An analysis of the self-excited torsional vibrations of the electromechanical drive system

Transcription

1 Vibration in Phyical Sytem Vol. 7 (16) An analyi of the elf-excited torional vibration of the electromechanical drive ytem Robert KONOWROCKI and Tomaz SZOLC Intitute of Fundamental Technological Reearch of the Polih Academy of Science ul. A. Pawińkiego 5B, -16 Warzawa, rkonow@ippt.pan.pl, tzolc@ippt.pan.pl Abtract Thi paper preent a dynamic analyi of torional vibration of the railway drive ytem. A dynamic electromechanical drive model ha been created and then integrated with the railway wheelet-rail ytem to imulate elf-excited torional vibration of the conidered ytem. Reult of thi analyi are ued in order to invetigate the drive ytem enitivity to torional ocillation. Here, the dynamic electromechanical interaction between the electric driving motor and the rotating wheelet i conidered. Thi invetigation ha proved that the torional tiffne and damping of drivetrain ytem trongly affect amplitude of the elf-excited vibration. A elf-excited vibration affecting on an energy conumption of the electric motor of the conidered ytem are tudied Keyword: torional vibration, electromechanical coupling, wheel-rail adheion, wheelet drivetrain dynamic 1. Introduction Mechanical vibration and deformation are phenomena aociated with an operation of majority of railway vehicle drivetrain tructure. The knowledge about torional vibration in tranmiion ytem of railway vehicle i of a great importance in the field dynamic of mechanical ytem [1]. Torional vibration in the railway vehicle drive train are generated by everal phenomena. Generally, thee phenomena are very complex and they can be divided into two main part. To the firt one belong the electromechanical interaction between of the railway drive ytem including the: electric motor, gear, the driven part of dic clutch and driving part of the gear clutch []. To the econd one belong torional vibration of the flexible wheel [3,4] and wheelet caued by variation of adheion force in the wheel-rail contact zone [5]. An interaction of the adheion force ha nonlinear feature which are related to the creep value and trongly depend on the wheel-rail zone condition and track geometry (when driving on a curve ection of the track). In many modern mechanical ytem torional tructural deformability play an important role. Often the tudy of railway vehicle dynamic uing the rigid multibody method without torionally deformable element are ued [6]. Thi approach doe not allow to analye elf-excited vibration which have an important influence on the wheelrail longitudinal interaction [7]. A dynamic modelling of the electrical drive ytem coupled with element of a driven machine or vehicle i particularly important when the purpoe of uch modelling i to obtain an information about the tranient phenomena of ytem operation, like a run-up, run-down and lo of adheion in the wheel-rail zone. In thi paper mot attention i paid to the modelling of an electromechanical interaction between the electric driving motor and the railway wheelet a well a to an influence of the elf-excited torional vibration in the conidered drive ytem.

2 188. Mathematical modeling of the wheelet and the electric motor In order to invetigate a character of elf-excited torional vibration in the electric railway vehicle powertrain and a dynamic mutual coupling between the wheelet and the electric motor, a poibly realitic and reliable electromechanical model of the railway drivetrain i applied. The mechanical drive ytem i repreented by a torionally vibrating ytem of four-dof. The cheme of the conidered model i hown in Figure 1. Fig. 1. Scheme of the dynamic model of the railway wheelet drive ytem A mathematical model of the ingle torionally deformable railway wheelet under torional vibration induced by the traction motor and variou adheion frictional effect occurring in wheel-rail contact zone ha been derived by mean of the econd-order Lagrange equation in the generalized coordinate φ i(t). Thee coordinate decribe angular diplacement of the drivetrain component of the wheelet. Here, there will be preented a torional dynamic analyi of the ingle wheelet running on a geometrically perfect traight ection under variou operational condition determined by longitudinal lip i of both wheel, vertical wheel force Q+m g and vehicle velocity v. The drive torque and the retarding one due to the creep force in contact of the rail with with the wheel complement the conervative railway drive model on the right ide (1) and it can be expreed a & (t) = ( K wheelet + K gear ) ϕ(t) + ( Cwheelet + Cgear ) & ϕ(t) = M drive M creep, (1) Iϕ where I denote the ma matrix containing ma moment of inertia of rotating element of the drive ytem, the matrice K wheel, K gear, C wheel and C gear expre the torional tiffne and damping propertie of the wheelet, dic-clutch and of the gearbox wheel, repectively. Vector M drive contain the electromagnetic torque generated by an aynchronou motor decribed in the following part of the paper and vector M creep contain the traction torque generated by longitudinal tangential load in the wheel-rail zone. Their form can be expreed a Tcreep _ i = µ i ( i ) ( Q + mi g), i = 1,, () where Q i the normal load impoed on the ingle wheel, r i the wheel radiu and µ( i) i the traction coefficient expreed in Eq. (4). It maximum value i called an adheion coefficient. The longitudinal creepage of the wheel i defined in the following form 3.6ωir 3.6ϕ& ir = ( -1), i = + ( -1), i = 1,, (3) v v

3 Vibration in Phyical Sytem Vol. 7 (16) where and i are the longitudinal creepage before and during diturbance, repectively. Symbol ωi i the angular peed of the i-th wheel, i-index mean the left and the right wheel and v denote forward wheelet velocity in km/h obtained by the equivalent angular peed of wheelet axle ϕ& at the contact point. In equation (1) the traction torque including torque M creep_1, on left and right wheel of the wheelet have nonlinear propertie. Thee propertie are dependent on a profile of adheion characteritic decribing a contact in the wheel-rail zone. Depending on the adopted variou maintenance, operation and weather condition, thi characteritic can take into conideration variou form of creepage curve, a hown in Fig.. The creepage curve applied for the carried out invetigation ha been plotted in Fig. 3 and it can be expreed by the following equation b µ ( i ) =.3*[(a + exp(-i ) + tanh( i )/ + (d atan(e i )) + exp(f i ))], i = 1,. (4) c For dry and wet weather condition in the wheel-rail zone parameter of Eq. 4 have numerical value contained in Table 1. Table 1. Parameter for traction coefficient in Eg. (4) Quantity a b/c d e f dry wet Fig.. Adheion-creep characteritic of the railway condition [8] table region untable region Fig. 3. Profile of adheion curve uing in invetigation The adheion curve can be divided into two region, ee Fig. 3. The firt region i characterized by a rapidly riing lope of the curve i the table region. The econd one, due to a negative damping reult in viible decreaing lope, can lead to elf-excited ocillation in the wheel-rail contact zone. Thi phenomenon make the driven wheelet lipping on the rail of a railway track. Conequently, when a tangential force between the wheel and the rail exceed an adheion force in the wheel-rail contact zone, the elf-excited torional vibration of the wheelet occur. Such a phenomenon ha a very large impact on the relative rotation between the wheel and the axle due to a lack of friction in pre fitting [9] and it can make vehicle derailed. Additional dynamic torional overload produce diturbance in the wheelet drive ytem, which ha a influence on the traction moment of a railway vehicle. Thi characteritic of the traction moment i alo dependent on electrical parameter of the motor, power upply and it regulation. A modeling of the

4 19 electrical part of a drivetrain i a very difficult and complex tak. For a imple olution it i poible to ue a linearization around of the working point tatic characteritic of the driving motor. But, in the cae of a more advanced analyi of tranient phenomena in the drivetrain an accurate circuit model of the electric motor i needed [1,11]. The aynchronou motor are very commonly applied a railway vehicle driving ource. From the viewpoint of electromechanical coupling invetigation, for an introductory approach the properly advanced circuit model of the electric motor eem to be required, imilarly a e.g. in [1]. In the cae of the ymmetrical three-phae aynchronou motor electric current ocillation in it winding are decribed by the ix circuit voltage equation. In order to implicity of their form they are tranformed into the ytem of four Park equation in the o called αβ-dq reference ytem U co( ω ) L Lm e t 3 U in( ω ) e t = 3 Lm R + 3 1( t) plm & ϕ R 3 pl 1 ( t) m & ϕ L 1 + L m 3 L m p & ϕ1( t) R r 1 Lr + Lm ( 1 Lr + Lm ) Lm i& ( ) α t 3 L i& ( ) m β t + i & r d ( t) 1 L + r r L ( ) m i& q t i ( ) α t i ( ) )( 1 ), 1 ( β t p & ϕ t L r + L m i r ( t) d Rr r iq ( t) where U denote the power upply voltage, ω e i the upply voltage circular frequency, L, L r are the tator coil inductance and the equivalent rotor coil inductance, repectively, L m denote the relative rotor-to-tator coil inductance, R, R r are the tator coil reitance and the equivalent rotor coil reitance, repectively, p i the number of pair of the motor magnetic pole, ϕ& 1 (t) i the current rotor angular peed including the average and vibratory component and iα, i β are the electric current in the tator winding reduced to the electric field equivalent axe α and β and i r r d, i q are the electric current in the rotor winding reduced to the electric field equivalent axe d and q, [1]. Then, the electromagnetic torque generated by uch a motor can be expreed by the following formula 3pL r r m i i i = β i d α q M. In our approach the interaction between the electromagnetic and mechanical ytem of the conidered powertrain coupled mutually through electromagnetic torque M and angular rotor velocity ϕ& 1 i hown in Eq. (5) and (6). In order to control the electric motor aumed in the applied drive ytem model the field-oriented control method ha been ued [13]. According to the above, thi et of coupled electromechanical Eq. (1), (5) and (6) i going to be imultaneouly olved by mean of a elected direct integration method for electric parameter including: reitance of the tator and the rotor equal R =.88 Ω, R r=.158 Ω. The relative inductance, inductance of the tator winding and (5) (6)

5 Vibration in Phyical Sytem Vol. 7 (16) inductance of the rotor winding are repectively equal to L m =,41 H, L =,45 H and L r =,418 H.The aynchronou motor ha 4 pole pair and it upply voltage i equal to 3 kv with 6 Hz upply frequency. In the conidered cae, the Runge-Kutta fourthorder method will be applied for motion equation of the electromechanical model aumed in thi way. 3. Numerical reult In the computational example railway drivetrain ytem with the torionally flexible wheelet i ued a an object of conideration. Thi wheelet of a total weight 15 kg and a load of the ingle wheel equal to Q=4 kn i driven by the aynchronou motor by mean of the dic-clutch with torional tiffne and damping coefficient k 1=3 knm, c 1=1 N/m. The pur gear tage of the ratio i=1:6 reduce a rotational peed of the * wheelet into & ϕ = & ϕ χ. There i aumed that the minimum radiu of the wheelet axle and the half of length of the axle are repectively equal to.8 and.75 m. Thi axle i made of teel P35G. The torional tiffne of thi axi ha been determined equal to k =k 3= 6.9e7 Nm/rad. More parameter applied in thi invetigation are alo given in the Table. Table Simulation bae parameter χ c c3 I Iz Ig Ikl Ikr.16 5 N/m 5 N/m.1 kgm. kgm 43 kgm 78 kgm 78 kgm The imulation model decribed above can be ued to imulate everal different condition of operation, i.e. motor acceleration, deceleration, load change, fault condition, etc. However, due the limited ize, only elected reult are preented here. An amplitude of elf-excited vibration i an important evaluating indicator to meaure the vibration magnitude. Some drive ytem parameter influencing the amplitude of the elf-excited torional vibration are hown in Fig. 4 and 5. Figure 4a and 4c preent the reult of the elf-excited vibration amplitude and the pectrum of them at variou damping between the drivetrain ytem and the wheelet wheel. A hown in thi figure, the vibration amplitude decreae with an increae of damping level where the dominant frequency of the vibration i kept contant. An increae of the damping can retrain thi amplitude and horten the convergence time of the torional vibration, but it i not affected whether the elf-excited vibration occur or not. The ame effect can be oberved on time-hitorie of the current of the tator winding hown in Fig. 4b and 4 d. Figure 5a how a reult of the elf-excited vibration amplitude at different equivalent tiffne between the drivetrain and the wheelet wheel of wheelet.

6 19 Fig. 4. Self-excited torional vibration amplitude of the mechanical and electric parameter at variou torional damping of wheelet drivetrain. Time-hitory (a) and amplitude pectrum (c) of the difference between the angular diplacement of the left and right wheelet wheel. Time-hitory (b) and amplitude pectrum (d) of the electric current in the tator winding Fig. 5. Self-excited torional vibration amplitude of the mechanical and electric parameter at variou torional tiffne of wheelet drivetrain. Time-hitory (a) and amplitude pectrum (c) of the difference between the angular diplacement of the left and right wheelet wheel. Time-hitory (b) and amplitude pectrum (d) of the electric current in the tator winding

7 Vibration in Phyical Sytem Vol. 7 (16) A hown in thi figure (Fig 5.), the vibration amplitude decreae with an increae of k and k 3. It indicate that where increaing the torional tiffne of drivetrain ytem, it i influence on the tability of the vibration and it hifting dominant frequency of the vibration in the higher range of the pectrum (Fig. 5c). Conidering the electrical parameter value of motor obtained from the above invetigation it i worth highlighting that the elf-excitation torional vibration affected on the entire electromechamical drivtrain ytem and they have a ignificant influence on the amount of theoretically expected electric energy P el conumed by the driving motor. In the cae of the invetigated ytem, thi energy can be determined by the electromotive force induced in the aynchronou motor phae by voltage and current in the tator winding. Thi energy can be defined [7]. P el tk 1 = [ U t k d ( t) iα ( t) + U q ( t) iβ ( t)] dt, (7) where U d(t)= co( ) 3 U ω e t, U q(t)= 3 U in( ω ) e t, and i α (t), i β (t), denote the voltage and current in the tator circuit of the electric motor phae tranformed into the reference ytem of Park equation, t k i the total duration time of the each variant of an analyi and the remaining ymbol have been already defined in Eq. (5) and (6). Table 3 illutrate the amount of electric energy conumed by the drivetrain motor during the conidered tet cenario at variou of parameter the drive ytem dicued above. Table 3. Amount of electric energy conumed by the aynchronou motor during the aumed four cenario of the invetigation uing the aumed drivetrain model tiffne of drivtrain [Nm/rad] 1e7 e7 3.5e7 6.9 e7 tiffne-energy conumed [kw] 6,35 7,44 76,65 78,93 damping of drivtrain [Nm/rad] 5, 1, 15,, damping-energy conumed [kw] 78,93 88,37 88,59 93, From a comparion of the reult hown in Table 3 it follow that, when the torional tiffne increae, more electric energy have been conumed. Thi fact can be ubtantiated by change amplitude of the time-hitorie of the difference between the angular diplacement of the left and right wheelet wheel characteritic of preented in Fig Final remark and preview In thi paper, an electromechanical model of the railway vehicle drive ytem ha been performed. Thi model ha been ued to invetigate elf-excited torional vibration occurring in thi ytem. In the invetigation their influence of the torional vibration on the electric parameter of the drive motor are alo conidered. From obtained reult it follow that a reduction of the elf-excited vibration amplitude by mean of increaing the damping and tiffne between the driving motor and the wheelet and torion tiffne of wheelet occur. The reult obtained uing numerical imulation indicated that the elf-

8 194 excited torional vibration in the conidered drive ytem are trongly dependent on the characteritic of the adheion coefficient in wheel-rail contact zone. A circuit model of the electric motor in the conidered drive ytem enable u to obtain value of electrical parameter characterizing the driving motor. The information concerning a frequency variation of the current in the driving motor tator can be ued for monitoring and identification of elf-excited vibration in the wheelet drivetrain ytem. The further work will be denoted to an aumption of the vehicle model with the drivetrain ytem and it will be carried out experimentally verification on real railway vehicle. Reference 1. R. Bogacz, T. Szolc, H. Irretier, An application of torional wave analyi to turbogenerator rotor haft repone, J.Vibr. Acou. -Tran. of the Ame, Vol (199) O. Ahmedov, V. Zeman, M. Byrtu, Modelling of vibration and modal propertie of electric locomotive drive, Eng. Mech., Vol. 19: /3 (1) S. Noga, R. Bogacz, T. Markowki, Vibration analyi of a wheel compoed of a ring and a wheel-plate modelled a a three-parameter elatic foundation, J.Sound Vib., Vol. 333:4, (14) R. Bogacz, R. Konowrocki, On new effect of wheel-rail interaction, Arch. Appl. Mech, Vol.8 (1) V. Zeman, Z. Hlavac, Dynamic wheelet drive load of the railway vehicle caued by hortcircuit motor moment, App. & Comp. Mech., Vol.3, No. (9) B.S. Branilav, Simulation of torion moment at the wheel et of the railway vehicle with the traction electromotor for wavy direct current, Mech. Tran. Com., Iue 3 (8) J. Liu, H. Zhao, W. Zhai, Mechanim of elf-excited torional vibration of locomotive driving ytem, Front. Mech. Eng.China, Vol.5:4 (1,) A.K. Kumar, Method and ytem of limiting the application of and to a railroad rail, U.S. Patent 7,9,87B, Nov. 6, (7). 9. S. Friedrich, M. Traupe, Dynamic Torional Load on Wheelet. Recent Finding for the Aement, Fatigue Strength and Simulation, T.TVI 4, Milan (14). 1. T. Szolc, R. Konowrocki, M. Michajłow, A. Pręgowka, An invetigation of the dynamic electromechanical coupling effect in machine drive ytem driven by aynchronou motor, Mech. Syt. Signal Proce., Vol.49 (14) R. Konowrocki, T. Szolc, A. Pochanke, A. Pręgowka, An influence of the tepping motor control and friction model on precie poitioning of the complex mechanical ytem, Mech. Syt. Signal Proce., Vol.7-71 (16) K.L., Shi, T.F., Chan, Y.K., Wong, L.S Ho, Modelling and imulation of the threephae induction motor uing Simulink, Int. J. Elect. Eng. Edu., 36 (1999) L. Lipińki, Applying imulation model for operation analyi and election of controller etting in traction drive with induction motor, Electrical Machine - Tranaction Journal, No. 1/ 94 (1) 75-8 (in Polih).