DRY FRICTION DAMPER FOR SUPERCRITICAL DRIVE SHAFT

Transcription

1 Jounal of KONES Powetain and Tanspot, Vol. 23, No DRY FRICTION DAMPER FOR SUPERCRITICAL DRIVE SHAFT Witold Pekowski Institute of Aviation Kakowska Avenue 110/114, Wasaw, Poland tel.: ext. 373, fax: witold.pekowski@ilot.edu.pl Abstact In this aticle the constuction and mathematical model of a dy fiction dampe with adial gap, designed fo dumping flexual vibations of a supecitical populsion shaft (developed peviously duing design woks made in Institute of Aviation on populsion system fo an ultalight IS-2 helicopte), while passing though the esonance is pesented. Some esults of mathematical and numeical analyses of such a system (supecitical shaft + dampe) behaviou ae also pesented among othe things, vey distinctive behaviou of the shaft, while passing though esonance, is shown. Fom a theoetical point of view, it is inteesting that cetain ange of dampe paametes, obtained in the couse of numeical analysis, lead to chaotic vibations of the system (they wee also obseved in pactice). Fom a pactical point of view, it will be inteesting that the aticle shows a way how to ceate a dimensionless (and theefoe geneal) paametes of the system: supecitical shaft and fiction dampe and also simple engineeing methodology fo selection of suitable (fo the coect opeation of the shaft) dampe paametes depending on the paametes of the shaft, developed duing seies of analyses. The pactical aspect of the poblem seems to be paticulaly impotant, as the supecitical populsion shaft equipped with a fiction dampe can be vey stuctually simple, light and inexpensive, and still it is not widely used, pobably because of cetain doubts aoused in the constuctos by the tem: esonant vibations. Keywods: supecitical dive shaft, supecitical populsion shaft, dy fiction dampe 1. Intoduction ISSN: e-issn: DOI: / Supecitical dive shaft, when eaching its opeating otational speed, at least once passes though the citical speed. Duing this pass, it significantly inceases the amplitude of its vibations, which, without sufficient vibation damping, can lead to damage o destuction of the shaft. This is one of the easons why supecitical shafts ae not widely adopted in the machiney, although it is known how to pevent excessive vibations in a dangeous aea: ensue adequate dampening o to coss the dangeous speed ange fast enough (the best is using both techniques simultaneously). Once dange aea is exceeded and appopiately distanced, chaacteistic selfstabilization of the shaft appeas and it otates fom now vey stable, and the amplitude of the vibations is small [5]. If we do not mind some ovehang of the shaft unde its own weight, welldesigned supecitical shaft is a geat option when we want to povide the dive fo a elatively lage distance to the powe eceive opeating mainly at constant speed with the use of supecitical shaft it can be achieved at low weight, simple design and technology, and at low cost of the shaft. A classic example of such application may be poviding powe fom the main geabox of the helicopte, to its tail geabox. The eplacement of the classical solution (the shaft divided into seveal sub-citical sections connected one to anothe by means of flexible couplings, and with many suppots on the helicopte tail beam) with one-piece supecitical shaft, can save not only a lot of mass and cost, but also impoves secuity by eliminating the dozens of so-called cucial elements, failue of which could lead to cash. The aticle descibes a simple-constuction dy fiction dampe fo damping supecitical shaft flexual vibations, descibes its model as well as the poposition of a methodology fo pe-selection of suitable dampes paametes depending on paametes of the shaft. This methodology povides simple, engineeing guideline, developed on

2 W. Pekowski the basis of many simulations of the shaft-dampe system dynamics, which, as fa as the autho s knowledge eaches, ae lacking in the liteatue. All values used in the aticle ae expessed with SI units. 2. The pimay shaft paametes The pimay shaft was designed as tail tansmission shaft fo a pototype of IS-2 helicopte. Fig. 1. Physical model of the pimay shaft equipped with a dy fiction dampe Pimay shaft paametes: length, L = 3.32 [m], density, ρ = 2700 [kg/m 3 ], coss-section aea, A = [m 2 ], Young modulus, E = [Pa], moment of inetia (geometical), I = [m 4 ], suppot stiffness, k0 = kn =10 6 [N/m], suppot damping, c0 = cn = 0 [Ns/m], extenal (viscous) damping coefficient, ct = 0.5 [Ns/m 2 ], eccenticity (constant along the shaft), e = [m]. Fo such a shaft (without dampe) the 1-st citical speed Ω0 is, due to: low moment of inetia, low damping and high suppots stiffness (elative to the stiffness of the shaft), pactically equal to the 1-st fequency of fee, undamped vibations of analogous beam on non-defomable suppots ω0. 3. The concept of the dampe The concept of the dampe [7, 8] was developed at the Institute of Aviation, fo coopeation with the pimay shaft, descibed befoe. Stuctual sketch of the dampe is shown in Fig. 2. Fig. 2. The dy fiction dampe stuctual sketch: 1 shaft, 2 movable disc, 3 dampe body, 4 pessue plate, 5 pessue sping 390

3 Dy Fiction Dampe fo Supecitical Dive Shaft The key assumptions fo the dampe wee following: 1) the dampe uses only dy fiction and its chaacteistic is a gap S (adial cleaance) in the movable disc (Fig. 2) selected so that the outside of the esonance egion shaft otates feely, without any contact with the dampe, 2) the dampe coopeates with a supecitical shaft of the 1-st ode (its opeating speed is located between the 1-st and the 2-nd citical speed), which has the fom of a cylindical beam, pivotally suppoted at the ends, and the dampe is located in the middle of the shaft (Fig. 1). 4. The mathematical model of the dampe It was assumed that the dampe disc pefoms the plane motion, and its mass is negligible. The disc movement is descibed in the fixed, left-handed coodinate system 0xyz (Fig. 1, Fig. 3). Fig. 3. The shaft-dampe inteaction model and its designation system An essential element of the dampe is movable disc, which has a cental hole of D diamete (though which the shaft of d diamete passes). By vaying the pessue spings (Fig. 2) foce, specified fiction against the side faces of the disc is set, giving the dampe esistance foce F. On the shaft, which is in contact with the dampe disc, opeates esultant foce F, which has two components: a nomal Fn and tangential Ft. The impact foces between the shaft and the dampe and basic designations used to descibe the model of the shaft-dampe inteaction, ae shown in Fig. 3. Vectos U and O ae position vectos espectively fo the shaft cente and the cente of the dampe disc in fixed coodinate system 0xyz, while the vecto is the position vecto fo the shaft cente in a moving coodinate system associated with the cente of the dampe disc (this system pefoms, along with the disc, plane motion in the system 0xyz). In ode to facilitate the detemination of the impact foce F between the shaft and the disc, an elastic zone of Ds diamete was intoduced into the disc. In the zone, the foce F vaies linealy fom zeo to F (Fig. 4). Fig. 4. The foce of shaft-dampe inteaction F() 391

4 W. Pekowski The intoduction of this zone allows avoiding answeing the question: what is actually the foce of shaft-dampe inteaction when they ae in contact? its pesence makes this foce selfdeveloping. This is vey impotant because knowledge of the dampe esistant foce F is not sufficient to detemine the impact of the shaft-dampe foce F, which can be shown though a thought expeiment: the fact that the dampe esistance is infinite does not mean that at the shaft, being in contact with the dampe, acts infinite foce (it s absud). The elastic zone is not only vey convenient tick fo easy calculation, but it also has a physical sense it can be linked with the disc own elasticity and elasticity of the whole dampe mounting. The thickness of the elastic zone is vey impotant in numeical modelling of the shaft vibations it cannot be too small and should be selected in pai with integation step fo equations of motion the shaft should need at least a few steps to tavese this zone, othewise it may lead to its oveshooting, and, consequently, to the wong wok of the model. The numeical model of the dampe assumes that the shaft, afte exceeding the elastic zone, moves the dampe disc by the amount of that excess. Dy fiction dampe paametes: esistant foce, F [N], the gap, S = (D d)/2 [m], thickness of the elastic zone, Hs = (Ds D)/2 [m], shaft-dampe fiction coefficient, µ [1], whee: d diamete of the shaft [m], D diamete of the disc hole [m], Ds diamete of the elastic zone [m]. The following concatenations wee used to detemine the vectos F, Fn, Ft: F = F + F, (1) F t n n t = µ F, (2) Fn () = F () 2 1+ µ, (3) Fn = Fn() (4) F U < 0. (5) t 5. Dimensionless paametes of the shaft-dampe system In ode to develop dimensionless dampe paametes a substitute shaft intoduced. The pimay shaft, with constant paametes (but not the only one), can be eplaced with a substitute shaft in which mass, eccenticity and damping ae concentated in the middle of the shaft (Fig. 5). Fig. 5. Physical model of the substitute shaft equipped with a dy fiction dampe 392

5 Dy Fiction Dampe fo Supecitical Dive Shaft In a limited ange of otational speed (up to about 3Ω0, egadless of the eccenticity distibution in the pimay shaft) the substitute shaft can be dynamically vey simila to the pimay shaft. It was assumed that the 1-st citical speed of Ω0 (and the 1-st fequency of own vibations ω0) ae the same fo the pimay and the substitute shaft, the same ae also all of the beam paametes (length, Young s modulus etc.). Simila dynamic deflections of the two shafts in the elevant speed ange ae povided by appopiately chosen: substitute eccenticity and substitute extenal damping. The stiffness of the substitute shaft in its midpoint can be obtained with the use of the known fomula fo homogeneous beam (suppoted in non-defomable suppots): 48EI k =. (6) 3 L Fo known Ω0, substitute (concentated) mass can be calculated as follows: k k Ω = m =. (7) Ω 0 2 m 0 Substitute eccenticity ε depends on the eccenticity distibution along the pimay shaft, but when it is constant (e(x) = e = const) we obtain: ε = 1.45 e. (8) Substitute extenal damping c depends on the damping distibution along the pimay shaft, but when it is constant (ct(x) = ct = const) we can obtain it fom simple elation: m m c= ct L = ct. (9) ρ AL ρ A Now we can define the dimensionless paametes of the shaft-dampe system (shaft speed, shaft deflection, dampe esistant foce, dampe gap and elastic zone thickness): Ω, U, c, F, S, H s β = u = = f = s = hs =. (10) Ω ε Ω Ω ε ε ε γ 2 0 m 0 m 0 6. Results of dynamics simulations of the shaft-dampe system In the simulations a FEM model of the pimay shaft was used the model was descibed peviously [2, 3, 5, 7], so only bief desciption hee: the shaft has been modelled by means of 4 beam elements of equal length, which defomations ae descibed by the espective Hemite polynomials. The foce coming fom the dampe was intoduced to the shaft model as exta edge foces fo the coesponding finite elements. The esults of the calculations ae displacements of the shaft nodes (finite element ends) in time especially, the most inteesting displacements of the cental node. The esults allowed dividing main dimensionless paametes of the dampe (f and s) into: lage geate than 1, and small less than 1. A complete pictue of chaacteistic behavio of supecitical shaft equipped with a gapped dy fiction dampe is shown in Fig. 6. With the inceasing otational speed of the shaft: gadually the shaft deflection in the midpoint eaches a size of s (at β1 shaft comes into the contact with the dampe) and becomes fixed at a cetain time, then at speed β2 shaft has enough powe to move the dampe disc, at β3 > 1 amplitude of the shaft eaches a maximum (esonance), then shaft deflection deceases, dampe disc is cented and at β4 shaft comes out fom woking with the dampe it is followed by a jump down, futhe the dimensionless amplitude of vibations is heading asymptotically to

6 W. Pekowski Fig. 6. The illustative pocess of dimensionless displacements in the midpoint of the shaft equipped with a dy fiction dampe of a big esistance (f > 1) and a big gap (s > 1) the aows indicate changes in shaft speed β When we educe the speed of otation: the shaft comes into contact with the dampe disc at β5 and it is immediately followed by a jump up, then the amplitude of the shaft vibation eaches a maximum at β6 <1 (esonance), late followed by centing of the disc at β7, then the shaft vibates with the amplitude of s, and finally, at β1 the shaft gently loses contact with the dampe. The chaacteistic jumps (up and down) shown hee ae not connected with the dy fiction dampe pesence they ae chaacteistic fo the supecitical shaft with an intemediate suppot (can be elastic o othewise), which has a gap [1, 4]. How effective can the use of well-chosen fiction dampe be is illustated by the esults obtained in simulations of dynamic deflection in the midpoint of the pimay shaft without a dampe (Fig. 7) and with the dampe (Fig. 8). As it can be seen, the dy fiction dampe almost completely tuned off esonant vibations of the shaft. In both cases, the angula acceleation of the shaft is constant and the same: dω/dt = 10 ad/s 2. In the simulations, in geneal, the effect of the shaft weight was omitted, but it was veified that to take this effect into account, displacements should be efeed to the line of fee deflection of the non-otating shaft, instead of to the 0x axis (accoding to theoy). Fig. 7. The couse of u(β) in the point x/l = 0.5 fo the pimay shaft without a dampe The mathematical analysis of the poblems as well as numeical simulations [7] showed that the dampe is, in pactice, sufficiently chaacteized by only 2 main paametes: the esistance and the gap (elastic zone only facilitates calculations and a modeate fiction between the shaft and the dampe disc e.g. μ < 0.2, has hadly no effect on the shaft behaviou). They also allowed to fomulate 3 pactical conditions fo the dampe popely chosen fo the supecitical shaft of nominal, dimensionless otational speed βnom: 394

7 Dy Fiction Dampe fo Supecitical Dive Shaft Fig. 8. The couse of u(β) in the point x/l = 0.5 fo the pimay shaft with the dy fiction dampe of following paametes: s = 2.07, f = 2.024, h s = (F = 16 N, S = m, H s = m), μ = 0 s > 1 (necessay fo the shaft feely otating afte leaving the dampe), (11) f > 1 (fo effectively educing the amplitude of the shaft vibations in esonance), (12) s+ f β4 < min( βnom, 2) (allows the shaft to leave the dampe befoe βnom), (13) s 1 and if the nominal speed of otation of the shaft βnom > 2, this condition amounts to: β 4 < 2 f < 3s 4. (14) These conditions (fo the case: βnom > 2) ae shown gaphically in Fig. 9 whee smax is the maximum allowable gap due to the dampe constuction and allowable shaft deflection, and the shaded aea indicates whee to look fo the appopiate dampe paametes. The gaph highlights also the point (2, 2) which, accoding to the autho, is close to optimum. Fig. 9. Recommended paametes of the dy fiction dampe when β nom > 2 7. Pactical guideline fo the selection of the dy fiction dampe paametes Fo a simple shaft simila to the pimay shaft descibed above (long cylindical beam, pivotally suppoted at the ends in elatively igid suppots) which opeating speed is above 2Ω0 (it is ecommended to set opeating speed in the ange of 2Ω0-3Ω0), the selection of the pope dampe paametes can be made in a few simple steps: 1) detemine the shaft stiffness k fom (6), 395

8 W. Pekowski 2) detemine the substitute mass of the shaft m fom (7), 3) estimate the shaft eccenticity e (it is caused mainly by staightness deviation, so take maximum value fom the manufactue specification, o, ideally, otate you suppoted shaft slowly and measue displacements in the middle) it s the hadest step, 4) detemine the substitute eccenticity ε fom (8), 5) take f and s fom Fig. 9 fo example f = 1.9 and s = 2.1 (just in case take f a little bit smalle and s a little bit geat then highlighted value of 2), 6) having: m, ε, f, s calculate dimensional paametes of the dampe, accoding to (10): Conclusions 2 0 F = f m ω ε, (15) S = s ε. (16) The conclusions ae summaized in a few points below: 1) The dy fiction dampe with a gap, popely sized, vey effectively educes supecitical shaft esonance vibations and allows it to otate feely, without contact with the dampe, at nominal otating speed. 2) The dampe is vey simple and inexpensive and can be easily build, even by amateus. 3) The dampe has 2 main paametes: the esistant foce F and the gap S. 4) Fo the supecitical shaft of simple design, simila to the pimay shaft descibed befoe (which can also be easily build by amateus), pope paametes of the dampe can be easily found with the guidance contained in the aticle (at least initially then they can be adjusted in the tials). 5) As the contact between the shaft and the dampe is limited in time (esonance egion), fiction wea should not be lage fo well-chosen mateials. 6) The dampe disk wea, in the fom of the incease of the gap, does not advesely affect the opeation of the dampe and causes only compaable incease in amplitude of the shaft vibations. 7) We should be awae when mounting the dampe that the flexible shaft hangs unde its own weight in the neutal position the dampe disc should be moe o less concentic to the shaft. 8) The supecitical shaft can be a geat solution when we need do povide the mechanical powe fo a long distance and we don t mind about shafts ovehang and elatively high amplitude of shaft vibations (in ode of millimetes). Such a shaft can be successfully used without balancing it at all o with static balancing only (dynamic balancing of a long, flexible shaft could be vey had) if the shaft is elatively staight (e.g. a well-manufactued tube o ode) it should be good enough. Refeences [1] Benay, B., Aicaft engine esponce due to Fan unbalance to the pesence of consumed gaps in the engine duing the phase of windmilling, ICAS 2000 papes. [2] Dżygadło, Z., Pekowski, W., Nonlinea dynamic model fo flexual vibations analysis of a supecitical helicopte s dive shaft, ICAS 2000 papes. [3] Dżygadło, Z., and othes, Zespoły winikowe silników tubinowych, WKiŁ [4] Ehich, F. F., O Conno, J. J., Stato whil with otos in beaing cleaence, Jounal of Engineeing fo Industy, [5] Gyboś, R., Dynamika maszyn winikowych, PWN, [6] Kuszewski, J. and othes, Metoda sztywnych elementów skończonych, Akady, [7] Pekowski, W., Analiza dynamiki wału pacującego w waunkach nadkytycznych, do napędu śmigła ogonowego ultalekkiego śmigłowca, paca doktoska, thesis, WAT, [8] Patent: PL B1. 396