Module 2: Dynamics of Electric and Hybrid vehicles Lecture 3: Motion and Dynamic Equations for Vehicles

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Violet Snow
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1 NPTE Electical Intoduction to Hybid and Electic Vehicles Module : Dynamics of Electic and Hybid vehicles ectue 3: Motion and Dynamic Equations fo Vehicles Intoduction The fundamentals of vehicle desin involve the basic pinciples of physics, specially the Newton's second law of motion. Accodin to Newton's second law the acceleation of an object is popotional to the net foce exeted on it. Hence, an object acceleates when the net foce actin on it is not zeo. In a vehicle seveal foces act on it and the net o esultant foce ovens the motion accodin to the Newton's second law. The populsion unit of the vehicle delives the foce necessay to move the vehicle fowad. This foce of the populsion unit helps the vehicle to ovecome the esistin foces due to avity, ai and tie esistance. The acceleation of the vehicle depends on: the powe deliveed by the populsion unit the oad conditions the aeoamics of the vehicle the composite mass of the vehicle In this lectue the mathematical famewok equied fo the analysis of vehicle mechanics based on Newton s second law of motion is pesented. The followin topics ae coveed in this lectue: Geneal desciption of vehicle movement Vehicle esistance Dynamic equation Tie Gound Adhesion and maximum tactive effot Geneal desciption of vehicle movement The vehicle motion can be completely detemined by analysin the foces actin on it in the diection of motion. The foces actin on a vehicle, movin up a ade, ae shown in Fiue 1. The tactive foce (F t ) in the contact aea between the ties of the diven wheels and the oad suface popels the vehicle fowad. The tactive foce (F t ) is poduced by the powe plant and tansfeed to the divin wheels via the tansmission and the final dive. When the vehicle moves, it encountes a esistive foce that ties to etad its motion. The esistive foces ae Joint initiative of IITs and IISc Funded by MHRD Pae 1 of 15

2 NPTE Electical Intoduction to Hybid and Electic Vehicles Rollin esistance Aeoamic da Uphill esistance V F w h w Msin(a) F a T f h T W f Mcos(a) a M b a W Fiue 1: Foces actin on a vehicle oin uphill [1] Usin the Newton's second law of motion, the vehicle acceleation can be expessed as (1) Rollin esistance dv dt whee V vehicle speed t t esis tan ce F M F esistac F total tactive effot [ Nm] F total esistance [ Nm] M total mass of the vehicle [ k] mass facto fo convetin the otational inetias of otatin components into tanslational mass The ollin esistance of ties on had sufaces is due to hysteesis in the tie mateial. In Fiue a tie at standstill is shown. On this tye a foce (P), is actin at its cente. The pessue in the contact aea between the tie and the ound is distibuted symmetically to the cente line and the esultin eaction foce (P z ) is alined alon P. Joint initiative of IITs and IISc Funded by MHRD Pae of 15

3 NPTE Electical Intoduction to Hybid and Electic Vehicles P z P z Fiue : Pessue distibution in contact aea [1] The defomation, z, vesus the load P, in the loadin and unloadin pocess is shown in Fiue 3. Fom this fiue it can be seen that, due to the hysteesis, the foce (P) fo the same defomation (z) of the tie mateial at loadin is eate than at duin unloadin. Hence, the hysteesis causes an asymmetic distibution of the ound eaction foces. Foce, P P 1 P Defomation, z Fiue 3: Foce actin on a tye vs. defomation in loadin and unloadin [1] The scenaio of a ollin tie is shown in Fiue 4. When the tie olls, the leadin half of the contact aea is loadin and the tailin half is unloadin. Thus, the pessue on the leadin half is eate than the pessue on the tailin half (Fiue 4a). This phenomenon esults in the ound eaction foce shiftin fowad. The shift in the ound eaction foce ceates a moment that opposes ollin of the wheels. On soft sufaces, the ollin esistance is mainly caused by defomation of the ound suface, (Fiue 4b). In this case the ound eaction foce almost completely shifts to the leadin half. Joint initiative of IITs and IISc Funded by MHRD Pae 3 of 15

4 NPTE Electical Intoduction to Hybid and Electic Vehicles P z a Fiue 4a: Foce actin on a tye vs. defomation in loadin and unloadin on a had suface [1] The moment poduced by fowad shift of the esultant ound eaction foce is called ollin esistance moment (Fiue 4a) and can expessed as T Pa Ma whee T ollin esistance [ Nm] P Nomal load actin on the cente of the ollin wheel [ N] M mass of the vehicle [ k] acceleation constant [ m / s ] a defomation of the tye [ m] P z () P P x z P z Fiue 4a: Foce actin on a tye vs. defomation in loadin and unloadin on a soft suface [1] To keeps the wheel ollin, a foce F, actin on the cente of the wheel is equied to balance this ollin esistant moment. This foce is expessed as Joint initiative of IITs and IISc Funded by MHRD Pae 4 of 15

5 NPTE Electical Intoduction to Hybid and Electic Vehicles F whee P Nomal load actin on the cente of the ollin wheel [ N] T Pa Pf T ollin esistance [ Nm] f amic adius of the tye [ m] ollin esistance coefficient (3) The ollin esistance moment can be equivalently eplaced by hoizontal foce actin on the wheel cente in the diection opposite to the movement of the wheel. This equivalent foce is called the ollin esistance and its manitude is iven by F whee P Nomal load actin on the cente of the ollin wheel [ N] f Pf ollin esistance coefficient (4) When a vehicle is movin up a adient, the nomal foce (P), in equation 4, is eplaced by the component that is pependicula to the oad suface. Hence, equation 4 is ewitten as F Pf cos( a) Mf cos( a) whee P Nomal load actin on the cente of the ollin wheel [ N] f ollin esistance coefficient a oad anle [ adians] (5) The ollin esistance coefficient, f, is a function of: tie mateial tie stuctue tie tempeatue tie inflation pessue tead eomety oad ouhness oad mateial pesence of absence of liquids on the oad Joint initiative of IITs and IISc Funded by MHRD Pae 5 of 15

6 NPTE Electical Intoduction to Hybid and Electic Vehicles The typical values of the ollin esistance coefficient (f ) ae iven in Table 1. Table 1: Refeence values fo the ollin esistance coefficient (f ) Conditions Rollin esistance coefficient (f ) Ca tie on smooth tamac 0.01 oad Ca tie on concete oad Ca tie on a olled avel 0.0 oad Ta macadam oad 0.05 Unpaved oad 0.05 Bad eath tacks 0.16 oose sand Tuck tie on concete o asphalt oad Wheel on ion ail The values iven in table 1 do not take into account the vaiation of f with speed. Based on expeimental esults, many empiical fomulas have been poposed fo calculatin the ollin esistance on a had suface. Fo example, the ollin esistance coefficient of a passene ca on a concete oad may be calculated as:.5 V f f0 fs 100 whee V vehicle speed [ km / h] (6) In vehicle pefomance calculation, it is sufficient to conside the ollin esistance coefficient as a linea function of speed. Fo most common ane of inflation pessue, the followin equation can be used fo a passene ca on a concete oad V f whee V vehicle speed [ km / h] Joint initiative of IITs and IISc Funded by MHRD Pae 6 of 15

7 NPTE Electical Intoduction to Hybid and Electic Vehicles (7) The equation 7 can pedict the values of f with acceptable accuacy fo speed up to 18km/h. Aeoamic da A vehicle tavelin at a paticula speed in ai encountes a foce esistin its motion. This foce is known as aeoamic da. The main causes of aeoamic da ae: shape da skin effect The shape da is due to the shape of the vehicle. The fowad motion of the vehicle pushes the ai in font of it. Howeve, the ai cannot instantaneously move out of the way and its pessue is thus inceased. This esults in hih ai pessue in the font of the vehicle. The ai behind the vehicle cannot instantaneously fill the space left by the fowad motion of the vehicle. This ceates a zone of low ai pessue. Hence, the motion of the vehicle ceates two zones of pessue. The hih pessue zone in the font of the vehicle opposes its movement by pushin. On the othe hand, the low pessue zone developed at the ea of the vehicle opposes its motion by pullin it backwads. The ai close to the skin of the vehicle moves almost at the speed of the vehicle while the ai away fom the vehicle emains still. Between these two layes (the ai laye movin at the vehicle speed and the static laye) the molecules move at a wide ane of speeds. The diffeence in speed between two ai molecules poduces fiction. This fiction esults in the second component of aeoamic da and it is known as skin effect. The aeoamic da is expessed as 1 Fw Af CDV whee A f 3 density of ai [ k / m ] vehicle fontal aea [ m ] V vehicle speed [ m / s] C D (8) da coefficient Joint initiative of IITs and IISc Funded by MHRD Pae 7 of 15

8 NPTE Electical Intoduction to Hybid and Electic Vehicles The aeoamic da coefficients and the fontal aea fo diffeent vehicle types ae iven in Table. Gadin esistance Table : Refeence values fo da coefficient (C D) and the fontal aea (A f in m ) fo some vehicle types Vehicle C D A f Motocycle ide with Open convetible Coach Tuck taile without imousine Tuck with taile Aticulated vehicle When a vehicle oes up o down a slope, its weiht poduces a component of foce that is always diected downwads, Fiue 5. This foce component opposes the fowad motion, i.e. the ade climbin. When the vehicle oes down the ade, this foce component aids the vehicle motion. The adin esistance can be expessed as F M sin( a) whee M mass of vehicle [ k] acceleation constant [ m / s ] a oad anle [ adians] (9) In ode to simplify the calculation, the oad anle a, is usually eplaced by the ade value, when the oad anle is small. The ade value is defined as (Fiue 5) H i tan( a) sin( a) Joint initiative of IITs and IISc Funded by MHRD Pae 8 of 15

9 NPTE Electical Intoduction to Hybid and Electic Vehicles (10) In some liteatue, the tie ollin esistance and the adin esistance taken toethe and is called oad esistance. The oad esistance is expessed as F F F M f cos( a) sin( a) d f whee M mass of vehicle [ k] f (11) acceleation constant[ m / s ] ollin esistance coefficient Msin(a) h H Mcos(a) a M a Fiue 5: Vehicle oin up a ade [1] Acceleation esistance In addition to the divin esistance occuin in steady state motion, inetial foces also occu duin acceleation and bakin. The total mass of the vehicle and the inetial mass of those otatin pats of the dive acceleated o baked ae the factos influencin the esistance to acceleation: Fa M whee ot M mass of vehicle [ k] ot intetia of otational components [ ] V speed of the vehicle [ km / h] J dv dt J k m amic adius of the tye [ m] Joint initiative of IITs and IISc Funded by MHRD Pae 9 of 15

10 NPTE Electical Intoduction to Hybid and Electic Vehicles (1) The otational component is a function of the ea atio. The moment of inetia of the otatin dive elements of enine, clutch, eabox, dive shaft, etc., includin all the oad wheels ae educed to the divin axle. The acceleation esistance can be expessed as dv Fa M dt whee otational inetia constant M mass of the vehicle [ k] V speed of the vehicle [ m / s] (13) Total divin esistance The taction foce (F t ) equied at the dive wheels is made up of the divin esistance foces and is defined as F F F F F esis tan ce w a (14) Substitutin the values of all the foces in equation 14, ives 1 Fesis tan ce Mf cos( a) Af CDV M sin( a) M dt (15) The equation 15 may be used to calculate the powe equied (P eq ): P F V eq esis tan ce (16) Dynamic equation In the lonitudinal diection, the majo extenal foces actin on a two axle vehicle (Fiue 1) include: the ollin esistance of the font and ea ties (F f and F ), which ae epesented by ollin esistance moment, T f and T the aeoamic da (F w ) ade climbin esistance (F ) acceleation esistance (F a ) dv Joint initiative of IITs and IISc Funded by MHRD 15 Pae 10 of

11 NPTE Electical Intoduction to Hybid and Electic Vehicles The amic equation of vehicle motion alon the lonitudinal diection is iven by dv M F F F F F F F dt tf t f w a (17) The fist tem on the iht side is the total tactive effot and the second tem is the total tactive esistance. To detemine the maximum tactive effot, that the tie ound contact can suppot, the nomal loads on the font and ea axles have to be detemined. By summin the moments of all the foces about point R (cente of the tie-ound aea), the nomal load on the font axle W f can be detemined as W f dv Mb cos( a) Tf T Fwh w Mh sin( a) Mh dt (18) Similaly, the nomal load actin on the ea axle can be expessed as dv Ma cos( a) Tf T Fwh w Mh sin( a) Mh dt W (19) In case of passene cas, the heiht of the cente of application of aeoamic esistance (h w ) is assumed to be nea the heiht of cente of avity of the vehicle (h ). The equation18 and 19 can be simplified as h b dv Wf M cos( a) Fw F Mf cos( a) M h dt (0) and h a dv W M cos( a) Fw F Mf cos( a) M h dt (1) Usin equation 5, 17, 0 and 1 can be ewitten as h a W M cos( a) Ft F 1 h Joint initiative of IITs and IISc Funded by MHRD 15 Pae 11 of

12 NPTE Electical Intoduction to Hybid and Electic Vehicles () h a W M cos( a) Ft F 1 h (3) The fist tem on the iht hand side of equation and equation 3 is the static load on the font and the ea axles when the vehicle is at est on level ound. The second tem is the amic component of the nomal load. The maximum tactive effot (F tmax ) that the tie-ound contact can suppot is descibed by the poduct of the nomal load and the coefficient of oad adhesion (m). In Table 3, the values of coefficient of adhesion ae iven fo diffeent speeds of the vehicle and diffeent oad conditions. Fo the font wheel dive vehicle, F tmax is iven by h b Ft max W f M cos( a) Ft max F 1 h (4) F t max M cos( a) b f h / 1 h / (5) Fo the ea wheel dive vehicle, F tmax is iven by h b Ft max W M cos( a) Ft max F 1 h (6) F t max M cos( a) a f h / 1 h / Joint initiative of IITs and IISc Funded by MHRD 15 Pae 1 of

13 NPTE Electical Intoduction to Hybid and Electic Vehicles (7) Table 3: Coefficient of oad adhesion Road speed Coefficient of oad Coefficient of oad [km/h] adhesion fo dy adhesion fo wet oads oads Adhesion, Dynamic wheel adius and slip When the tactive effot of a vehicle exceeds the maximum tactive effot limit imposed by the adhesive capability between the tye and ound, the diven wheels will spin on the ound. The adhesive capability between the tye and the ound is the main limitation of the vehicle pefomance especially when the vehicle is diven on wet, icy, snow coveed o soft soil oads. The maximum tactive effot on the diven wheels, tansfeed fom the powe plant thouh the tansmission should not exceed the maximum values iven by equation 5 and equation 7. Othewise, the diven wheels will spin on the ound, leadin to vehicle instability. The slip between the tyes and the suface can be descibed as: dive slip S whee T R anula speed of the tye [ ad / s] R R V Joint initiative of IITs and IISc Funded by MHRD 15 Pae 13 of

14 NPTE Electical Intoduction to Hybid and Electic Vehicles (8) The amic wheel adius ( ) is calculated fom the distance tavelled pe evolution of the wheel, ollin without slip. The amic wheel adius is calculated fom a distance tavelled at 60km/h. The inceasin tye slip at hihe speeds ouhly offsets the incease in. The values of fo diffeent tye sizes ae iven in table 4. Rollin Cicumfeence Tye Size [m] Passene cas Table 4: Dynamic wheel adius of common tye sizes R Rollin Cicumfeence [m] Tye Size [m] R [m] Passene cas 05/ R R / R R /60 R 155 R /70 R iht commecial vehicles 155/70 R R /70 R R /70 R R /75 R 175 R /75 R 185 R /70 R Tucks and buses 185/65 R R /80 R 185/60 R /80 R 195/60 R /75 R 195/70 R Joint initiative of IITs and IISc Funded by MHRD 15 Pae 14 of

15 NPTE Electical Intoduction to Hybid and Electic Vehicles 185/65 R /65 R /70 R /70 R Refeences: [1] M. Ehsani, Moden Electic, Hybid Electic and Fuel Cell Vehicles: Fundamentals, Theoy and Desin, CRC Pess, 005 Suested Readin: [1] I. Husain, Electic and Hybid Electic Vehicles, CRC Pess, 003 [] C. C. Chan and K. T. Chau, Moden Electic Vehicle Technoloy, Oxfod Science Publication, 001 [3] G. echne and H. Naunheime, Automotive Tansmissions: Fundamentals, Selection, Desin and Application, Spine, 1999 Joint initiative of IITs and IISc Funded by MHRD 15 Pae 15 of

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