N Popplewell

C N Bapat~

K Mclachlan

STABLE PERIODIC VIBROIMPACTS OF AN OSCILLATOR

The asymptotically stable vibrations of a loaded oscillator colliding periodically with a rigid mass are described. Comparison of the numerical results with the few existing examples is encouraging but inconclusive. Better overall agreement is demonstrated with fairly comprehensive measurements from a specially built experimental rig. Impact motions are shown to be very sensitive to small fluctuations in the clearance between masses and the stiffness and loading of the oscillator near its linear, or collisionless, resonant frequency. The rigid mass is quite an effective damper at or just above this frequency condition.

INTRODUCTION

A general theory to describe the stable periodic motions of two colliding rigid bodies called an impact-pair has been developed and verified experimentally in a companion paper [l]. The theory will be extended by letting one of the bodies have flexibility and damping in order to assess resonant effects. Such effects are likely important in high speed mechanisms or machinery where clearances arise from tolerances, wear, etc. On the other hand, an extra impacting body may be introduced deliberately into a resonant system to reduce excessive vibration excursions. The device is normally termed an "impact damper" and can be quite effective despite the high transient accelerations generated by the impacts [2-211. The basic theory of these applications will be checked against previous, rather restricted, theoretical results and new, exhaustive experimental data. A closed form solution has been derived for impacts consistently happening twice for every cycle of a sinusoidal load [2-5, 7, 8, 10, 11, 14, 22, 231. The impacts have been assumed generally instantaneous and distributed primarily evenly in time. Masri [7, 131 and the Kobrinskii's

[23] subsequently applied the concept of error propagation in difference equations to ascertain whether these periodic motions were asymptotically stable or not. These approaches are fundamental but the practical application of the solutions is quite restricted. For example Sadek [6] and Dittrich [9] suggested from limited experimental and theoretical studies that uneven temporal distributions are most likely at frequencies close to where the collisionless, flexible system is resonant. Sadek also inferred after careful experimentation that "equally spaced impacts hardly ever occur" even away from this resonance.

Therefore it is not surprising that Sadek [6] and Masri cycle were emphasized; only a single, probably analog computed illustration of three impacts was given by Masri alone. The lone three impact example may well be the consequence of a slowly converging and, hence, computationally time consuming iteration scheme for non-linear equations. An efficient algorithm will be used here for a comparable set of linearized equations to facilitate the computation of stable periodic motions involving any number of impacts. Numerical examples however will be confined to five or fewer impacts in each loading cycle. The theory basically is an elaboration of Masri's work which is itself based on the concept of a coefficient of restitution and the conservation of linear momentum. Previous experiments suggest that answers may not be unique in those instances when the theoretically neglected transients cannot be ignored [l].

GENERAL THEORY

The idealized vibroimpact device is shown in Figure 1. Its primary system consists of a linear spring with stiffness K, a viscous dashpot having damping constant C, and mass M excited by the external harmonic load, FO sin Lb. The secondary system is composed of a rigid mass, m, which can move uniaxially in a slot inside the usually much heavier mass h4. The supposed frictionless motion of m is instigated by collisions with M which occur intermittently because of clearance d. Any one impact, typically i, is assumed, reasonably for metals [l, 22,231, to be instantaneous and to be described by a coefficient of restitution, Ri. This coefficient may vary periodically in the general formulation but constant values, RI and R2 corresponding to collisions on the right-and left-hand sides of the slot in Figure 1, will be stipulated in illustrative examples.

Equations of motion and their solutions will be developed in a manner similar to that in reference [l]. The main difference is that the conservation of linear momentum now has to be employed along with the definition of Ri to accommodate the discontinuous velocities at a collision. Displacements, conversely, are invariably continuous and the sinusoidal load sustains motions of the primary and secondary systems which bear similar periodicity conditions to before. However, the periodicity is related more conveniently to the temporal behaviour of the load rather than the displacement of the primary mass. This relationship is illustrated in Figure 2. A motion's periodicity is shown in Figure 2(b) to be To during which time the load in the particular example of Figure 2(a) has undergone two cycles. Meanwhile N impacts between m and M, represented by the spikes in Figure 2(c), have happened. Consequently this motion will be labelled N impacts/two cycles (of the load) with the generalization being obvious. The motion is periodic so that spike YcNtlj at instant t(N+l), equal to (tl + To), iS identical to Y1, YcN+z) at t(N+Z), or (h + To), is identical to Y2, etc. The traditional approach will be taken of considering the periodicity, To, the number of impacts, N, and their occurrence at instants tl, t2, . . . , tN as known. The load's amplitude and phase with respect to the first impact on the other hand will be treated together with the absolute displacement of M at each of the N collisions as unknown. Relationships between these (N + 2) unknowns will be developed next. The equation of motion of mass M between impacts is MX+CX+KX=F,sinRt,

where X(t) represents the absolute displacement of the primary mass M. A dot superscript indicates differentiation with respect to time. The solution of equation ( 1) can be obtained by superimposing free and forced motions [5, 241. A typical solution between two consecutive collisions, say the i and the (i + 1) impacts happening at ti and tci+l) respectively, is (

X(t) = exp (-~[.flt-~i]}[~i
) 2
The temporal origin is taken at tl so that t in equation ( 2) is the convenient abbreviation of (t -tl). Quantities just after and just before an impact are represented by subscripts a and b respectively. Consequently tia, for example, is the instant immediately after the ith impact. Variables 5, o, r, 7, T and A are given conventionally by

6 = C/2(KM)'j2, w = (K/M)'12, r = O/w, q = (1 -t2)l'*, ( 3
) 7 = Rtl -tan-' {2&/(1-r2)}, A = (Fo/K){(l -r2)' + (2@)2}-"2. ( 4
)
Remaining variables are defined as

(a~ is zero because the temporal origin has been taken as the instant of the first impact.)

The displacement of M is continuous at collisions so that equation (2) remains valid then. Consequently, the unknown displacement of M at the N collisions within To, Xi, may be obtained by merely substituting their instants of occurrence, ti where i = 1,2, . . . , N, into equation (2). For example, the displacement at the (i + 1)th impact is

X(ti+l) 'Xci+l) = Cliai +Czibi +A sin [(Y(i+1)+T],
i=l,2 ,..., N, The Xci+l) and, from equation (4), A and T are certainly not known in these last three expressions. The (Y~, i = 1, 2, . . . , N, may be calculated from the given driving frequency, 0, and impact instants, ti, by using equation (7). Then the Cl; and Czi may be evaluated by substituting these ai and the determinable parameters of equation (3) into equations ( 9) and (10). Similar substitutions into equations ( 5) and ( 6) leave the final two variables in equation ( 8), ai and hip requiring Xi, other than the Xi, A and T. The discontinuity in the velocity of A4 at the ith impact may be obtained from the conservation of linear momentum and the definition of Ri.

Linear momentum conservation gives [START_REF] Warburton | [END_REF]13,231 MXib + &ib = MXia + t&i,, i=1,2 ,..., N, (11) for the ith impact whereas the definition of the coefficient of restitution is

Ri = -(i, -gia)/(kib -Z!ib)p i=1,2 ,..., N. ( 12
)
The Pib and ii, are the absolute velocities of the secondary mass just before and after the ith impact, respectively. Equations ( 11) and ( 12) may be solved straightforwardly to give Xiii;., ={(l+~~)Ril(l+Ri)Iiib +{(l-ccRi)l(l+Ri)>ii,, i=1,2 ,..., N, (13) p=m/M [START_REF] Babitskii | On the dynamics of a system with an impact vibration absorber[END_REF] being the (known) mass ratio of the secondary and primary systems. The velocity of m must remain constant between two consecutive impacts because no forces are assumed to act upon it then. Consequently, its velocity ii, between the i and (i + 1) impacts, for example, is simply the ratio of the absolute distance travelled by m and the time elapsed between these two impacts. (18)

Consequently, the only unknowns in expression (16) for & are the xi. Similarly, the velocity of the secondary mass just before the ith impact, &, equals its velocity between the (i -1)th and ith impacts. Therefore, iib = 0[Xi + Yi -Xci-lJ-Yci-r,]/{ai -aci-1,}, i=1,2 ,..., N.

Equations ( 13), ( 16) and ( 19) may be combined to give This last relationship may be substituted into equation ( 5) to produce an expression for ai in terms of the unknown Xi, A and T. Consequently,

(l+PUW ai = (1 +Ri)?jU(LYi -CX(i_l)) [Xi -X(i-l)+ Yi -Y(i-111 C1-CLRiW +(1+Ri)7)~(~(i+l)-ai) [X(i+l)-Xi + Y(i+l)- Yi]-ArCOS[ai+7]+'6i, rl T i = 1,2,. . . ,N, (21)
where, the bi of equation ( 6) are already in the desired form, bi=Xi-Asin[ai+T], i=l,2 )...) N. ( 22)

Equations ( 9), (lo), ( 17), ( 21) and ( 22) can be substituted into equation ( 8) to give N linear, simultaneous equations. A perusal of relation ( 21) should indicate that terms like a(i-1) and a(i+i) have not yet been defined at the two extreme values of i. They have to be obtained from the requirement for periodic motion. It has been seen already that periodicity implies f(N+i) = tl + rr,, t(Nc2) = t2 + TO, . . . , or, in general, if the load undergoes k cycles in time To. Consequently, when i equals N, CY((+~) is simply ((~1 +L?To), or l2T0 in view of equation (7). When i is unity on the other hand, terms like ((Wi -(Yci-i)) in relation ( 21) may be Written as (a(N+i) -"(N+i-1)) by eII'@Oying equation ( 25). These last expressions can be evaluated by using the known driving frequency, 0, the assumed periodicity, To, and presumed contact instants ti, i = 1,2, . . . , N. Terms such as Y(N+i-1) can be computed by further presuming the side of it4 at which any particular impact happens. Then equation ( 18) can be used straightforwardly.

All other variables except for the Xi, i = 1, 2, . . . , N, A and T are understood to be given or calculable from equations ( 3), ( 14) and (24). Consequently (N + 2) unknowns exist. If trigonometric terms like sin [cz(i+i) +r] in equations ( 8) and ( 22) are expanded and the resulting coefficients of the unknowns unified, the N simultaneous equations can be shown to be [25]

Wlixi + W2iX(i+l) + W3i-X (N+i-1) + WJiA COS T + WSiA Sill 7 = Whi, i = 1,2, . . . , N. ( 26
)
The often cumbersome but calculable coefficients Wriy r = 1, 2, . . . , 6 and i = 1, 2, , . . , N, are given more conveniently in the Appendix. The Xi must still satisfy periodicity condition (24) so that only X1, X2, . . . , XN are actually involved in equation ( 26). An additional N linear simultaneous equations may be developed by considering the velocity rather than primarily the displacement of M. The velocity of M just before the 5 (i +l)th impact, for example, is simply the derivative of equation ( 2) at t(i+i)b. An alternative expression may be found like that for the velocity just after an impact from linear momentum and the definition of the coefficient of restitution. Relating the two expressions produces the desired equations. From equation ( 2)

X(i+l)b = exp 1 -Fta(i+l) -ai))[ [ 4

coS ff ((Y(i+l) -cWi)-bi

sin y (aci+i) -ai) I rln 7

Ui sin? (a(i+i

) -Lyi) +bi COS p ((Y(i+l) -(Yi) II +AR COS [(Y(i+l) +T], i = 1,2, . . . , N, (27)
with equation ( 7) and Xci+ijb rather than X(fci+l)b)) being employed. Solving the momentum and restitution equations ( 11) and ( 12) for X;.b rather than Xi, gives

J;riib={(Ri-CL)/(l+Ri)}iib+{(l+E.L)/(l+Ri)}ii,, (28)

where p is defined by equation ( 14). Expressions ( 16) and ( 19) for ii, and .& may be substituted into equation ( 28) to yield Exact closed form solutions are possible when the number of equations and unknowns are identical (N equal to two). Then the Xi, i = 1,2, . . . , N, can be eliminated completely leaving only A and T as unknowns [25]. The two resulting equations are independent except in the special, essentially symmetric case when coefficients of restitution are identical and the durations between any three consecutive impacts are the same. This situation is analogous to that observed previously for exclusively rigid impacting mechanisms [l]. Consequently the same procedure as before was adopted: i.e., assume some value for A then calculate the ensuing T. More generally however, the IMSL subroutine LEQTlF was used to solve the four independent equations ( 26) and (30) numerically by utilizing Gaussian elimination [26]. This course was felt more expedient than solving the equations analytically to produce extremely lengthy expressions. A numerical techt Alternatively, the A and r may have been assumed rather than the (Y~ and the 2N unknowns of Xi and air i = 1,2,. . . , N, calculated, maybe somewhat more precisely, from the 2N equations. However, the resulting equations are linear no longer so that numerical computations are very likely to be much more time consuming.

nique appears virtually essential on the other hand when N exceeds two. Then the least squares subroutine LLSQF of the IMSL library [26] was employed. Values of the unknowns were finalized when the square of the difference between the left and right sides of relations ( 26) and (30), g2, was reduced so that E was less than 3% of these values. (The 3% was decreased to 1.5% with virtually no effect in several arbitrarily chosen examples.) All computations were performed throughout the present work with double precision arithmetic on an AMDAHL V7 digital computer. Periodic solutions are grouped conventionally into asymptotically stable or unstable subsets. A periodic motion is asymptotically stable if, after a small perturbation, the original motion is restored eventually [7,13,231. Consequently asymptotic stability would enable machinery performance to be predictable [22,27-291. On the other hand, asymptotic instability implies that a small perturbation will change some aspect of the periodic motion for all time. The asymptotic stability of periodic motions was determined by the Kobrinskii's [23] and Masri [7,131 for contact surfaces having a single constant coefficient of restitution. Small perturbations at or just after the first impact were related to corresponding changes at or just after the second, third, etc., impacts. This technique was adapted in a companion paper to a rigid impact-pair with different, maybe periodically varying, coefficients of restitution. A similar, somewhat minor extension is possible here. The absolute displacement and velocity of M are no longer prescribed here so they need perturbing along with the previously changeable 2 and 2. Some of the presumed collision instants, and hence the ai, may be modified in accordance with the equations of motion and periodicity conditions. However, these adjustments must be transitional for asymptotically stable motion because the timing sequence must revert eventually to the claimed periodic form. Masri [13] adopted a slightly different approach but with the same end results. He replaced 2, which equation ( 16) indicates depends upon (yi, by the (Y, of equation (7). Then the initial perturbation and subsequent changes, each denoted by A, were shown to be related by if terms involving a single perturbation are significantly larger than expressions containing products of perturbations. Nomenclatures of previous references have been adjusted to be compatible with earlier formulations in this section. The a in a subscript for instance still implies an instant immediately after an impact. Different coefficients of restitution have been accommodated in the elements of the typical component matrix [Pia] given by equations (A5) through (A7) of the Appendix. These coefficients and variables like Xi and Y, must fulfil periodicity condition (24) and the ai must similarly comply with equation (25). In addition, the Y, must satisfy equation (18). All elements of the component matrices may be calculated once the unknown Xi, i = 1, 2,. . . , N, A and r have been computed in the fashion detailed previously. Then [P] may be formed according to equation (32). If all the moduli of the eigenvalues of [P] are strictly less than unity, then a periodic motion is asymptotically stable [13]. Eigenvalues were computed by employing Hessenberg's method in the readily available EIGRF of the IMSL library [26].

COMPARISON WITH PREVIOUS THEORETICAL RESULTS

Computations were performed by using the theory of the last section in order to compare the results with more limited theoretical data available for several particular A numerical comparison of previous [13] and

7 140 , , , , , , , , , ,

present theoretical results for two equispaced and two unequispaced impacts/cycle of the external load

Parameters of primary and secondary system 6 = 0.005, is the maximum deflection of the primary system with impacts. 3 and Table 1 or, regardless of stability, to two unequispaced impacts/cycle in Figure 4. More comprehensive calculations based upon the general theory are presented conventionally in Figure 3 as continuous curves relating values of dK/Fo to the frequency ratio r or 0/w. Particulars of the parameters employed are given at the top of each figure or in Table 1. In addition the periodicity, location, number and distribution of impacts were presupposed from values assumed by previous authors. Consequently, two values of r were considered for the equispaced impact cases of Table 1 [13]. Stability zones displayed in Figure 3 are similar to ones measured for the analogous impact-pair. Associated boundaries are indicated by two curves having a common symbol. Any point within compatible boundaries corresponds to a stable motion with the same periodic form. Further comparisons are possible for two equispaced impacts/cycle alone but they do not add to conclusions from present communications [25]. Table 1 suggests that agreement is normally closest with Masri's results for two unequispaced impacts/cycle. Then the otherwise strong correlations intimate that the single major change in X,,, from 13.79 to 14.77 arises most likely from a printing error in reference [13]. Similar conformity is demonstrated initially for the equispaced impacts but significant differences seem to develop in the eigenvalues. These differences have t Equispaced indicates that the durations between any three consecutive impacts are identical. More general, unequispaced impacts where consecutive durations are unequal will be implied when the distribution of impacts is not stated. The future abbreviation of cycle will infer cyck of the sinusoidal load. 9

r = O-94, p = O-042, RI = R2 = O-75, d/(F,/K) = 37

Ratlo of durattans between three camecuttve Impacts

been shown [25] to stem largely from a slight slip in former manipulations of the simultaneous equations which affects only the equispaced case. The slip also, plausibly, .explains the premature termination of Masri's equispaced results at r equal to 1.04 on the line d = 37(F0/K) in Figure 3. It does not account on the other hand for Masri's not obtaining the very narrow two unequispaced impacts/cycle stability zone near r equal to 1.00. The maximum eigenvalue modulus is near unity in this zone. Consequently the "all-or-nothing" stability criterion could easily produce opposite conclusions from even minor variations in numerical data. Such variations could well arise from Masri's, apparently, using single rather than, presumably more accurate, double precision arithmetic in numerical computations [7]. Apart from these discrepancies, agreement is reasonable overall in Figure 3 and also at point "a" corresponding to the lone three impacts/cycle example. The absence in this figure of stable equispaced impacts between 0.960 < r < 0.985 supports the previous observation that unevenly spaced impacts are most likely when r is near one. An additional comparison is given in Figure 4 of the results from the degeneration of the general theory into two unequispaced impacts/cycle and those from a Fourier series approach proposed by Sadek [6]. Sadek disregarded the question of stability so that the information is presented without such a determination. Figure 4(a) indicates that values of dK/Fo correlate quite closely whereas large differences occur generally in Figure 4(b) for sin r at frequency ratios, r, near unity. A detailed examination of Sadek's theory indicates that the discontinuity in the velocity of the primary mass at an impact is treated improperly. Consequently, equation ( 13) of reference [6] implies that the phase angle T is independent of the amplitude F. of the external force. This assertion is refuted clearly by Sadek's own results presented as the solid curves in In summary, the theory's conformity with previously meagre results is promising but not convincing. More complete verification is needed. Therefore a mechanical system was built to simulate the mathematical model as far as possible in order to provide a further independent test.

EXPERIMENTAL DETAILS

The experimental analog of the ideal primary system in Figure 1 was composed of items 4, 6 and 18 in Figure 5. Slotted mass 6 with adapters, 4, was supported on two sides by flexible, spring steel strips interposed by a very stiff but light hollow beam, 18. These supports were made so long, 12.00 * 0.08 in, that their ends closest to the slotted mass never moved more than 0*38"* 0.05" from the vertical. Consequently gravitational effects should be negligible. The supports were also much stiffer in torsion than flexure so that the experimental movement was almost uniaxial to conform to conditions imposed by theory. Tightened screws and bolts were used to connect structural components and enable the primary system to move integrally. It is practically impossible to prevent all relative motions, however, so that friction developed at joints. Dissipation was found from conventional free decay and sinusoidal resonance tests [24, 301 to be equivalent to a viscous damping ratio, 6, of 0.0114 f 0.0005. In addition the first and second natural frequencies of the primary system were found to be 19.87 f 0.03 Hz and 430* 2 Hz. Further measurements of select amplitudes and phases of the displacement of the primary solid frame; @ Wayne-Kerr feed back amplifier TE MK II; @J Hewlett-Packard multimeter 5306A; @ cold rolled plate; @J Tektronix double beam storage oscilloscope 7313; @ concrete block; @ hollow beams; @ weak spring; impedance head, type amplifier type 2626; 5306A.

riiel & Kjaer conditioning amplifier type 2626. @ Briiel& Kjaer conditioning Kiaer mass compensation unit type 5565; @ Hewlett-Packard multimeter mass also indicated good linearity over extreme load characteristics. Hence, the primary system fairly represented a linear oscillator for the frequencies between 10 and 30 Hz employed subsequently. The total weight was equivalent to the tip weights of brackets, slotted mass and accelerometer, labelled 7 in Figure 5, plus 33/140 times the weight of each support for an essentially fundamental cantilevered action [24]. This total was calculated from straightforward measurements to be l-61 lb(f). The corresponding stiffness was approximated for light damping as the square of the measured fundamental radian frequency multiplied by the equivalent weight divided by the gravitational constant. This procedure gave K in Figure 1 as 65.01 lb(f)/in. The secondary system consisted of a dumbbell-like mass, item 5 in Figure 5, strung by two light threads, 10, from a solid frame. It was identical to the mature stainless steel secondary mass described previously in similar impact-pair experiments [l]. The dumbbell reduced extraneous lifting which happened increasingly at higher loading amplitudes but the threads alleviated rotations about the vertical. Hence, the theoretically assumed free, uniaxial motion was duplicated reasonably for the secondary mass which weighed 0,044 f 0.002 lb(f). This mass was effectively rigid too because its lowest natural frequency was 500 times, or more, greater than any loading frequency. High speed photography measurements of coefficients of restitution have been reported earlier [l, 251. Experiments were performed only with magnetically insensitive, mature steel which gave the most consistent results for impact-pairs. As before, unequal coefficients of restitution were achieved by taping one side of the primary mass.

The primary system was connected as in Figure 5 through an impedance head and spring to the electromagnetic shaker. The spring was much weaker than the primary system's effective stiffness so that a force rather than a displacement-like input was obtained. The sinusoidally time varying input, generated to within 2% of the nominal amplitude, was monitored with the aid of the directly coupled impedance head. Compensation for the extraneous mass of this head was achieved by employing the electrical compensation circuit recommended by its manufacturer [31]. Measurements were taken 11 at night to avoid extraneous building vibrations. Absolute displacements and accelerations of the slotted primary mass were observed separately by using a capacitance displacement transducer, 8, and accelerometer, 7. Displacements were determined relative to one of the aluminium brackets, 4. The other bracket maintained symmetry and prevented unbalance. Experimental procedures have been detailed previously [l, 251 and, hence, will not be described again. The only additional limitation was imposed by the restricted force of the shaker [32] which curtailed measurements at the highest periodic impact numbers. Stability had to be demonstrated conclusively again within an arbitrarily imposed but previously satisfactory time limit of 2 min [l].

COMPARISON OF EXPERIMENTAL AND THEORETICAL RESULTS

Experimental and theoretical stability zones are compared in Figures 678with the inverse of dK/Fo in Figure 3 taken as the ordinate. Vertical lines correspond invariably to experimental data whilst the continuous curves present related theoretical results. Coefficients of restitution both average 0.75 in Figures 6 and7 whereas Figure 8 shows the results of taping one contact surface to produce dissimilar coefficients of 0.75 and 0.61. In actuality, a coefficient of restitution was not constant but decreased generally by about 9% as the frequency ratio, I, increased within the experimental range indicated in the abscissa. Variations from the constant coefficients assumed can be anticipated from analogous impact-pairs to give various but on the whole small fluctuations for the different stability zones considered [ 1, 251. Other parameters observed experimentally and employed in the numerical computations are reported in the figure captions. at the start of calculations were obtained from experimental histories like those displayed in Figure 9. (The parts (a)-(c) of Figure 9 correspond to the identically lettered points in Figure 8.) A collision is identified by a sudden acceleration in Figure 9 followed by a gradually decaying envelope as the resulting elastic stress waves diminish. These later transients appeared unimportant providing they were attenuated completely before the subsequent collision [33]. Then the idealization of effectively an acceleration pulse with infinitesimal duration is reasonable. Furthermore, the number and timing of impacts in, for example, Figures 9(a) and (bj can be determined straightforwardly. Difficulties arose, however, when collisions occurred repeatedly and very rapidly on the taped side of the primary mass. This phenomenon is illustrated in Figures 9(c) and (d). It is analogous to the sliding observed for the impact-pair with unequal coefficients of restitution [l]. Such behaviour is outside the scope of the present theory. Figures 678indicate that the non-sliding experimental and theoretical stability zones generally agree well. The largest differences occur in the highest numbered zones and, somewhat surprisingly, in the two equispaced impacts/cycle zone where coefficients of restitution are equal. A close inspection of the equispaced impact zone revealed that the maximum absolute eigenvalue of [P] in equation (32) varied between 0.996 and 0.999. Computations were repeated with single precision arithmetic for this case alone and the ensuing upper stability boundary can be seen from Figure 6(a) to be clearly different. Correlation with experiment is usually poorer with single precision although double precision arithmetic gives an apparently superfluous spike in the practically significant region of r equal to unity. It was impossible, however, to control and load sufficiently finely there to obtain reliable experimental results which might confirm the spike's existence. A similar difficulty was experienced previously [START_REF] Arnold | Proceedings of International Congress of AppliedMechanics[END_REF] which suggests that this point is worthy of further investigation. These experiences together with the narrowness and closeness of all stability zones in the peaks or troughs in Figures 3 and6-8 indicate that the system's behaviour in the vicinity of r equal to one is very sensitive to small fluctuations in the parameters. Consequently, the traditional two equispaced impacts/cycle representation would appear to have limited applicability in such circumstances. Remaining differences between experiment and theory are similar to those noted previously for an impact-pair [ 11. Consequently, sources of experimental error introduced by the new spring coupling to the shaker and interactions between closer modes of the primary system seem acceptable. Major discrepancies at the largest loads which excite the highest impact rates are largely created again by the uncontrolled lifting of the experimental secondary mass.

Information presented virtually conventionally in Figures 6-8 cannot be applied easily to what is, probably, the most important practical application where vibroimpacts act as a "damper". Then, the secondary mass should oppose and, hence, reduce the originally resonant displacement of the primary system. Attenuation happens only when _&,,,/A, the ratio of the maximum deflections of the primary system with and without impacts, is less than unity. Consequently the alternative representation of Figure 10, where FO/(dK) is plotted for a constant r near unity against &,,,/A rather than a variable r, is most appropriate. The figure suggests that the primary system's maximum periodic deflection is attenuated most when r equals unity. It then seems beneficial to have equal coefficients of restitution and two stable equispaced impacts/cycle. The attenuation at a given FO/(dK) diminishes for a small increase in r from unity but, very importantly, amplifications always occur and may be fairly large for a slight decrease in r. Therefore the performance of a traditionally designed, virtually viscous-less impact damper [7, 211 will deteriorate noticeably if r falls consistently below one in practice. Consequently the driving frequency or speed will require close controls with the addition of an impact damper.

15

  [ 131 attempted an extension to unevenly spaced impacts. Two impacts per loading t Present address: Department of Mechanical Engineering, Concordia University, 1455 De Maisonneuve Blvd West, Montreal, Canada H3G lM8. 41 0022-460X/83/050041 + 19 $03.00/O 0 1983 Academic Press Inc. (London) Limited

1

 1

Figure 1 .

 1 Figure 1. Single degree of freedom system with a second impacting mass.

Figure 2 .

 2 Figure 2. General periodic motion of N impacts happening during period To. (a) Load F0 sin 01; (b) distribution in time of the N impacts; (c) presumed sequence of impacts.

  sin (: [Rt-oi])+bicos(~[nt-~iI)]+Asin[~t+~I, tin s t c f(i+l)b, i=l,2,3 ,..., N.

  ai =(l/n){(l/~)Xi, -Arcos [ai +r]+[bi}r i=l,2 ,..., N, (5) bi =Xi -A sin [ai + T], i=l,2 ,...

  Mathematically, Zia = (z,i+l,-zi)l(tci+l,-ti), i=1,2 ,..., N. (15) Substituting equation (7) into equation (15) produces ii, =n[X~i+I,+ Y(i+l)-Xi -Yi]/{a(i+l)-ai), Yi = Zi -Xi, i=l,2,...,N (16, 17) being the relative displacement between m and AI. The Yi is introduced because impacts occur only when the relative displacement is +d/2 for a collision on the right side of M in Figure 1 and -d/2 for a collision on the left. Hence, at impact, y, = d/2, collision on right of M ' {-d/2, collision on left of M 1 ' i=l,2 ,..., N.

  23) by 0 and using equation (7) leads to (Y (N+i) = (Yi + C!To = (Yi + 2rky i=l,2,...,N (25)

1 [

 1 [xi -X(i-1) + Yi -Y(i-I). . . ) N. (29) NOW Xci+l)b is needed rather than Xib. It is obtained by replacing every i in equation (29) with (i + 1). Identifying the resulting relationship to equation (27) leads to the following simultaneous equations [25]: VI&i + V;?iX(i+l) + V3iX(i+2) + V4iXcN+i_l) + VSiA COS T + VeiA sin r = Vyi, i = 1,2,. . . , N, (30) after expanding trigonometric terms and accounting for periodicity in the manner needed to generate equation (26). The computable coefficients Vri, r = 1, 2, . . . , 7 and i = 1, 2 **, N, are again detailed for a given i in the Appendix. The (N + 2) unknowns of Xi, ill, 2,..., N, A and T are common to both equations (26) and (30) because of the periodicity condition (24). Consequently 2N linear simultaneous equations have been formed in (N + 2) unknowns?.

  [Ewll[~u-lkzl . . * [Pial * . * [J'~cz], (31932) la Aala

Figure 3 .

 3 Figure 3. Comparison of predicted two equispaced, two and three unequispaced impacts/cycle stability zones with the previously most comprehensive results at d =37(F,/K) alone. ---, Masri's results [ 131 including the single case "a" of three impacts/cycle, stable motion. t, Implies equispaced, e, or unequispaced, u, impacts/cycle. g=O.OOS; h =0.042; R,=R2=0.75. TABLE 1

Figure 4 .

 4 Figure 4. Comparison of two unequispaced impacts/cycle theoretical results with those of Sadek [6]. Both data sets were computed without regard for stability. -, Sadek's results; ----, particular results of general theory. 5=0.004, *=0.0484, R1=R2=06

Figure 4 .

 4 An independent increase in F. reduces dK/F,, but r remains unaltered at a value, for example, of 0.999. Consequently point B is transposed leftward along the solid curve r = O-999 to position B', say, if the periodic motion, despite a change in the ratio of durations, is still to have two unequispaced impacts/cycle. Points B and B' respectively correspond to points A and A' in Figure 4(b) which obviously relate to different T.

Figure 5 .

 5 Figure 5. Details of experimental apparatus and instrumentation. 0 Electromagnetic shaker, Ling Model 400 Series 192; 8 Hewlett-Packard function generator 3110B; 0 amplifier Ling Model PA300; @ aluminium adapter; 0 secondary mass made from stainless steel; @ slotted primary mass of die tool steel; 8 Briiel & K'aer accelerometer d @ Wayne-Kerr capacitance transducer; 8 support for capacitance probe; 10 two light cotton threads;solid frame; @ Wayne-Kerr feed back amplifier TE MK II; @J Hewlett-Packard multimeter 5306A; @ cold rolled plate; @J

  Figures

Figure 6 .

 6 Figure 6. Comparison of experimental and theoretical stability zones for two impacts/cycle with identical coefficients of restitution. Gap size is 0.0105 f 0.001 in. (a) Equispaced impacts; (b) unequispaced impacts. 1, Range of experimental results. Values used theoretically: [=0.0114, ~=0.028, Ri=R2=0.75. Symbols 0,

Figure 7 . 6 Figure 8 .

 768 Figure 7. Comparison of experimental and theoretical stability zones for three, four and five impacts/cycle with identical coefficients of restitution. Gap size is 0~0105+0~001 in. 1, Range of experimental results. Values used theoretically: [=0.0114, p =0.028, R1=R2=0.75. 0 0, Theoretical stability boundary; ?? ---@, experimental stability zone. Other symbols as in Figure 3.

13

 13

Figure 9 .

 9 Figure 9. Typical experimental motions of primary mass with unequal coefficients of restitution. Top and bottom trace in a figure represent the external load and absolute acceleration of the primary mass, respectively. (a) Two unequispaced impacts/cycle with one impact on steel and one on tape; (b) three impacts/cycle with one impact on steel and two on tape; (c) one impact on steel and many on tape; (d) two impacts on steel and many on tape. Horizontal time scale is 16.7 ms/division.

Figure 10 .

 10 Figure 10. Comparison of the theoretical and experimental displacement performance of an impact damper with (a) identical coefficients and (b) different coefficients of restitution. Values used theoretically: .$'=0.0114, CL =0.028. (a) R,=R,=0.75; (b) R1=0.61, RZ=0.75. Symbols as in Figure 3.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support by the Natural Sciences and Engineering Research Council of Canada. The second author was sujpported also by an Indian Government Scholarship.

C. CEMPEL 1974 Journul ofSound and Vibration 34,199-209. The multi-unit impact damper: equivalent continuous force approach. S. F. MASRI and K. KAHYAI 1974 International Journal of Non-linearMechanics 9,451-462. Steady-state motion of a plate with a discontinuous mass. C. CEMPEL 1975 Journal of Sound and Vibration 40, 249-266. Receptance model of the multi-unit vibration impact neutralizer-"MUVIN". R. K. ROY, R. D. ROCKE and J. E. FOSTER 1975 Journal of Engineering for Industry, Transactions of the American Society of Mechanical Engineers 97,13 17-1324. The application of impact dampers to continuous systems. E. SKIPOR and L. J. BAIN 1980 Journal of Mechanical Design, Transactions of the American Society of Mechanical Engineers 102, 338-343. Application of impact damping to rotary printing equipment. A. YE. KOBRINSKII 1969 NASA Technical Translation TTF-534. Mechanisms with elastic couplings, dynamics and stability. A. E. KOBRINSKII and A. KOBRINSKII 1973 Vibroimpact System. Moscow: Nauka Press (in Russian). W. T. THOMSON 1965 Vibration Theory and Applications. New Jersey: Prentice-Hall, Inc. C. N. BAPAT 1981 Ph.D. Thesis, University of Manitoba. A study of vibroimpact systems. IMSL Library 0007 1979 International Mathematics and Statistical Library, Inc. C. C. FU and B. PAUL 1968 International Journal of Solids and Structures 4,897-905. Stability of motion of impact tools. C. C. FU 1969 Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers 91, 743-749. Dynamic stability of an impact system connected with rock drilling. I. I. BYKHOVSKY 1980 Fundamentals of Vibration Engineering. New York: Robert E. Krieger Publishing Company. C. C. KENNEDY and C. D. P. PANCU 1947 Journal of the Aeronautical Sciences 14,603-625. Use of vectors in vibration measurement and analysis. Instructions and applications manual for Impedance heads Type 8000 and 8001,197l Copenhagen: Briiel & Kjaer. Instruction manual for 400 series vibrator and amplifier model PA300. 1972 Hertfordshire, England: Ling Dynamic Systems Limited. Y. MUZYKA 1981 B.Sc. (Engineering) Thesis, University of Manitoba. An experimental investigation of multiple unit impact dampers.

N. POPPLEWELL, C. N. BAPAT AND K. McLACHLAN 5. CONCLUSIONS

A more general theory has been developed to accommodate any number of repetitive impacts influencing a, formerly, resonant system. Although most incongruences with earlier, sparse predictions of stability may be rationalized, a completely convincing case is not presented for the theory's accuracy. Credibility is enhanced greatly therefore by the close correlation demonstrated between the results from the general theory and comprehensive experiments. The recommended procedure for designing impact dampers is seen to be reasonable for one example of a lightly damped primary system with external sinusoidal loading. This particular example, however, suggests that the operation of the primary system should be maintained at or slightly above its individual fundamental resonance. Careful control of the driving speed may be needed in practice because of the extreme sensitivity displayed by the response in this frequency region. APPENDIX Coefficients in the 2N linear simultaneous equations ( 26) and (30) and elements of the component stability matrix [Pia] in equation (32) are given explicitly here. The following variables are needed in addition to the Cri and Czi, i = 1, 2, . . . , N, defined in equations ( 9) and ( 10) of the main text: "ariables 5, w, r and n used in the above expressions are defined in equation ( 3). The ai, fi and Yi are given, respectively, by equations ( 7), ( 14) and ( 18). Periodicity conditions (24) and ( 25) are employed to bring subscripts within the range 1,2, . . . , N and then all the coefficients are calculable. Explicit expressions given by Masri [13] for elements of the component matrix [Pna] in equation ( 32) were modified slightly to accommodate different coefficients of restitution. For example, coefficients k,, r = 1, 2, 3, 4, in reference [13] have to be generalized to kti, r = 1,2, 3, 4 andi (A7)

Variables can be computed in the same manner as the coefficients of equations (A2) and (A3) with the additional terms *ia, Zia, A and T determined by the particular periodic solution under consideration. Subscript i in equations (A6) and (A7) varies from 1 to iV.


Mariko Matsui
email: mmatsui@pasteur.nc

Louise Roche

Marie-Estelle Soupé-Gilbert

Milena Hasan

Didier Monchy

Cyrille Goarant

High level of IL-10 expression in the blood of animal models possibly relates to resistance against leptospirosis

Keywords: leptospirosis, in vivo, cytokine, IL-10, IL-1β, TNF-α Highlights

Leptospirosis is a severe zoonosis which immunopathogenesis is poorly understood. We evaluated correlation between acute form of the disease and the ratio of the anti-inflammatory cytokine IL-10 to the pro-inflammatory TNF-α and IL-1β expression during the early phase of infection comparing resistant mice and susceptible hamsters infected with two different species of virulent Leptospira. The IL-10/TNF-α and IL-10/IL-1β expression ratios were higher in mouse compared to hamster independently of the Leptospira strain, suggesting a preponderant role of the host response and notably these cytokines in the clinical expression and survival to leptospirosis. Using an IL-10 neutralization strategy in Leptospira-infected mouse model, we also showed evidence of a possible role of this cytokine on host susceptibility, bacterial clearance and on regulation of cytokine gene expression.

Introduction

Mainly reported in tropical and developing countries, leptospirosis is a zoonosis caused by pathogenic Leptospira, and recent estimations revised the annual number of human cases over one million infection cases with almost 60,000 deaths [START_REF] Costa | Global Morbidity and Mortality of Leptospirosis: A Systematic Review[END_REF]. Contamination occurs after contact with bacteria shed in the urine of reservoir animals, mostly rodents [START_REF] Ko | Leptospira: the dawn of the molecular genetics era for an emerging zoonotic pathogen[END_REF] and human leptospirosis is characterized by various clinical symptoms that can degenerate to severe forms with haemorrhages and multiple organ failure [START_REF] Haake | Leptospirosis in humans[END_REF].

Severe human leptospirosis presents evidence of a "cytokine storm" process characterized by increased levels of pro-(TNF-α, IL-6) and anti-(IL-10) inflammatory cytokines in serum of patients [START_REF] Haake | Leptospirosis in humans[END_REF] and several inflammatory mediators were investigated as biomarkers of severity in human cases. Interestingly, high ratio of IL-10 to TNF-α circulating level was correlated to severity of human leptospirosis [START_REF] Tajiki | The Ratio Of Plasma Levels Of Il-10/Tnf-Alpha And Its Relationship To Disease Severity And Survival In Patients With Leptospirosis[END_REF][START_REF] Mikulski | Severity markers in severe leptospirosis: a cohort study[END_REF]. However, contradictory results were also reported showing that higher ratio of IL-10 to TNF-α serum levels was positively correlated with fatal outcome [START_REF] Reis | Cytokine response signatures in disease progression and development of severe clinical outcomes for leptospirosis[END_REF][START_REF] Kyriakidis | Serum TNF-alpha, sTNFR1, IL-6, IL-8 and IL-10 levels in Weil's syndrome[END_REF]. Interestingly, the last study of Chirathaworn and coll. did not show significant difference of IL-10/TNF-a ratio in leptospirosis patients with or without organ involvement [START_REF] Chirathaworn | Cytokine levels as biomarkers for leptospirosis patients[END_REF]. Previously, we demonstrated that gene expression levels of TNF-α and IL-10 in blood were significantly higher and related to lethality in a susceptible infection model using hamsters [START_REF] Vernel-Pauillac | Differential cytokine gene expression according to outcome in a hamster model of leptospirosis[END_REF]. We also studied the cytokine gene expression in the blood of resistant mice, and, while pro-inflammatory cytokines were restored to basal levels during the early days postinfection, IL-10 was rapidly overexpressed [START_REF] Matsui | Gene expression profiles of immune mediators and histopathological findings in animal models of leptospirosis: comparison between susceptible hamsters and resistant mice[END_REF].

In this study, we performed further experiments in vivo to clarify the regulation of cytokines IL-10, IL-1β and TNF-α, in animal models of experimental leptospirosis, also investigating tissue lesions and bacterial burden. Mice were used as asymptomatic models, either regarded as resistant when infected with L. interrogans serovar Icterohaemorrhagiae strain Verdun (LiVV) or as a typical reservoir model when inoculated with a L. borgpetersenii serogroup Ballum. Hamsters were used to reproduce severe leptospirosis when infected with either Leptospira strain. As infecting serovars can affect the severity of human leptospirosis [START_REF] Adler | Leptospira and leptospirosis[END_REF], two different species of virulent leptospires were used for infection. Moreover, to better assess the acute phase of infection in the susceptible model, gene expression was investigated between 3 and 4 days (D) postinfection in the blood of animals, also corresponding to high leptospiremia in susceptible hamsters. The ratios of IL-10/TNF-α and IL-10/IL-1β were investigated according to animal survival or fatal outcome, histological changes in organs and bacterial burden in blood. We then investigated the possible involvement of the immunosuppressive IL-10 in leptospirosis pathogenesis using an IL-10 neutralization strategy in Leptospira-infected mouse model to monitor its possible effect on host susceptibility, bacterial clearance and on regulation of cytokine gene expression.

Material and Methods

Animals and bacterial strains

Outbred OF1 mice (Mus musculus) and golden Syrian hamsters (Mesocricetus auratus), initially obtained from Charles River Laboratories, were bred in animal facility in Institut Pasteur in New Caledonia. Virulent L. interrogans serovar Icterohaemorrhagiae strain Verdun (LiVV) or L. borgpetersenii serogroup Ballum isolate B3-13S (B3-13S) were cultured in EMJH medium as previously described [START_REF] Matsui | Experimental Hamster Infection with a Strain of Leptospira borgpetersenii Ballum Isolated from a Reservoir Mouse in New Caledonia[END_REF]. Virulence of leptospires was maintained and verified by monthly passages in 6-to 8-week-old golden Syrian hamsters after intraperitoneal injection before re-isolation from blood. Animal manipulations were conducted according to the guidelines of the Animal Care and Use Committees of the Institut Pasteur and followed European Recommendation 2007/526/EC.

Comparative experimental infection

For comparative study between Leptospira strains, 6-to 8-week-old healthy animals (n ≥ 5 individuals per condition) were infected and experiments were carried out as previously described [START_REF] Matsui | Experimental Hamster Infection with a Strain of Leptospira borgpetersenii Ballum Isolated from a Reservoir Mouse in New Caledonia[END_REF]. Briefly, mice and hamsters were injected intraperitoneally with 2 x 10 8 leptospires in EMJH medium. Non-infected animals served as controls. After euthanasia with chloroform or by inhalation of CO2, whole blood was collected between D3 and D4 postinfection by cardiac puncture and conserved in PAXgene blood RNA tubes (PreAnalytiX, Qiagen) for molecular biology. Sera of LiVV-infected mice were collected at D3 and D7 after infection for protein quantitation. Organs were fixed in 10% neutral buffered formalin for histology.

IL-10 neutralization challenge

For evaluation of IL-10 neutralization in murine models, 5-week old OF1 mice were injected with 200 µg of anti-IL-10 antibodies [START_REF] Arai | Effects of in vivo administration of anti-IL-10 monoclonal antibody on the host defence mechanism against murine Salmonella infection[END_REF] in 200 µL of PBS (LEAF Purified anti-mouse IL-10, rat IgG1, κ from clone JES5-2A5, or rat IgG2b, κ from clone JES5-16E3, BioLegend, Australia) 3 hours prior to infection with 2 x 10 8 leptospires of the strain LiVV in EMJH medium (αIL-10+LiVV group). Infected animals without antibodies treatment were injected with 200 µL of PBS (PBS+LiVV group) to reproduce double injection. Mice injected with anti-IL-10 and EMJH medium were also included (αIL-10+EMJH group). Animals without injection served as control. One hamster was also infected with the same dose of bacteria to confirm virulence and lethality of the strain during experiment. Body weight of mice was monitored prior to injections and until D7 postinfection. Animals were observed for survival and for any signs of clinical illness (loss of appetite or reactivity, prostration, ruffled fur, external hemorrhage).

Blood samples (50 µL) were non-lethally collected by caudal vein puncture at D3 postinfection and processed as detailed above using the PAXgene blood RNA system (PreAnalytiX, Qiagen). Animals were humanely killed at D7 postinfection and kidneys were rapidly collected for molecular bioassays (within 5 minutes of euthanasia) for extraction of total DNA. Briefly, 25 mg samples were placed into MagNA Lyser Green Beads tubes (Roche Applied Sciences) containing 360 μL lysis buffer (QIAamp DNA Mini kit, Qiagen).

Histology

Paraffin-embedded sections were stained with hematoxylin-erythrosin (HE). Morphological changes in liver, lungs and kidneys were analyzed and lesions were graded blindly according to previously reported criteria [START_REF] Matsui | Gene expression profiles of immune mediators and histopathological findings in animal models of leptospirosis: comparison between susceptible hamsters and resistant mice[END_REF]. Average scores were totalized ranging from normal (0) to severe [START_REF] Taga | IL-10 inhibits human T cell proliferation and IL-2 production[END_REF] for each individual.

Purification of DNA

Total DNA was extracted using QIAamp DNA Mini kit (Qiagen) from renal tissues. Tissue samples in MagNA Lyser Green Beads tubes were first disrupted and homogenized using the MagNA Lyser Instrument (Roche Applied Science) during 50 s at 7,000 r/min. Lysates were incubated at 56°C for 1 h with 190 μL of PBS and 50 μL of proteinase K. After washing steps, the eluted DNA was quantified by spectrophotometry (NanoDrop 2000, ThermoFisher).

Reverse Transcription

For gene expression, total RNA was purified using the PAXgene blood RNA system before treatment with DNase (Turbo DNA-free kit; Ambion, Applied Biosystems), and cDNA synthesized using Transcriptor First Strand cDNA synthesis kit (Roche Applied Science) as previously described [START_REF] Matsui | Gene expression profiles of immune mediators and histopathological findings in animal models of leptospirosis: comparison between susceptible hamsters and resistant mice[END_REF].

Quantitative PCR

Primers and probes were purchased from Eurogentec (Seraing, Table 1). Renal DNA was amplified using the LightCycler FastStart DNA Master SYBR Green I and targeted the lfb1 gene [START_REF] Merien | A rapid and quantitative method for the detection of Leptospira species in human leptospirosis[END_REF] on a LightCycler 2.0 (Roche Applied Science). Alternatively, bacterial load was assessed using the LightCycler 480 Probe Master targeting lipL32 gene on a LightCycler 480 II instrument (software v.1.5.0; Roche Applied Science) [START_REF] Stoddard | Detection of pathogenic Leptospira spp. through TaqMan polymerase chain reaction targeting the LipL32 gene[END_REF]. Genomic DNA from corresponding leptospires was used as a positive control. Quantitative PCR for IL-10, IL-1β, TNF-α, Leptospira 16S rRNA and the reference genes glyceraldehyde-3-phosphodeshydrogenase (GAPDH) and β-actin were conducted on the LightCycler 480 II using the cDNA previously generated and the LightCycler 480 SYBR Green I Master kit as previously described [START_REF] Matsui | Gene expression profiles of immune mediators and histopathological findings in animal models of leptospirosis: comparison between susceptible hamsters and resistant mice[END_REF]. Normalization of expression was processed using qbase PLUS software (Biogazelle) by extracting the expression levels of the reference genes. The normalized expression ratio of Leptospira 16S rRNA was calculated for evaluation of bacteria level. For cytokine gene expression, the relative normalized expression ratio was then calculated as the ratio of the expression level in infected animals to the expression level in control animals.

Quantification of cytokines (xMAP technology)

Interleukin-10, IL-1β and TNF-α levels in serum of mouse were quantified using a multiplex magnetic bead-based immunoassay (MyCytoMag 70k-10.Mouse kit, Merck-Millipore), according to manufacturer's instructions. The specific bead fluorescence was measured by a MAGPIX® instrument (Luminex, Austin, USA). The sample concentrations were calculated using a 10-parameter logistic standard curve using the MILLIPLEX® Analyst 5.1 software (Merck Millipore, Billerica, USA).

Statistical analyses

Statistical studies were performed using GraphPad Prism v4 (GraphPad Software Inc.). Mann-Whitney test was used for analysis of significant differences in histopathological score, bacterial load and cytokine gene expression between animals. A Two-way ANOVA test was used to compare difference in body weight variation depending on treatment between mice during IL-10 neutralization challenge.

Results

Tissue lesions and leptospiremia in Leptospira-infected animals

Animals were infected with LiVV or B3-13S strain and fatal outcome was observed in hamsters within four to six days after inoculation as previously shown [START_REF] Vernel-Pauillac | Differential cytokine gene expression according to outcome in a hamster model of leptospirosis[END_REF][START_REF] Matsui | Experimental Hamster Infection with a Strain of Leptospira borgpetersenii Ballum Isolated from a Reservoir Mouse in New Caledonia[END_REF]. Severe alterations were observed in B3-13S-infected hamster organs with diffuse alveolar haemorrhages, oedema and pulmonary congestion with inflammatory infiltrates in lungs (Figure 1A), interstitial nephritis, glomerular congestion and tubular haemorrhages in kidneys (Figure 1B), and hepatocytic necrosis, sinusoidal congestion (Figure 1C), portal inflammation and hyperplasia of Kupffer cells in liver. Similar observations were reported in LiVV-infected hamsters (data not shown).

In contrast, no or mild morphological changes were observed in Leptospira-infected mice (data not shown).

Histopathological scores (Figure 1D) showed significantly more severe lesions in hamsters compared to mice infected with LiVV (mean score ± SE at 5.33 ± 0.95 and 13.17 ± 1.33 in mice and in hamsters, respectively) or with B3-13S (0.15 ± 0.15 and 7.0 ± 3.41 in mice and in hamsters, respectively). Bacterial burden was investigated in the blood of animals during the onset of the symptoms between three and four days postinfection (Figure 1E) and, as expected, bacterial load was significantly higher in hamster blood (mean ± SE at 17.06 ± 6.31 and 0.57 ± 0.18 for infection with LiVV and B3-13S, respectively) compared to mice (3.41 ± 1.20 and <1E-4 for infection with LiVV and B3-13S, respectively) regardless of the infecting strain.

Cytokine gene expression in blood of animals depending on infecting strains

Gene expression level of cytokines was quantified in the blood of animals and IL-10 gene expression (Figure 2A) was significantly higher at D3-D4 postinfection in the blood of infected mice compared to hamsters (mean ratios of 18.95 ± 7.15 compared to 5.07 ± 1.51 respectively for LiVV infection or 80.19 ± 21.67 compared to 17.34 ± 3.33 respectively for B3-13S infection). Significant overproduction of IL-10 was confirmed in sera of mice infected with LiVV 3 days after infection (265.0 ± 89.72 pg/mL) compared to control level (7.23 ± 0.48 pg/mL; Figure 2D). Blood IL-10 returned to basal level at D7 postinfection (8.12 ± 1.38 pg/mL). Unfortunately, commercial kits are not available for quantification of cytokines in the blood from Syrian hamster.

Gene expression of TNF-α was also significantly up-regulated at D3-D4 in both mice and hamsters (Figure 3A) infected with LiVV (means ± SE of expression ratio at 2.46 ± 0.84 and 3.31 ± 0.40 in mice and hamsters, respectively) or with B3-13S strain (9.26 ± 2.59 and 15.07 ± 2.70 in mice and hamsters, respectively). Interestingly, no difference was observed between animals irrespective of strains. All IL-1β expression levels were up-regulated and significantly different compared to control in animals infected with LiVV (ratio at 3.75 ± 1.8 in hamsters) or with B3-13S strain (10.06 ± 4.66 in mice and 9.95 ± 5.17 in hamsters, respectively), except for LiVV-infected mice (Figure 3B). However, contrasting with TNF-α, significant difference was observed in IL-1β expression between mice and hamsters during LiVV infection, with higher expression in hamsters.

Both IL-10/TNF-α and IL-10/IL-1β ratios of gene expression values in blood (Figure 3C-D) were significantly higher in mice (25.13 ± 11.90 and 11.91 ± 3.44 with LiVV and B3-13S, respectively for IL-10/TNF-α, and 13.99 ± 7.49 and 7.99 ± 4.21 with LiVV and B3-13S, respectively for IL-10/IL-1β) than in hamsters (1.44 ± 0.29 and 2.39 ± 0.99 with LiVV and B3-13S, respectively for IL-10/TNF-α ratio, and 1.31 ± 0.51 and 2.626 ± 1.19 with LiVV and B3-13S, respectively for IL-10/IL-1β). Interestingly, cytokine gene expression level was significantly different depending on the strain used for infection among mice or hamsters (Figure 3A-B) but no effect of strain was observed when comparing expression ratios IL-10/TNF-α or IL-10/IL-1β (Figure 3C-D). Cytokine concentrations in sera of mice were also quantified using multiplex technology, and TNF-α and IL-1β were not different between control and infected mice at D3 postinfection possibly related to low expression level (data not shown).

Characterization of LiVV infection after IL-10 neutralization in mice

In order to investigate the role of IL-10 in pathogenesis, anti-IL-10 neutralizing antibodies were injected mice prior to their infection. LiVV strain was used and animals were double injected either with PBS or with EMJH to evaluate potential activity of solutions. All mice survived during experimental infection. However, lower reactivity and mobility, and loss of appetite were observed in LiVV-infected mice challenging IL-10 neutralization during the first 2 days postinfection while no clinical signs were observed for the other animal groups. Body weight of animals (Figure 4A) was monitored to assess the effect of IL-10 neutralization on leptospirosis severity and bacterial load. No significant difference was observed for double injected PBS+LiVV or for αIL-10+EMJH mice compared to non-injected control animals, confirming natural resistance of OF1 mouse to LiVV infection. In contrast, αIL-10+LiVV mice showed a significant decrease in body weight (percentage of initial body weight of 94.14 ± 2.80 % and 98.43 ± 3.17 % at D1 and D2 postinfection, respectively) when compared to PBS+LiVV injected animals (99.42 ± 2.78 and 103.3 ± 3.60 at D1 and D2 postinfection, respectively).

Leptospiremia was also measured in blood of infected mice and no difference was observed during IL-10 neutralization challenge at D3 postinfection (Figure 4B). Presence of leptospires was investigated and bacterial load was also quantified in kidneys of animals at D7 postinfection (Figure 4C). Interestingly, leptospires were detected in 1 / 7 mice in kidneys from LiVV-infected mice pre-injected with anti-IL-10 antibodies while 5 / 7 animals were positive in PBS+LiVV group. Moreover, leptospiral load was lower in αIL-10+LiVV mice (6.27 ± 16.59) compared to PBS+LiVV animals (552.20 ± 760.80) supporting a possible effect of IL-10 neutralization on bacterial dissemination or clearance in mice kidneys.

Effect of IL-10 neutralization on Leptospira-dependent cytokine gene expression in mouse

Regulation of cytokine gene expression was investigated in blood of LiVV-infected mice challenging IL-10 neutralization at D3 postinfection (Figure 5). Up-regulation of IL-10 gene expression was confirmed in infected animals compared to control group (mean expression ratio at 14.98 ± 3.87 in PBS+LiVV group; Figure 5A) without significant effect of anti-IL-10 antibodies treatment on LiVV-dependent gene overexpression (ratio at 10.84 ± 4.72). Similarly, TNF-α was overexpressed in blood of infected mice compared to control but no significant difference was observed between LiVV-infected animals with or without anti-IL-10 antibodies pre-injection (Figure 5B). A different expression pattern was observed for IL-1β transcript level (Figure 5C) with no significant overexpression of this cytokine in blood from PBS+LiVV group (mean expression ratio at 1.45 ± 0.32) compared to control animals, comforting previous results (Figure 3B). In contrast, IL-1β gene expression was significantly upregulated in blood from LiVV-infected mice challenging IL-10 neutralization (ratio at 2.12 ± 0.24) compared to control group, and significantly different compared to PBS+LiVV animals.

Discussion

Cytokines constitute key inflammatory mediators that orchestrate immune responses to infection [START_REF] Chong | Pro-inflammatory mechanisms in sepsis[END_REF] and differential regulation of cytokine expression was related to clinical outcomes in human leptospirosis. Expression levels of cytokines including IL-10, IL-1β and TNF-α were previously evaluated as potential markers of leptospirosis severity in human, and studies showed contrasting results regarding ratio of IL-10 to TNF-α correlated with disease severity [START_REF] Tajiki | The Ratio Of Plasma Levels Of Il-10/Tnf-Alpha And Its Relationship To Disease Severity And Survival In Patients With Leptospirosis[END_REF][START_REF] Mikulski | Severity markers in severe leptospirosis: a cohort study[END_REF][START_REF] Reis | Cytokine response signatures in disease progression and development of severe clinical outcomes for leptospirosis[END_REF][START_REF] Kyriakidis | Serum TNF-alpha, sTNFR1, IL-6, IL-8 and IL-10 levels in Weil's syndrome[END_REF][START_REF] Chirathaworn | Cytokine levels as biomarkers for leptospirosis patients[END_REF]. This might be explained by difference in infecting strains. Indeed, using a hamster model, Fujita and collaborators showed differential regulation of cytokine gene expression depending on the infecting serovars [START_REF] Fujita | Comparison of Bacterial Burden and Cytokine Gene Expression in Golden Hamsters in Early Phase of Infection with Two Different Strains of Leptospira interrogans[END_REF]. Thus, we analysed regulation of IL-10, IL-1β and TNF-α in blood from susceptible hamsters and resistant mice infected with two different species of leptospires to investigate effect of strains on cytokine expression, and relation between animal resistance and cytokine regulation. Except for IL-1β in LiVV-infected mice, gene expression of cytokines was up-regulated in blood of animals at D3-4 postinfection compared to control level.

Interestingly, IL-10, IL-1β and TNF-α expression levels but not the IL-10/TNF-α nor IL-10/IL-1β expression ratios were differentially regulated within the same animal model depending on inoculated leptospires. Infecting strains might differentially affect each cytokine gene expression while IL-10/TNF-α and IL-10/IL-1β ratios appear independent of leptospire serovars, suggesting these ratios as possible indicators of leptospirosis severity.

Less tissue lesions and lower leptospiremia with higher cytokine expression levels were observed in B3-13S-compared to LiVV-infected animals and it is noticeable that B3-13S leptospires belong to the serovar Ballum which natural reservoirs are mice [START_REF] Adler | Leptospira and leptospirosis[END_REF]. Thus, infection of mice with their host-coadapted serovar also promotes inflammatory responses with upregulation of cytokine levels during the early phase of infection. This also suggests virulence of strain as a preponderant factor of cytokine regulation possibly affecting observations and prognosis in human cases as previously shown [START_REF] Tubiana | Risk Factors and Predictors of Severe Leptospirosis in New Caledonia[END_REF]. Conversely, cytokine regulation could be regarded as a major actor of leptospirosis pathophysiology. Pro-inflammatory TNF-α and anti-inflammatory IL-10 are amongst the main biological mediators exerting adverse effects during bacterial infection and both cytokines are induced during experimental leptospirosis as previously shown [START_REF] Matsui | Gene expression profiles of immune mediators and histopathological findings in animal models of leptospirosis: comparison between susceptible hamsters and resistant mice[END_REF]. However, no difference was observed in TNF-α expression level between hamster and mouse when infected with the same strain. This correlates with observations in leptospirosis patients of whom TNF-α although induced in blood, was not different depending on organ involvement or severity [START_REF] Mikulski | Severity markers in severe leptospirosis: a cohort study[END_REF][START_REF] Chirathaworn | Cytokine levels as biomarkers for leptospirosis patients[END_REF]. Oppositely, IL-10 level was higher in mouse blood compared to hamster blood irrespective of the strain. This suggests a preponderant role of this anti-inflammatory cytokine in the resistance of mice to leptospirosis.

That is reflected by a higher IL-10/TNF-α and IL-10/IL-1β ratios in resistant mouse compared to susceptible hamster. As a limitation, evaluation of gene expression at the transcript level might not correlate with protein regulation due to possible post-transcriptional regulation mechanisms. Thus, we performed quantification of cytokine concentration in blood of LiVVinfected mice and confirmed the increase of IL-10 levels in this resistant mouse model. Assessment of cytokine concentration in hamster blood would also be of interest but specific antibodies are hardly available for hamsters.

Interleukin-10 plays a major role in the resolution of inflammation during sepsis and infection avoiding appearance of deleterious lesions but is also involved in persistence of pathogens by interfering with innate and adaptive immunity. Indeed, this molecule acts on several types of leucocytes including antigen-presenting cells (APCs), by downregulating expression of co-stimulatory markers and production of proinflammatory cytokines [START_REF] Gregori | The cellular and molecular mechanisms of immunosuppression by human type 1 regulatory T cells[END_REF]. It also directly inhibits T-cell response and proliferation while down-regulating expression of Th1 type cytokines, IL-2 and interferon-γ (IFN-γ) [START_REF] Taga | IL-10 inhibits human T cell proliferation and IL-2 production[END_REF]. Interestingly, in an IL-10 deficient murine model of Lyme disease, sustained IFN-γ production was observed resulting in development of disease related arthritis [START_REF] Sonderegger | Localized production of IL-10 suppresses early inflammatory cell infiltration and subsequent development of IFN-gamma-mediated Lyme arthritis[END_REF]. In parallel, IL-10 also has the potential to stimulate B cells contributing to immunoglobulin class switching in these leucocytes [START_REF] Meiler | Distinct regulation of IgE, IgG4 and IgA by T regulatory cells and toll-like receptors[END_REF]. Considering that IL-10 is a multipotent anti-inflammatory cytokine with different facets closely associated with susceptibility or resistance to infections and microbial persistence, we investigated the role of IL-10 in leptospirosis pathogenesis using a pre-neutralization strategy by injecting anti-IL-10 antibodies in LiVV-infected mouse. Although not lethal, IL-10 pre-neutralization in Leptospira-infected mice led to the development of a disease in this usually resistant model with observations of clinical signs and weight loss during onset of the disease.

To better assess the role of IL-10 in the regulation of immune response occurring during leptospirosis, effect of IL-10 previous neutralization was also evaluated on the regulation on Leptospira-dependent cytokine gene expression. Surprisingly, no effect of IL-10 neutralization was observed on LiVV-induced TNF-α transcript level. Experimental infection of C57BL/6 mice deficient for TNF-α receptor (TNFR) injected with pathogenic L. interrogans serovar Copenhageni exhibited more severe renal lesions suggesting this cytokine as an important mediators in the early control of infection [START_REF] Athanazio | Experimental infection in tumor necrosis factor alpha receptor, interferon gamma and interleukin 4 deficient mice by pathogenic Leptospira interrogans[END_REF]. Similarly, no difference was observed in TNFα expression level between resistant mouse and susceptible hamster during Leptospira infection (Figure 3A). Contrasting with TNF-α results, IL-10 neutralization seemed to potentiate IL-1β overexpression in LiVV-infected mice as observed for LiVV-infected susceptible hamsters.

Previous studies highlighted importance of pro-inflammatory cytokine IL-1β through overstimulation of the inflammasome system that could contribute to tissue damages [START_REF] Petrilli | The inflammasome: a danger sensing complex triggering innate immunity[END_REF].

Leptospiral glycolipoprotein was also shown to activate the inflammasome system through modulation of potassium transport and, together with Leptospira lipopolysaccharide, contribute to the IL-1β overexpression and production induced by L. interrogans infection in susceptible murine models [START_REF] Lacroix-Lamande S, D'andon | Downregulation of the Na/K-ATPase pump by leptospiral glycolipoprotein activates the NLRP3 inflammasome[END_REF]. Similarly, high IL-1β expression was associated with fatal outcome from pulmonary haemorrhages in canine leptospirosis [START_REF] Maissen-Villiger | Expression Profile of Cytokines and Enzymes mRNA in Blood Leukocytes of Dogs with Leptospirosis and Its Associated Pulmonary Hemorrhage Syndrome[END_REF]. Considering that IL-1β overexpression might be related to susceptibility and organ failures in Leptospira-infected host, the immuno-protective role of IL-10 might be effective by regulating inflammatory response during leptospirosis counterbalancing deleterious activities of pro-inflammatory IL-1β.

Effect of IL-10 neutralization was also evaluated on bacterial clearance and leptospiral burden in blood and kidneys from LiVV-infected mice. Although no impact was observed in leptospiremia, results showed evidences of IL-10 role in bacterial clearance in renal tissues.

Indeed, neutralization of IL-10 in Leptospira-infected mice led to a decrease in bacterial burden in kidneys at D7 postinfection. Interestingly, contribution of IL-10 in the attenuation of the inflammatory lesions during Lyme disease was shown but, interestingly, inhibition of endogenous IL-10 also enhanced bacterial elimination [START_REF] Brown | Dual role of interleukin-10 in murine Lyme disease: regulation of arthritis severity and host defense[END_REF]. Thus, IL-10 might be a major component in the maintenance of Leptospira in the kidneys of reservoir mouse in its natural environment.

This study suggests that the regulation of cytokine gene expression, especially the antiinflammatory IL-10, could be of importance in the leptospirosis pathogenesis. Indeed, lower levels of IL-10 were shown in the blood of hamsters lethally infected regardless of the Leptospira species used for inoculation. Moreover, inhibition of endogenous IL-10 in Leptospira-infected mouse presented evidences of the dual role of this cytokine by regulating immune response occurring during leptospirosis but exerting inhibitory effect on effective bacterial clearance. addressed to the Anatomic Pathology Laboratory staff, especially Evelyne Tuheiava and Karine Maguet, for their skilful assistance. Significant difference was evaluated compared to basal level in uninfected animals (above asterisks)

Figures and captions

or between treatments (solid lines) using a non-parametric Mann-Whitney test. *P<0.05, **P<0.001, ns: not significant.

Figure 1 .Figure 2 .Figure 3 .CFigure 4 .

 1234 Figure 1. Characterization of leptospirosis infection in resistant and susceptible animal models

Figure 5 .

 5 Figure 5. Regulation of cytokine gene expression in blood of LiVV-infected mice challenging IL-10

Acknowledgements

This work was supported by the French Ministry of Research and Technology, Institut Pasteur de Nouvelle-Calédonie, and Institut Pasteur de Paris. Position of M.M. is financed by the Government of New Caledonia. We are grateful to Jean-Marc Cavaillon for his expert assistance and helpful suggestions during the preparation of manuscript. Warm thanks are