Theory of Machines : Gears and Gear Trains

Gears- Gears are used to transmit motion from one shaft to another or between a shaft and slide. It is a higher pair mechanism. Gear use to intermediate link or connector and transmit the motion by direct contact.

Gear Terminology

• Circular Pitch (p): It is a distance measured along the circumference of the pitch circle from a point on one tooth to the corresponding point on the adjacent tooth.

       where, p = circular pitch

          d = pitch diameter

          T = number of teeth

• Diametrical Pitch (P): It is the number of teeth per unit length of the pitch circle diameter in inches.

• Module (m): It is the ratio of pitch diameter in mm to the number of teeth. The term is used SI units in place of diametrical pitch.

• Gear Ratio (G): It is the ratio of the number of teeth on the gear to that on the pinion.

        where, T = number of teeth on the gear

            t = number of teeth on the pinion

• Velocity Ratio: The velocity ratio is defined as the ratio of the angular velocity of the follower to the angular velocity of the driver gear.

• Addendum Circle: A circle bounding the ends of the teeth, in a normal section of the gear.
• Dedendum Circle: The circle bounding the spaces between the teeth, in a normal section of the gear.
• Addendum: The radial distance between the pitch circle and the addendum circle.
• Dedendum: The radial distance between the pitch circle and the root circle,
• Standard Proportions:  The following standard proportions are used in 20° full depth system
  • Addendum h_a = m
  • Clearance C = 0.25 m
  • Whole depth h = 2.25 m
  • Tooth space = 1.5700 m
  • Dedendum h_f = 1.25 m
  • Working depth h_k = 2m
  • Tooth thickness (s) = 1.5708 m
  • Fillet radius = 0.4 m
• Form of Teeth: The form of teeth in a gear system can be as follows.
• Conjugate Teeth: Two curves of any arbitrary shape that fulfill the law of gearing can be used as the profiles of teeth. Such a gear The profiles of tooths which are generally used given as
• Cycloidal Profile Teeth: As cycloid is the locus of a point on the circumference of a circle that rolls without slipping on a fixed straight line In this type, the faces of the teeth are epicycloids and the flanks the hypocycloids.
• Epicycloid Teeth: An epicycloid is the locus of point on the circumference of a circle that rolls without slipping on a fixed straight line.

Arc PK = arc KJG

• Hypocycloid Teeth: A hypocycloid is the locus of a point on the circumference of a circle that rolls without slipping inside the circumference of another circle.

Arc AC = Arc CD

• Involute Profile Teeth: An involute is defined as the locus of a point on a straight line which rolls without slipping on the circumference of a circle. From figure at the start the tracing point is at A as the rolls on the circumference of the circle, the path ABC traced out by the point A is the involute.D is regarded as the instantaneous centre of rotation of B. The motion of B is perpendicular to BD. Since, BD is tangent to the base circle, the normal to the involute is a tangent to the base circle.

Comparison between Cycloidal Teeth and Involute Teeth

• Cyclodial Teeth
  • Pressure angle maximum at beginning of engagement, then reduce to zero at pitch point and again maximum at end of engagement.
  • Less smooth running of gears.
  • Involve double curve, epicycloid and hypocycloid for the teeth
  • Costier manufacturing
  • Exact centre distance is reuired to transmit a constant velocity ratio.
  • Phenomenon of interference does not occur at all.
  • Spreading flank teeth and strong.
  • Convex flank always has contact with a concave face resulting in less wear.
• Involute Teeth
  • Pressure angle is constant throughtout the engagement of teeth.
  • Smooth running of gears.
  • Involve single curve
  • Cheaper manufacturing
  • A little variation in centre distance does not effect the velocity ratio
  • Interference can occur if the condition of minimum number of teeth on a gear is not followed.
  • Raidal flank teeth and weaker compare to cycloidal.
  • Two convex surfaces are in contact and thus there is more wear.

Methods of Gear Manufacturing

• Casting, blanking, machining.
• For power transmitting gears (made of steel), the following methods generally used are milling, rack generation, hobbing, fellow gear shaper method.
• For good accuracy and surface finish following finishing process are used grinding, lapping, sharing, barnishing.

The followings are the parameters used in manufacturing of gears as

• Path of Contact (CD): According to the figure, the path of the contact is given by the relation

      where, r = pitch circle radius of pinion

          R = pitch circle radius of wheel

          r_a = addendum circle radius of pinion

         R_a = Addendum circle radius of wheel

• Arc of Contact: From the figure, the arc of the contact as given by

Angle of action δp = arc of contact 

• Interference in Involute Gear
  • Meshing of two non-conjugate (non-involute) teeth is known as interference because the two teeth do not slide properly and thus rough action and binding occurs.
  • For equal addenda of the wheel and pinion, the addendum radius of the wheel decides whether the interference will occur or not.
• Minimum Number of Teeth
  • Maximum value of the addendum radius of wheel to avoid interference can be upto BE

• Addendum value of wheel

        where, G = gear ratio= 

                  T and t are number of teeth on the wheel and pinion respectively.

         When addendum is equal to one module A = 1m

• Interference between Rack and Pinion: The interference between rack and pinion is shown in the figure.

To avoid interference GE ≥ mA_r

For standard addendum, 

    To eliminate interference, there are following methods

1. By using more teeth on gears (however, it can make gear noiser).
2. By larger pressure angle.
3. By using gears with shorter teeth.
4. To eliminate the undercutting or interference, the pinion addendum may be enlarged until the addendum circle of the pinion passes through the interference point of gear.

• Undercutting: When gear teeth are produced such that interference is eliminated then this effect is called undercutting.
• Long and Short addendum System: In the long and short addendum system, there is no change in pitch circle and consequently not in pressure angle. The pinion addendum enlarge until the addendum circle of the pinion passes through the point of gear. In long and short addendum system, recess action is obtained more than approach action.

Gear Train: A gear train is a combination of gears used to transmit motion from one shaft to another. Gear trains are used to speed up or stepped down the speed of driven shaft. The following are main types of gear trains.

• Simple Gear Train: Series of gears, capable of receiving and transmitting motion from one gear to another is called a simple gear train.

Train value 

Speed ratio 

        The intermediate gears have no effect on the speed ratio and therefore they are known as idlers.

• Compound Gear Train: When a series of gears are connected in such a way that two or more gears rotate about an axis with the same angular velocity.

Train value 

• Reverted Gear Train: If the axis of first and last wheel of a compound gear coincide, it is called as reverted gear train.

Train valve, 

        If r is the pitch circle radius of a gear

r₁ + r₂ = r₃ + r₄

• Planetary or Epicyclic Gear Train: A gear train having a relative motion of axes is called a planetary or an epicyclic gear train. In an epicyclic train, the axis of at least one of the gears also moves relative to the frame. If the arm a is fixed the wheels S and P constitute a simple train. However if the wheel S is fixed so that arm a can rotate about the axis of S. The P would be moved around S therefore it is an epicyclic train.

• In general, gear trains have 2 degree of freedom.

• Torque in epicyclic Trains: Let NS, Na, Np, and NA be the speeds and TS, Ta, Tp, and TA the external transmitted by S, a, P and A

∑T = 0

T_S + T_a + T_p + T_A = 0

       Since planet P can rotate on its our pin fixed to a but is not connected to anything outside.

T_p = 0

T_S + T_a + T_A = 0

       Assuming no loss in power transistor

∑Tω = 0

T_SN_S + T_aN_a + T_AN_A = 0

       If efficiency η to deliver power to S.

• Tabular Method: In tabular method, we apply following procedure. Assume gear makes m revolution and arm makes n revolution. Consider clockwise rotation is positive and anticlockwise rotation is negative. Analysis of Tabular Method

        On the basis of given condition, we can determine the values m and n. For example, if gear A is at rust, then