[Pg / 2] Gears Objectives. 2. 3. 4. 5. Cmpute the frces exerted n gear teeth as they rtate and transmit pwer. Use apprpriate stress analyses t determine the relatinships amng the applied frces, the gemetry f the gear teeth, the precisin f the gear teeth and ther factrs specific t a given applicatin, in rder t make final decisins abut thse variables. Describe varius methds fr manufacturing gears and levels f precisin and quality t which they can be prduced. Describe suitable metallic materials frm which t make the gears, in rder t prvide adequate perfrmance fr bth strength and pitting resistance. Use the Lewis Equatin and the standards f the American Gear Manufacturers Assciatin (AGMA) as the basis fr cmpleting the design f the gears. Types f Gear Pg xxx Spur Gear Teeth is parallel t axis f rtatin Can transmit pwer between parallel shaft The simplest frm fr gear Helical gear Teeth is inclined t the axis f rtatin Smther than spur Develp thrust lad (helix angle) Can transmit pwer frm ne shaft t a parallel and nn-parallel shaft Bevel gear Teeth n cnical surfaces Transmit pwer between tw intersecting shafts Wrm gear Transmit pwer between tw intersecting shafts Extremely lw gear rati
[Pg 2 / 2] Pinin and Gear Accrding t the layman terminlgy, the pinin is the smaller gear. Hwever, in the prper terminlgy, pinin is the driver and gear is driven. Althugh in mst applicatin, pinin is usually smaller than gear, it can be als be the larger f the tw. A pair f gears (pinin and gear) can be represented by 2 circles. The diameter f the tw circles are based n the pitch circle diameters f the mating gears. Pitch circle diameter, d The pitch circle is the theritical diameter upn which all gear dimensins are based. Similarly, almst all gear calculatins are als based n the pitch circle diameter. METRIC: d Nm (mm) IMPERIAL: N d (inch) P d where N number f gear teeth P d diametral pitch (tth per in) m mdule (mm) "Mating gears must have same mdule (m) r diametral pitch P d " Facewidth : width f the gear tth addendum : distance between the tp land t the pitch circle dedundum : distance between the pitch circle t the bttm f the gear
[Pg 3 / 2] Interference T avid interference, the minimum number f teeth fr a ne t ne gear rati is N p 4k 6sin 2 ( ϕ) + + 3sin 2 ( ϕ) where k.0 fr full-depth, φ pressure angles ( 20, 22 / 2, 25 ) k 0.8 fr stub teeth N G If m G > indicating external gearset then N P minimum number f teeth in the pinin, 2k 2 N P ( + 2m G )sin 2 m G + m G + ( + 2m G )sin 2 ( ϕ) ( ϕ) t mate with the maximum number f teeth in the gear, 2 2 2 N P sin ( ϕ) 4k N G 4k 2N P sin 2 ( ϕ) If the abve equatin is nt adhered t, then undercutting will ccur. Frming f Gear teeth Machined by varius methds including sand casting, shell mlding, investment casting, die casting, etc. Mst ppular methds Frm milling: mainly used fr large gears. A milling cutter that has the shape f the tth space is used. See diagram. Shaping: usually used fr internal gears. Cutter used reciprcates n a vertical spindle. See diagram. Hbbing: similar prcess t milling except that bth the wrkpiece and the cutter rtate in a crdinated manner.
[Pg 4 / 2] Tth Systems The tth system is a standard that specifies the relatinships invlving addendum, dedendum, wrking depth, tth thickness, and pressure angle. Refer t pg 676: Table 3- and Table 3-2 fr spur gears specs and sizes.
[Pg 5 / 2] Gear Train Fr mating gears, m m 2 then r d N d 2 N 2 n N n 2 N 2 n 2 n π d n V 60 where n is in rpm The gear-train rati is given by e nl nf rtatinal_speed_f_last_gear rtatinal_speed_f_first_gear Gear-train rati fr a set f 2 gears e n 2 n N N 2 Idler gear n N n 2 N 2 N n 2 n N 2 n 2 N 2 n 3 N 3 N 2 n 3 n 2 N 3 N N 2 N n 3 n n n N n 3 N 3 N 2 N 3 N 3 Fr cmputing gear train with an idler gear, n 3 e (bth gear rtating same directin) n
[Pg 6 / 2] Multi gearset system The gear-train rati fr the first gearset N e N 2 5 50 n 2 n n 2 20 n 2 20 5 336rpm 50 Speed n 3 n 2 336rpm The gear-train rati fr the 2nd gearset N 3 e 2 N 4 25 75 n 4 n 3 n 4 336 n 4 336 25 2rpm 75 The gear-train rati fr the multi-gearset system: e Pdt_f_driving_gears Pdt_f_driven_gears N N 3 N 2 N 4 where als e nl nf n 4 n Example: See diagram abve e N N 3 N 2 N 4 5 25 50 75 0 e n 4 n n 4 20 Final shaft utput speed n 4 20 2 rpm 0
[Pg 7 / 2] Gear teeth Frce analysis The pinin tth applied frce F against the gear teeth 2 at an angle equal t pressure angle. Fr equilibrium, gear teeth 2 reacts with equal frce F 2 against pinin teeth and s F F 2 The frce F R and F T is the cmpnents f F. The radial cmpnent, F R acts tward the center f the gear causing bending stress n the gear shaft. F T which is the tangential cmpnent f F is the frce which turn the gears. F R F T tan( ϕ) F T W t H where Wt is the tangential frce V d Trque at shaft T W t (Nm, lbf-in) 2 M E T R I C S V π d n (m/s) ; d (mm), n (rpm) 60( 000) H W t V 2π nt (watts) ; V (m/s), T (Nm) 60 30H T (Nm) ; H (watts), Wt (N), n (rpm) π n I M P E R I A L d (in), n (rpm) V (ft/min), T (lbf-in) H (hp), W t (lbf) V π d( in)n( rpm) 2 (fpm feet per min) H W t ( lbf) V 33000 (hp) T 63025H n (lbf-in)
[Pg 8 / 2] Lewis Bending Equatins Dynamic effects are present when a pair f gear rtate at mderate and high speed. These effects are expressed by using the Velcity factr K V pg 727. Unit V meter/sec Unit : V ft/min r m/sec I M P E R I A L 600 + V 3.05 + V K v K v cast irn, cast prfile 600 3.05 M E T 200 + V 6. + V K v K v cut r milled prfile 200 6. R I C S 50 + V 3.56 + V K v K v hbbed r shaped prfile 50 3.56 78 + V 5.56 + V K v K v shaved r grund prfile 78 5.56 Bending stress K v W t P d Allwable Bending stress number σ (imperial) σ FY K v W t FmY Y n s t < s at SF K r Cntact stress number W t K K s K m K v s c C p Fd p I Z n C H s c < s ac ( SF)K R
[Pg 9 / 2] AGMA Stress Equatins Thrugh Hardening AISI 020, 040, 050, 340, 440, 450, 4340, 650, 8650 nt recmmended fr HB > 400 Case Hardening use flame hardening, inductin hardening, carburizing and nitriding n surface available steel grade, grade 2 and grade 3 (see Table 9-3) Gear quality. 2. 3. Cmpsite variatin: allwable amunts f variatins f the actual tth frm frm the actual tth frm Specified by AGMA as quality numbers. Quality numbers range frm 5 t 5 with increasing precisin. Bending stress (Imperial units) W t P d Bending stress number s t FJ K K s K m K B K v Y n s t < s at SF K r W t K K s K m K v Cntact stress number s c C p Fd p I Z n C H s c < s ac ( SF)K R
[Pg 0 / 2] Sample Yu are required t design a gearbx fr 2 level speed reductin. At bth level there are 8T gear as shwn in diagram belw. Prpse 2 gear cmbinatins fr Gear 3 and 5 if the ttal reductin rati is 24. Yu are cnstrained t a maximum speed rati nt exceeding 0 at each gearset. Yur answers must be tabulated in the frmat given belw. Jmlh gigi G3 Jmlh gigi G5 Gabungan Gabungan 2 Sample 2 A mtr prducing 8kW at 000 rpm clckwise is used t pwer a drilling machine via a gearset f 5 gears as shwn in diagram belw. If all gear have mdule size (m 4mm) and pressure angle 20. i. Calculate the drilling machine rtatinal speed in rpm. ii. Calculate the trque T and speed (RPM) acting n gear 3 and 5. iii. Calculate and draw the frces F, Fr and Ft acting n gear teeth.
[Pg / 2] Sample Given e (because 24X speed reductin in the gear set) 24 e 24 Np Np3 Ng2 Ng4 Np Ng2 Np3 Ng4 6 4 8 3 2 2 G3 G4 C 08 72 rati (:6 & :4) r vice-versa C2 44 54 rati (:8 & :3) r vice-versa C3 62 48 rati (:2 & :2) (wrng! why???) Sample 2 i ) e e Np Np4 Ng3 Ng5 22 72 8 6 5.5 Rtatin at input shaft: n 000rpm Output rtatinal speed: n 5 en n 5 5500 rpm ii ) Analysis gear ϕ 20 deg N 22 H 8 kw m 4mm d N m d 88 mm W t 60H W t.736 0 3 N π d n d T W t T 76.394 N m 2 Analysis gear 3 (Mating gear &2&3) V 3 V 2 V W t3 W t W t3.736 0 3 N N 3 8 d 3 N 3 m d 3 72 mm
[Pg 2 / 2] d 3 T 3 W t3 T 3 62.504 N m 2 N n 3 n n 3.222 0 3 rpm N 3 F t3 F t3.736 0 3 N W t3 F r3 F t3 tan( ϕ) F r3 63.938 N F 3 F t3 F 3.848 0 3 N cs( ϕ) speed n 3.222 0 3 rpm trque T 3 62.504 N m frce acting n teeth : F 3 847.7 N F t3 736.2 N F r3 63.9 N Analysis gear 4 T 4 T 3 T 4 62.504 N m N 4 72 d 4 m N 4 d 4 288 mm 2T 4 W t4 W t4 434.059 d 4 Analysis gear 5 W t5 W t4 N 5 6 W t5 434.059 F t5 W t5 m N 5 T 5 W t5 T 5 3.89 N m 2 F 5 F r5 W t5 F 5 46.96 N cs( ϕ) W t5 tan( ϕ) F r5 57.985 N speed n 5 5.5 0 3 rpm trque T 5 3.89 N m frce acting n teeth : 46.96 N F t5 434.059 N F r5 57.985 N F 5