MOTION ANALYSIS OF PLANETARY GEAR AND VERIFICATION OF RESULTS BY CODING ON MATLAB
MOTION ANALYSIS OF PLANETARY GEAR
AIM:
To create a 3D model and to perform motion analysis of Planetary Gear Train by plotting Angular velocities of output and input gears by fixing different gears to obtain different speed reductions using SOLIDWORKS MOTION and verifying the data by coding on MATLAB.
DESCRIPTION:
Sometimes, two or more gears are made to mesh with each other to transmit power from one shaft to another. Such a combination is called a gear train. The nature of the train used depends upon the velocity ratio required and the relative position of the axes of the shafts.A gear train may consist of the spur, bevel, or spiral gears.
Epicyclic gearing or planetary gearing is a gear system consisting of one or more outer, or planet, gears or pinions, revolving about a central sun gear or sun wheel.
The four basic components of the epicyclic gear are:
Sun Gear: The central gear, it rotates about its own axis.
Arm/Carrier: Holds one or more peripheral Planet gears, all of the same size, meshed with the sun gear. We can provide input power or take power either from planetary gear, sun gear, or the Ring gear.
Ring or Annulus Gear: An outer ring with inward-facing teeth that mesh with the planet gear or gears. It rotates about its own axis.
Planetary Gear: it is positioned between the sun and ring gear and as imagined its rotation axis is going to change.
We can provide input power or take power either from planetary gear, sun gear, or the Ring gear.
APPLICATIONS:
- They are popular for automatic transmissions, usually seen in high power applications, high torque transmissions in automobiles.
- These are used in the back gear of the lathe, hoists, pulley, blocks, wristwatches, etc.
- They are also used in bicycles for controlling the power of pedaling automatically or manually.
- They are also used for power train between the internal combustion engine and electric motor.
- Early pencil sharpeners had a small planetary gear train.
CALCULATION:
GIVEN DATA:
Ring Gear:
Module = 2.5 (You must use the metric system in design modeler)
Number of teeth = 46
Sun Gear:
Number of teeth = 14
Input Speed of the Gear = 200 rpm.
Number of planet gears = 4
To run a simulation for the following cases:
|
Sr.No |
Input |
Output |
Fixed |
|
1 |
Sun Gear |
Carrier |
Ring Gear |
|
2 |
Ring Gear |
Carrier |
Sun Gear |
|
3 |
Sun Gear |
Ring Gear |
Carrier |
CALCULATION OF PITCH CIRCLE DIAMETER AND NUMBER OF TEETHS:
Module = (Pitch circle diameter/Number of teeth)`=D/T`
Diametrical pitch = 1/m
RING GEAR:
m= module =2.5
`T_R`= Number of teeth on Ring Gear = 46 teeth
`D_R`= Pitch circle diameter for Ring gear =`m*T_R` = 2.5*46 = 115mm.
SUN GEAR:
m= module =2.5
`T_S`= Number of teeth on Sun Gear = 14 teeth
`D_s`= Pitch circle diameter for Sun gear = `m*T_s` = 2.5*14 = 35mm.
PLANET GEAR:
m= module =2.5
`D_R=D_s+2*D_P`
`m*T_R=m*T_s+2*m*T_P`
`T_R=T_s+2*T_P`
46 = `2*T_P`+ 14
`T_P`= Number of teeth on Planet Gear =16 teeth,
`D_P`= Pitch circle diameter for Planet gear= 2.5 * 16 = 40mm.
|
S No |
Input |
Number of teeth |
Pitch circle diameter (mm) |
|
Case 1 |
Sun Gear |
14 |
35 |
|
Case 2 |
Planet Gear |
16 |
40 |
|
Case 3 |
Ring Gear |
46 |
115 |
CREATION OF 3D CAD MODELS:
To create gear from the toolbox
Design library > Toolbox > ANSI metric > Power transmission >
Choose internal spur gear for ring gear and external spur gear for sun and planet gear.
In order for the two gears to mate correctly and transfer power correctly the pitch circles of both the gears should be tangent to each other in addition to this the module should be the same for the two gears.
The dimensions of the carrier are taken based on the nominal shaft diameter of the gears.
Face width fall all gears = 12mm,
Pressure angle for all gears= 14 degrees,
PLANET GEAR:
SUN GEAR:
RING GEAR:
CARRIER:
ASSEMBLY:
MOTION ANALYSIS:
SOLIDWORKS motion is selected and motion analysis in enabled.
The first step is to add a motor to the input gear, for each case the input angular speed given = 200 rpm.
The second step is to define the contacts between the sun gear & planet gears, planet and ring gear using the section of the gears and the material chosen is dry steel for all.
Make sure there are no gear mates added as we are doing the motion study we need the gears motion to be determined by actual gear contact.
The plots are added for the angular velocity of the Sun gear, Planet gear, Ring gear, and Carrier.
In the motions settings, 200 fps is entered, and in additional settings, the WSTIFF integration method is chosen.
Now the calculation is Run.
RESULTS AND PLOTS OBTAINED FROM SOLIDWORKS MOTION:
CASE 1:
When Ring Gear is fixed, Sun gear is input (Angular velocity = 200 rpm), Carrier is output.
Angular velocity of Carrier (output):
The data from the graph below was exported to Excel and Average value obtained= 279.8966242 deg/s
Angular Velocity of Sun gear (input):
Input angular velocity = 1200 deg/s
Angular Velocity of Planet gear:
The data from the graph below was exported to Excel and Average value obtained= 524.920899
CASE 2:
When Sun Gear is fixed, Ring gear is input (Angular velocity = 200 rpm), Carrier is output.
Angular velocity of Carrier (output):
The data from the graph below was exported to Excel and Average value obtained= 919.8487474 deg/s
Angular Velocity of Ring gear (input):
Input angular velocity = 1200 deg/s
Angular Velocity of Planet gear:
The data from the graph below was exported to Excel and Average value obtained= 1724.367573 deg/s
CASE 3:
When Carrier is fixed, Sun gear is input (Angular velocity = 200 rpm), Ring is output.
Angular velocity of Ring (output):
The data from the graph below was exported to Excel and Average value obtained= 365.0539571
Angular velocity of Sun (input):
Input angular velocity = 1200 deg/s
Angular velocity of Planet:
The data from the graph below was exported to Excel and Average value obtained=
1049.558404
CODE FOR MOTION ANALYSIS ON MATLAB:
Here we have used tabular method to find the angular velocities
Let
`N_s`= Angular velocity of the Sun gear,
`N_r`= Angular velocity of the Ring gear,
`N_p`= Angular velocity of the planet gear,
`N_c`= Angular velocity of the Carrier,
`T_s`= Number of teeth in Sun gear= 14,
`T_r` = Number of teeth of the Ring gear=46,
`T_p` = Number of teeth of the planet gear=16,
`T_s`= Angular velocity of the Carrier.
`(N_p )/( N_s )=-(T_s )/( T_p )`
`N_p =-(14 )/( 16 )*N_s`
`(N_r )/( N_s )=-(T_s )/( T_r )`
`N_r =-(14 )/(46 )*N_s`
Here -ve sign indicates both gears are rotating in opposite direction.
|
Conditions of motion |
`N_s` | `N_p` | `N_r` |
|
Arm fixed-Sun gear rotates through +1 revolution i.e. 1 rev. anticlockwise |
+1 |
`-(14)/(16)` | `-(14)/(46)` |
|
Arm fixed-Sun gear rotates through +x revolution |
+x |
`-(14)/(16)*x` | `-(14)/(46)*x` |
|
Add +y revolutions to all elements |
+y |
+y |
+y |
|
Total motion |
x+y |
`y-14/( 16)*x` |
`y-14/(46)*x` |
MATLAB CODES AND RESULTS:
CODE 1:
CASE 1: When Ring Gear is fixed, Sun gear is input (Angular velocity = 200 rpm), Carrier is output.
%%%%%%%%%%%%%%%%%%% Motion analysis to find angular velocity of PLANETARY GEAR TRAIN %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clc
clear all
close all
syms x y % Defining two symbols x and y to solve simulations equations
%%%%%%%%%%%%%%%%%%% CASE_1 %%%%%%%%%%%%%%%%%%%%%%%%%%%
%Here the rotation is CCW is taken as positive (+) and
%CW rotation is taken as negative (-)
%Sun gear is input and is given Ns=200rpm angular velocity clockwise
%Ring gear is fixed,angular velocity is Nr=0
%Carrier is the ouput
%Here y is the angular velocity of the carrier or arm
Nr = 0; %since the Ring gear is fixed
Ns = -200; %Sun gear is given angular velocity of 200rpm (CW)
Ns_1 = x+y == Ns; %Speed or Angular velocity of sun gear
Nr_1 = y-14/46*x == Nr; %Speed or Angular velocity of ring gear
[x y]= solve([Ns_1,Nr_1],[x,y]) %Solving simulations equations using solve
Np_1 = y-14/16*x %Speed or Angular velocity of planet gear
% As we know that 1 rpm = 6 deg/s hence multiplying to the solution for conversions for verification of solution from SOLDIWORKS MOTION
Angular_velocity_of_Ring_gear_fixed_in_deg_per_sec = Nr*6
Angular_velocity_of_Carrier_output_in_deg_per_sec_Nc = double(y*6)
Angular_velocity_of_Sun_gear_input_in_deg_per_sec_Ns = (x+y)*6
Angular_velcoty_of_Planet_in_deg_per_sec_Np = (Np_1)*6
Speed_reduction_obtained = double((Angular_velocity_of_Carrier_output_in_deg_per_sec_Nc)/(Angular_velocity_of_Sun_gear_input_in_deg_per_sec_Ns))RESULTS FOR CODE 1:
CODE 2:
CASE 2: When Sun Gear is fixed, Ring gear is input (Angular velocity = 200 rpm), Carrier is output.
clc
clear all
close all
syms x y % Defining two symbols x and y to solve simulations equations
%%%%%%%%%%%%%%%%%%% CASE_2 %%%%%%%%%%%%%%%%%%%%%%%%%%%
%Here the rotation is CCW is taken as positive (+) and
%CW rotation is taken as negative (-)
%Ring gear is input and is given Nr=200rpm angular velocity clockwise
%Sun gear is fixed,angular velocity is Ns=0
%Carrier is the ouput
%Here y is the angular velocity of the carrier or arm
Nr = -200; %Ring gear is given angular velocity of 200rpm (CW)
Ns = 0; %since the Sun gear is fixed
Ns_1 = x+y == Ns; %Speed or Angular velocity of sun gear
Nr_1 = y-14/46*x == Nr; %Speed or Angular velocity of ring gear
[x y]= solve([Ns_1,Nr_1],[x,y]) %Solving simulations equations using solve
Np_1 = y-14/16*x %Speed or Angular velocity of planet gear
% As we know that 1 rpm = 6 deg/s hence multiplying to the solution for conversions for verification of solution from SOLDIWORKS MOTION
Angular_velocity_of_Ring_gear_input_in_deg_per_sec = Nr*6
Angular_velocity_of_Carrier_output_in_deg_per_sec_Nc = double(y*6)
Angular_velocity_of_Sun_gear_fixed_in_deg_per_sec_Ns = (x+y)*6
Angular_velcoty_of_Planet_in_deg_per_sec_Np = (Np_1)*6
Speed_reduction_obtained = (Angular_velocity_of_Carrier_output_in_deg_per_sec_Nc)/(Angular_velocity_of_Ring_gear_input_in_deg_per_sec)RESULTS FOR CODE 2:
CODE 3:
CASE 3 :When Carrier is fixed, Sun gear is input (Angular velocity = 200 rpm), Ring is output.
clc
close all
close all
%%%%%%%%%%%%%%%%%%% CASE_3 %%%%%%%%%%%%%%%%%%%%%%%%%%%
%Here the rotation is CCW is taken as positive (+) and
%CW rotation is taken as negative (-)
%Sun gear is input and is given Ns=200rpm angular velocity clockwise
%Carrier gear is fixed,angular velocity is Nc=0
%Ring gear is the ouput
%Here y is the angular velocity of the carrier or arm
Ns = -200; %Sun gear is given angular velocity of 200rpm (CW)
y=0; %since the Carrier gear is fixed
x= Ns; %Ns=x+y where y=0, hence Ns=x
Ns_1 = x; %Speed or Angular velocity of sun gear
Nr_1 = -14/46*x; %Speed or Angular velocity of ring gear
Np_1 = -14/16*x; %Speed or Angular velocity of planet gear
% As we know that 1 rpm = 6 deg/s hence multiplying to the solution for conversions for verification of solution from SOLDIWORKS MOTION
Angular_velocity_of_Ring_gear_output_in_deg_per_sec_Nr = double(Nr_1)*6
Angular_velocity_of_Carrier_fixed_in_deg_per_sec_Nc = double(y*6)
Angular_velocity_of_Sun_gear_input_in_deg_per_sec_Ns = (x)*6
Angular_velcoty_of_Planet_in_deg_per_sec_Np = (Np_1)*6
Speed_reduction_obtained=(Angular_velocity_of_Ring_gear_output_in_deg_per_sec_Nr)/(Angular_velocity_of_Sun_gear_input_in_deg_per_sec_Ns)RESULTS FOR CODE 3:
RESULTS COMPARISON:
CASE 1: When Ring Gear is fixed, Sun gear is input (Angular velocity = 200 rpm=1200 deg/s), Carrier is output.
|
Angular velocity (deg/s) |
FROM SOLIDWORKS MOTION |
FROM MATLAB CODE |
| `N_s` |
1200(CW) |
1200(CW) |
| `N_p` |
524.92(CCW) |
525(CCW) |
| `N_r` |
0 |
0 |
| `N_c` |
279.896(CW) |
280(CW) |
|
SPEED REDUCTION RATIO |
0.2333 |
0.2333 |
CASE 2: When Sun Gear is fixed, Ring gear is input (Angular velocity = 200 rpm=
1200 deg/s), Carrier is output.
|
Angular velocity (deg/s) |
FROM SOLIDWORKS MOTION |
FROM MATLAB CODE |
| `N_s` |
0 |
0 |
| `N_p` |
1724.367(CW) |
1725(CW) |
| `N_r` |
1200(CW) |
1200(CW) |
| `N_c` |
919.8474(CW) |
920(CW) |
|
SPEED REDUCTION RATIO |
0.7665 |
0.7667 |
CASE 3: When Carrier is fixed, Sun gear is input (Angular velocity = 200 rpm=1200deg/s), Ring is output.
|
Angular velocity (deg/s) |
FROM SOLIDWORKS MOTION |
FROM MATLAB CODE |
| `N_s` |
1200(CW) |
1200(CW) |
| `N_p` |
1049.558(CCW) |
1050(CCW) |
| `N_r` |
365.059(CCW) |
365(CCW) |
| `N_c` |
0 |
0 |
|
SPEED REDUCTION RATIO |
-0.3042 |
-0.3043 |
YOUTUBE ANIMATION LINKS:
CASE 1:
CASE 2:
CASE 3:
CONCLUSION:
1) As seen from the Results plots of SOLIDWORKS MOTION there are some fluctuations or wiggles as we cannot get power transmissions with 100% efficiency there will always be some losses that are accounted in form of gear efficiency.
2) The results plot of SOLIDWORKS MOTION can be improved by using a precise contact option but it will increase the computation time.
3) We have obtained the plots of angular velocities because from these plots, we can determine the torque transmitted and we can then find the power transmissions.
4) We can see from the results that the higher speed reduction from input to output is obtained in case 2 which is 0.7667 and lower speed reductions can be obtained from case 1 which is 0.2333 we can select any one of the configurations based on the applications.
5) The results obtained from the SOLIDWORKS MOTION are consistent with the results obtained from MATLAB code and hence we can say that the simulation that we performed is correct and hence this verifies the results.
GOOGLE DRIVE LINK OF THE CAD FILES:
https://drive.google.com/drive/folders/1e6h-WRE_7gKNnubZScUDn9SJLM2OgbSG?usp=sharing