渐开线齿轮滚刀设计

A NOVEL HO

B DESIGN FOR PRECISION INVOLUTE GEARS: PART II

38By Stephen P. Radzevich, Ph.D.

Abstract

This pa per is a imed a t the development of a novel design of precision gea r hob for the ma chining of involute gea rs on a conventiona l gea r-hobbing ma chine. The reported resea rch is ba sed on the use of funda menta l results obta ined in a na lytica l mecha nics of gea ring. For solving the problem, both the descriptive-geometry-ba sed methods (further DGB-methods) together with pure a na lytica l methods ha l problems, which consequently ha ve a n a na lytica l solution. These a na lytica l methods provide a n exa mple of the a pplica tion of the DG/K-method of surfa ce genera tion ea rlier developed by the a uthor. For interpreta tion of the results of resea rch, severa l computer codes in the commercia l softwa re Ma thCAD/Scientific

stra ight-line la tera l cutting edges of the hob with the stra ight-line cha ra cteristics of its genera ting surfa ce elimina tes the ma jor source of devia tions of the hobbed involute gea rs. The rela tionship between ma jor

I can be downloaded at [].

• MAY 2007 • GEAR SOLUTIONS 39

W R

= pitch plane of the auxiliary phantom rack R

λψg N = number of gear teeth

h N = number of starts of the involute hob

g O = gear axis of rotation h O = hob axis of rotation

.x h P = axial pitch of the involute hob

R

= auxiliary phantom rack of the involute hob

h S = involute hob feed-rate

T = the generating surface of the involute hob

U = idle distance in gear hobbing operation

g a = gear tooth addendum

a R

= the auxiliary rack tooth addendum

g b = gear tooth dedendum

b R

= the auxiliary rack tooth dedendum

.b g d = base diameter of a gear

.b h d = base diameter of an involute hob

.f g d = gear root diameter

h d = gear hob pitch diameter

.o h d = outside diameter of the involute hob

.t g h = gear tooth whole depth

m = gear modulus

.b h p = base pitch of the involute hob

c t = normal tooth thickness

/g h C = center distance

g D = pitch diameter of the gear .o g D = outside diameter of the gear

G = gear tooth surface being machining

Σ= cross-axis angle

h ζ= hob-setting angle

.b h λ = involute hob base lead angle (=90deg −)n φ= normal pressure angle .b h ψ= involute hob base helix angle g ψ= gear pitch helix angle

h ψ= involute hob pitch helix angle ψR

= auxiliary rack pitch helix angle

g ω= gear rotation h ω= involute hob rotation

g = gear to be machined h = involute hob to be applied

b.h g.h Nomenclature

Greek Symbols

Subscripts

The pitch diameter neither of the new hob, nor of the completely worn hob, could be used for the computation of parameters of design of the gear hob. For accurate computations it is rec-ommended to use pitch diameter of the partly worn gear hob that correlates with outside diameter of the cutting tool. Outside diameter of the new gear hob is equal to d o.h (Fig. 10), while outside diameter of the completely worn gear hob could be computed from the equation (d o.h − Δd o.h ). For computation of the outside diameter reduction Δ d o.h , the following approxi-mate equation

.tan 2tan 5.585rh

o h rh h

d L n ααΔ≅⋅⋅=⋅

is derived. H ere it is designated that: L is a

distance between two neighboring hob teeth that is measured along the helix on the outside cylinder of the hob (Fig. 10), αrh is clearance angle at the top cutting edge of the hob tooth, and n h is effective number of the hob teeth.For involute hobs with straight slots, n h is always an integer number, and it is always equal to the actual hob teeth number n (a)h

which is usually in the range of n (a)

h = 8~16 [because of this, the distance L can be computed from equation ∪L =(π⋅d o.h)/n (a)h

]. For gear hobs with helical slots, the effec-tive hob teeth number n h is always a number with fractions. Moreover, the actual value of n h depends upon the hand of helix of slots. This is due to that in the last case the distance L is computed from the equation ∪L =(π⋅d o.h )/n (a)h ]+P x.h ⋅N h ⋅ cos λrf ⋅ sin λrf [1], [13], [14] and oth-ers. Here is designated: d o.h is outside diameter of the hob, P x.h is axial pitch of the hob, N h is the hob starts number, λrf is lead angle of the hob rake face (λrf is the signed value).

Equation (25) is a simple one. It is an approximation, which returns reasonably accu-rate results of computation.

The performed analysis of the gear hob design reveals that decrease of number of starts N h of the hob (Fig. 11) (a), increase of normal pressure angle φn (Fig. 12) (b), increase of the hob-setting angle ζh (Fig. 13) (c), and increase of the hob pitch diameter d h (Fig. 11 through Fig. 13) (d) result in reduction of the angle ξ of the rake surface orientation.

The plots (Fig. 11 through Fig. 13) are cre-ated using commercial software MathCAD/Scientific . Unfortunately, the lack of capabilities of MathCAD/Scientific imposed restrictions on graphical interpretation of the functions ξ = ξ (N h ) , ξ =ξ(φn ), ξ =ξ(ζ h ), and ξ = ξ (d h ). The lack of capabilities is the sole reason that the listed functions are interpreted as a function ξ = ξ (d h ) under various values of the gear hob number starts N h (Fig. 11), normal pressure angle φn (Fig. 12), and the gear hob-setting angle (Fig. 13). However, Fig. 11 through Fig. 13 provide clear understanding of the impact of the above mentioned parameters of the gear hob design onto the rake face inclination (ξ).

BEEN REGROUND.

h OF

THE INVOLUTE HOB ONTO THE ACTUAL ORIENTA-TION OF THE RAKE PLANE DETERMINED BY THE ANGLE ξ ( M = 10 MM, φn = 20 DEG , ζh = 3 DEG , n h = 10 , αt = 12 DEG ).

n h FIG. 13. IMPACT OF THE HOB-SETTING ANGLE

ζ h ONTO THE ACTUAL ORIENTATION OF THE RAKE PLANE DETERMINED BY THE ANGLE ξ ( M = 10 MM, 20 DEG φn = 20 DEG, N g = 1, n h =10, αt = 12 DEG ).