INTERDEPENDENCE OF TORSIONAL VIBRATION DAMPER PARAMETERS ON CRANKSHAFT S TORSIONAL VIBRATIONS

Transcription

1 Journal of KONES Powertran and Transport, Vol. 15, No INTERDEPENDENCE OF TORSIONAL VIBRATION DAMPER PARAMETERS ON CRANKSHAFT S TORSIONAL VIBRATIONS Wadysaw Mtanec Konrad Buczek Cracow Unversty of Technology Al. Jana Pawla II 37, Krakow, Poland tel.: ; fax: e-mal: wmtane@usk.pk.edu.pl buczek@fev.com Abstract Torsonal vbraton of crankshaft s a harmful phenomenon n every type of engne. It can easly cause a fatgue falure n engnes wth relatvely long crankshafts and bg bore dameters (usually above 9 mm). For these types of engnes, the resonance caused by specfc low order harmonc of gas force and nerta force of oscllatng masses occurs n engne speed range, thus resonant torsonal moment of bg ampltude s generated. Ths moment, addtonally actng on crankshaft, s harmful and can easly cause ts fatgue falure. In order to reduce the resonance effects, the torsonal vbraton damper (TVD) has to be used. That s the reason why applcaton of torsonal vbraton dampers s almost a rule n these types of engnes. The vscose TVD s the most popular type of vbraton damper n heavy duty desel engne. Moment of nerta of ts plunger s the man crteron of damper selecton. It s the basc factor, whch determnes the amount of dsspaton work, decreasng the addtonal torsonal moment caused by resonance. Increasng plunger s moment of nerta nduce the rse of dsspaton work but smultaneously decrease natural frequences of a crank tran, what allows to resonate wth low order harmonc of tangental force actng on crankpn. The paper concerns an analyss of nfluence of vscose TVD parameters on crankshaft s torsonal vbraton for sx cylnder nlne heavy duty desel engne. Keywords: transport, combuston engnes, HD engnes, crank drve, torsonal vbraton 1. Introducton Torsonal vbraton of crankshaft presents a sgnfcant problem n bg bore mult-cylnder engnes (D > 9 mm). For these types of engnes, the resonance caused by specfc low order harmonc of gas force and nerta force of oscllatng masses occurs n engne speed range, thus resonant torsonal moment of bg ampltude s generated. Ths moment, addtonally actng on crankshaft, s harmful and can easly cause ts fatgue falure. In order to reduce the resonance effects, the torsonal vbraton damper (TVD) has to be used. Analyss of torsonal vbraton can be carry out by dfferent calculaton methods. Nevertheless all of them consst of: - calculaton of natural frequences of crank tran, - determnaton of dfferent vbraton modes, - Fourer analyss of excter forces, - determnaton of excter work, - calculaton of dampng work, ncludng dsspaton work of TVD, - determnaton of resonant moment ampltude, - frng order analyss. Calculaton of natural frequences of crank tran and determnaton of dfferent modes s usually made by takng advantage of crank tran reducton to mult-mass model (Fg. 1). The model conssts of elements wth specfed moments of nerta and zero-mass elements wth determned flexbltes e,+1. The parameters take nto consderaton masses of specfed parts of

2 W. Mtanec, K. Buczek crankshaft, connectng rod assembles and pston assembles, when e,+1 descrbes flexbltes of crankpns, man journals, and cranks. Fg. 1. Mult-mass model of 6 cylnder nlne engne The soluton of the physcal problem can be obtaned by solvng system of dfferental equatons provded by Lagrange formula: d E V, (1) d B B E - knetc energy of whole model, V - potental energy of whole model, B - nstantaneous angle deflecton of element, - tme. Results of foregong problem return values of natural frequences and relatve ampltudes of dfferent modes of resonance. Excter forces, whch act on each crank pn, consst of cylnder gas forces and tangental forces actng on specfc crank pn caused by oscllatng masses. In order to calculate excter work, above mentoned forces have to be subjected to Fourer analyss. The excter work can be then expressed by dependence (2): L c 1 [ M ] sn, (2) [M] - ampltude of torque of consdered harmonc of gas forces (cylnder ), A - torsonal vbraton ampltude of consdered crank, I - angular delay of torson relatng to torque harmonc of cylnder, c - number of cylnders. In connecton wth equalty of [M] values for all cylnders, and addtonally defnng ampltudes usng relatve ampltudes as follows: A A, (3) The summaton n (2) can be reduced to: c 1 A 1 sn sn, (3) The relaton (3) ntroduces mathematcal quantty called resultant relatve ampltude whch depends on order of consdered harmonc and assumed frng order. The excter work durng resonance becomes maxmal (sn = 1), and fnally t can be determned as: L res [ M ] A1, (4) 352

3 Interdependence of Torsonal Vbraton Damper Parameters on Crankshaft s Torsonal Vbratons Dampng forces durng oscllatons are manly caused by pston frcton. Other source of dampng s crankshaft materal, but only cast ron hysteress has sgnfcant nfluence of dampng work. Consderng steel crankshafts, dampng nto materal can be neglected. Remndng that pston frcton s assumed as sem-flud, the moment of dampng can be expressed as: db M t, (5) d M t - moment of torsonal vbraton dampng, - dampng coeffcent, B - torson, - tme. After mathematcal transformatons, dampng work caused by pstons frcton s expressed as: L tc A - frequency of crankshaft oscllatons, A 1 - ampltude of frst mass oscllaton, ( 2 ) w - sum of squares of relatve ampltudes. c A1 ( ) w 1, (6) The dampng coeffcent s usually determned by Holtzer formula:,4, (7) Addtonal dampng s provded by torsonal vbraton damper. There are many dfferent types of TVD and nevertheless the vscose type s the most popular as effcent and durable. The dampng work of ths type of TVD can be expressed as: L T 2 ob 2 t A 1 t p t - vscose dampng coeffcent nto TVD, A ob - ampltude of torson of TVD s housng, p - moment of nerta of TVD s plunger. Optmum dampng coeffcent for specfc plunger s moment of nerta amounts to:, (8), (9) t p and can be vared by changng dampng flud vscosty and gap between damper s housng and plunger. Applcaton of TVD sgnfcantly reduces resonant ampltudes but smultaneously decreases natural frequences of crankshaft. The addtonal moment of nerta appled on front of crankshaft by TVD, assumng that optmum vscose dampng coeffcent s used, s equal to sum of moment of nerta of damper s housng and half of moment of nerta of damper s plunger. The possblty of determnaton of ampltudes durng resonance occurs by equate excter work wth dampng work: L res L L, (1) tc T 353

4 W. Mtanec, K. Buczek The foregong equaton has only one unknown - ampltude of frst mass, whch thus can be easly calculated. Determnaton of frst mass ampltude leads to defnng the rest of ampltudes, twstng angles of specfc elements of crankshaft and, usng known flexbltes, torsonal moments caused by resonance. Frng order analyss takes nto consderaton, that dfferent frng orders produces dfferent excter work whch s caused by varyng of resultant relatve ampltude (3). Optmum frng order can only be found by evaluatng full calculatons for dfferent frng orders. 2. Analyss of TVD applcaton n sx cylnder heavy duty nlne engne The analyss has been performed on desgned sx cylnder engne crank tran of four stroke heavy duty desel engne wth 12 dm 3 engne dsplacement. The determnaton of natural frequences wthout TVD has been calculated takng advantage of mult-mass model analyss and usng FEM, what gves possblty to determne the accuracy of results obtaned from mult-mass model (Tab. 1). Tab. 1. Comparson of natural frequences obtaned by mult-mass model analyss and FEM analyss Natural frequences (multmass model results) Natural frequences (FEM calculaton results) Relatve error Mode n mm n FEM n [1/mn] [1/mn] % Analyses wth dfferent TVDs (vscose type) have been performed only on mult-mass model. The moment of nerta of plunger p has been assumed as ndependent varable (8), keepng the vscose dampng coeffcent t on ts optmum value accordng to (9). The dependency between moment of nerta of plunger and crankshaft natural frequency of 1 st mode s shown n Fg. 2. Applcaton of TVD strongly reduces the natural frequency of frst mode of crank tran, whch maxmal speed reaches 19 rpm, and also moves the node of oscllaton to front of crankshaft (Fg. 3) n [1/mn] 6 4 2,1,2,3,4,5,6,7,8 p [kg m 2 ] Fg. 2. Crankshaft natural frequency of frst mode for dfferent moment of nerta of TVD s plunger 354

5 Interdependence of Torsonal Vbraton Damper Parameters on Crankshaft s Torsonal Vbratons Fourer analyss of excter forces (tangental forces) has been performed up to sx order harmonc (h = 6) - Fg. 4. It can be notced, that low order harmoncs has the bggest ampltude and thus the resonance wth ths harmoncs has been assumed as the most dangerous. In order to determne excter work, the resultant relatve ampltudes for dfferent frng orders have been calculated (Fg. 5). 1,8 relatve ampltude [-],6,4,2 p [kg m 2 ],15,3,45,6,8 -, mass number [-] Fg st mode of torsonal vbraton as a functon of TVD parameter 7 [N] T Sum_of_12_harm. a h=,5 h=1 h=1,5 h=2 h=2,5 h=3 h=3,5 h=4 h=4,5 h=5 h=5,5 h= [deg] Fg. 4. Fourer analyss of excter force (T) 355

6 W. Mtanec, K. Buczek Frng order: [-] 2,5 2 1,5 1,5 p [kg m 2 ],15,3,45,6,8,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 h - harmonc order Frng order: ,5 p [kg m 2 ] [-] 2 1,5 1,5,15,3,45,6,8,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5 6 h - harmonc order Fg. 5. Resultant relatve ampltudes of 1st mode of oscllatons as a functon of TVD parameters for two possble frng orders Analysng natural frequences of crankshaft (Fg. 2) we can notce, that resonance occurs wth 5 th and hgher orders of harmonc for small TVDs, and reach range of 3.5 th and hgher orders of harmonc for the bggest damper. Consderng dependences n Fg. 5 we can decde, that untl 3.5 order harmonc ddn t resonate and the frng order would be an optmal soluton. Applcaton the TVD wth p >.625 kg m 2 causes foregong harmonc to resonate and thus the optmal frng order would be In addton, resonance of 3.5 order harmonc causes sgnfcant ncreasng of excter work. Thus maxmal resonant moment rses up n comparson wth smaller dampers. The dependency between maxmal ampltude of resonant moment of crankshaft and TVD parameters s shown n Fg. 6. There s no reason to apply TVD wth hgher moment of nerta of plunger then.625 kg m 2 because the ampltude of resonant moment rapdly ncreases. Fgure 7 shows the torsonal moment n crankshaft after 5 th cylnder n resonance wth 6 th order harmonc whch occurs under 112 rpm engne speed. 356

7 Interdependence of Torsonal Vbraton Damper Parameters on Crankshaft s Torsonal Vbratons 3 25 Mres [Nm] ,1,2,3,4,5,6,7,8,9 p [kg m 2 ] Fg. 6. Maxmal ampltude of resonant moment of crankshaft as a functon of TVD parameter Wthout TVD [Nm] [deg] M_eng M_resonant M_resultant Wth TVD (p =,625 kg m 2 ) [Nm] [deg] M_eng M_resonant M_resultant Fg. 7. Torsonal moment after between 5th and 6th cylnder caused by tangental forces (M_eng), resonance and superposton of ths loads n crankshaft wth and wthout TVD. 357

8 W. Mtanec, K. Buczek In order to realze the damper effcency, the fgure also shows the torsonal moment durng resonance wth the same harmonc wthout TVD. Conclusons 1) Torsonal vbratons are especally harmful for mult-cylnder bg bore engnes, 2) Applcaton of vscose type TVD hghly reduces the resonant effects, 3) Increasng moment of nerta of damper s plunger changes natural frequences and modes of crankshaft, 4) There s optmal sze of damper s plunger whch gves the lowest values of ampltude of resonant moment. References [1] Wajand, J. A., Tokowe slnk spalnowe redno szybkoobrotowe, WNT, Warszawa 25. [2] Jdrzejowsk, J, Mechanka ukadów korbowych slnków samochodowych, WK, Warszawa [3] May poradnk mechanka, Tom 1 2, WNT, Warszawa1994. [4] Orow, P. L., Zasady konstruowana w budowe maszyn. [5] Basshuysen, R., Internal Combuston Engne Handbook, SAE Internatonal, 24. [6] Matzke, W., Projektowane mechanzmów korbowych slnków szybkobenych, WK, "The author s reward was sponsored by Socety of Collectve Management of Copyrghts of Creators of Scentfc and Techncal Works KOPIPOL wth regstered offce n Kelce wth dutes obtaned on the ground of the art. 2 and art. 2 1 of law on copyrghts and related rghts." 358