written 7.2 years ago by | • modified 6.9 years ago |
Determine:
sliding velocities at engagement at an disengagement of pair of a teeth, and
Contact ratio.
Subject: Kinematics of Machinery
Topic: Gears and Gear Trains
Difficulty: Medium
written 7.2 years ago by | • modified 6.9 years ago |
Determine:
sliding velocities at engagement at an disengagement of pair of a teeth, and
Contact ratio.
Subject: Kinematics of Machinery
Topic: Gears and Gear Trains
Difficulty: Medium
written 7.1 years ago by |
Given ϕ=20∘; t= 30; T= 50; m=4; N= 1000 rpm; or ω1=2π×(1000/60)=104.7rad/s.
1.Sliding velocities at engagement and at disengagement of pair of a teeth
First of all, let us find the Radius of the addendum circles of the smaller gear and the larger gear.
We know that,
Addendum of the smaller gear,
=mt2[√1+Tt(Tt+2)sin2ϕ−1]
=4×302[√1+5030(5030+2)sin220∘−1]
=60(1.31−1)=18.6mm
And Addendum of the larger gear,
=mt2[√1+tT(tT+2)sin2ϕ−1]
=4×502[√1+3050(3050+2)sin220∘−1]
=100(1.09−1)=9 mm
Pitch circle Radius of the smaller gear,
r= mt/2=4(30/2)=60 mm
Radius of the Addendum circle of the smaller gear,
rA=r+Addendum of the smaller gear= 60+18.6=78.6 mm
Pitch circle Radius of the larger gear,
R=mt/2=4(50/2)=100 mm
Radius of addendum circle of the larger gear,
RA=R+Addendum of the larger gear = 100+9=109 mm
We know that the path of approach(i.e path of contact when engagement occurs),
KP=√(RA)2−R2cos2ϕ−Rsinϕ
=√(109)2−(100)2cos220∘−100sin20∘=55.2−34.2=21mm
and the path of recess(i.e path of contact when disengagement occurs),
PL=√(rA)2−r2cos2ϕ−rsinϕ
=√(78.6)2−(60)2cos220∘−60sin20∘=54.76−20.52=34.24mm
Let ω2 = Angular speed of the larger gear in rad/s
We know that ω1ω2=Tt or ω2=ω1×tT=10.47×3050=62.82rad/s
Sliding velocity at engagement of a pair of teeth
=(ω1+ω2)KP=(104.7+62.82)21=3518 rad/s
=3.518 m/s
And Sliding velocity at Disengagement of a pair of teeth
=(ω1+ω2)PL=(104.7+62.82)34.24=5736 rad/s
=5.736 m/s
2. Contact Ratio: We know that the length of the arc of contact
=Length of the path of contact/ cosϕ
=KP+PLcosϕ=21+34.24cos20∘=58.78 mm
And circular pitch=π×m=3.142×4=12.568mm
Contact Ratio= Length of arc of contact / circular pitch= 58.78/12.568=4.67 say 5