On Pressure Distributions of Drum Brakes

Transcription

1 Yuan Mao Huang Professor e-mal: J S Shyr Research Assstant Department of Mechancal Engneerng, Natonal Tawan Unversty, Tape, Tawan, Republc of hna On Pressure Dstrbutons of Drum Brakes Based on the assumptons that there s perfect contact at the nterface between the brake drum and the lnng plates, the frcton coeffcent s constant, the thermal effect s neglected, and the brake drum s a rgd body, the pressure dstrbutons of drum brakes were studed by usng the boundary element method The constant element s used n the two-dmensonal model of the drum brake for smplcty and economy The frcton force versus the effectve lft at the actuaton edge and the locaton of the maxmum pressure are compared and ndcate a good correlaton wth exstng data The effects of the Young s modulus of elastcty of the metal shoe, the arc lengths of the metal shoe and lnng plate, the locaton, the thckness, the frcton coeffcent, the Young s modulus of elastcty of the lnng plate, and the angle of actuaton force on the pressure dstrbutons were then studed By selectng proper values of these parameters, a drum brake can be desgned to have a more unform pressure dstrbuton and a longer lfe DOI: / Introducton Drum brakes, whch are an mportant devce and are nstalled n vehcles and motor cycles, are used to reduce the speed and to assure the safety of the drvers and passengers on the road The schematc drawng of a drum brake s shown n Fg 1 It prmarly conssts of a brake drum and brake shoes wth lnng plates Qualtes t should have nclude: good brakng performance; unform brakng on all wheels; good endurance wthout falure; and easy operaton, nspecton, adjustment and mantenance Advantages offered by the drum brake are ts capablty from the selfenergzng effect to reduce the requred actuatng force and ts relatvely lesser cost Day et al 1 used the fnte element method to analyze a twodmensonal model of the drum brake Thereafter, the thermal expanson, mperfect contact, the decay of brakng capablty and all knds of transent phenomena were analyzed by usng the fnte element method 2 6 Watson and Newcomb 7 argued that t s more expensve to use the fnte element program n the man frame and that the pressure dstrbuton on the lnng plate does not vary sgnfcantly along the axs of the brake The measurement of the torque or the wear can be used to calculate the pressure dstrbuton 8 Nevertheless, these methods are ndrect methods It s dffcult to obtan drectly the pressure dstrbuton from an experment 2,5 Recently, the applcatons of the boundary element method BEM have been ncreased and showed that the BEM can reduce the tme requred to prepare numercal data and can provde good results n the lnear problems 9 14 The purpose of ths study s to analyze the effect of the effectve lft at the actuaton end on the frcton force, effects of the Young s modulus of elastcty for metal shoes, the angles of the arc lengths of metal shoes and lnng plates, respectvely, the leadng locaton of the lnng plates, the thckness of the lnng plate, the frcton coeffcent, the Young s modulus of elastcty for lnng plates, and the locaton of the actuatng force of the drum brake on the pressure dstrbutons A two-dmensonal model of the drum brake s developed by usng the BEM, and ndependent and less expensve software s generated for the personal computer to reduce the cost for desgnng drum brakes By selectng proper values of these parameters, a drum brake can be desgned to have a more unform pressure dstrbuton and a longer lfe ontrbuted by the Desgn Automaton ommttee for publcaton n the JOUR- NAL OF MEHANIAL DESIGN Manuscrpt receved Nov 1999 Assocate Edtor: H Lankaran Method of Approach Durng the dervaton of the boundary ntegraton equatons, the soluton of an nfnte doman subjected to a concentrated load at any nternal locaton s requred If a unt force acts on an nternal pont M, determnaton of the effect of ths force on any pont Q must satsfy two requrements The frst one s that all of the stresses must be zero when the dstance between the ponts M and Q becomes nfnte The second one s that the stress becomes nfnte, or the pont M s a sngular pont when the dstance between the ponts M and Q becomes zero ombnng the dfferental equaton of the stress, the relatonshp of the stress and the stran and the relatonshp of the stran and the dsplacement yelds 2 u x x 2 u x 2 y 2 2 u y x 2 2 u y y u x x 2 u x u y x 2 u y 2 xy f x G s xy f y G s (1) whch s the Naver two-dmensonal dfferental equatons and can be solved by usng the complementary functon and the specal ntegraton If the dsplacements are expressed n terms of the Galerkn vector, u x 2 Ḡ x x 2 Ḡ x 2 y Ḡ x x 2 Ḡ y 2 u y 2 Ḡ y x 2 Ḡ y 2 y the Galerkn vector can be determned as Ḡ y y 2 xy xy (2) 2 Ḡ x Ḡ x Ḡ y 1 r 2 8G s ln 1 (3) r and the dsplacement s u U j e j (4) Substtutng Eq 4 nto the relatonshp of the stress and the dsplacement yelds the tracton force t T j e j (5) Journal of Mechancal Desgn opyrght 2002 by ASME MARH 2002, Vol 124 Õ 115

2 Fg 1 Schematc drawng of drum brake s a rgd body The two-dmensonal model of the brake shoe that s used n ths analyss can be dvded nto two parts based on materals, the metal shoe and the lnng plate They are dvded nto 28 elements, respectvely, due to the lmted sze of the computer memory The dsplacement and the tracton force at the center pont n the constant element are used for the element to smplfy the ntegraton Applyng Eq 10 to the metal shoe and the lnng plate, respectvely, combnng the ponts on the contact surface of the lnng plate and the metal shoe and the ndependent porton of the lnng surface, and combnng the ponts on the contact surface of the metal shoe and the lnng plate and the ndependent porton of the metal shoe wth the tracton equlbrum condton To mnmze the error of numercal approxmaton, the basc soluton of the dsplacement as a weghtng factor s multpled to every term and ntegrated to yeld the Somglana equaton u l T lk u k U lk t k du lk f k d (6) d The dsplacement of any nternal pont can be determned f the dsplacements of the boundary ponts and the tracton forces are determned The Somglana equaton can also be appled to the boundary ponts and yelds P L B P S and the dsplacement compatblty condton yelds H A B L H L 0 H S U L B U S G L B G S D A U S B 0 U L DUL H S P S (11) (12) c lk u k T lk u k U lk t k du lk f k d (7) d Four varables u 1, u 2, t 1 and t 2 exst for each nodal pont n a two-dmensonal problem; therefore, there wll be 4N varables for N nodal ponts If there s only one soluton for an elastc problem, only two values of these four varables are known for each nodal pont and 2N equatons are requred to solve the unknown varables The geometrc confguraton, the dsplacements and the tracton forces must be descrbed If there s a unt force actng on the pont and the effect of body force can be neglected, the two ntegrated terms related to the nodal ponts and j form the nfluence coeffcents, G j and Ĥ j If Eq 7 becomes H j Ĥ j j (8) H j Ĥ j c j N jl N H j u j jl G j t j (9) Applyng to all boundary ponts and usng local coordnates as the bass of the boundary value yelds HTUGTP (10) where the matrces H and G are 2N2N matrces The values of the dsplacement matrx U and the tracton force matrx P are all relatve to global coordnates The boundary condton should be gven after calculaton of all coeffcents for the matrces H and G n order to obtan the unque soluton Snce t s not a symmetrc matrx, a drect method such as the Gaussan elmnaton method can be used Model and Numercal Analyss It s assumed that the varatons of the pressure and the deformaton n the axal drecton are neglgble for smplcty, that no deformaton occurs durng brakng, that there s the perfect ntal contact between the contact surfaces of the lnng plate and the crcular profle of brake drum, that the frcton coeffcent of the lnng plate and the brake drum s constant, that the wear s neglgble, that the thermal effect s neglected and that the brake drum G A L D A DPL (13) G S 0 P S where the matrces H and G have been multpled by the transfer matrx, the superscrpt B stands for the common porton of the lnng plate and the metal shoe and the superscrpt A stands for the ndependent porton of the lnng plate, the superscrpt stands for the common part of the metal shoe and the lnng plate and the superscrpt D stands for the ndependent porton of the metal shoe In the local coordnates, four varables for each nodal pont are the dsplacements U n and U t n the normal drecton and the tangental drecton, respectvely, and the tracton forces P n and P t n the normal drecton and the tangental drecton, respectvely The boundary condtons along the frcton surface are U n 0 P t P n (14) Two varables U t and P n are unknown Equaton 13 can be wrtten G n G t H t P n U t U n Hn 0 0 The boundary condtons for the fxed anchor are (15) U n 0 U t (16) 0 The supportng porton for the sldng abutment can be rewrtten G n H t G 1 n H 1 t P n U t P 1 n U 1 t T H n G t H 1 n G 1 t U n P t U 1 n P 1 n T (17) 116 Õ Vol 124, MARH 2002 Transactons of the ASME

3 where the superscrpt stands for the number of the nodal pont and the subscrpt stands for the drecton Smlarly, the supportng porton for the fxed anchor can be rewrtten G n G t G 1 n G 1 t P n P t P 1 n P 1 t T H n H t H 1 n H 1 t U n U t U 1 n U 1 t T (18) After consderng the boundary condton, the matrx can be rewrtten HUGP (19) where the matrces H and G are the matrces after arrangement, and the matrx U s obtaned from the unknown dsplacement and the tracton force The matrx P s obtaned from all the boundary condtons After obtanng the tracton forces and the dsplacements of nodes, the torque can be obtaned from where A s the area of the element m Torque n1 PA (20) Table 2 Data of second drum brake Result The data of the frst drum brake wth the supportng end fxed by a pn as shown n Fg 1 are lsted n Table 1 The angle s measured from the lne of the pn center and the brake drum center to the lne of the brake drum center to near edge of the lnng plate The angle s measured from the lne of the actuaton force to the normal vector of the surface at the load pont The frcton force versus the effectve lft at the actuaton edge that s normal to the actng surface s shown n Fg 2 The data obtaned from Day and Hardng 15 by usng the fnte element method are also shown for comparson The data of the second drum brake are shown n Table 2 If the Fg 3 Pressure dstrbuton wth hgh Young s modulus of elastcty for rgd metal shoe Table 1 Data of frst drum brake Fg 4 Pressure dstrbuton versus Young s modulus of elastcty of metal shoe Fg 2 Frcton force versus effectve lft at actuaton end Fg 5 Pressure dstrbuton versus lnng arc for varous arc lengths of metal shoe Journal of Mechancal Desgn MARH 2002, Vol 124 Õ 117

4 Fg 6 Pressure dstrbuton versus lnng arc for varous arc lengths of lnng plate Fg 9 Pressure dstrbuton versus lnng arc for varous frcton coeffcents of lnng plate Young s modulus of elastcty ncreases, the materal becomes stffer At frst, the Young s modulus of elastcty of the brake shoe n Table 2 s replaced by a value four tmes larger, N/mm 2, and the result of the pressure dstrbuton s compared wth the snusodal curve of the rgd brake obtaned from Offner 16 as shown n Fg 3 Based on the data shown n Table 1 and the dsplacement of the surface along the normal drecton of the surface to be 01 mm, the pressure dstrbutons versus the lnng arc for varous Young s modul of elastcty of metal shoes are shown n Fg 4 The effects of the arc lengths of the metal shoe and the lnng plate wth the same center locaton on the pressure dstrbuton are shown n Fgs 5 and 6, respectvely Shftng the lnng plate, the pressure dstrbuton s shown n Fg 7 The pressure dstrbutons versus the lnng arc for the thckness of the lnng plate to be 3, 4 and 5 mm are shown n Fg 8 It s well known that the frcton coeffcent of materal can affect the brakng performance The effect of the frcton coeffcent of the lnng plate on the pressure dstrbuton s shown n Fg 9 hangng the Young s modulus of elastcty of the lnng plate, the pressure dstrbuton s shown n Fg 10 If the rato of the Young s modulus of elastcty of the lnng plate to the Young s modulus of elastcty of the metal shoe remans constant and both Young s modul of elastcty of the lnng plate and the metal shoe ncrease Fg 7 Pressure dstrbuton versus lnng arc for varous leadng locatons of lnng plate Fg 10 Pressure dstrbuton versus lnng arc for varous Young s modul of elastcty of lnng plate Fg 8 Pressure dstrbuton versus lnng arc for varous lnng plate thcknesses Fg 11 Pressure dstrbuton versus lnng arc for varous Young s modul of elastcty of metal shoe and lnng plate 118 Õ Vol 124, MARH 2002 Transactons of the ASME

5 Fg 12 Pressure dstrbuton versus lnng arc for varous angles of actuaton force 20% and decrease 20%, respectvely, the pressure dstrbutons are shown n Fg 11 The effect of the angle measured from the lne of the actuatng force to the normal vector of the surface at the load pont wth the tracton pressure 50 N/mm 2 on the pressure dstrbuton s shown n Fg 12 Dscusson Watson and Newcomb 7 argued that t s more expensve to use the fnte element program n the man frame Ths study uses the boundary element method n the personal computer Nevertheless, the computer runnng tme s not avalable for comparson Usng the lnear elements or the second order elements creates dscontnuous phenomena along the rregular boundary The common nodal pont has the dfferent normal vector and boundary condtons It s necessary to have an extra equaton to provde a unque soluton for the fnal lnear equaton 12 Therefore, these elements are napproprate to treat the corner problem Usng the constant element can get rd of ths problem In addton, snce there s no shape functon and no need of the Jacobn coordnate transform, ntegratng the equaton s easer and wrtng a computer code s smpler for the constant element than the second order element Watson and Newcomb 7 also argued that the pressure dstrbuton on the lnng plate does not vary sgnfcantly along the axs of the brake Therefore, two-dmensonal model s used n ths analyss to reduce the computer tme It s dffcult to obtan drectly the pressure dstrbuton from an experment 2,5 Day17 studed two dstnct brake operatng condtons expermentally to nvestgate the effect of vehcle speed on the pressure dstrbuton wth tme However, the dmensons of the brake are not specfed completely, and the results are not approprate for comparson wth ths study The frcton force s proportonal to the effectve lft at the actuatng edge as shown n Fg 2 The devaton of the calculated frcton force from that obtaned by Day and Hardng 15 ncreases when the effectve lft ncreases It may come from the nsuffcent number of the elements and the accumulatve error from the numercal calculaton, and t can be reduced f the number of the elements s ncreased The maxmum pressure on the lnng plate that occurs at 60 n Fg 3 s located at 85 from the center of the pn The conventonal angular locaton of the maxmum pressure from the supportng pont for the rgd drum brake s at Fgure 4 shows that the angular locaton of the maxmum pressure shfted further from the supportng pont when the Young s modulus of elastcty of metal shoe ncreases The locaton of the maxmum pressure should be close to 90 from the supportng pont f the Young s modulus of elastcty ncreases to smulate the rgd brake Therefore, the mathematcal model s feasble for the study of the pressure dstrbuton of lnng plates When the Young s modulus of elastcty of the metal shoe decreases, the pressure at the center of the metal shoe decreases, but the pressure ncreases toward the supportng pont Ths result may come from the bucklng effect that the pressure at the center of the brake shoe s lowered The pressure close to the supportng pont decreases and the pressure dstrbuton s more unform f the arc length of the metal shoe ncreases as shown n Fg 5 The smlar result of a more unform pressure can be obtaned f the arc length of the lnng plate ncreases as shown n Fg 6 If the arc length of lnng plate decreases, a local hgh pressure may occur Fgure 7 shows that the maxmum pressure decreases f the lnng plate s shfted toward the supportng pont and the pressure dstrbuton becomes more unform If the thckness of the lnng plate ncreases, the maxmum pressure decreases and a more unform pressure dstrbuton can be obtaned as shown n Fg 8 Fgure 9 shows that the maxmum pressure decreases and the pressure dstrbuton becomes more unform f the frcton coeffcent decreases If the Young s modulus of elastcty of the lnng plate decreases, the maxmum pressure decreases and a more unform pressure dstrbuton can be acheved as shown n Fg 10 Fgure 11 shows that the maxmum pressure and the pressure dstrbuton decrease f both Young s modul of elastcty of the metal shoe and the lnng plate decrease whle ther rato remans the same The shapes of the pressure dstrbuton reman the same Ths result dentfes that obtaned by Wntle 4 The maxmum pressure decreases and the pressure dstrbuton becomes more unform f the angle of the actuaton force ncreases as shown n Fg 12 Reducng the angle of the actuaton force decreases the pressure at the actuaton edge and may result n the separaton of the lnng plate at the actuaton end from the brake drum Frctonal work n a brake s prmarly converted nto heat that ncreases the temperature of shoes and lnng plates of the brake and changes the propertes, partcularly n the thermal expanson The crcumferental dstrbuton of pressure along the brake lnng arc, that plays an mportant role n determnng the torque generated, depends upon the thermal expanson, flexural deflecton and contact at the nterface Heat generated at the sldng nterface between the frcton materal and the matng surface of the brake s not unformly dstrbuted and leads to pronounced varaton n hgh surface temperature and the hgh nterface pressure 2 Frcton nterface temperature n the brake s affected by the nterface pressure dstrbuton, so that regons of local hgh nterface pressure wll also be regons of hgh surface temperature 6 The effects n the areas of frctonal heat generaton and dsspaton wth thermal consderaton on the dstrbuton of pressure, that s very mportant n mnmzng brake thermal problems, s requred n order to ncorporate the non-lnear characterstcs of frcton materals to obtan the unform frcton nterface pressure The dstrbuton of pressure s sgnfcantly affected by drum flexblty, whch relates to the Young s modulus of elastcty The pressure generated for the flexble drum s reduced because a large actuaton dsplacement s absorbed by drum deflecton thereby reducng both effectve cam lft and actuaton force oncluson Based on the smlarty between the calculated results and the avalable data, the results suggest that the software developed can be used to predct the pressure dstrbuton of the lnng plate on the drum brake If the deflectons of the metal shoe and the lnng plate and the thermal effect are neglected, the pressure dstrbuton s a snusodal curve If these deflectons are consdered n the analyss, the pressure dstrbuton wll be changed In order to desgn a drum brake wth a more unform pressure actng on the lnng plate and to prolong the lfe of the frcton materal the arc lengths of the brake shoe and the lnng plate should be longer; the Journal of Mechancal Desgn MARH 2002, Vol 124 Õ 119

6 locaton of the lnng plate should be close to the supportng pont; the lnng plate should be thcker; the frcton coeffcent should be smaller; the Young s modulus of elastcty for the materal used n the brake shoe and the lnng plate should be smaller; and the angle of the actuaton force should be larger However, t should be noted that the materal wth the small Young s modulus of elastcty mght wear easly and faster Although the materal wth a small frcton coeffcent can provde more unform pressure actng on the lnng plate, t should not be too small Otherwse, t wll lower the brakng effect The frcton coeffcent of materal about 02 s a good selecton for applcaton In addton, a larger actuaton force should be appled wth the actuaton angle of 35 to compensate and ncrease the brakng effort Acknowledgment The authors would lke to express ther sncere apprecaton for the grant no NS E from the Natonal Scence ouncl of the Republc of hna for ths study and to Grant D Huang for comments and revsons made on ths manuscrpt Nomenclature A area e unt vector n j drecton F actuaton force f body force n the drecton Ḡ Galerkn vector G j,h j nfluence coeffcent matrx regardng ponts and j G S shear modulus P tracton force matrx r dstance between ponts M and Q T tracton bass soluton matrx or transverse matrx T coordnate transform matrx t tracton force basc soluton U dsplacement matrx u dsplacement basc soluton Posson s rato x,y coordnate ntegraton doman frcton coeffcent angle measured from lne of centers of drum brake and pn to lne of center of drum brake and near edge of lnng plate angle measured from the lne of actuaton force to the normal vector of the surface at the load pont angle of arc length of lnng plate boundary Subscrpts j th row and jth column n matrx n normal component l n l drecton t tangental component L lnng plate S metal shoe x,y drecton Superscrpts pont j dsplacement vector of Q n j drecton wth a unt force acted on pont M n drecton References 1 Day, A J, Hardng, P R J, and Newcomb, T P, 1979, A Fnte Element Approach to Drum Brake Analyss, Proc ImechE, 193, pp Day, A J, Hardng, P R J, and Newcomb, T P, 1984, ombned Thermal and Mechancal Analyss of Drum Brakes, Proc ImechE, No 15, 198D, pp Mashnostroenya, V, 1986, alculaton of the Shoe of Drum Brakes, Sovet Engneerng Research, 6, No 7, pp Wntle, J B, 1978, Torque Varatons of Drum Brakes, MSc Thess, Loughborough Unversty of Technology 5 Day, A J, 1991, Drum Brake Interface Pressure Dstrbuton, Proc ImechE, 205, pp Day, A J, Trovc, M, and Newcomb, T P, 1991, Thermal Effects and Pressure Dstrbuton n Brakes, Proc ImechE, 205D, pp Watson,, and Newcomb, T P, 1990, A Three-dmensonal Fnte Approach to Drum Brake Analyss, Proc ImechE, 204, pp Sceszka, S F, and Bareck, Z, 1984, Geometry of ontact Between Brake Shoes and Drums, The South Afrca Mech, Engr, pp Becker, A A, 1992, The Boundary Element Method n Engneerng, McGraw- Hll, Inc, pp 62-90, Rzzo, F J, 1967, An Integral Equaton Approach to Boundary Value Problems of lasscal Elastostatcs, Q Appl Math, 25, pp ruse, T A, 1968, A Drect Formulaton and Numercal Soluton of the General Transent Elastodynamc Problems-II, J Math Anal Appl, 22, pp Brebba, A, 1980, Boundary Element Technques n Engneerng, Butterworth & o Ltd, pp Rzzo, F J, and Shppy, D J, 1968, A Formulaton and Soluton Procedure for the General Non-homogeneous Elastc Incluson Problem, Int J Solds Struct, 4, pp Swedlow, J L, and ruse, T A, 1971, Formulaton of Boundary Integral Equatons for 3-D Elastoplastc Flow, Int J Solds Struct, 7, pp Day, A J, and Hardng, P R J, 1983, Performance Varaton of am Operated Drum Brake, Proc R Soc London, Ser A, pp Offner, D, 1969, Generalzng the Analyss of Shoe-type Brake-lutch Systems, ASME J Eng Ind, pp Day, A J, 1988, An Analyss of Speed, Temperature, and Performance haracterstcs of Automotve Drum Brakes, ASME J Trbol, 110, p Õ Vol 124, MARH 2002 Transactons of the ASME