Design of helical gear

1. Helical and Bevel Gears
Types of helical and Bevel gears, Terminology, Virtual
number of teeth, and force analysis of Helical and Straight
Bevel Gear. Design of Helical and Straight Bevel Gear
based on Beam Strength, Wear strength and estimation
of effective load based on Velocity factor (Barth factor)
and Buckingham’s equation. Mountings of Bevel Gear.
(No numerical on force analysis of helical & Bevel Gear)
By S M Gujrathi

2. Helical Gear
By S M Gujrathi

3. Helix angle: It is defined as the
angle between the axis of the
shaft and the centre line of the
tooth taken on the pitch
plane(ψ)
1.XX is the plane of rotation,
2.YY is a plane perpendicular
to the tooth elements.
transverse circular pitch=A1A2
Normal circular
pitch(pn)=B1B2=A1C
Formative or Virtual
Gear Concept
By S M Gujrathi

4. where Pn and P are normal and
transverse diametral pitches
respectively.
Substituting (P = 1/m) in
the above expression,
we have
mn =m*cos(ψ)
mn = normal module
(mm)
m = transverse module
(mm)
By S M Gujrathi

5. By S M Gujrathi

6. VIRTUAL NUMBER OF TEETH
Semi
major(a)=
semi-minor(b)=
Radius of curvature r’ at the point
B is given by,
Substituting the values of a and b in
the expression for r’,
By S M Gujrathi

7. In the design of helical gears, an imaginary spur gear is considered
in the plane A–A with centre at O’ having a pitch circle radius of r’
and module mn. It is called a ‘formative’ or ‘virtual’ spur gear. The
pitch circle diameter d’ of the virtual gear is given by,
By S M Gujrathi

8. The number of teeth z’ on this imaginary spur gear is
called the virtual number of teeth
where z is the actual number of teeth.
By S M Gujrathi

9. TOOTH PROPORTIONS
• In helical gears, the normal module mn should be
selected from standard values. The first preference
values of the normal module are
mn (in mm) = 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8 and 10
• The standard proportions of the addendum and the
dedendum are,
addendum (ha) = mn
dedendum (hf) = 1.25 mn
clearance (c) = 0.25 mn
The addendum circle diameter da is given by
By S M Gujrathi

10. • Similarly, the dedendum circle diameter df is
given by
The normal pressure angle an is always 20°. The
helix angle varies from 15 to 25°. A portion of the
top view of a helical gear is shown in Fig
By S M Gujrathi

11. For this rotation, the point A1 will be the fi rst point to come in contact with its
meshing tooth on the other gear.
It is called the ‘leading’ edge of the tooth.
Also, the point A2 will be the last point to come in contact with its meshing tooth on
the other gear. It is called the ‘trailing’ edge of the tooth.
In order that the contact on the face of the tooth shall always contain at least one
point, the leading edge of the tooth should be advanced ahead of the trailing end by a
distance greater than the circular pitch. Or, x ≥ p
By S M Gujrathi

12. FORCE ANALYSIS
• The resultant force P acting on the tooth of a helical gear is
resolved into three components, Pt, Pr and Pa as shown in
Fig.
By S M Gujrathi

13. How to find force
magnitude?
How to find directions of forces?
First find:
(i) Find driven and driving element
(ii) Direction of rotation of pinion(CW/CCW)
(iii)Right hand or left hand helix
(i) Tangential Component (Pt)
(a) The direction of tangential component for a driving gear
is opposite to the direction of rotation.
(b) The direction of tangential component for a driven gear
is same as the direction of rotation.
(ii) Radial Component (Pr )
(a) The radial component on the pinion acts towards the
centre of the pinion.
(b) The radial component on the gear acts towards the
centre of the gear.
(iii) Thrust Component (Pa)
(a) Select the driving gear from the pair.
(b) Use right hand for RH-helix and left hand for LH-helix.
(c) Keep the fingers in the direction of rotation
of the gear and the thumb will indicate the direction of the
thrust component for the driving gear.
(d) The direction of the thrust component for the
driven gear will be opposite to that for the driving gear.
By S M Gujrathi

14. • A pair of parallel helical gears is shown in Fig. A 5 kW power at 720
rpm is supplied to the pinion A through its shaft. The normal module
is 5 mm and the normal pressure angle is20°. The pinion has right-
hand teeth, while the gear has left-hand teeth. The helix angle is 30°.
The pinion rotates in the clockwise direction when seen from the left
side of the page. Determine the components of the tooth force and
draw a free-body diagram showing the forces acting on the pinion
and the gear.
By S M Gujrathi

15. BEAM STRENGTH OF HELICAL
GEARS
• In order to determine beam strength, the
helical gear is considered to be equivalent to a
formative spur gear.
EFFECTIVE LOAD ON GEAR TOOTH
By S M Gujrathi

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33. Bevel Gears
• Bevel gears are used to transmit power between two intersecting
shafts.
Types of Bevel Gears: (a) Straight Bevel Gear (b) Spiral Bevel Gear
Three types of bevel gears that are based on pitch angle are as follows:
(i) When the pitch angle is less than 90°, it is called external bevel gear.
(ii) When the pitch angle is equal to 90°, it is called crown bevel gear.
(iii) When the pitch angle is more than 90°, it is called internal bevel gear.
(i) Miter Gears When two identical bevel gears are
mounted on shafts, which are intersecting at right
angles, they are called ‘miter’ gears.
By S M Gujrathi

34. Terminologies of bevel gear
(i) Pitch Cone: Pitch cone is an imaginary cone the
surface of which contains the pitch lines of all teeth in
the bevel gear.
(ii) Cone Centre The apex of the pitch cone is called the
cone centre. It is denoted by O.
(iii) Cone Distance Cone distance is the length of the
pitch-cone element. It is also called pitch-cone radius. It
is denoted by A0.
(iv) Pitch Angle The angle that the pitch line makes with
the axis of the gear, is called the pitch angle. It is
denoted by γ. The pitch angle is also called centre angle.
(v) Addendum Angle It is the angle subtended by the addendum at the cone centre. It is denoted by
α.
(vi) Dedendum Angle It is the angle subtended by the dedendum at the cone centre. It is denoted by
δ.
(vii) Face Angle It is the angle subtended by the face of the tooth at the cone centre.
Face angle = pitch angle + addendum angle = γ +α.
(viii) Root Angle It is the angle subtended by the root of the tooth at the cone centre.
Root angle = pitch angle – dedendum angle = γ – δ
(ix) Back Cone The back cone is an imaginary cone and its elements are perpendicular to the elements
of the pitch cone.
(x) Back Cone Distance It is the length of the back cone element. It is also called back cone radius. It is
denoted by r .
By S M Gujrathi

35. Cross section of the tooth decreases in size as it
approaches towards the apex point O
Pitch circle diameter, module, addendum, and
dedendum decreases and there is no single value for
these parameters
The dimensions of the bevel gear are always specified
and measured at the large end of the tooth
Observe ha, hf and D in a figure.
An imaginary spur gear is considered in a plane
perpendicular to the tooth at the large end.
rb is the pitch circle radius of this imaginary spur
gear(formative gear) and z’ is the number of teeth on
this gear.
By S M Gujrathi

36. where m is the module at the large end of
the tooth.
If z is the actual number of teeth on the
bevel gear ,then
A pair of bevel gears is shown in Fig. 19.8. Dp
and Dg are the pitch circle diameters of pinion
and gear respectively. γ is the pitch angle of
the pinion, while Г is the pitch angle of the
gear. Line AB is perpendicular to the line OB.
Consider the triangle OAB,
By S M Gujrathi

37. FORCE ANALYSIS
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39. (i) Tangential Component (Pt)
(a) The direction of tangential component for
the driving gear is opposite to the direction
of rotation. (b) The direction of tangential component
for the driven gear is same as the direction of rotation.
(ii) Radial Component (Pr)
(a) The radial component on the pinion acts towards
the centre of the pinion.
(b) The radial component on the gear acts towards the
centre of the gear.
(iii) Thrust Component (Pa) The following
guidelines can be used to determine the direction of the thrust component:
(a) The thrust component on the pinion is equal and opposite of the radial component on the
gear.
(b) The thrust component on the gear is equal and opposite of the radial component on the
pinion.
By S M Gujrathi

40. A pair of bevel gears transmitting 7.5 kW at 300 rpm is shown
in Fig. 19.11(a). The pressure angle is 20°. Determine the
components of the resultant gear tooth force and draw a free-
body diagram of forces acting on the pinion and the gear.
By S M Gujrathi

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42. BEAM STRENGTH OF BEVEL GEARS
Consider an elemental section of the tooth at a distance x from the
apex O and having a width dx. Applying the Lewis equation to a
formative spur gear at a distance x from the apex,
By S M Gujrathi

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44. The face width of the bevel gear is generally
taken as 10 m or (Ao/3), whichever is smaller, i.e.,
b = 10 m or b = Ao /3 (whichever is smaller)
By S M Gujrathi

45. WEAR STRENGTH OF BEVEL GEARS
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46. pinion or the gear is generally
overhanging. It has been found that to
transmit the load, only three quarters of
the face width is effective
By S M Gujrathi

47. EFFECTIVE LOAD ON GEAR TOOTH
By S M Gujrathi

48. Bevel gears are usually made of steel and the
deformation factor C is 11 400 N/mm2.
In practice, it is necessary to contact the
manufacturer and find out the expected error
between meshing teeth. In the absence of
such information. Table 19.1 may be used to
get the values of error e. The classes of
gears mentioned in the table indicate the
following manufacturing methods,
Class-1 Well cut commercial gear teeth
Class-2 Gear teeth cut with great care
Class-3 Ground and lapped precision gear
teeth
By S M Gujrathi

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56. Reference
• V.B.Bhandari,Design of Machine Elements,3rd
Edition,Chpter 18 and 19,pp:694-728
By S M Gujrathi