Simple and Planetary gear trains(2)

Reverted Gear Train

When the axes of first gear i.e. first driver and last gear i.e. last driven or follower are co-axial then the gear train is known as reverted gear train which is shown in fig. We see that gear 1 i.e. first driver drives the gear 2 i.e. first driven in opposite direction. Since the gears 2 and 3 are mounted on the same shaft. Therefore they form a compound gear and gear 3 will rotate in the same direction as that of gear 2. The gear 3 which is now second driver drives the gear 4 i.e. the last driven in the same direction as that of the gear 1. So that in reverted gear train the motion of first gear and the last gear is like.

Fig: Reverted Gear Train

Let

T₁ = Number of teeth on the gear 1

r₁= Pitch circle radius of gear 1

N₁ = Speed of gear 1 in r.p.m.

Similarly,

T₂, T₃, T₄ = Number of teeth of respective gears

r₂, r₃, r₄ = Pitch of circle radii of respective gears

N₂, N₃, N₄ = Speed of the respective gears in r.p.m.

Since distance between the centres of the shafts of gears 1 and 2 as well as gears 3 and 4 is same. Therefore

r₁ + r₂ = r₃ + r₄ ....................(a)

Also, the circular pitch or module of all the gears is supposed to be same. Therefore number of teeth on each of gear is directly proportional to its circumference radius.

∴ T₁ + T₂ = T₃ + T₄ ........................(b)

\[Speed{\rm{ }}ratio = \frac{{Speed{\rm{ }}of{\rm{ }}the{\rm{ }}first{\rm{ }}driver}}{{Speed{\rm{ }}of{\rm{ }}last{\rm{ }}driven}} = \frac{{\Pr oduct{\rm{ }}of{\rm{ }}No.{\rm{ }}of{\rm{ }}teeth{\rm{ }}on{\rm{ }}driven}}{{\Pr oduct{\rm{ }}of{\rm{ }}No.{\rm{ }}of{\rm{ }}teeth{\rm{ }}on{\rm{ }}driver}}\]

N₁/N₄ = (T₂/T₁)*( T₄/T₃)..................................(c)

From equations (a), (b) and (c) that we can find the number of teeth on each gear for given centre distance with speed ratio and module only when there number of teeth on one gear is chosen. The reverted gear trains are applicable in the field of automotive transmissions, industrial speed reducers, lathe back gears and in clocks where minute and hour hand shafts that which are co-axial.

Epicyclic Gear Train

In the epicyclic gear train the axes of the shafts over which the gears are mounted that may move relative to a fixed axis. A simple epicyclic gear train is shown in the fig where a gear A and arm C have a common axis at O₁ about they rotate. The gear B mesh with gear A and has its axis on the arm at O₂ about which the gear B rotate. If the arm is fixed then the gear train is simple and gear A can drive gear B or vice versa. But if gear A is fixed and the arm is rotated about axis of gear A i.e. O₁ then the gear B is forced to rotate upon on it and around gear A. Such type of motion is called epicyclic and the gear trains arranged in that a manner that one or more than one of their members shift upon on it and around another member which are known as epicyclic gear trains i.e. epi. means upon and cyclic means around. The epicyclic gear trains may be the simple or compound.

The epicyclic gear trains that are applicable for transmitting high velocity ratios with the gears of definite moderate size in a comparatively lesser space. The epicyclic gear trains are also used in back gear of the lathe, differential gears of automobiles, pulley blocks, hoists and wrist watches etc.

Velocity Ratio of an Epicyclic Gear Train

The two methods may be used for determine the velocity ratio of an epicyclic gear train.

• Tabular method and
• Algebraic method

Tabular method

Consider an epicyclic gear train shown in fig.

Let

T_A = Number of teeth of gear A

T_B = Number of teeth on gear B

First of all let suppose that arm is fixed. Therefore the axes of both the gears are also fixed relative to each other. When the gear A makes one revolution in the anticlockwise direction then the gear B will make T_A / T_B revolutions clockwise. Assuming anticlockwise rotation as positive and clockwise as negative. We may say that when gear A makes + 1 revolution then the gear B make (–T_A / T_B) revolutions. This relative motion is entered in the first row of the table.

Secondly if gear A makes + x revolutions then gear B will make – x × T_A / T_B revolutions. This Inside view of a car engine. This statement is in the second row of the table. In other words multiply the each motion entered in first row by x.

Thirdly each element of given epicyclic train is given + y revolutions and entered in third row. Finally the motion of each element of gear train is added up and entered in fourth row.

Table: Montion table

A little supposition will show that when two conditions of about the motion of rotation of any two elements are known then the unknown speed of the third element obtained by substituting the given data in the third column of the fourth row.

Algebraic method

In this method the motion of each element of given epicyclic train relative to the arm is set down in form of equations. The number of equations depends upon the number of elements in given gear train. But two conditions are mostly supplied in any epicyclic train viz. some element is fixed and other has specified motion. These two conditions are enough to solve all the equations and hence to determine the motion of any element in epicyclic gear train.

Let an arm C is fixed in an epicyclic gear train which is shown in fig. Therefore speed of the gear A relative to the arm C

= N_A – N_C

and speed of the gear B relative to the arm C,

= N_B – N_C

Since the gears A and B are meshing directly, therefore they will revolve in opposite directions.

(N_B – N_C )/(N_A – N_C) = (–T_A / T_B)

Since the arm C is fixed, therefore its speed, N_C = 0.

∴ N_B/ N_A = –T_A / T_B

If the gear A is fixed, then N_A = 0.

(N_B – N_C )/(0 – N_C) = (–T_A / T_B)

∴ N_B/ N_A = 1 + T_A / T_B

We know that the tabular method is easier and hence mostly used in solving problems on epicyclic gear train.

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.