MBD Simulation of a Planetary Gear System

AIM

MBD Simulation of a Planetary Gear System

OBJECTIVES

• To design a Planetary Gear System with the given inputs.

To run a simulation for the following cases.

• Giving Input speed of 200 RPM to Sun Gear and taking Output from Carrier, keeping Ring Gear fixed.
• Giving Input speed of 200 RPM to Ring Gear and taking Output from Carrier, keeping Sun Gear fixed.
• Giving Input speed of 200 RPM to Sun Gear and taking Output from Ring Gear, keeping Carrier fixed.

Plotting the Angular Velocity for Output

INTRODUCTION

A Planetary Gear System,(also known as Epicyclic Gear Train/System) consists normally of a centrally pivoted sun gear, a ring gear and several planet gears which rotate between these.A carrier connects the centers of the gears(sun and planets) and rotates to carry the planet gears or planet pinions, around the sun gear or sun wheel. The planet and sun gears mesh so that their ptch circles roll without slip. A point on the pitch circle of the planet gear traces an epicycloid curve.

This assembly concept explains the term planetary transmission, as the planet gears rotate around the sun gear as in the astronomical sense the planets rotate around our sun.

The advantage of a planetary transmission is determined by load distribution over multiple planet gears. It is thereby possible to transfer high torques utilizing a compact design.

THEORY

A gear train is a mechanical system formed by mounting gears on a frame so the teeth of the gears engage.Gear teeth are designed to ensure the pitch circles of engaging gears roll on each other without slipping, providing a smooth transmission of rotation from one gear to the next.

Some important gear terminologies are needed in order to further understand the simulation-

• Face Width- Length of the Gear teeth in axial direction.
• Base Circle- It is the circle from where involute teeth profile is generated.
• Dedendum / Root Diameter- Root Diameter is the diameter of the gear at root circle or bottom teeth.
• Pitch Circle- It is the imaginary circle on a gear where mating gears meet with each other.
• Pitch Circle Diameter (D)- Diameter of pitch circle is known as pitch circle diameter.
• Addendum / Outside Diameter- It is the diameter of gear at the top tip of gear teeth.
• Addendum- It is the height of gear teeth above pitch circle.
• Dedendum- Height of gear teeth below pitch circle is known as Dedendum.
• Diametral Pitch (p)- Diametral pitch of the gear is the number of teeth per inch of pitch circle diameter. It defines the spacing of teeth along the pitch circle.

p = number of teeth / pitch circle diameter

• Circular Pitch- Circular pitch is the distance between two gear teeth at pitch circle.
• Module- Module of a gear is inverse of diametric pitch. Mathematically it is equal to the ratio of pitch circle diameter and number of teeth. Gears with equal module can only mate with each other.

Module (m) = Pitch circle diameter / Number of teeth=D/T

• Tooth Thickness- It is the width of the tooth at pitch circle diameter.
• Tooth Space- It is the distance between two adjacent teeth at the pitch circle.
• Tooth Face - Surface of the tooth above the pitch surface.
• Flank of Tooth - It is the surface of the tooth below the pitch surface.
• Top Land - Top tooth surface is known as gear top land .
• Pitch Point - In mating gears, pitch point is point of contact between two pitch circles.
• Pressure Angle- Pressure angle is the angle between gear center line and a line perpendicular to line of action. It defines tooth profile. Generally pressure angle value is kept 20°, 25°, 22.5°or 14.5°. Higher pressure angle increases gear strength and lower tooth tip interference. But they cause higher noise and bearing load.
• Clearance - Mathematically clearance is equal to the difference between the dedendum of one gear and the addendum of the mating gear.

Clearance = Dedendum of Gear2 – Addendum of Gear1

• Backlash - It is the tangential space between gear mating teeth at pitch circles.

Backlash = Teeth Space – Teeth Width

Given data-

Ring Gear

Module,m = 2.5 mm

Number of teeth,T_R = 46

Sun Gear

Number of teeth,T_S = 14

Input Speed of the Gear = 200 rpm.

Number of planet gears = 4

So, pitch circle diameters are-

Pitch Circle Diameter of the Ring Gear, D_R = m*T_R = 2.5*46=115mm

Pitch Circle Diameter of the Sun Gear, D_S = m*T_S = 2.5*14=35mm

As, there are 4 planet Gears the arrangement will be as shown in the diagram.

And for proper meshing of gear their pitch circles will be tangent to each other.

So, D_R=D_S+2D_P(where D_P is the pitch circle diameter of planet gears)

or D_R=D_R-D_S/2 =115-35/2=80/2=40mm

As all the gears have same module, so number of teeths in planet gears, T_P=D_P/m=40/2.5=16.

So, the final data required for all the elements are-

Sun Gear- Module=2.5, Number of Teeth=14, Pressure Angle=20, Face width=20, Nominal Shaft Diameter=14

Ring Gear- Module=2.5, Number of Teeth=46, Pressure Angle=20, Face width=20 ,Outside Diameter=130

Planet Gears- Module=2.5, Number of Teeth=16, Pressure Angle=20, Face width=20, Nominal Shaft Diameter=20

The Nominal Shaft Diameter for the Planet gear is the diameter for the extruded part of the carrier.

The Google Drive links for the Parts and Assemblies are given below-

Sun Gear-

https://drive.google.com/file/d/1hYFSkRPEyclx9cgALEINsPRw53cXAXDy/view?usp=sharing

Ring Gear-

https://drive.google.com/file/d/1aoHq3F5ZnJoHx5HazqwOxnkwmqPFM_TT/view?usp=sharing

Planet Gear-

https://drive.google.com/file/d/1hKt4KMXsJj6maPjlumw2jiDEJfgUoqEs/view?usp=sharing

Carrier-

https://drive.google.com/file/d/10a73AiGEderuIMo-eZG0tbnII7COey4r/view?usp=sharing

Assembly-

https://drive.google.com/file/d/1XU2GQZKI7a9nAQCdT-94SrtsFPVRdhE3/view?usp=sharing

https://drive.google.com/file/d/1ORQErsIY8nlZ9BxLvgGP2Bd7Z_tW-YsV/view?usp=sharing

https://drive.google.com/file/d/1LtflgdZALC397a9omNL4zhpXOr89CO7-/view?usp=sharing

MOTION STUDY SETUP

• The Carrier is designed as a part file using the obtained dimensions.
• In SolidWorks Assembly, from Design Library, the required number and types of Gears(Internal Spur Gear and Spur Gear) are chosen from the ANSI Metric System(then Power Transmission) and their required dimensions are input.
• Then, the gears are constrained by the following mates-

1. All the front faces of the gears are made coincident to the Right Plane by using ‘Multiple mate mode’.
2. The inner face of Sun Gear and the outer face of Ring Gear are made concentric
3. The Carrier is inserted into the Planet gears by using the coincident mate on the inner curved surfaces of the gears.
4. The axis of the Carrier and the Sun Gear is made coincident

• Then, Motion Study is conducted in the following steps
• SOLIDWORKS Motion is enabled from the SOLIDWORKS Add-Ins.
• In each of the 3 cases, the Input element is provided a rotary motion by using a Rotary Motor with a constant speed specified as 200RPM
• Then the element to be fixed is fixed by using the ‘Fix’ option.
• Then the contacts between the gears is specified as Solid Body Contact by using ‘Contact’ option and enabling ‘Use contact groups’ for easier selection of contacts for planet gears.Also the materials were selected as ‘Steel(Dry)’.
• A frame rate of 1200 frames per second was used
• Then, the ‘Calculate’ option is used and the simulation was done.
• The required plots were obtained.

RESULTS

Case 1: Input-Sun Gear(200 RPM), Output-Carrier, Ring Gear-Fixed

Plot for Angular Velocity of Carrier

Plot for Angular Velocity of Planet Gear

Case2: Input-Ring Gear(200 RPM), Output-Carrier, Sun Gear-Fixed

Plot for Angular Velocity of Carrier

Plot for Angular Velocity of Planet Gear

Case 3: Input-Sun Gear(200 RPM),Output- Ring Gear, Carrier-Fixed

Plot for Angular Velocity Ring Gear

Plot for Angular Velocity of Planet Gear

EXPLANATION

1. In Case 1, as the Ring Gear is fixed and input is given to Sun Gear, the Sun Gear rotates and as a result rotates all the Planet Gears. As the Ring Gear in contact with all Planet Gears is fixed, the Planet Gears rotate about their own axis as well as around the Sun Gear making the Carrier rotate.So, output is obtained from Carrier.The Angular Velocity of the Carrier is around 287 deg/sec or 47.8 RPM .
2. In Case 2, as the Sun Gear is fixed and input is given to Ring Gear, the Ring Gear rotates and as a result rotates all the Planet Gears. As the Ring Gear in contact with all Planet Gears and the Planet Gears rotate about their own axis as well as around the Sun Gear, but the Sun Gear is fixed.So, the Carrier rotates and output is obtained from Carrier.Angular Velocity of the Carrier is around 942 deg/sec or 157 RPM .
3. In Case 3,the Carrier is fixed and input is given to the Sun Gear.As a result the Sun Gear rotates and provides rotation to all the Planet Gears; but the Planet Gears cannot move around the Sun Gear as Carrier is fixed, so they rotate the Ring Gear. And, output is obtained from Ring Gear.Angular Velocity of the Ring Gear is around 378 deg/sec or 63 RPM .

The results and values obtained from the motion study are checked against the theoretical results.

Step No.	Conditions of Motion	Revolutions of Elements
		Carrier	Sun Gear	Planet Gears	Ring Gear
1.	Carrier is fixed, Sun Gear is given +1 revolution(i.e. 1 rev counter clockwise)	`0`	`+1`	`-frac{T_S}{T_P}`	`-frac{T_S}{T_P}timesfrac{T_P}{T_R}=frac{T_S}{T_R}`
2.	Carrier is fixed, Sun Gear is given +x revolutions	`0`	`+x`	`-xtimesfrac{T_S}{T_P}`	`-xtimesfrac{T_S}{T_R}`
3.	Adding +y revolutions to all elements	`+y`	`+y`	`+y`	`+y`
4.	Total motion	`+y`	`x+y`	`y-xtimesfrac{T_S}{T_P}`	`y-xtimesfrac{T_S}{T_R}`
5.	Total motion with given data	`+y`	`x+y`	`y-frac{14}{16}x`	`y-frac{14}{46}x`

Case 1: Input-Sun Gear(200 RPM), Output-Carrier, Ring Gear-Fixed

As input is given to Sun Gear i.e. 200 RPM(counter clockwise, as positive)

So, `x+y=200`

Also Ring Gear is Fixed

So, `y-frac{14}{46}x=0`

Solving these equations, we get

x= 460/3=153.33 RPM or 920 deg/sec

y= 140/3=46.67 RPM or 280 deg/sec

So, speed of Planet Gears N_P= `y-frac{14}{16}x`= -87.5 RPM or -525 deg/sec( clockwise)

Speed of Carrier, N_C= y =46.67 RPM or 280 deg/sec(counter clockwise)

Case2: Input-Ring Gear(200 RPM), Output-Carrier, Sun Gear-Fixed

As input is given to Ring Gear i.e. 200 RPM(counter clockwise, as positive)

So, `y-frac{14}{16}x=200`

Also Sun Gear is Fixed

So,`x+y=0`

Solving these equations, we get

x= -460/3=-153.33 RPM or -920 deg/sec

y= 460/3=153.33 RPM or 920 deg/sec

So, speed of Planet Gears N_P= `y-frac{14}{16}x`=287.5 RPM or 1725 deg/sec(counter clockwise)

Speed of Carrier, N_C= y =153.33 RPM or 920 deg/sec(counter clockwise)

Case 3: Input-Sun Gear(200 RPM), Output- Ring Gear, Carrier-Fixed

As input is given to Sun Gear i.e. 200 RPM(counter clockwise, as positive)

So, `x+y=200`

Also Carrier is Fixed

So, `y=0`

Solving these equations, we get

x=200 RPM or 1200 deg/sec

So, speed of Planet Gears N_P= `y-frac{14}{16}x`= -175 RPM or -1050 deg/sec(clockwise)

Speed of Ring Gear, N_R= `y-frac{14}{46}x`= -60.87 RPM or -365.217 deg/sec(clockwise)

CONCLUSION

So, the Motion Analysis was successfully conducted and the working of the Planetary Gear System was studied.