Lecture 13 BEVEL GEARS CONTENTS 1. Bevel gear geometry and terminology 2. Bevel gear force analysis 3. Bending stress analysis 4. Contact stress analysis 5. Permissible bending fatigue stress 6. Permissible contact fatigue stress BEVEL GEARS Bevel gears transmit power between two intersecting shafts at any angle or non- intersecting shafts. They are classified as straight and spiral tooth bevel and hypoid gears in Fig.1
BEVEL GEARS GEOMETRY AND TERMINOLOGY When intersecting Shafts are connected by gears, the pitch cones (analogous to the pitch cylinders of spur and helical gears) are tangent along an element, with their apexes at the intersection of the shafts Fig.2. BEVEL GEARS GEOMETRY AND TERMINOLOGY The size and shape of the teeth are defined at the large end, where they intersect the back cones. Pitch cone and back cone elements are perpendicular. The tooth profiles resemble those of Spur gears having pitch radii equal to the developed back cone radii r bg and r bp, Fig. 3.
2πrb1 Z1 2πrb2 Z2 Z v1 = = (1) & Z v2 = = (2) p cosγ1 p cosγ2 where Z v is called the virtual number of teeth, p is the circular pitch of both the imaginary spur gears and the bevel gears. Z 1 and Z 2 are the number of teeth on the pinion and gear, γ and 1 γ2 are the pitch cone angles of pinion and gears. It is a practice to characterize the size and shape of bevel gear teeth as those of an imaginary spur gear appearing on the developed back cone corresponding to Tredgold s approximation. a) Bevel gear teeth are inherently noninterchangeable. b) The working depth of the teeth is usually 2m, the same as for standard spur and helical gears, but the bevel pinion is designed with the larger addendum ( 0.7 working depth). c) This avoids interference and results in stronger pinion teeth. It also increases the contact ratio. d) The gear addendum varies from 1m for a gear ratio of 1, to 0.54 m for ratios of 6.8 and greater.
The gear ratio can be determined from the number of teeth, the pitch diameters, or the pitch cone angles: Gear ratio i : ω n Z d 1 1 2 2 i = = = = = tan γ 2 = cot γ1 (3) ω2 n2 Z1 d1 Accepted practice usually imposes two limits on the face width L b 10m and b 3 Where L is the cone distance. Smaller of the two is chosen for design. (4) The Fig.4 illustrates the measurement of the spiral angle ψ of a spiral bevel gear. Bevel gears most commonly have a pressure angle of 20 o, and spiral bevels usually have a spiral angle ψ of 35 o.
Fig.5 The Fig.5 on the right illustrates Zerol bevel gears, which are having curved teeth like spiral bevels. But they have a zero spiral angle.
DIFFERENT TYPES OF BEVEL GEARS BEVEL GEARS FORCE ANALYSIS
In the Fig.10, the resolution of resultant tooth force F n into its tangential (torque producing), radial (separating), and axial (thrust) components, is designated F t, F r and F a, respectively. An auxiliary view is needed to show the true length of the vector representing resultant force F n (which is normal to the tooth profile). Resultant force F n is shown applied to the tooth at the pitch cone surface and midway along tooth width b. It is also assumed that load is uniformly distributed along tooth width, despite the fact that the tooth width is larger at the outer end. dav = d bsin γ (5) πd n 6000 av V av = (6) 1000W F t = (7) v av Where V av is in meters per second, d av is in meters, n is in revolutions per minute, F t is in Newtons, and W is power in kilo Watts. Fn = F t /cos φ (8) Fr = Fncosγ = Ft tanφ cos γ (9) F = F sin γ = F tanφ sin γ (10) a n t
For spiral bevel gear Ft F r = ( tanφn cos ψ cos γ sinψ sin γ) (11) Fa = F t ( tanφn sin γ ± sinψcos γ) (12) Where or ± is used in the preceding equation, the upper sign applies to a driving pinion with right-hand spiral rotating clockwise as viewed from its large end and to a driving pinion with left-hand spiral rotating counter clock-wise when viewed from its large end. The lower sign applies to a left-hand driving pinion rotating clockwise and to a driving pinion rotating counter clockwise. Similar to helical gears, φ n is the pressure angle normal measured in a plane normal to the tooth. BEVEL GEARS TOOTH BENDING STRESS The equation for bevel gear bending stress is the same as for spur gears: Ft σb = KvKoK m (13) bmj Where, F t =Tangential load in Newtons m = module at the large end of the tooth in mm b = Face width in mm J = Geometry form factor based on virtual number of teeth, Fig. 13 & 14. K v = Velocity factor, from Fig. 15. K o = Overload factor, Table 1. K m = Mounting factor, depending on whether gears are straddle mounted (between two bearings) or overhung (outboard of both bearings), and on the degree of mounting rigidity. Table 2.
BEVEL GEARS GEOMETRY FACTOR J FOR STRAIGHT BEVEL GEARS BEVEL GEARS GEOMETRY FACTOR J FOR SPIRAL BEVEL GEARS
BEVEL GEARS DYNAMIC LOAD FACTOR K V BEVEL GEARS OVER LOAD FACTOR K o Table 1 -Overload factor K o Driven Machinery Source of power Uniform Moderate Shock Heavy Shock Uniform 1.00 1.25 1.75 Light shock 1.25 1.50 2.00 Medium shock 1.50 1.75 2.25
BEVEL GEARS MOUNTING FACTOR K m TABLE 2 BEVEL GEARS PERMISSIBLE TOOTH BENDING STRESS (AGMA) Fatigue strength of the material is given by: σ e = σ e k L k v k s k r k T k f k m (14) Where, σ e endurance limit of rotating-beam specimen k L = load factor, = 1.0 for bending loads k v = size factor, = 1.0 for m < 5 mm and = 0.85 for m > 5 mm k s = surface factor, is taken from Fig. 16 based on the ultimate strength of the material and for cut, shaved, and ground gears. k r = reliability factor given in Table 3. k T = temperature factor, = 1 for T 120 o C more than 120 o C, k T < 1 to be taken from AGMA standards
BEVEL GEARS SURFACE FACTOR k s BEVEL GEARS RELIABILITY FACTOR k R k f = fatigue stress concentration factor. Since this factor is included in J factor, its value is taken as 1. k m = Factor for miscellaneous effects. For idler gears subjected to two way bending, = 1. For other gears subjected to one way bending, the value is taken from the Fig.17. Use k m = 1.33 for σ ut less than 1.4 GPa.
Permissible bending stress σe [σ b ] = (15) s Hence the design equation from bending consideration is : σ b [ σ b ] (16) BEVEL GEARS SURFACE FATIGUE STRESS Bevel gear surface fatigue stress can be calculated as for spur gears, with only two modifications. Ft σ H=Cp KVKoK m (17) bdi BEVEL GEARS CONTACT STRESS C p values of 1.23 times the values given in the table are taken to account for a somewhat more localized contact area than spur gears.
BEVEL GEARS GEOMETRY FACTOR I FOR STRAIGHT BEVEL GEAR
BEVEL GEARS GEOMETRY FACTOR I FOR SPIRAL BEVEL GEAR BEVEL GEARS DYNAMIC LOAD FACTOR K v
BEVEL GEARS OVER LOAD FACTOR K o Table 5 -Overload factor K o Driven Machinery Source of power Uniform Moderate Shock Heavy Shock Uniform 1.00 1.25 1.75 Light shock 1.25 1.50 2.00 Medium shock 1.50 1.75 2.25 BEVEL GEARS MOUNTING FACTOR K m TABLE 6 BEVEL GEARS SURFACE FATIGUE STRENGTH Surface fatigue strength of the material is given by: σ sf = σ sf K L K H K R K T (18) Where σ sf = surface fatigue strength of the material given in Table 7 K L = Life factor given in Fig. 21
BEVEL GEARS SURFACE FATIGUE STRENGTH, LIFE FACTOR KL
K H = Hardness ratio factor, K the Brinell hardness of the pinion by Brinell hardness of the Gear. Given in Fig.22. K H = 1.0 for K < 1.2 K R = Reliability factor, given in Table 8. BEVEL GEARS SURFACE FATIGUE STRENGTH, HARDNESS FACTOR K H Table 8. Reliability factor K R Reliability (%) K R 50 1.25 99 1.00 99.9 0.80
K T = temperature factor, = 1 for T 120 o C based on Lubricant temperature. Above 120 o C, it is less than 1 to be taken from AGMA standards. BEVEL GEARS ALLOWABLE SURFACE FATIGUE STRESS (AGMA) Allowable surface fatigue stress for design is given by [ σ H ] = σ Sf / s (19) Factor of safety s = 1.1 to 1.5 Hence Design equation is: σ H [ σ H ] (20)