Motion Analysis of a Planetary Gear System

OBJECTIVE

To perform motion analysis of the planetary gear with a sun gear, a ring gear, a carrier and four planet gears.

INTRODUCTION

A planetary gear train (also known as epicyclic gear) consists of two gears mounted so that the centre of one gear revolves around the centre of the other. A carrier connects the centres of the two gears and rotates to carry one gear, called the planet gear or planet pinion, around the other, called the sun gear or sun wheel. The planet and sun gears mesh so that their pitch circle roll without slip. An epicyclic gear train can be assembled so the planet gear rolls on the inside of the pitch circle of a fixed, outer gear ring, or ring gear sometimes called an annular gear.

Planetary gear trains provide high power density in comparison to standard parallel axis gear trains. They provide a reduction in volume, multiple kinematic combinations, purely torsional reactions, and coaxial shafting. The efficiency loss in a planetary gear train is low. This type of efficiency ensures that a high proportion (about 97%) of the energy being input is transmitted through the gearbox, rather than being wasted on mechanical losses inside the gearbox. The load in a planetary gear train is shared among multiple planets; therefore, torque capability is greatly increased. The more planets in the system, the greater the load ability and the higher the torque density.

The method of motion of a planetary gear structure is different from traditional parallel gears. Traditional gears rely on a small number of contact points between two gears to transfer the driving force. In this case, all the loading is concentrated on a few contacting surfaces, making the gears wear quickly and sometimes crack. But the planetary speed reducer has multiple gear contacting surfaces with a larger area that can distribute the loading evenly around the central axis. Multiple gear surfaces share the load, including any instantaneous impact loading, evenly, which make them more resistant to damage from higher torque. The housing and bearing parts are also less likely to be damaged and crack from high loading.

THEORY

Gear Terminologies:

Face Width: It is the length of teeth in the axial direction.

Pitch circle: When gear sets are visualized as a pair of smooth contacting discs, the pitch circles of the gear corresponds to the diameter of the discs. Pitch circles of mating gears are tangent to each other and have identical linear velocities. For this reason, pitch circles, rather than base circles, are used in many gear calculations.

Addendum: It is the height of the tooth above the pitch circle.

Dedendum: It is the depth of the tooth below the pitch circle.

Outside Diameter od Addendum Circle: The outside diameter of a gear is the diameter of the addendum (tip) circle.

Root Circle: The root circle coincides with the bottoms of the tooth spaces. It is the diameter of the dedendum circle.

Base Circle: It is the circle from which the involute portion of the tooth profile is generated.

Diametrical Pitch: The Diametral Pitch (DP) is the number of teeth per inch of diameter of the pitch circle.

`\"Diametrical Pitch = DP=\" \"Number of teeth(n)\"/\"Pitch Circle Diameter(PD)\"`

Circular Pitch: The circular pitch defines the width of one tooth and one gap measured on an arc on the pitch circle; in other words, this is the distance on the pitch circle from a point on one tooth to the corresponding point on the adjacent tooth.

`\"Circular Pitch = CP=\" pi/\"DP\"`

Module: The module is the measure of gear tooth size which is normally used for metric system gears. The module is the pitch circle diameter divided by the number of teeth. For two gears to mesh, modules must be equal.

`\"Module= MM =\" \"Pitch Circle Diameter(PD)\"/\"number of teeth(n)\"`

Face: That part of the tooth surface lying outside the pitch surface.

Flank: The part of the tooth surface lying inside the pitch surface.

Pressure Angle: Pressure angle in relation to gear teeth, also known as the angle of obliquity, is the angle between the tooth face and the gear wheel tangent. It is more precisely the angle at a pitch point between the line of pressure (which is normal to the tooth surface) and the plane tangent to the pitch surface.

To calculate the speed:

Step No.	Condition of Motion	Revolutions of elements
Arm	Sun Gear	Planet Gear	Ring Gear
1	Arm fixed-sun gear rotates through + 1 revolution (i.e. 1 rev. anticlockwise)	0	+1	`-T_(SUN)/T_(PLANET)`	`-T_(SUN)/T_(PLANET)*T_(PLANET)/T_(RING) =-T_(SUN)/T_(RING)`
2	Arm fixed-gear C rotates through + x revolutions	0	+x	`-x.T_(SUN)/T_(PLANET)`	`-x.T_(SUN)/T_(RING)`
3	Add + y revolutions to all elements	+y	+y	+y	+y
4	Total motion	+y	x+y	`y-x.T_(SUN)/T_(PLANET)`	`y-x.T_(SUN)/T_(RING)`

TABLE 1: Table of Motions

METHODOLOGY

Ring Gear:

Module= 2.5 mm

Number of teeth= 46

Pitch Circle Diameter of Ring Gear= D1= Number of teeth X Module= 46 x2.5= 115 mm

Sun Gear:

Module= 2.5 mm

Number of teeth= 14

Pitch Circle Diameter of Ring Gear= D3= Number of teeth X Module= 14 x2.5= 35 mm

Planet Gear:

Module= 2.5 mm

Pitch Circle Diameter of Planet Gear= D2= (D1/2)-(D3/2)= 57.5-17.5= 40 mm

Number of teeth= Pitch Circle Diameter/ Module= 40/2.5= 16

Number of Planetary gear= 4

Input Speed= 200 rpm

RESULTS

Case 1: Fixed: Ring Gear, Input: Sun Gear=200 rpm, Output: Carrier

Figure: Angular Velocity of Sun Gear in deg/s

Figure: Angular Velocity of Planet Gears in deg/s

Figure: Angular Velocity of Ring Gear in deg/s

Figure: Angular Velocity of Carrier in deg/s

Case 2: Fixed: Sun Gear, Input: Ring Gear=200 rpm, Output: Carrier

Figure: Angular Velocity of Sun Gear in deg/s

Figure: Angular Velocity of Planet Gears in deg/s

Figure: Angular Velocity of Ring Gear in deg/s

Figure: Angular Velocity of Carrier in deg/s

Case 3: Fixed: Carrier, Input: Sun Gear=200 rpm, Output: Ring Gear

Figure: Angular Velocity of Sun Gear in deg/s

Figure: Angular Velocity of Planet Gears in deg/s

Figure: Angular Velocity of Ring Gear in deg/s

Figure: Angular Velocity of Carrier in deg/s

CALCULATIONS

From Table 1,

Step No.	Condition of Motion	Revolutions of elements
Arm`(N_(ARM))`	Sun Gear `(N_(SUN))`	Planet Gear`(N_(PLANET))`	Ring Gear`(N_(RING))`
1	Arm fixed-sun gear rotates through + 1 revolution (i.e. 1 rev. anticlockwise)	0	+1	`-14/16`	`-14/46`
2	Arm fixed-gear C rotates through + x revolutions	0	+x	`-x*14/16`	`-x*14/46`
3	Add + y revolutions to all elements	+y	+y	+y	+y
4	Total motion	+y	x+y	`y-x*14/16`	`y-x*14/46`

Table 2: Table of Motions

Case 1: Fixed: Ring Gear, Input: Sun Gear=200 rpm, Output: Carrier

Speed of Sun Gear= 1200 deg/s, Speed of Ring Gear=0(Fixed).

`N_(SUN)=1200=x+y`

`N_(RING)=0=y-x*14/46`

Solving we get

x=920, y=280

`N_(ARM)=y=280\" deg/s\"`

`N_(SUN)=x+y=1200\" deg/s\"`

`N_(PLANET)=y-x*14/16=280-920*14/16= -525 \" deg/s (-ve indicates opposite direction)\"`

`N_(RING)=y-x*14/46=0\" deg/s\"`

Case 2: Fixed: Sun Gear, Input: Ring Gear=200 rpm, Output: Carrier

Speed of Sun Gear=0(Fixed), Speed of Ring Gear=1200 deg/s.

`N_(SUN)=0=x+y`

`N_(RING)=1200=y-x*14/46`

Solving we get

x=-920, y=920

`N_(ARM)=y=920\" deg/s\"`

`N_(SUN)=x+y=0\" deg/s\"`

`N_(PLANET)=y-x*14/16=920+920*14/16= 1725 \" deg/s (-ve indicates opposite direction)\"`

`N_(RING)=y-x*14/46=920+920*14/46=1200\" deg/s\"`

Case 3: Fixed: Carrier, Input: Sun Gear=200 rpm, Output: Ring Gear

Speed of Carrier=0(Fixed), Speed of Sun Gear=1200 deg/s.

`N_(ARM)=y=0\" deg/s\"`

`N_(SUN)=x=1200\" deg/s\"`

`N_(PLANET)=y-x*14/16=0-1200*14/16= -1050 \" deg/s (-ve indicates opposite direction)\"`

`N_(RING)=1200=y-x*14/46=0-1200*14/46=-365.21\" deg/s (-ve indicates opposite direction)\"`

CONCLUSION

Thus both theoretical and simulation results are concurring.