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Differential Gear Mechanism
Introduction
This model simulates the mechanism of a differential gear used in cars and other wheeled vehicles. The model is built using the gears functionality in the Multibody Dynamics Module in the COMSOL Multiphysics.
A differential allows the outer drive wheel to rotate faster than the inner drive wheel during a turn. This is necessary when a vehicle turns in order to allow the wheel that is traveling along the outside of the turning curve to roll faster and to cover greater distance than the wheel on the inside of the turning curve. The average of the rotational speed of the two driving wheels is simply the input rotational speed of the drive shaft. An increase in the speed of one wheel is balanced by a decrease in the speed of the other.
Model Definition
The geometry of a differential is shown in Figure 1.
Figure 1: Geometry of a differential mechanism.
Geometry and connections
A differential has the following components:
•
A drive shaft with pinion
•
A ring gear
•
Two spider gears
•
Left and right axles with side gears
A drive shaft is connected to the ring gear through a pinion. Two spider gears are mounted on the ring gear in such a way that they are free to rotate about their own axis. These spider gears are connected to side gears that are mounted on the inner end of the two axles.
The wheels, which are not modeled, are supposed to be mounted on the outer ends of the two axles. In this model, it is assumed that the wheel mounted on the right axle is on the outer side while vehicle takes a turn.
Working of a differential
A differential performs the following tasks:
•
Allows the left and the right axle to rotate at different speeds.
•
Transfers power at .
•
Reduces speed of wheels compared to the drive shaft.
Under normal conditions, with small tire slip, the ratio of the speeds of the two driving wheels is determined by the track-width of the vehicle (the distance between the driving wheels) and the radius of the turn.
Straight Path
When a vehicle moves on a straight path, both wheels of the vehicle roll with the same speed. In this situation, both axles rotate with the same speed while the spider gears follow a planetary motion and thus do not spin about their own axes.
Curved Path
In case of a curved path, the outer wheel has to cover a longer distance. It must therefore roll faster than the inner wheel in order to avoid any slip. If the differential was absent in the vehicle, both wheels would have to rotate with the same speed causing slip between the wheels and the ground.
A differential allows the two wheels of the vehicle to rotate with different speeds. In this situation the spider gears start spinning together with their planetary motion. Spinning of the spider gears creates a difference in the speed of both wheels.
Model Parameters
•
The gear ratio between the pinion and the ring gear is 4.
•
The gear ratio between the spider gear and the side gear is 1.5.
•
To avoid any slip on a curved path, speed of the outer wheel is supposed to be 6 times higher than that of the inner wheel.
•
The vehicle enters a curved path after one revolution of the drive shaft and returns back to a straight path after three revolutions of the drive shaft.
Constraints and loads
•
The drive shaft is rotating about the x-axis with an angular velocity of 200 rad/s.
•
The ring gear is hinged about the y-axis.
•
The side gears (or axles) are hinged about the y-axis.
An external torque is applied on both axles while vehicle is taking a turn. This torque, which represents a friction between the wheels and the ground, forces the two wheels to rotate in the desired velocity ratio.
This external torque (T) is to be defined with
(1)
where
•
cf is the frictional damping coefficient,
•
ωo and ωi are the angular velocities of the outer and the inner wheels,
•
ωr is the velocity ratio between the outer and the inner wheels (set to 1 on a straight path and assumed to be 6 on a curved path), and
•
ω is the angular velocity of the drive shaft.
Results and Discussion
Figure 2 shows a velocity magnitude of the different components in a differential at t = 0.125 sec. At this time instance the vehicle is moving along a straight path. This is also predicted by the model since both axles rotate with the same speed.
Figure 2: Speed of the different components in a differential after 0.125 s.
Figure 3 shows the velocity y-component of different differential parts at t = 0.08 sec. At this time instance, the vehicle is moving on a curved path. This is also predicted by the model since the spider gears have a nonzero velocity along the y-axis, which indicates that besides a planetary motion they are also spinning. The velocity y-component of the remaining parts is zero as they all rotate about the y-axis.
The spinning of the spider gears when the vehicle travels on a curved path can be visualized by plotting the relative velocity of the spider gears with respect to the ring gear as shown in Figure 4.
Figure 5 shows the angular velocity of the inner and outer wheels for the two cases when a vehicle moves on a straight path and on a curved path. It can be seen that as soon as the vehicle starts taking a turn, the outer wheel velocity increases and the inner wheel velocity decreases. It happens in such a way that the average is kept constant while the ratio between the two velocity components changes to 6, which is the desired ratio for no slip. Once the vehicle comes back on a straight path, both wheels start rotating with the same speed.
Figure 3: Velocity y-component in different parts of a differential after 0.08 s.
Figure 4: Relative velocity of spider gears with respect to ring gear at 0.08 s.
Figure 5: Time variation of the angular velocity of an inner and an outer wheel.
Figure 6: Time variation of the spinning velocity of a spider gears.
Figure 6 shows the spinning velocity of the spider gears. It can be seen that as soon as the vehicle starts taking a turn, the spider gears start spinning. Through this motion it transfers speed from inner wheel to the outer wheel in such a way that both wheels roll without slip. The spider gears stops spinning, once the vehicle comes back on a straight path.
Notes About the COMSOL Implementation
•
To build a gear geometry, you can import a gear part from the Parts Library and customize it by changing its input parameters. Alternatively, you can also create an equivalent disc or cone to represent the gear.
•
All the gears are assumed rigid. The elasticity of a gear mesh can be included on Gear Pair nodes using the Gear Elasticity subnode.
•
All the Gear Pair nodes are assumed ideal and frictionless. You can add Transmission Error, Backlash, or Friction subnodes when required.
•
To constrain gear motion, you can use Prescribed Displacement/Rotation or Fixed Constraint subnodes. Alternatively, you can mount gears on a shaft or on the ground through various Joint nodes.
•
The contact force on a Gear Pair is computed using Weak constraints or Penalty method. By default, the contact force computation is turned off. Use the weak constraints method for more accurate contact forces. However you can switch to the penalty method for large rigid body systems.
Application Library path: Multibody_Dynamics_Module/Automotive_and_Aerospace/differential_gear
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  3D.
2
In the Select Physics tree, select Structural Mechanics > Multibody Dynamics (mbd).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Time Dependent.
6
Click  Done.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file differential_gear_parameters.txt.
Geometry 1
Import 1 (imp1)
1
In the Geometry toolbar, click  Import.
2
In the Settings window for Import, locate the Source section.
3
Click  Browse.
4
Browse to the model’s Application Libraries folder and double-click the file differential_gear.mphbin.
5
Click  Import.
Form Union (fin)
1
In the Model Builder window, under Component 1 (comp1) > Geometry 1 click Form Union (fin).
2
In the Settings window for Form Union/Assembly, locate the Form Union/Assembly section.
3
From the Action list, choose Form an assembly.
4
Clear the Create pairs checkbox.
5
In the Geometry toolbar, click  Build All.
Disable the analysis of the geometry as the remaining small geometric details can be kept.
6
In the Model Builder window, click Geometry 1.
7
In the Settings window for Geometry, locate the Cleanup section.
8
Clear the Automatic detection of small details checkbox.
Add Material
1
In the Materials toolbar, click  Add Material to open the Add Material window.
2
Go to the Add Material window.
3
In the tree, select Built-in > Structural steel.
4
Click the Add to Component button in the window toolbar.
5
In the Materials toolbar, click  Add Material to close the Add Material window.
Multibody Dynamics (mbd)
Add a Bevel Gear node and specify its properties.
Bevel Gear: Pinion
1
In the Physics toolbar, click  Domains and choose Bevel Gear.
2
In the Settings window for Bevel Gear, type Bevel Gear: Pinion in the Label text field.
3
Select Domain 6 only.
4
Locate the Gear Properties section. In the n text field, type n_pn.
5
In the dp text field, type dp_pn.
6
In the α text field, type alpha.
7
In the γ text field, type gamma_pn.
8
Locate the Gear Axis section. Specify the eg vector as
-1
x
0
y
0
z
9
Locate the Center of Rotation section. From the list, choose User defined.
10
Specify the Xc vector as
dp_rg/2
x
-dp_pn/2
y
0
z
Bevel Gears
Similarly add 5 more Bevel Gears using the information given in the following table:
Name
Selection (Domains)
Number of Teeth
Pitch Diameter
Pressure Angle
Cone Angle
Bevel Gear: Ring Gear
1
n_rg
dp_rg
alpha
gamma_rg
Bevel Gear: Spider Gear 1
5
n_sp
dp_sp
alpha
gamma_sp
Bevel Gear: Spider Gear 2
4
n_sp
dp_sp
alpha
gamma_sp
Bevel Gear: Side Gear 1
3
n_sd
dp_sd
alpha
gamma_sd
Bevel Gear: Side Gear 2
2
n_sd
dp_sd
alpha
gamma_sd
Bevel Gears
1
Use the axis and center information given in the following table:
Name
Gear Axis
Center of Rotation
Bevel Gear: Ring Gear
(0,-1,0)
(0,0,0)
Bevel Gear: Spider Gear 1
(0,0,-1)
(0,-d_sp,dp_sd/2)
Bevel Gear: Spider Gear 2
(0,0,1)
(0,-d_sp,-dp_sd/2)
Bevel Gear: Side Gear 1
(0,-1,0)
(0,-d_sp+dp_sp/2,0)
Bevel Gear: Side Gear 2
(0,1,0)
(0,-d_sp-dp_sp/2,0)
2
In the Model Builder window, under Component 1 (comp1) > Multibody Dynamics (mbd) click Bevel Gear: Pinion.
Prescribed Displacement/Rotation 1
1
In the Physics toolbar, click  Attributes and choose Prescribed Displacement/Rotation.
2
In the Settings window for Prescribed Displacement/Rotation, locate the Prescribed Displacement at Center of Rotation section.
3
Select the Prescribed in x direction checkbox.
4
Select the Prescribed in y direction checkbox.
5
Select the Prescribed in z direction checkbox.
6
Locate the Prescribed Rotation section. From the By list, choose Prescribed rotation.
7
Specify the Ω vector as
-1
x
0
y
0
z
8
In the ϕ0 text field, type omega*t.
Bevel Gear: Ring Gear
In the Model Builder window, under Component 1 (comp1) > Multibody Dynamics (mbd) click Bevel Gear: Ring Gear.
Prescribed Displacement/Rotation 1
1
In the Physics toolbar, click  Attributes and choose Prescribed Displacement/Rotation.
2
In the Settings window for Prescribed Displacement/Rotation, locate the Prescribed Displacement at Center of Rotation section.
3
Select the Prescribed in x direction checkbox.
4
Select the Prescribed in y direction checkbox.
5
Select the Prescribed in z direction checkbox.
6
Locate the Prescribed Rotation section. From the By list, choose Constrained rotation.
7
Select the Constrain rotation around x-axis checkbox.
8
Select the Constrain rotation around z-axis checkbox.
Bevel Gear: Pinion, Bevel Gear: Ring Gear, Bevel Gear: Side Gear 1, Bevel Gear: Side Gear 2, Bevel Gear: Spider Gear 1, Bevel Gear: Spider Gear 2
1
In the Model Builder window, under Component 1 (comp1) > Multibody Dynamics (mbd), Ctrl-click to select Bevel Gear: Pinion, Bevel Gear: Ring Gear, Bevel Gear: Spider Gear 1, Bevel Gear: Spider Gear 2, Bevel Gear: Side Gear 1, and Bevel Gear: Side Gear 2.
2
Right-click and choose Group.
Bevel Gears
In the Settings window for Group, type Bevel Gears in the Label text field.
Gear Pair 1
1
In the Physics toolbar, click  Global and choose Gear Pair.
2
In the Settings window for Gear Pair, locate the Gear Selection section.
3
From the Wheel list, choose Bevel Gear: Pinion.
4
From the Pinion list, choose Bevel Gear: Ring Gear.
Gear Pairs
Similarly add 4 more Gear Pairs using the information given in the following table:
Name
Wheel
Pinion
Gear Pair 2
Bevel Gear: Spider Gear 1
Bevel Gear: Side Gear 1
Gear Pair 3
Bevel Gear: Spider Gear 1
Bevel Gear: Side Gear 2
Gear Pair 4
Bevel Gear: Spider Gear 2
Bevel Gear: Side Gear 1
Gear Pair 5
Bevel Gear: Spider Gear 2
Bevel Gear: Side Gear 2
Hinge Joint 1
1
In the Physics toolbar, click  Global and choose Hinge Joint.
2
In the Settings window for Hinge Joint, locate the Attachment Selection section.
3
From the Source list, choose Bevel Gear: Ring Gear.
4
From the Destination list, choose Bevel Gear: Spider Gear 1.
5
Locate the Center of Joint section. From the list, choose Centroid of selected entities.
6
From the Entity level list, choose Point.
Center of Joint: Point 1
1
In the Model Builder window, click Center of Joint: Point 1.
2
In the Settings window for Center of Joint: Point, locate the Point Selection section.
3
Click  Paste Selection.
4
In the Paste Selection dialog, type 1528, 1531 in the Selection text field.
5
Click OK.
Hinge Joint 1
1
In the Model Builder window, click Hinge Joint 1.
2
In the Settings window for Hinge Joint, locate the Axis of Joint section.
3
Specify the e0 vector as
0
x
0
y
1
z
Hinge Joints
Similarly add 3 more Hinge Joints using the information given in the following table:
Name
Source
Destination
Center of Joint (Points)
Axis of Joint
Hinge Joint 2
Bevel Gear: Ring Gear
Bevel Gear: Spider Gear 2
1282, 1341
(0,0,1)
Hinge Joint 3
Fixed
Bevel Gear: Side Gear 1
1060, 1061
(0,1,0)
Hinge Joint 4
Fixed
Bevel Gear: Side Gear 2
748, 749
(0,1,0)
Add a damping force on both side gears (or axles) to prevent the slip on a curved path.
Hinge Joint 3
In the Model Builder window, click Hinge Joint 3.
Applied Force and Moment 1
1
In the Physics toolbar, click  Attributes and choose Applied Force and Moment.
2
In the Settings window for Applied Force and Moment, locate the Applied On section.
3
From the list, choose Joint.
4
Locate the Applied Force and Moment section. In the M text field, type cf*omega_d.
Hinge Joint 4
In the Model Builder window, under Component 1 (comp1) > Multibody Dynamics (mbd) click Hinge Joint 4.
Applied Force and Moment 1
1
In the Physics toolbar, click  Attributes and choose Applied Force and Moment.
2
In the Settings window for Applied Force and Moment, locate the Applied On section.
3
From the list, choose Joint.
4
Locate the Applied Force and Moment section. In the M text field, type -cf*omega_d.
Hinge Joint 1, Hinge Joint 2, Hinge Joint 3, Hinge Joint 4
1
In the Model Builder window, under Component 1 (comp1) > Multibody Dynamics (mbd), Ctrl-click to select Hinge Joint 1, Hinge Joint 2, Hinge Joint 3, and Hinge Joint 4.
2
Right-click and choose Group.
Hinge Joints
In the Settings window for Group, type Hinge Joints in the Label text field.
Define the wheel slip in terms of the difference in angular velocity between the outer and the inner wheel.
Definitions
Variables 1
1
In the Model Builder window, expand the Component 1 (comp1) > Definitions node.
2
Right-click Definitions and choose Variables.
3
In the Settings window for Variables, locate the Variables section.
4
In the table, enter the following settings:
Name
Expression
Unit
Description
omega_i
d(mbd.hgj3.th,t)
rad/s
Angular velocity of inner wheel
omega_o
d(mbd.hgj4.th,t)
rad/s
Angular velocity of outer wheel
omega_r
1+5*rect1(omega*t)
 
Velocity ratio of outer and inner wheel
omega_d
omega_o-omega_r*omega_i
 
Velocity difference of outer and inner wheel
Rectangle 1 (rect1)
1
In the Definitions toolbar, click  More Functions and choose Rectangle.
2
In the Settings window for Rectangle, locate the Parameters section.
3
In the Lower limit text field, type 2*pi.
4
In the Upper limit text field, type 6*pi.
5
Click to expand the Smoothing section. In the Size of transition zone text field, type pi/4.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
From the Element size list, choose Coarse.
4
Click  Build All.
Study 1
Step 1: Time Dependent
1
In the Model Builder window, under Study 1 click Step 1: Time Dependent.
2
In the Settings window for Time Dependent, locate the Study Settings section.
3
In the Output times text field, type range(0,5e-4,0.125).
4
In the Study toolbar, click  Compute.
5
Click the  Show Grid button in the Graphics toolbar.
Results
Displacement (mbd)
Use the following instructions to plot the speed of the different gears in the differential as shown in Figure 2.
Velocity: Magnitude
1
Right-click Displacement (mbd) and choose Duplicate.
2
In the Settings window for 3D Plot Group, type Velocity: Magnitude in the Label text field.
Surface
1
In the Model Builder window, expand the Velocity: Magnitude node, then click Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type mbd.vel.
4
In the Velocity: Magnitude toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Use the following instructions to plot the rotation of the spider gears about their own axis as shown in Figure 3.
Study 1/Solution 1 (2) (sol1)
1
In the Model Builder window, expand the Results > Datasets node.
2
Right-click Results > Datasets > Study 1/Solution 1 (sol1) and choose Duplicate.
Selection
1
In the Results toolbar, click  Attributes and choose Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Domain.
4
Select Domains 1–5 only.
Velocity: Y Component
1
In the Model Builder window, right-click Velocity: Magnitude and choose Duplicate.
2
In the Settings window for 3D Plot Group, type Velocity: Y Component in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 1/Solution 1 (2) (sol1).
4
From the Time (s) list, choose 0.08.
5
Locate the Plot Settings section. From the View list, choose New view.
6
In the Velocity: Y Component toolbar, click  Plot.
This generates a dedicated view for this plot.
Surface
1
In the Model Builder window, expand the Velocity: Y Component node, then click Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type mbd.u_tY.
4
Locate the Coloring and Style section. From the Color table list, choose AuroraAustralis.
5
In the Velocity: Y Component toolbar, click  Plot.
In order to visualize the motion of the spider gears with respect to the ring gear, you can use the option of defining a reference frame available in the Multibody Dynamics interface and plot the postprocessing variables for displacement and velocity with respect to the reference frame.
Multibody Dynamics (mbd)
1
In the Model Builder window, under Component 1 (comp1) click Multibody Dynamics (mbd).
2
In the Settings window for Multibody Dynamics, click to expand the Results section.
3
From the Body defining reference frame list, choose Bevel Gear: Ring Gear.
You can update the solution to plot the postprocessing variables available for the new reference frame.
Study 1
In the Study toolbar, click  Update Solution.
Results
Study 1/Solution 1 (3) (sol1)
In the Model Builder window, under Results > Datasets right-click Study 1/Solution 1 (2) (sol1) and choose Duplicate.
Selection
1
In the Model Builder window, expand the Study 1/Solution 1 (3) (sol1) node, then click Selection.
2
In the Settings window for Selection, locate the Geometric Entity Selection section.
3
In the list, choose 2 and 3.
4
Click  Remove from Selection.
5
Select Domains 1, 4, and 5 only.
Velocity: Ring Gear Reference
1
In the Model Builder window, right-click Displacement (mbd) and choose Duplicate.
2
In the Model Builder window, click Displacement (mbd) 1.
3
In the Settings window for 3D Plot Group, type Velocity: Ring Gear Reference in the Label text field.
4
Locate the Data section. From the Dataset list, choose Study 1/Solution 1 (3) (sol1).
5
From the Time (s) list, choose 0.08.
6
Locate the Plot Settings section. Clear the Plot dataset edges checkbox.
Surface
1
In the Model Builder window, click Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type mbd.vel_ref.
Deformation
1
In the Model Builder window, expand the Surface node, then click Deformation.
2
In the Settings window for Deformation, locate the Expression section.
3
In the X-component text field, type u_ref.
4
In the Y-component text field, type v_ref.
5
In the Z-component text field, type w_ref.
Velocity: Ring Gear Reference
1
In the Model Builder window, under Results click Velocity: Ring Gear Reference.
2
In the Settings window for 3D Plot Group, locate the Plot Settings section.
3
From the View list, choose New view.
4
In the Velocity: Ring Gear Reference toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Follow the instructions below to plot the angular speed of both the wheels and the spider gears shown in Figure 5 and Figure 6 respectively.
Angular Velocity: Wheels
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Angular Velocity: Wheels in the Label text field.
Global 1
1
Right-click Angular Velocity: Wheels and choose Global.
2
In the Settings window for Global, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Multibody Dynamics > Hinge joints > Hinge Joint 3 > mbd.hgj3.th_t - Relative angular velocity - rad/s.
3
Click Add Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Multibody Dynamics > Hinge joints > Hinge Joint 4 > mbd.hgj4.th_t - Relative angular velocity - rad/s.
4
Click to expand the Coloring and Style section. From the Width list, choose 2.
5
Click to expand the Legends section. From the Legends list, choose Manual.
6
In the table, enter the following settings:
Legends
Inner wheel
Outer wheel
Angular Velocity: Wheels
1
In the Model Builder window, click Angular Velocity: Wheels.
2
In the Settings window for 1D Plot Group, click to expand the Title section.
3
From the Title type list, choose None.
4
Locate the Plot Settings section.
5
Select the y-axis label checkbox. In the associated text field, type Angular velocity (rad/s).
6
In the Angular Velocity: Wheels toolbar, click  Plot.
7
Click the  Zoom Extents button in the Graphics toolbar.
Angular Velocity: Spider Gear
1
Right-click Angular Velocity: Wheels and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Angular Velocity: Spider Gear in the Label text field.
3
Locate the Title section. From the Title type list, choose Manual.
4
In the Title text area, type Spider gear.
Global 1
1
In the Model Builder window, expand the Angular Velocity: Spider Gear node, then click Global 1.
2
In the Settings window for Global, click Replace Expression in the upper-right corner of the y-Axis Data section. From the menu, choose Component 1 (comp1) > Multibody Dynamics > Hinge joints > Hinge Joint 1 > mbd.hgj1.th_t - Relative angular velocity - rad/s.
3
Locate the Legends section. Clear the Show legends checkbox.
4
In the Angular Velocity: Spider Gear toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Animation 1
1
In the Results toolbar, click  Animation and choose Player.
2
In the Settings window for Animation, locate the Scene section.
3
From the Subject list, choose Velocity: Magnitude.
4
Locate the Frames section. In the Number of frames text field, type 100.
Animation 2
1
Right-click Animation 1 and choose Duplicate.
2
In the Settings window for Animation, locate the Scene section.
3
From the Subject list, choose Velocity: Y Component.
Animation 3
1
Right-click Animation 2 and choose Duplicate.
2
In the Settings window for Animation, locate the Scene section.
3
From the Subject list, choose Velocity: Ring Gear Reference.