VIBRATIONS OF THE BICARDANIC TRANSMISSIONS WITH ELASTIC SUPPORTS

Transcription

1 VITIONS OF THE ICNIC TNSMISSIONS WITH ELSTIC SUPPOTS Ion ULC octorand, Universitatea din Pitești, bstract: In order to study the lateral vibrations o the olicardanic transmissions, they assimilate with the elastic systems o bars articulated between them and susended by elastic cantilivers, the bars having in their comonent dierent sections.in this aer some bases o comutation or own ulsations are elaborated and numerical calculations are erormed or this urose. Keywords: cardan, eak revolutions.. FIEL MTIX.STTUS VECTO The artial dierential equation o the oened transverse vibration o an average iber bar (see Figure ) is [], [6], [0] : Fig.. w w 0 x EI t () where: w - is the delection; - normal section area; ρ - the density; E - longitudinal elasticity moment; I- is the geometrical inertial moment o a bar in regard to the normal main central axis on the XY lane. articular solution o the equation () is [], [6], [0] : w ( x, t) ( x)cos( t ) () where (x) is the harmonic vibrations amlitude, and this unction ulils the conditions: d d M ( x) d F( x) (x) ; ; () dx dx EI dx EI θ, M, F being the amlitudes, resectively o the section rotation, o the bending moment and cutting orce. 5 Fiabilitate si urabilitate - Fiability & urability Sulement no / 0 Editura cademica râncuşi, Târgu Jiu, ISSN 8 60X

2 One considers the notations: - F(0) ; (0) ; M M (0) ; F F(0) ; () F(l) ; (l) ; M M (l) ; F F(l) or the amlitudes o the dislacements rom the end sections and ; - ; ; or the status vectors, deined by the relations:,, M, F T ;,, M, F T ;,, M, F T (5) i, i=,,, or the Krâlov unctions, deined by the relations: - ( ch( cos( sh( sin( ( ; ( ; (6) ch( cos( sh( sin( ( ; ( z z z z e e e e where : ch( ; sh( ; F or the Krâlov matrix deined by the relation : - ( ( ( ( ( ( ( ( ( F ( (7) ( ) ( ) ( ) ( ) z z z z ( ( ( ( - α or the arameter deined by the relation: -, or the diagonal matrix: (8) EI EI EI - the ield matrix deined by the relation: - where was noted with: z x (9) 0 0 EI EI ; F( x) (0) With the relations [], [6], [0] are obtained the relations between the status vectors: F( x) ; () In the case where the bar is made out o segments with dierent sections (see Figure ), then the ield matrices o the segments are given by the relations: 6 Fiabilitate si urabilitate - Fiability & urability Sulement no / 0 Editura cademica râncuşi, Târgu Jiu, ISSN 8 60X

3 F( x ) Fig.. i i i i i, in the resented case, i=,, () and between the status vectors are the relations: ; C ; C () rom which results: () and the ield matrix or the entire bar is: (5). THE ISTIUTE MSS MOEL OF THE OF THE OULE IVE SHFT TNSMISSION One associates the equivalent model rom to the constructive model o the double drive shat transmission (see Figure.) and in the sections and are laced the elastic bearers that are connected to the mounting rame, that have the elastically constants: Fig.. ssimilating the bonds rom and with the articulations, one will obtain: 7 Fiabilitate si urabilitate - Fiability & urability Sulement no / 0 Editura cademica râncuşi, Târgu Jiu, ISSN 8 60X

4 M M 0 (6) The cutting orces rom and are given by the relations: F k ; F k (7) Next are obtained the status vectors: T F / k,,0, F ; T,,0, k (8) I one uses the notations: 0 k 0 T ; T (9) then the status vectors rom oints and can be written under: T T, F (0) and the equality : () becomes: 0 () F k and rom here is obtained the homogenous equation system in, F, : F ; 0 F ; k F () In order that the system () to have a solution dierent then zero is needed that the determinant to be equal with zero. ( ) 0 () where : ( ) 0 k (5). THE EPESENTTION OF THE SPECIFIC WYS OF VITION In the grahical reresentation o a vibration mode there are calculated the amlitudes in dierent sections o the deections or the ulse that corresonds to this mode o vibration. So it is assigned to the delection the numerical value equal with the unity and then rom the system () results: F k ; ; k ( ) (6) 8 Fiabilitate si urabilitate - Fiability & urability Sulement no / 0 Editura cademica râncuşi, Târgu Jiu, ISSN 8 60X

5 The delections rom sections and C are the irst elements o the column matrices, that are obtained rom the relations:, C F T ; F T C (7) Fig.. The delections rom the intermediary oints (see Figure.) are the irst elements o the column matrices, and are calculated with the relations:, T F ( x ) T, F F( x) F( x (8) (9) ) C (0). NUMEICL PPLICTION One considers the double drive shat transmission o an o-road vehicle or which: 6 6 k 85 0 N / m ; k 0 0 N / m ; l 0, 07m ; l 0, 59m ; l 0, 5m ; 9,6 0 m ;,6 0 m ; 7,060 m ; 7800kg / m For the irst aroximation the double drive shat transmission is relaced with a constant section bar, yielding seats at both ends, made out o three same length arts. In this case the ollowing values or the characteristic ulses are obtained: 78, 5s ; n crt 75,5rot 59, 79s ; n crt 07,rot considering the bar with the comlete section with mm ext 50 06, 5s ; n crt 05,56rot 00, 9s ; n crt 86,7rot considering the bar with the circular section with ext 50mmand int 5mm For the real double drive shat transmission with the data resented above, one obtains these values or the characteristic ulses: 876, 66s ; n crt 875,78rot 87, 7s ; n crt 70,5rot ccordingly to the characteristic ulses the characteristic modes o vibration are obtained as in Figure.c,d 9 Fiabilitate si urabilitate - Fiability & urability Sulement no / 0 Editura cademica râncuşi, Târgu Jiu, ISSN 8 60X

6 5. CONCLUSIONS. Calculating the eak revolutions with this method, even i it is more comlicated and hard, it is closest to reality. y comaring the value o the irst critical calculated revolution with the relation : n d 7 ncrt,0 [rot/min] () L with the value obtained with this method, or the same double drive shat transmission, a dierence o 7% is obtained between the two values.. In automotive construction, the most requent case is the one where the bearer rom is rigid (the cardan shat with the ront axle) and the one rom is elastic (the connection o the cardan with the engine-gearbox ensemble). Considering those two, the calculation rogram elaborated on this method can determine the inluence o the elastic constants on the eak revolutions.. In a matter that concerns the characteristic modes o vibration, the amlitude o the medium iber bar calculated in the characteristic oints dros as the rigidity rom increases. For an increase in the elastically constant rom N/m to N/m, the oints amlitude dros our times or the irst characteristic ulsation. The values o the eak revolutions decrease as the rigidity or bearer decreases.. To avoid the disturbing eects o the eak revolutions it is wanted that the irst characteristic requency n to be as lowest as it can be and the second characteristic requency n to be as highest as it can be, so that the transition to be smooth. 5. The increase o the second eak revolution can be done by using the cardan shats with high rigidity that are obtained by reducing the length or by increasing the exterior diameter. In the construction o automotive is usually used the irst constructive method, through which the long bio-cardan transmissions are relaced with multi-cardan transmissions with short shats. EFEENCES. UZUGN, GH. FETCU, LUCI., EȘ, M., Vibrațiile sistemelor,.s.. cademy Publishing House, ucharest, UMITU, N., NHU, GH., VINTILĂ, NIEL., Mecanisme și transmisii mecanice, idactic and Pedagogical Publishing House, ucharest, UIțĂ, FL., Transmisii cardanice, Technical Publishing House, ucharest, UIțĂ, FL., ICONESCU,., OHN, C., NEGOE, M., SĂULESCU,., Transmisii cardanice,transilvania Exres Publishing House, rașov, HISS, C., CUU, GH., Șocuri și vibrații, Technical Publishing House, ucharest, PNE, N., PÂLC, S., Vibrații mecanice, University o Pitești Publishing House, Pitești, PNE, N., PÂLC, S., POP,., Modele entru studiul vibrațiilor automobilelor, Tiarg Publishing House, Pitești, PNE, N., Elemente de mecanica solidelor în coordinate lucheriene, omanian cademy Publishing House, ucharest, IPINU,., CĂCIUN, I., Osii, arbori dreți și arbori cotiți, Technical Publishing House, ucharest, VOINE,., VOICULESCU,, SIMION, FL., Introducere în mecanica solidului cu alicații în inginerie,.s.. cademy 0 Fiabilitate si urabilitate - Fiability & urability Sulement no / 0 Editura cademica râncuşi, Târgu Jiu, ISSN 8 60X