Gears and Gear Trains Introduction Gears are used to transmit motion from one shaft to another or between a shaft and a slide. This is accomplished by successfully engaging teeth. Types of Gears 1. Parallel Shafts Gear (large gear) Spur gears – tooth profile is parallel to the axis of rotation, transmits motion between parallel shafts.
Pinion (small gear) Internal gears Spur Rack and Pinion sets – a special case of spur gears with Pinion the gear having an infinitely large diameter, the teeth are laid flat. Parallel Shafts Helical gears – teeth are inclined to the axis of rotation, the angle provides more gradual engagement of the teeth during meshing, transmits motion between parallel shafts. Herringbone gears- To avoid axial thrust, two helical gears of opposite hand can be mounted side by side, to cancel resulting thrust forces Herringbone gears are mostly used on heavy machinery. Intersecting shafts Bevel gears – teeth are formed on a conical surface, used to transfer Straight motion between non-parallel and bevel intersecting shafts.
Straight bevel gear gears make line of contact similar to spur gears Spiral bevel gears- smoother in Spiral action and quieter than straight bevel bevel gears. gear Zero Bevel Gear- In this bevel gear spiral angle is zero at the middle of the face width 3. Skew shafts (non- parallel and non- intersecting) In case of Skew shafts (Non- parallel- non-intersecting) a uniform rotary motion is not possible as in case of parallel and intersecting shafts which has pure rolling contact Worm gear sets – consists of a helical gear and a power screw (worm), used to transfer motion between non-parallel and non- intersecting shafts. Crossed Helical gears- Applicable to light load conditions.
Used to drive feed mechanisms in machine tools, camshafts, and small IC engines 3. Skew shafts (non- parallel and non- intersecting) Nomenclature Smaller Gear is Pinion and Larger one is the gear In most application the pinion is the driver, This reduces speed but it increases torque. Terminology Pitch circle, theoretical circle upon which all calculation is based Pc, Circular pitch is the distance from one teeth to the next, along the pitch circle. Note : If D1 and D2 are the diameters of the two meshing gears having the teeth T1 and T2 respectively, then for them to mesh correctly, m, module=d/T , pitch circle diameter/number of teeth Therefore Circular pitch, Pc= πm Pd, Diametral Pitch , Number of teeth per unit length, Pd =T/D= t/d=1/m Pc Pd = π Backlash It is the difference between the tooth space and the tooth thickness, as measured along the pitch circle.
Theoretically, the backlash should be zero, but in actual practice some backlash must be allowed to prevent jamming of the teeth due to tooth errors and thermal expansion. Velocity Ratio (VR) The velocity ratio is defined as the ratio of the angular velocity of the driven gear to the angular velocity of the driving gear VR= ω2 / ω1 (D= diameter of driven gear; d= diameter of driving gear) = N2 / N1 (ω= 2πN) =d /D (V=π d N1 = π D N2) =t/T (p= π d/ T1 = π D / T2 ) Pressure line and pressure angle •The standard pressure angles are 14 1/2 ° and 20° Pressure line CP- path of approach PD- Path of recess Arc of contact- arc APB or EPF Arc of approach- AP or EP Arc of recess- PB or PF Angle of action (δ)= Angle of approach(α) + Angle of recess (β) Angle of action (δ) Arc of contact Contact ratio= = Pitch angle (𝛾) Circular pitch [Pitch angle (𝛾) - Angle subtended by circular pitch at the center of gear] Contact ratio Number of pairs of teeth in contact Angle of action (δ) Arc of contact Contact ratio= = Pitch angle (𝛾) Circular pitch Contact ratio should be always greater than unity to ensure continuous transmission of motion, for at least one pair of teeth should be in meshed condition for the mating gears. If contact ratio is 1.6, it means that one pair of teeth is always in contact whereas two pairs of teeth are in contact for 60% of the time. FORMS OF TEETH 1.Involute profile Cycloid Epicycloid Hypocycloid Epicycloidal and hypocycloidal teeth profile Construction of cycloidal teeth for Rack In Fig.
(a), the fixed line or pitch line of a rack is shown. When the circle C rolls without slipping above the pitch line in the direction as indicated in Fig. (a), then the point P on the circle traces epi-cycloid PA. This represents the face of the cycloidal tooth profile.
When the circle D rolls without slipping below the pitch line, then the point P on the circle D traces hypo-cycloid PB, which represents the flank of the cycloidal tooth. The profile BPA is one side of the cycloidal rack tooth. Similarly, the two curves P' A' and P'B' forming the opposite side of the tooth profile are traced by the point P' when the circles C and D roll in the opposite directions. Construction of cycloidal teeth for gear In the similar way, the cycloidal teeth of a gear may be constructed as shown in Fig.
The circle C is rolled without slipping on the outside of the pitch circle and the point P on the circle C traces epi-cycloid PA, which represents the face of the cycloidal tooth. The circle D is rolled on the inside of pitch circle and the point P on the circle D traces hypo-cycloid PB, which represents the flank of the tooth profile. The profile BPA is one side of the cycloidal tooth. The opposite side of the tooth is traced as explained above.
Involute profile Involute Teeth An involute of a circle is a plane curve generated by a point on a tangent, which rolls on the circle without slipping or by a point on a taut string which in unwrapped from a reel as shown in Fig. In connection with toothed wheels, the circle is known as base circle. The involute is traced as follows : A3, the tangent A3T to the involute is perpendicular to P3A3 and P3A3 is the normal to the involute. In other words, normal at any point of an involute is a tangent to the circle.
Comparison Between Involute and Cycloidal Gears In actual practice, the involute gears are more commonly used as compared to cycloidal gears, due to the following advantages : Advantages of involute gears The most important advantage of the involute gears is that the centre distance for a pair of involute gears can be varied within limits without changing the velocity ratio. This is not true for cycloidal gears which requires exact centre distance to be maintained. In involute gears, the pressure angle, from the start of the engagement of teeth to the end of the engagement, remains constant. It is necessary for smooth running and less wear of gears.
But in cycloidal gears, the pressure angle is maximum at the beginning of engagement, reduces to zero at pitch point, starts decreasing and again becomes maximum at the end of engagement. This results in less smooth running of gears. The face and flank of involute teeth are generated by a single curve where as in cycloidal gears, double curves (i. epi-cycloid and hypo-cycloid) are required for the face and flank respectively.
Thus the involute teeth are easy to manufacture than cycloidal teeth. In involute system, the basic rack has straight teeth and the same can be cut with simple tools. Note : The only disadvantage of the involute teeth is that the interference occurs with pinions having smaller number of teeth. This may be avoided by altering the heights of addendum and dedendum of the mating teeth or the angle of obliquity of the teeth.
Advantages of cycloidal gears Following are the advantages of cycloidal gears : Since the cycloidal teeth have wider flanks, therefore the cycloidal gears are stronger than the involute gears, for the same pitch. Due to this reason, the cycloidal teeth are preferred specially for cast teeth. In cycloidal gears, the contact takes place between a convex flank and concave surface, whereas in involute gears, the convex surfaces are in contact. This condition results in less wear in cycloidal gears as compared to involute gears.
However the difference in wear is negligible. In cycloidal gears, the interference does not occur at all. Though there are advantages of cycloidal gears but they are outweighed by the greater simplicity and flexibility of the involute gears. Law of Gearing Type equation here.
𝜔1 𝐵𝐹 𝐵𝑃 𝐹𝑃 = = = 𝜔2 𝐴𝐸 𝐴𝑃 𝐸𝑃 Velocity of sliding Path of contact Arc of contact Arc of approach PP’ + Arc of recess PP” Arc of approach PP’ Arc of recess PP” Interference in involute gears • Mating of two non- conjugate (non- involute)teeth is known as interference • Two teeth does not slide properly and thus rough action and binding occurs • Contacting teeth have different velocities which can lock the two gears • The points E and F are called interference points Maximum possible Addendum value for teeth to avoid interference Minimum Number of Teeth Required to avoid interference Interchangeable gears Gears are interchangeable if they have: • The same module • The same pressure angle • The same addendum and dedendum • The same thickness Non-Standard Gears • Centre- distance Modifications - no. of teeth on a pinion can be reduced by increasing C-C distance and by changing marginally the tooth proportions and pressure angle. - Reduction in interference and improvement in contact ratio • Clearance Modifications - can be increased to 1.3 times module to 1.4 times module to have a larger fillet at the root of the tooth. As a result fatigue strength is increased • Addendum modifications - if this modification is carried out then there has to be no change in pitch circle radius and pressure angles Helical and spiral gears - Depending on the direction in which helix slopes away 1.
Left handed Terminology of helical gears Velocity ratio of Helical gears Centre to centre distance in helical gears In case of helical gears of parallel shafts: Worm and worm gear To transmit a higher load than usual spiral gears a worm and worm gear can be used. Large speed reduction can be also possible. Worm and Worm Gear Terminologies ψ Single start Double start In case of worms, the lead angle is very small and the helix angle approaches 900. As the shaft axis of the worm and worm gear are 90 deg., lead angle of worm = helix angle of the gear wheel i., Axial pitch of worm = Circular pitch of wheel Velocity Ratio Lead is also the distance turned by the pitch circle of the worm gear.