# The theory of error of gear hob shaping is analysed

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Summary: Derivation analysed the shaping error between the tooth form of the tooth form of involute hob and Archimedes hob, found out sum of errors of gear hob shaping the relation between each element. Gear hob is a kind of cutting tool with straight tine of the clench the teeth outside treatment and the most commonly used gear of involute of inclined tine cylinder. From theoretic tell, the hob that machines involute gear is basic worm should be involute worm, such hob calls involute hob. Face of the knife after the side of involute hob is rolling the cut form in cutter shaft section plane is a curve, this kind of hob is more difficult when make and be measured (if cannot use radial to shovel tine to replace axial to shovel tine,wait for) , although it is reasonable,talk so going up is correct, but be used rarely in actual production. What be used extensively is Archimedes the hob of basic worm, call Archimedes hob. The axial cut form of face of the knife after side of this kind of hob is linear, when if use it,replacing involute hob angrily, the gear tooth form that cuts is not involute, be in consequently theoretic caused certain teeth shape error, call the shaping error of gear hob. To reduce this kind of error, the axial cut form that should make the axial cut form of face of the knife after Archimedes hob side is separating side of the knife after round place and involute hob side is tangent, even so, the other site that two kinds of hob manage except cent circle still has F of Δ of teeth shape error to exist. Affect the factor that this error is worth to make clear further, can do the theoretical analysis that be as follows to its. Setting CD is a on the left whorl face of involute worm straight generatrix, the rectangular coordinates that M nods arbitrarily on its is (X, y, z) , go to the lavatory to discuss an issue, introduce cylinder coordinate (X, ρ , θ ) , among them ρ is the distance that M nods X axis, the included angle that θ is the plane that decides by M dot and X axis place and Xoy plane (when θ =0, CD and Y axis are vertical) . If the lead of worm is P, basic circle radius is R, criterion the coordinate of direction of axes of worm of M drop edge is: X=AC+ABtan λ because AC=P(θ - α M) , cos α M=r, π of M2 of α of Tan of · of =P of λ of Tan of · of M of α of =rtan of λ of RABtg of ρ of π of 2-r22 of ρ of √ of Tan α M= so X=P(θ + θ of M)=P[of α of Tan α M- + 2 π of π of √ ρ 2-r2-arccosr]2 R ρ accordingly, equation of left lateral of whorl of dextral involute worm is: X=P[θ + θ of Sin of ρ of Z= of θ of Cos of ρ of Y= of ρ of R of π of √ ρ 2-r2-arccosr]2 (1) makes theoretical analysis for the axial teeth shape error to two kinds of hob, beg the equation of curve of axial tooth form that gives involute worm above all, for simple for the purpose of, xoz is planar (namely θ = π / the equation of axial tooth form on 2) undertakes discussion. By type (1) reachs the equation of the L1 of curve of tooth form of axial section plane of worm of the involute on Xoz plane is X=P[π + ρ of Z= of ρ of 2r of π of √ ρ 2-r2-arccosr]2 (2) namely Rz(3)L1 of π of X=P+P[√ Z2-r2-arccosr]42 is separating the aspect on θ of columnar face {y=Rcos is M(x1, y1, z1) , among them: Rz of π of X1=P+P[√ Z2-r2-arccosr]42, y1=0, z1=R. Z=Rsin θ by type (the equation that the tangent slope that the tangent slope that 2) can get the random on L1 to was in is Z2-r2 of √ of Z(x)=2 π RzP to be in L1 in M dot is R2-r2 of √ of K=2 π RRP to cross the linear L0 with M dot and tangent L1 is (of L0- - the straight generatrix of the left helicoid that R(4) of π of Rr42 of π of R2-r2Z+P-Parccosr2 of √ of RRX=P of π of Arccosr))]P √ R2-r242 is Archimedes worm with L0, criterion the axial tooth form of Archimedes worm is linear the curve of axial tooth form with involute worm is nodded in M place is tangent. When Z and Z take identical cost, (X-X) is two kinds of hob the tooth form line in axial section plane is in basic worm the F of error value Δ on same tooth depth. R of π of Rr42 of π of Rz2 of π of R2-r2z+P-Parccosr42 of √ of Z2-r2-arccosr)-(P of Δ F=x=X=P+P(√ namely the cost that π of R2-r2+arccosr-arccosr2 of R2-r2-z √ of Δ F=P(√ faces Δ F below RRrRz(5) is analysed further. Still can get Z"(x)=-4 π 2r2zP2(z2-r2)2 to want Z>0 only further by Z(x) , have Z"(x)<0, consequently curvilinear L1 is protruding curve when Z>0. Nod in M as a result of L0 and L1 again tangent, the slope of L0 is more than 0, consequently L1 always is in of L0 on the right side of, always have Δ F=x-X ≥ so 0. Only the equality sign when Z=R just holds water. Can beg piece (Rrz of π of Z2R2-z2r2)2 of √ R2z2-R2r2- √ is like Δ F)z=1(P, R, R is certain, become only when Z=R, (Δ F)z=0; is become when Z>R, (Δ F)z>0, Δ F is about Z add function, z is bigger, bigger; becomes Δ F when Z<R, (Δ F)zz<0, Δ F is about Z decrease function, z is smaller, Δ F is bigger. This shows, | Z-R | Bigger, Δ F is bigger. Can beg piece (Δ F)z=P × (0 equality sign of ≥ of F)z of Δ of R2z2-R2r2(of √ of R2 of π of Z2R2-(z2r2+R2r2)+r4)2 of √ Z2R2-2zRr2+r4- √ are only when Z=R hold water, when Z, R, P is certain, Δ F is about R add function, r is smaller, Δ F is smaller. By type (5) still can see directly, when Z, R, R is certain, p is smaller, namely the helix of worm promotes role smaller, Δ F is smaller. Analyse by above knowable, to the Archimedes hob that is used extensively, to reduce its shaping error, the cent cylinder helix of the basic worm of Archimedes hob rises horn to take small cost as far as possible. And the error that hob tooth distributes columnar part is 0, and value of the error when support to tine more or aging the root is in is greater. CNC Milling CNC Machining