Spur Gears(2)

Involute Teeth

An involute of a circle is defined in a plane curve generated by a point on a tangent which rolls on the circle without slipping also by a point on a taut string which in unwrapped from a reel as shown in fig. In connection with the toothed wheels then the circle is known as base circle. The involute is traced as follows.

Fig: Construction of involute curve

Let A be the starting point of involute. The base circle is now divided into the equal number of parts e.g. AP₁, P₁P₂, P₂P₃ etc. The tangents at the P₁, P₂, P₃ etc. are drawn and length P₁A₁, P₂A₂, P₃A₃ equal to the arcs AP₁, AP₂ and AP₃ are set off. Joining the points like A, A₁, A₂, A₃. We obtain involute curve AR. A little supposition show that at any instant A₃ the tangent A₃T to involute is normal to P₃A₃ and P₃A₃ is the normal to the involute. Normal at any point of an involute is tangent to the circle.

Fig: Involute teeth

Let O₁ and O₂ be the fixed centres of the two base circles as shown in fig (a). Let corresponding involutes AB and A₁B₁ be in contact at point Q. MQ and NQ are normals to involutes at Q and are tangents to base circles. Since the normal of an involute at a given point is tangent drawn from that point to given base circle. Therefore the common normal MN at Q is also the common tangent to the two given base circles. We see that common normal MN intersects the line of centres O₁O₂ at fixed point P called pitch point. Therefore the involute teeth satisfy fundamental condition of constant velocity ratio.

From similar triangle O₂NP and O₁MP

O₁M/O₂N = O₁P/ O₂P = ω₂/ω₁ ... (i)

Which determines the ratio of the radii of the two base circles. The radii of the base circles is given by

O₁M = O₁Pcosφ and O₂N = O₂ P cosφ

Also the centre distance between the base circles

O₁O₂ = O₁P + O₂P = O ₁M/ cosφ + O₂N/ cosφ

Where φ is the pressure angle or the angle of obliquity. It is the angle which the common normal to the base circles i.e. MN makes with the common tangent to the pitch circles. When the power is being transmitted then the maximum tooth pressure is exerted along the common normal through pitch point. This force resolved into tangential and radial or normal components. These components act along and at right angles to common tangent to pitch circles.

If F is the maximum tooth pressure as shown in fig (b), then

Tangential force, F_T = F cos φ and

Radial or normal force, F_R = F sin φ.

∴ Torque exerted on the gear shaft = F_T × r

Where r is pitch circle radius of the gear.

The tangential force provides the driving torque and the radial or normal force produces radial deflection of rim and bending of shafts.

Effect of Altering the Centre Distance on Velocity Ratio for an Involute Gears teeth

The velocity ratio for the involute teeth gears is given by

O₁M/O₂N = O₁P/ O₂P = ω₂/ω₁... (i)

Let, in fig (a), the centre of rotation of one of the gears is shifted from O₁ to O₁'. Then the contact point shifts from Q to Q '. The common normal to the teeth at point of contact Q ' is tangent to base circle because it has a contact between two involute curves and they are generated from base circle. Let tangent M'N' to base circles intersects O₁′ O₂ at pitch point P’. As a result of this the wheel continues to work correctly.

Now from similar triangles O₂NP and O₁MP,

O₁M/O₂N = O₁P/ O₂P ... (ii)

From similar triangles O₂N'P' and O₁'M'P',

O₁′M′/ O₂N' = O′₁P₁′/ O₂P′ ... (iii)

But O₂N = O₂N', and O₁M = O₁' M'.

Therefore from equations (ii) and (iii),

O₁P/ O₂P= O₁′P′/ O₂P′

Thus we see that if the centre distance is changed within limit then the velocity ratio remains unchanged. However, the pressure angle increases (from φ to φ′) with the increase in the centre distance.

Length of Path of Contact

Fig: Lenth of path of contact

Consider a pinion driving the wheel as shown in fig. When pinion rotates in clockwise direction the contact between a pair of involute teeth begins at K on the flank near base circle of pinion or outer end of tooth face on the wheel and ends at L outer end of the tooth face on pinion or on flank near base circle of wheel. MN is the common normal at the point of contacts and common tangent to base circles. Point K is the intersection of the addendum circle of wheel and common tangent. The point L is intersection of addendum circle of pinion and common tangent. If the wheel is made to act as a driver and directions of motion are reversed then the contact between a pair of teeth begins at L and ends at K.

The length of path of contact is the length of common normal cutoff by the addendum circles of the wheel and pinion. So length of path of contact is KL which is the sum of the parts of the path of contacts KP and PL. The part of path of contact KP is known path of approach and part of the path of contact PL is known as path of recess.

Let

r_A = O₁L = Radius of addendum circle of pinion,

R_A = O₂K = Radius of addendum circle of wheel,

r = O₁P = Radius of pitch circle of pinion, and

R = O₂P = Radius of pitch circle of wheel.

From fig we find that radius of the base circle of pinion,

O₁M = O₁P cos φ = r cos φ

And radius of the base circle of wheel,

O₂N = O₂P cos φ = R cos φ

Now from right angled triangle O₂KN,

\[KN = \sqrt {{{\left( {{O_2}K} \right)}^2} - {{\left( {{O_2}N} \right)}^2}} = \sqrt {{{\left( {{R_A}} \right)}^2} - {{\left( {R\cos \varphi } \right)}^2}} \]

And PN = O₂Psinφ = R sinφ

∴ Length of the part of the path of contact, or the path of approach,

\[KP = KN - PN = \sqrt {{{\left( {{R_A}} \right)}^2} - {{\left( {R\cos \varphi } \right)}^2}} - R\sin \varphi \]

Similarly from right angled triangle O₁ML,

\[ML = \sqrt {{{\left( {{O_1}L} \right)}^2} - {{\left( {{O_1}M} \right)}^2}} = {\sqrt {{{\left( {{r_A}} \right)}^2} - {{\left( {r\cos \varphi } \right)}^2}} ^{}}\]

And MP = O₁Psin φ = rsin φ

∴ Length of the part of the path of contact, or path of recess,

\[PL = ML - MP = \sqrt {{{\left( {{r_A}} \right)}^2} - {{\left( {r\cos \varphi } \right)}^2}} - r\sin \varphi \]

∴ Length of the path of contact,

\[{\rm K}L = KP + PL = \sqrt {{{\left( {{R_A}} \right)}^2} - {{\left( {R\cos \varphi } \right)}^2}} + \sqrt {{{\left( {{r_A}} \right)}^2} - {{\left( {r\cos \varphi } \right)}^2}} - \left( {R + r} \right)\sin \varphi \]

References:
1. H.H. Mabie and C. F. Reinholtz, “Mechanism and Dynamics of Machinery”, Wiley.
2. J.S. Rao & R.V. Dukkipati Mechanisms and Machine Theory, New Age International (P) Limited..
3. J.E. Shigley and J.J. Uicker, Jr., “ Theory of Machines and Mechanisms”, McGraw Hill.
4. B. Paul, “Kinematics and Dynamics of Planar Machinery”, Prentice Hall.
5. C. E. Wilson, J.P. Sadler and W.J. Michels, “Kinematics and Dynamics of Machinery”, Harper Row.