Energy Consideration

Transcription

1 Energy Considertion It hs been noted tht the most common brkes employ friction to trnsform the brked system's mechnicl energy, irreversibly into het which is then trnsferred to the surrounding environment - Kinetic energy is bsorbed during slippge of either clutch or brke, nd this energy ppers s het. If the het generted is fster thn it is dissipted, then the temperture rises. Thorough design of brke therefore requires detiled trnsient therml nlysis of the interply between het generted by friction, het trnsferred through the lining nd the surrounding metlwork to the environment, nd the instntneous temperture of the surfce of the drum s well s the lining. For given size of brke there is limit to the mechnicl power tht cn be trnsformed into het nd dissipted without lthe tempertures reching dmging levels. Temperture of the lining is more criticl nd the brke size is chrcterized by lining contct re, A. The cpcity of clutch or brke is therefore limited by two fctors:. The chrcteristics of the mteril nd,. The bility of the brke to dissipte het. Het Generted In Brking During decelertion, the system is subjected to n essentilly constnt torque T exerted by the brke, nd in the usul sitution this constncy implies constnt decelertion too. Appliction of the work or energy principle to the system enbles the torque exerted by the brke nd the work done by the brke, U, to be clculted from:-

2 U = E = T θ () Where E is the loss of system totl energy which is bsorbed by the brke during decelertion, trnsformed into het, nd eventully dissipted. The elementry equtions of constnt rottionl decelertion pply, thus when the brke drum is brought to rest from n initil speed ω o :- Decelertion = ω o / () θ = ω m. t ; ω m = ω o / = θ/ t where ω m is the men drum speed over the decelertion period. The men rte of power trnsformtion by the brke over the brking period is :- P m = U / t = T ω m ( 3 ) which forms bsis for the selection or the design of the necessry brke dimensions. The rise in temperture in the lining mteril is lso importnt s rte of wer is lso function of the temperture. Further for ny lining mteril, the mximum llowble temperture is lso nother performnce criteri. Temperture Rise The temperture rise of the brke ssembly cn be pproximted by the clssic expression,, E T = C.m Where is temperture T is rise in temperture in o C, C is the specific het of the brke drum mteril (5J/Kg for steel or Cst Iron) nd m is the mss (kg) of the brke prts dissipting the het into the surroundings. Though the eqution ppers to be simple, there re so mny vribles involved tht it would be most unlikely tht such n nlysis would even pproximte experimentl results.

3 On the other hnd the temperture-rise equtions cn be used to explin wht hppens when clutch or brke is operted frequently. For this reson such nlysis re most useful, for repetitive cycling, in pin pointing those design prmeters tht hve the gretest effect on performnce. An object heted to temperture T cools to n mbient temperture T ccording to the exponentil reltion Time-temperture reltion T T (T T )e (AU / WC)t i = Where T = instntneous temperture t time t, C A= het trnsfer re, m U= Het Trnsfer coefficient, W/(m.s. C) T = Initil temperture, C T = Ambient temperture, C C - Specific het t - time of opertion, s

4 T T A T T B T t t B t A Time t C C Figure 3..6 Figure shows n ppliction of Eq. (). At time t A clutching or brking opertion cuses the temperture to rise to T t A. Though the rise occurs in finite time intervl, it is ssumed to occur instntneously. The temperture then drops long the decy line ABC unless interrupted by nother brking opertion. If second opertion occurs t time t B, the temperture will rise long the dshed line to T nd then begin n exponentil drop s before. About 5 - % of the het generted t the sliding interfce of friction brke must be trnsferred through the lining to the surrounding environment without llowing the lining to rech excessive tempertures, since high tempertures led to hot spots nd distortion, to fde (the fll-off in friction coefficient) or, worse, to degrdtion nd chrring of the lining which often incorportes orgnic constituents

5 In order to determine the brke dimensions the energy need to be bsorbed during criticl brking conditions is to be estimted. Energy to be Absorbed If t is the time of brke ppliction nd ω m the men or verge ngulr velocity then the energy to be bsorbed in brking E E = T. ω m.t = Ek+ Ep+ Ei where E k is the kinetic energy of the rotting system E p is the potentil energy of the moving system E i is the inertil energy of the system Energy to be bsorbed E k ω = g = mv ω = ( v v) = I ( ω ω) g E = mgh =ωh E p i = Iω Frictionl Mteril A brke or clutch friction mteril should hve the following chrcteristics to degree, which is dependent upon the severity of the service. A high nd uniform coefficient of friction. Imperviousness to environmentl conditions, such s moisture. The bility to withstnd high tempertures together with good therml conductivity. Good resiliency. High resistnce to wer, scoring, nd glling.

6 Linings The choice of lining mteril for given ppliction is bsed upon criteri such s the expected coefficient of friction; fde resistnce, wer resistnce, ese of ttchment, rigidity or formbility, cost, brsive tendencies on drum, etc. The lining is scrificil - it is worn wy. The necessry thickness of the lining is therefore dictted by the volume of mteril lost - this in turn is the product of the totl energy dissipted by the lining throughout its life, nd the specific wer rte R w (volume scrificed per unit energy dissipted) which is mteril property nd strongly temperture dependent. The chrcteristics of typicl moulded sbestos lining mteril is illustrted in the figure below. The coefficient of friction, which my be tken s.39 for design purposes, is not much ffected by pressure or by velocity - which should not exceed 8 m/s. The mximum llowble temperture is 4 C. However t this temperture the wer is very high. From lower wer or higher life point, the mximum temperture should not exceed bout o C.5 o Temperture ( C ) 4 Figure 3..7

7 Linings trditionlly were mde from sbestos fibers bound in n orgnic mtrix, however the helth risks posed by sbestos hve led to the decline of its use. Non-sbestos linings generlly consist of three components - metl fibers for strength, modifiers to improve het conduction, nd phenolic mtrix to bind everything together. Brke Design Section The brked system is first exmined to find out the required brke cpcity tht is the torque nd verge power developed over the brking period. - The brke is then either selected from commercilly vilble rnge or designed from scrtch ff drum brke hs to be designed for prticulr system (rther thn chosen from n vilble rnge) then the slient brke dimensions my be estimted from the necessry lining re, A, together with drum dimeter- tolining width rtio somewhere between 3: nd :, nd n ngulr extent of C sy for ech of the two shoes. Worked out Exmple An improved lining mteril is being tried on n existing pssenger cr drum brke shown in Figure. Qulity tests on the mteril indicted permissible pressure of. MP nd friction co-efficient of.3. Determine wht mximum ctuting force cn be pplied for lining width of 4 mm nd the corresponding brking torque tht could be developed. While cruising on level rod t kmph, if it is to decelerted t.5g nd brought to rest, how much energy is bsorbed nd wht is the expected stopping distnce?

8 While cruising on level rod t kmph, if it is to decelerted t.5g nd brought to rest, how much energy is bsorbed nd wht is the expected stopping distnce? 3 3 F.9 F R=5 Pin Pin AUTOMOTIVE DOUBLE SHOE BRAKE Figure 3..8 Anlysis bsed on leding shoe P = MP f =.3 = 87.5 mm b = 4 mm θ 9 mx = d = +86. = mm θ 5 = θ = r =5 mm

9 B B B n f 3 6 p brd θ θ M (sin sin ) M n = θ θ sin θ 4 n *4* *5 * *. 5 π = * ( sin 4 sin ) 8 4 =4*5*..3- (.3) 4 = N.m ( ) f.b.rpm d M r(cos - cos ) - sin - sin f = sin θ θ θ θ θ =.3*4* -3 *5* -3 * 6.5(cos5 - cos ) -.4(sin - sin 5 ) M f = 4.85 N - m F*= M N M f Mn Mf F = = = 74.3N.87 Mx. ctutting force p F T = fbr ( cos θ cos θ )) + sin α M + M T =.3 * 4 * * (.5) * (Cos5 Cos) T = N-m Running t kmph =*5/8 = 7.7 m/s

10 U= 7.7 m/s Decelertion =.5*9.8=4.9 V U = S (7.7) = *( 4.9) *S 7.7 S = = 78.9 m *4.9 E = T. ω v.t = = 386 = 38.KJ Worked out exmple A spring set, hydruliclly relesed double shoe drum brke, schemticlly shown t Fig is to be designed to hve torque cpcity of 6 N.m under lmost continuous duty when the brke drum is rotting t 4 rpm in either direction. Assume tht the brke lining is to be molded sbestos hving friction coefficient of.3 nd permissible pressure of.8 MP. The width of the brke shoe is to be third of drum dimeter nd the remining proportion's re s shown in figure. = 4. D b= D 3 6. D d = = 693. D cos3

11 .4 D 9 ο.6 D 3 ο Double block brke Figure 3..9 Determine the required brke drum dimeter, width of the lining nd the spring force required to be set. =.4D b = D 3.6D d = =.693D cos3

12 6 D π sin( ) sin 3 =.8* * D * *.68D M = D *.35 N θ θ 3 3 Pbrf MJ = ( r cosθ) sinθdθ d ( ) ( θ) = Pbrf r cos θ cos θ sin θ sin 6 D D D.698 =.8* * * *.3 (.5) (.866) 3 = 5D P 'for the triling shoe Torque due to triling nd leding shoe= totl τ= ' MN MF P = P M N + M F = P =.7687P fdn.r P sinθ = f.r.rd θ.b sin θ fbr θ θ = P sin θdθ+ P ' sin θdθ sin θ θ θ fbr =.P + P'.cosθ cos sin θ ( ) ( θ ) D D.3* * = = 4334 D T *.8* 6 cos5 cos.5 3 = 6 Nm Therefore D = 4.4mm ( )

13 Actuting force due to spring MN MF 836 *.4 F = = = 343.N.4 * D.4 Actuting force =343. N D Living width = 3 = 8.4 mm