive revolute joints while the second one is an actuated prismatic joint. i  , (i=1, 2 and 3) represents the passive variable that corresponds to the angles between the leg axis and the x-axis. The input variables are defined by i  (i=1, 2 and 3); the lengths of legs. The output variables are usually defined by the Cartesian coordinates ( , ) xy of 1 B and ;

the orientation of the platform in the plane with respect to the x-axis. In this paper, the position coordinates will be more conveniently defined by ( 1  , 1  ) i.e., the cylindrical coordinates of 1 B . With these parameters, indeed, it is possible to simplify the problem: we consider 2dimensional slices of the joint space and of the workspace by fixing the joint parameter 1

 . The projection of a ( 1

 , )-slice onto the ( 2  , 3 
) plane is consistent with the projection of the configuration space onto the joint space [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF].

The geometric parameters of the illustrative manipulator are the same as those used in [3, 10,

SINGULARITIES

Singularities and aspects

The 3-RPR parallel manipulator is in a singular configuration whenever the axes of its three legs are concurrent (the platform orientation gets out of control) or parallel (the translation along a direction orthogonal to the two parallel legs is out of control).

1  1 A x y 2 A 3 A 2  3  3  1 d  1  2  2 d 3 d 2 B 1 B 3 B
The singular configurations of the 3-RPR planar parallel manipulator can be calculated without any difficulty. They define surfaces both in the workspace and in the joint space. The singular surfaces divide the workspace into two singularity-free domains called aspects [START_REF] Husty | Non-singular assembly mode change in 3-RPRparallel manipulators[END_REF][START_REF] Wenger Ph | Definition Sets for the Direct Kinematics of Parallel Manipulators[END_REF], referred to as 1 WA and 2 WA . When mapped into the joint space, these two aspects coincide and thus define two coincident sets.

Since we are considering slices by fixing 1  , the singularity locus can be depicted as curves in the workspace (resp. the joint space) in the ( 1 , )-plane (resp. in the ( 2 , 3  )plane). At a point on a singularity curve in the workspace, the manipulator is necessarily in a singular configuration.

At a point on a singularity curve that bounds the joint space, the manipulator is in a singular configuration. On an internal singularity curve, the point admits several nonsingular DKSs in addition to the singular solution. Near a singular point, there are two "mirrored" DKSs in the workspace on each part of the singular curve [START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators[END_REF]. When a singular point of a singular curve is met in the joint space, the two mirrored DKSs coincide and one DKS is then lost in each aspect (see [START_REF] Innocenti | Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators[END_REF][START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF] for example).

Plots of the singularity curves for 1

 =17 are shown in figure 2. A 3-RPR planar parallel manipulator has up to 6 assembly modes [START_REF] Hunt | Structural Kinematics of In Parallel-Actuated Robot-Arms[END_REF]. For a given point in the joint space that is not on a singular curve, there are 2, 4 or 6 DKSs in the workspace. The number of DKSs in each region of the joint space is indicated in figure 5.

Cusps and nodes

Cusps and nodes appear on the singular curves as a consequence of the projection of the manipulator configuration space folds onto its joint space [START_REF] Mcaree | An explanation of never-special assembly changing motions for 3-3 parallel manipulators[END_REF]. As described by Whitney [START_REF] Whitney | On singularities of mappings of euclidean spaces. I. Mapping of the plane into the plane[END_REF], a cusp arises when a fold is "folded". Figure 3 shows the local model of a cusp where the singular locus is shown in red lines. Three DKSs coincide at a cusp point. A node appears on the singular curves in the joint space at a crossing point. A node arises from the projection of two distinct folds. There are two pairs of coincident DKSs at a node.

Characteristic surfaces, basic regionsbasic components

The notion of characteristic surfaces was introduced in [START_REF] Wenger Ph | Definition Sets for the Direct Kinematics of Parallel Manipulators[END_REF].

Its definition is recalled thereafter. , are defined as follows:

i i i c WA WA g g WA S     )) ( ( ) ( 1 (1)
where:

g maps all points of a given set of the workspace into the joint space through the manipulator inverse kinematics.

1  g maps all points of a given set of the joint space in the workspace through the manipulator direct kinematics.

Note that since we are considering slices by fixing 1  , the characteristic surfaces are defined by curves in this paper.

) ( i c WA S divides i WA into different basic regions ik WAb , ) ( i c i ik K k WA S WA WAb     (2)
where K is the set indexing the basic regions.

The basic components ik QAb are defined as the images of ik WAb in the joint space:

) ( ik ik WAb g QAb  (3)
Since the 3-RPR planar parallel manipulator has 2 aspects and up to 3 DKSs in each aspect, as many as 3 distinct points in each aspect map onto one unique point in the joint space. Thus, as many as 3 of the basic regions in the workspace map onto 3 coincident basic components in the jointspace. In figure 2a, the basic components are the domains bounded by parts of the singularity curves. Some of the basic regions are shown in figure 6.

The following notation will be used throughout the paper.

k D : direct kinematic solution number k. Let us label 1, 2, 3 the solutions in aspect k S : singular curve image of k C in the workspace.

Figure 2a shows the curve segments k C for our 3-RPR planar parallel manipulator when 1  =17.

SOLUTION LOSS ON THE SINGULARITY CURVES

Let take a first point 4). From Whitney [START_REF] Whitney | On singularities of mappings of euclidean spaces. I. Mapping of the plane into the plane[END_REF] and Corvez [START_REF] Corvez | Études de systèmes polynomiaux: contributions à la classification d'une famille de manipulateurs et au calcul des intersections de courbess A-splines[END_REF], the local model of a cusp always defines a point of tangency between the singular curve f C associated with f and the set

Case of a cusp point

f f C C f f \ )) ( ( 1 
, as illustrated in figure 3. In our example, the sets

Example on the cusp point

The 3-RPR planar parallel manipulator described in section 2 is chosen as illustrative example. We will explain the correspondence, in the workspace, of cusp point It is known that, in the reduced configuration space, the different DKSs corresponding to same input variables represent the intersection points of a vertical line with the configurations space [START_REF] Zein | Non-singular assembly-mode changing motions for 3-RPR parallel manipulators[END_REF][START_REF] Macho | Singularity free change of assembly mode in parallel manipulators: Application to the 3RPR planar platform[END_REF]. For a cusp point, the vertical line is tangent on the singular curve folded in a specific way, where three DKSs coalesce. The projection of this fold on the joint space yields the cusp point, also the others images of the cusp points will be projections of this fold on the configuration space, where these images correspond to a cusp point formed by the characteristic surfaces. Figure 6 shows that the other images of C , marked by (*), (**) and (***). As we have 6 cusp points in the section of the joint space represented in figure 5 (the red dots), figure 6 shows 6 tangent points (large red dots) between the DKP singular and the curves as-sociated with the characteristic surfaces. Also, the studied (see figure 7). Therefore, the second correspondence of a node is also a crossing point between the curves associated with the characteristic surfaces and the singular curves. We have 6 double points in the joint space (figure 5), three of them have 4 distinct images and the three remaining ones have 2 distinct images. Figure 8 shows 12 intersection points (the large blue dots) between the singular curves and the curves associated with the characteristic surfaces, and 6 intersection points (the small blue dots) between the curves associated with the characteristic surfaces.

Case of a node

Example on the node Node

CONCLUSION

A detailed analysis of singularities in the joint space and in the workspace was presented in this paper.

Figure 1 .

 1 Figure 1. The 3-RPR planar parallel manipulator.

1 WA:

 1 and 4, 5, 6 those in 2 WA . k C : curve segment on the singularity curves in the joint space where solution k D is lost when crossing k C . Note that since one solution is lost in each of the two aspects, there will be always a mirrored solution n D in the second aspect that is lost simultaneously. Before disappearing, n D and k D coincide on k C . m k CP  : the cusp point that enables a nonsingular assembly-mode change between k D and m D . m k N  : node defined at the crossing point between k C and the nonsingular curve image of k C in aspect i WA at the boundary of the basic region im WAb .

3 .

 3 each aspect. There are 3 coincident basic components associated with each aspect in this domain. Let take =20.39). Figure2bshows why solutions 1 We can determine the solution loss on all the other curve segments in a similar manner.

Figure 2 .

 2 Figure 2. Loss of solutions when crossing singular curves in the joint space

1 WADWAb 1 1 WAWAb 1

 1111 As shown in figure 4, a cusp point m k CP  appears where curve segments k C and m C meet. Both are parts of the boundary of coincident basic components. Since m k CP  allows a non singular AM change between solutions k D and m D , these two solutions lie necessarily in the same aspect [4], say is lost, and defines a part of the singular boundary of the basic region k and one singular curve segment m S defining a part of the singular boundary of the basic region m

  images in the workspace belong to

   in the workspace.

Figure 3 .

 3 Figure 3. Tangency at a cusp point (from [19]).

Figure 4 .

 4 Figure 4. The correspondence of the three coincident solutions of CPk-m.

5 . 4  2 . 6 CFigure 5 .

 54265 Figure 5. Cusp point CP4-6 and node N1-2 in the joint space. Number of DKSs in each region is indicated (orange disks)

  DKSs and the other 5 cusps have only 2 distinct DKSs. There are formed by singular curves that separate two regions in the joint space, where one region has 4 DKSs and the other has 2 DKSs. Figure 6 show 3+5=8 cusp points formed by the characteristic surfaces in the workspace (small red dots).

Figure 6 .

 6 Figure 6. The images of CP4-6 . Red points show the images of the all cusp points. The largest points are those associated with a tangency point.

Figure 7

 7 In the joint space, a point where two singular curves cross is called a node. Such a point corresponds to the simultaneous coalescence of two couples of DKSs. For the illustrative manipulator, the image in the workspace of a node yields two couples of coincident DKSs lying on the singular curves, plus, possibly, two nonsingular DKSs. Assume that curve segments h C = m C and k C = n C intersect in the joint space at node k h N  . Since a node comes from the projection of two distinct folds of the configuration space, generically the distribution of DKSs in the four domains when turning around k h N  is n, n-2, n-4 and n-2, where n is 4 or 6 for the manipulator under study (

2 CFigure 7 .N

 27 Figure 7. The correspondence of the two couples of coincident DKSs of Nh-k in the workspace.

N

  figure 8. The curve images colored in blue and light brown are the images of the parts of curves 1 C and

Figure 8 .N

 8 Figure 8. Images of N1-2 in the workspace.

  It was shown that the image of a cusp point corresponding to the coalescence of three DKSs, and the image of a double point corresponding to the coalescence of two couples of the DKSs are, respectively, a tangent point and a crossing point between a singular curve and a curve associated with the characteristic surfaces. The demonstration was based on a comprehensive description of the images of the singular curves reflecting the coalescence of the DKSs on it. This preliminary work will help study in detail non-singular assembly-mode changing motions in parallel manipulators.


Mickael Laumonier

L Arbaret

A Burgisser

R Champallier

Porosity redistribution enhanced by strain localization in crystalrich magmas

Magma degassing, characterized by changes in permeability and porosity distribution, has a crucial control on the style of eruption. During ascent, magma might develop large porosities and crystallise while it is subjected to shear. Shear, in turn, enhances complex fabrics that result from the reorganization of the different phases (crystals, gas, melt). Such fabrics have not yet been evaluated experimentally on a 3-phase system. We performed torsion experiments on a synthetic crystal-rich hydrous magma at subsolidus conditions with 11 vol.% porosity to establish a link between strain partitioning and porosity redistribution. Crystals induce non-Newtonian deformation, resulting in localization of the shear strain. 3-D microtomography and 2-D Scanning Electron Microprobe (SEM) imaging show gas accumulation in local microstructures caused by shearinduced crystal fabric. Our data show that strain localization is a mechanism that could enable magma degassing at very low vesicularity.

INTRODUCTION

The eruption style of hydrous magmas is mainly controlled by their ability to degas (e.g. [START_REF] Eichelberger | Non-explosive silicic volcanism[END_REF]. During magma transfer to the surface, the decompression causes bubble exsolution. If these bubbles remain in the magma, the over-compressed gas can exceed the tensile strength of magma [START_REF] Zhang | A criterion for the fragmentation of bubbly magma based on brittle failure theory[END_REF], causing fragmentation that may lead to catastrophic explosive eruption [START_REF] Villemant | Transition from dome-forming to plinian eruptive styles controlled by H2O and Cl degassing[END_REF][START_REF] Sparks | Forecasting volcanic eruptions[END_REF]. The most efficient mechanism for the gas to escape the magma is the creation of an interconnected bubble network [START_REF] Eichelberger | Non-explosive silicic volcanism[END_REF]. Shearing is an efficient process to stretch and coalesce bubbles, which can start at 20 to 30 % of porosity [START_REF] Okumura | Evolution of bubble microstructure in sheared rhyolite: formation of a channel-like bubble network[END_REF]. A connected bubble network can be generated in a sheared crystalfree rhyolite with 30 vol% of bubbles (Okumura et al. 2009) by a modest shear strain (γ = 8). In natural felsic pumices, [START_REF] Wright | Strain localization in vesicular magma: Implications for rheology and fragmentation[END_REF] recognised strain localization at microscopic scale. The three phases composing magma (bubbles, crystals and melt) can be organized by strain localization and associated structures, and conversely. Such structural evolutions rule the mechanical behaviour of the magmas and influence their transfer to the surface [START_REF] Saar | Numerical models of the onset of yield streggh in crystal-melt suspensions[END_REF][START_REF] Picard | HP-HT deformation of silicic magmas: experimental constrains on the structural evolution and the rheological threshold at middle and high cristallinities[END_REF]. The rheological properties of crystal-bearing magmas have been the subject of several experimental studies [START_REF] Caricchi | Non-Newtonian rheology of crystal-bearing magmas and implications for magma ascent dynamics[END_REF][START_REF] Champallier | Experimental investigation of magma rheology at 300MPa: From pure hydrous melt to 75 vol. % of crystals[END_REF][START_REF] Cordonnier | Rheological properties of dome lavas: Case study of Unzen volcano[END_REF] and are closely linked to microstructures development [START_REF] Arbaret | Microstructures and rheology of hydrous synthetic magmatic suspensions deformed in torsion at high pressure[END_REF][START_REF] Lavallée | Seismogenic lavas and explosive eruption forecasting[END_REF][START_REF] Picard | HP-HT deformation of silicic magmas: experimental constrains on the structural evolution and the rheological threshold at middle and high cristallinities[END_REF]. [START_REF] Picard | HP-HT deformation of silicic magmas: experimental constrains on the structural evolution and the rheological threshold at middle and high cristallinities[END_REF] experimentally reproduced S/C' type textures [START_REF] Berthé | Orthogneiss, mylonite and non-coaxial deformation of granite: the example of the South Armorican Shear Zone[END_REF] similar to those encountered in natural, felsic, crystal-rich magmas [START_REF] Smith | Structural analysis of flow-related textures in lavas[END_REF]). Many experimental and numerical studies focused on the behaviour of two-phase systems (bubble-melt or crystal-melt), which left out the potential coupled effects of the crystal network on the evolution of the bubble network.

In order to establish a link between strain-induced microstructures and magma outgassing, we experimentally synthesized and deformed three phase felsic magma at subsolidus conditions. The relationship between strain localization structures and bubble network geometry has been evidenced by 3-D microtomography and 2-D Scanning Electron Microprobe (SEM) imaging.

SUMMARY OF EXPERIMENTAL METHODS

A synthetic haplotonalitic suspension composed of plagioclases (crystal fraction Φc = 0.52, calculated on bubble-free assemblage), hydrated melt (Φm = 0.48) and water gas bubbles (Φg = 0.11) was deformed by torsion in a Paterson gas-medium apparatus (ASI Instruments, [START_REF] Paterson | Rock deformation tests to large shear stress in torsion[END_REF] at pressure and temperature of 200 MPa and 600°C, respectively (see [START_REF] Champallier | Experimental investigation of magma rheology at 300MPa: From pure hydrous melt to 75 vol. % of crystals[END_REF]. Strain rate γ r ranged from 3•10 -5 to 1•10 -3 s -1 , and bulk finite strains reached γ bulk = 1.3.

Before and after experiments, the size, proportion and shape-preferred orientation of the bubbles have been characterized in 3D by microtomographic imaging and 2D by SEM imaging. The starting material has an isotropic distribution of crystals and bubbles, determined by SPO2003 software [START_REF] Launeau | Mise en évidence des écoulements magmatiques par analyse d'image 2-D des distributions 3-D d'Orientations Préférentielles de Forme[END_REF]. Crystals are 25 µm long plagioclases with a shape ration of 2.5 (see supplementary data for more information). Water content in the saturated melt is around 6.5 wt%.

STRAIN LOCALISED STRUCTURES

Shearing causes a pervasive deformation that affects the entire sample and a localisation of strain that creates normal bands, synthetic in relation to the sense of shear with varied local strain γ loc visible on the jacket (Fig. 1). We identified three types of localized, planar structures (Fig. 2A).

Types 1 and 2 consist in normal bands, synthetic with respect to the direction of shear, and along which voids accumulate. We distinguish bubbles from voids, which are gas-filled cavities that result from gas phase reorganization. Bubbles are local exsolution of volatiles from the melt. Type 1, roughly 400 µm thick, dips ~22°. These shear bands consist in a large cavity that forms a continuous open space bounded by two bubble-free zones, which are each up to 200 µm thick (Fig. 2B). Type 2 shear bands dip ~60° and join Type 1 bands (Fig. 2A). They are thinner (<150 µm) than Type 1 bands and also feature zones where bubbles are concentrated. In some areas, such as in the vicinity of Type 1 bands, they seem dense. The Type 3 structure is composed of bubble-free bands (Fig. 2A) that fluctuate from flat-lying to -6° clockwise with respect to shear direction. They are synthetic reverse, therefore mainly transpressive. All planar structures are connected lengthwise. In order to get more detailed information about the development of the structures, 2-D SEM images were selected in areas having undergone various local strains γ loc (Fig. 3). At low strain (γ loc ≈ 0.4, Fig. 3B), the gas voids cover a cumulative surface identical to that outside strained structures (Fig. 3A) but with a smaller individual size (53 µm² vs. 110 µm² outside). Whereas most of the surface is composed of large voids, a population of numerous little voids develops. It is difficult to determine the nature of this new population from post-experiment observations. Isobaric quench excludes that bubbles form at the end of experiment when the temperature decreases. The small voids might be issued from the break up of large bubbles. They could also possibly come from the dehydration of the melt due to local underpressure in transtensive zones. At γ loc ≈ 1.7, the gas phase consists only of small voids (~35 µm 2 ) preferentially oriented (void long axis at 28° below the shear direction, Fig. 3C). The porosity (~16 %) is larger than in zones that suffered lower strain. This suggests that the initial bubbles were not only dismantled by crystal displacement, but also that they migrated from adjacent regions subjected to lower strains. At γ loc ≈ 9.4, the gas phase covers around 29 % of the surface and voids aggregate in a large cavity oriented around +25° with the plane of shear (Fig. 3D). These transtensive zones typify Type 1 in 3-D observations. In addition to these transtensive zones, different sets of transpressive, slightly tilted bands were generated by shear (Fig. 3E). These bands feature cataclasis (fragmented crystals), shape fabric (oriented crystals in a direction parallel to the shear band) and only some small voids (~6 µm²) that cover less than 4% of the surface. The bubbles in the vicinity of the bands suffered little deformation as they remain nearly equant, contrary to sheared zones (e.g. fig. 3C). The cataclased shear bands are Y-like bands, as described by [START_REF] Bauer | Strain localization and fluid pathways in mylonite: Inferences from in situ deformation of a water-bearing quartz analogue (norcamphor)[END_REF]references therein). Crystal cataclasis establishes the transition from ductile S/C geometry to brittle behaviour where structures match Riedel system characterised by R shears associated with P compressive shears [START_REF] Riedel | Zur Mechanik Geologischer Brucherscheinungen[END_REF] which crosscut the pervasive fabric to allow for volume conservation. In addition, the bands of Type 2, locally bubble-free, fit R' fracture orientations [START_REF] Riedel | Zur Mechanik Geologischer Brucherscheinungen[END_REF]. This is consistent with our observations showing that all these bands are connected lengthwise to form a continuous anastomosed shear system.

Observations made in zones that suffered different finite local strains evidence the link between strain localization and gas phase organization. Along normal shear zones, initial bubbles are dislocated and then accumulate in large voids connecting the gas phase (γ loc ≈ 9.4; fig. 3D). On the contrary, bubbles tend to disappear in reverse shear bands where transpressive conditions prevail.

MAGMA RHEOLOGY AND POROSITY SEGREGATION

The synthetic suspension was designed to reproduce natural magmas in terms of chemical composition, relative phase proportions and textural organisation. Experimental subsolidus temperature, pressure and applied strain rates are consistent with conditions expected along the margins of volcanic conduits. Unlike natural systems, however, the experiments occurred over a limited range of finite strain because runs were stopped when strain localisation became so important that it caused a significant drop in the stress we measured, with potential risk for the jacket integrity. Although we evidenced porosity concentration along transtensional shear zones that lead to open fractures, associated measures of local and global permeabilities are beyond the scope of the present work.

In dilute crystal suspensions, the shear flow is accommodated by crystal rotation, which leads to pervasive fabrics with a maximum density direction almost parallel to the shear direction [START_REF] Arbaret | Microstructures and rheology of hydrous synthetic magmatic suspensions deformed in torsion at high pressure[END_REF]. At a critical crystal fraction, Φ*, shear flow is prevented by the formation of a crystal network. The value of Φ* depends on both crystal fraction and shape; the higher the crystal aspect ratio, the lower the critical crystal fraction is [START_REF] Saar | Numerical models of the onset of yield streggh in crystal-melt suspensions[END_REF]. Our plagioclase suspension has a crystal fraction (Φc = 0.52) that exceeds the Φ* expected for the corresponding plagioclase shape (Φ*<38%; [START_REF] Picard | HP-HT deformation of silicic magmas: experimental constrains on the structural evolution and the rheological threshold at middle and high cristallinities[END_REF]. This implies the suspension has a non-Newtonian behaviour (e.g. [START_REF] Caricchi | Non-Newtonian rheology of crystal-bearing magmas and implications for magma ascent dynamics[END_REF][START_REF] Lavallée | Seismogenic lavas and explosive eruption forecasting[END_REF][START_REF] Picard | HP-HT deformation of silicic magmas: experimental constrains on the structural evolution and the rheological threshold at middle and high cristallinities[END_REF]. Whereas the strain rate has been evidenced to be a possible cause of the brittle-ductile transition [START_REF] Lavallée | Seismogenic lavas and explosive eruption forecasting[END_REF], our experiment shows the development of both brittle and ductile structures while the global stress conditions are constant.

Locally, however, the strain increases in shear bands, resulting in an increase of the strain rate. As evoked by [START_REF] Caricchi | Rheological properties of magma from the 1538 eruption of Monte Nuovo (Phlegrean Fields, Italy): An experimental study[END_REF], the presence of bubbles favours strain localization, locally reducing stress. The bulk deformation over the entire sample is complex, in spite of a simple mechanical stress-strain curve from rheological measurements (Supplementary data B). The stable strength measured is the sum of several different comportments resulting in various structure developments. The presence of discrete cataclasis means that the crystal suspension locally behaved in a brittle fashion.Outside of the bands, the strain marker (Fig. 1) and pervasive fabrics indicate a pseudo-plastic behaviour [START_REF] Champallier | Experimental investigation of magma rheology at 300MPa: From pure hydrous melt to 75 vol. % of crystals[END_REF]. The mechanical transition, as revealed by these two different behaviours, has commonly been observed in ash flow tuffs [START_REF] Best | Origin of broken phenocryst in ash-flow tuffs[END_REF] and reproduced experimentally [START_REF] Picard | HP-HT deformation of silicic magmas: experimental constrains on the structural evolution and the rheological threshold at middle and high cristallinities[END_REF]. The existence of this transition has also been inferred from similar natural observations in tuffisites [START_REF] Tuffen | Repeated fracture and healing of silicic magma generate flow banding and earthquakes[END_REF], in a rhyolite flow [START_REF] Smith | Ductile-brittle transition structures in the basal shear zone of a rhyolite lava flow, eastern Australia[END_REF], and in a dacitic lava dome [START_REF] Smith | Interpretation of porosity in dacite lava domes as ductile-brittle failure textures[END_REF]. Our results show that, even above the solidus and for relatively low bulk strain rates, crystals are largely responsible for strain localization, which can increase the strain rate enough to cause local brittle cataclasis.

The redistribution of the porosity depends on the orientation and the kinematics of the bands, which is essentially controlled by crystal motion. Large voids are exclusively observed along transtensive bands whereas transpressive bands are almost gas-free (Fig. 3). A possible mechanism controlling gas segregation is the relative stress induced by the deformation-induced localizations. Gas moves away from local compressed domains to less stressed surroundings, mostly because the gas phase itself is mobile within the connected voids. Although not observed in our experiments, it can freely flow out of transpressive P shear bands and partly along R' fractures, which causes porosity to vanish. The localization of the deformation enhances such transfer by displacing the gas phase from transpressive shear bands to other transtensive structures. In the transtensive R-like bands, bubbles concentrate and fill the band, increasing the porosity locally. The melt does not seem to be segregated from the crystals, at least over the strain imposed experimentally. Strain localisation has also been recognized in natural, felsic crystal-poor pumice by [START_REF] Wright | Strain localization in vesicular magma: Implications for rheology and fragmentation[END_REF]. Despite high porosity (> 60%) and high inferred strain, they do not evidence bubble coalescence. In crystal-free magmas, Okumura et al. (2009) demonstrate that the same result can be achieved with only 30 vol% of gas if the deformation reaches γ = 8. Although our study reveals the coalescence of bubbles by strain localization at low bulk strain and low strain rates in crystal-rich magmas, it does not imply individual bubble elongation as proposed by Okumura et al. (2009). We conclude that highly crystallized magmas are able to redistribute the porosity in shear bands by strain localization and reorganization of the crystal assemblage.

IMPLICATION FOR SHEAR BANDING AND MAGMA OUTGASSING

Local shear structures are hardly observable in place at depth. However, indirect evidences in eruptive products suggest the common development of such structures at high pressure. In the products from Sycamore Canyon, Arizona, [START_REF] Seaman | The effects of heterogeneity in magma water concentration on the development of flow banding and spherulites in rhyolitic lava[END_REF] recognised layers with different water contents. They attributed this layering to a repeated fracturing which enhanced magma outgassing because of depressurisation in the fragmented zones. Another example of tuffisite from Mule Creek, New Mexico, presents heterogeneous permeability and may be the result of a sheared magma [START_REF] Stasiuk | Degassing during magma ascent in the Mule Creek vent (USA)[END_REF]. [START_REF] Rust | Magma degassing buffered by vapour flow through brecciated conduit margins[END_REF] observed host fragments mixed with the erupted magma that they interpreted as the consequence of the simultaneous fragmentation of the ascending magma and the conduit walls. Banded low-porosity textures in the product of the Vulcanian eruptions of the Soufrière Hills, Montserrat, suggests that the bands were formed along the conduit wall [START_REF] Kennedy | Conduit implosion during Vulcanian eruptions[END_REF]. Our own sampling of the 1997 Vulcanian products at Montserrat [START_REF] Burgisser | Preexplosive conduit conditions of the 1997 Vulcanian explosions at Soufrière Hills Volcano, Montserrat: I. Pressure and vesicularity distributions[END_REF] allows us to complete the possibility that the banded low porosity textures were formed during a localised sheared deformation within the magma prior to eruption. Along volcanic conduit walls, such a network is supposed to connect to fractures developed along the conduit margins (Fig. 4). Through torsion experiments of crystal-rich magma, we demonstrated that low shear strains trigger gas segregation from randomly distributed large bubbles to a connected, porous Riedel network. We thus expect the connected network to efficiently segregate the exsolved water from the adjacent hydrous melt and favour its migration through open fractures, thereby causing a local and heterogeneous outgassing of the magmatic plug (Fig. 4).

Finally, our experiments evidence the significant control that shear strain localization exerts on porosity segregation to form open and therefore permeable fractures. Our results show that crystalrich magma outgassing along conduit margins can occur at much lower porosity and strain than expected from crystal-free bubbly melts (Okumura et al., 2009). Since crystal-rich situations prevail in shallow regions of conduits, where slow ascent rates promote microlite crystallisation, our experiments provide a likely mechanism of fast, shallow outgassing in highly viscous, three-phase magmas.

FIGURE CAPTIONS

  cript re to download Manuscript: LAUMONIER_geologyRevised.doc

  3-D imaging shows that, overall, these three types of structures form an anastomosing pattern of shear bands along which the porosity is clearly redistributed. Outside of these structures, vesicles are slightly oriented by the low pervasive deformation (Fig. A, supplementary data).

Figure 1 .

 1 Figure 1. External appearance of the sample in its copper jacket after torsion experiment. The

Figure 2

 2 Figure 2. 3-D image from microtomographic analysis. A) Column showing the gas phases as shade

Figure 3 .

 3 Figure 3. (A-D) SEM pictures showing bubble fabrics at 4 different local strains. Gas voids are

Figure 4 .

 4 Figure 4. Cartoon depicting the possible natural context where shear develops along the walls of a

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  1column width, grey colors re to download high resolution image -1column width, grey colors re to download high resolution image

ACKNOWLEDGEMENTS

The authors are grateful to S. Okumura and Y. Lavallée for their constructive reviews. We benefited from discussions with Michel Pichavant, Luca Caricchi and David Picard. Olivier Rouer helped us to use the SEM.The project was partly funded by the Agence Nationale de la Recherche (ANR-EXPLANT, Contract No ANR-05-CATT-003 to C. Martel) and by the ERC grant 202844 under the European FP7 attributed to A. Burgisser.

re to download high resolution image ental file re to download Supplemental file: supplementary file.pdf