Unit ii bevel gears

Pimpri Chinchwad Education Trust’s
Pimpri Chinchwad College of Engineering and
Research, Pune
Course- Theory of Machines-II
Topic – Bevel Gear
By Prof. Fodase G.M.

Bevel Gears
• Terminology Of Bevel gears
• Pitch cone angles & its
geometrical relationships
• Force Analysis of bevel gears

Bevel Gears
• Used for intersecting shafts (900) in same plane
• Straight Bevel Gear / Spiral Bevel Gear
• Used to change direction

Bevel Gears
Straight Bevel Gear

• Pitch cones :
It is an imaginary cones in a bevel gear which
rolls without slipping when they are in peripheral contact
with each other. (∆OAB)
• Pitch cone angle ( γ) :
It is the angle subtended by pitch line of pitch
cone with the axis of gear.
• Cone distance ( Da ) :
It is the distance measured along the pitch
line from PC diameter to the apex of cone.
i.e. distance AO or BO
• Back cone :
It is an imaginary cone and its elements are
perpendicular to the elements of pitch cone.
(∆O1 AB)
Terminology Of Bevel Gears

• Back cone Distance (Db) :
It is the distance measured along the pitch
line from PCD to the apex of back cone .
i.e. distance AO1 or BO1
• Back cone angle ( β ) :
It is the angle subtended by pitch line of back
cone to the axis of gear.
• Shaft angle ( θ ) :
It is the angle between two intersecthg axis of
bevel gears. It is equal to sum of the pitch cone angles of
both bevel gears. (900)
• Face width (b) :
It is the length of tooth measured along the
pitch line of pitch cone.
Terminology Of Bevel Gears

Pitch Cone Angles & Its Geometrical Relations

Let,
θ = Shaft angle =γ1+γ2
γ1 = Pitch cone angle of bevel gear 1
γ 1 = Pitch cone angle bevel gear 2
T1 = No. of teeth on bevel gear 1
T2 = No. of teeth on bevel gear 2
ω1 = Transverse module for bevel gear 1
ω 2 = Transverse module for bevel gear 2
r1 = PC radius for bevel gear 1

11
2
1
1
211
2
1
1
2
2
1
1
2
1
2
1
2
2
1
1
sincoscossinsin
].....[).........sin(sin
sinsin
sin
sin
)1(..........
sinsin











r
r
r
r
r
r
r
r
rr
OP
G = Gear ratio = T2/T1= r2/r1 = ω1/ ω2
Frm fig;

Dividing above eq. by cosγ1;
 
 









sincostan
sintancostan
tancossintan
tancossintan
cos
sin
cossin
cos
sin
1
2
1
11
1
2
11
1
2
1
2
1
1
1
1
2
1
1
1

















r
r
r
r
r
r
r
r
r
r
































cos
sin
tan
,
cos
sin
tan
cos
sin
tan
1
2
2
2
1
1
1
2
1
Similarly
r
r
OR
Eq. to find out Pitch Cone Angles of bevel pinion & gear

G
T
T
And
GT
T














1
2
1
1
0
1
2
0
2
2
1
1
2
0
2
1
0
1
90cos
90sin
tan
1
90cos
90sin
tan










When Shaft angle θ=900

Force Analysis Of Bevel Gears
Considering pitch cone of bevel gear 1
The resultant force F is acting at the
point of contact C on the tooth which is at
the mid-point along the face of tooth
Let,
Φ = Pressure angle
γ1 = Pitch cone angle of bevel gear 1
rm = PC radius for bevel gear at the mid-point of face width
b = Face width





2
sin 1
1
b
rrmFrm fig.

)1.(..........tants FF 
Consider plane ABCD……From ∆ABC
Consider plane BEGC……From ∆BEC
1
1
cos
sin


sr
sa
FF
FF


From eq.(1), we get Fa & Fr as
)3.(..........costan
)2.(..........sintan
1
1




tr
ta
FF
FF
Eq(1), (2) & (3) are used to determine verious
forces acting on driving gear 1

The forces acting on driven gear 2 can be
determined by considering actions & reactions in equal &
opposite directions.
Therefore,
axial force of driving gear 2 = radial force of driving gear 1
i.e., Fa2 = Fr1
radial force of driving gear2 = axial force of driving gear
i.e., Fr2 = Fa1

Unit ii bevel gears

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Unit ii bevel gears