PDF
Dynamics of Helical Gears
Introduction
This model illustrates the dynamics of helical gears. It is built using the gears functionality in the Multibody Dynamics interface in COMSOL Multiphysics.
A transient study is performed to analyze the effect of constant gear mesh stiffness, varying gear mesh stiffness, and the transmission error on the angular velocity of the driven gear and the contact force. An eigenfrequency analysis is performed to compute the natural frequencies and mode shapes of the gear pair for rigid and for elastic gear mesh.
Model Definition
The geometry of helical gears shown in Figure 1.
Figure 1: Model geometry.
Gear Properties
The properties of the wheel and pinion are given in the table below:
Table 1: Gear properties.
Properties
 
Wheel
pinion
Number of teeth
 n
20
20
Pitch diameter
 dp
100 mm
100 mm
Pressure angle
 α
25°
25°
Helix angle
 β
30°
-30°
Center of rotation
 xc
(0, 0, 0) mm
(100, 0, 0) mm
Axis of rotation
 eg
(0, 1, 0)
(0, 1, 0)
Time-Dependent Analysis
The time-dependent analysis is performed to analyze the dynamics of helical gears. The following gear meshes are considered while computing the contact force and the speed of the driven gear:
•
Rigid gear mesh
•
Elastic gear mesh with constant stiffness
•
Elastic gear mesh with varying stiffness
•
Elastic gear mesh with constant stiffness and transmission error
In all the cases, the driver gear rotates with an angular velocity of 100 rad/s, and a resisting torque of 10 Nm is applied to the driven gear. The analysis is performed for 3 mesh cycles, and the number of time steps per mesh cycle is 50.
Case-1: Rigid Gear Mesh
In this case, the gear mesh is assumed rigid, and there is no flexibility in the system.
Case-2: Elastic Gear Mesh with Constant Stiffness
In this case, the gear mesh is assumed elastic. The stiffness of a gear tooth is 107 N/m. The contact ratio in a mesh cycle is assumed constant. The gear mesh damping is also added, and it is 0.05 % of the mesh stiffness.
Case-3: Elastic Gear Mesh with Varying Stiffness
This case is similar to the case-1. However, in this case the contact ratio is varying in a mesh cycle. The maximum contact ratio in a mesh cycle is 2, and the next tooth engagement position in mesh cycle is 0.8. The variation of mesh stiffness of the gear pair in a mesh cycle is shown in Figure 2.
Figure 2: The variation of mesh stiffness in a mesh cycle (case-3).
Figure 3: The variation of transmission error in a mesh cycle (case-4).
Case-4: Elastic Gear Mesh with Constant Stiffness and Transmission Error
This case is similar to the case-1. However, in this case the static transmission error is also added on both gears. The maximum static transmission error on each gear is 0.01 mm. The variation of transmission error on each gear in a mesh cycle is shown in Figure 3.
Eigenfrequency Analysis
An eigenfrequency analysis is performed to compute the natural frequencies and mode shapes of the helical gear pair. The following gear meshes are considered:
•
Rigid gear mesh
•
Elastic gear mesh with constant stiffness
In all cases, both gears are free to rotate about their axis. The driven gear is mounted on an elastic joint. The translational and rotational stiffness of the elastic joint is 2·107 N/m and 2·107 Nm/rad, respectively.
Results and Discussion
Time-Dependent Analysis
The angular velocity of the pinion or driven gear for various gear meshes is shown in Figure 4. It can be seen that for the rigid gear mesh, the angular velocity of the driven gear is constant. For constant stiffness case, the angular velocity of the driven gear fluctuates in the beginning, and it settles down to the mean value after a while.
For varying stiffness or transmission error cases, the angular velocity of the driven gear keeps changing with the gear rotation. The variation of angular velocity is periodic, and it is repeated in the next mesh cycle.
Figure 5 shows the variation of contact force for various gear meshes. It can be seen that for the rigid gear mesh, the contact force has a constant value of 254.8 N. For constant stiffness case, the contact force fluctuates in the beginning, and it settles down to the mean value after a while.
For the varying stiffness case, the contact force keeps changing with the gear rotation. The variation of contact force is periodic, and it is repeated in the next mesh cycle. The maximum and minimum values of the contact force in a mesh cycle are approximately 440 N and 150 N, respectively. For the transmission error case, the behavior of the contact force is similar to that in the varying stiffness case.
Figure 6 and Figure 7 show the variation of the reaction forces and reaction moments at the center of the driver gear for the gear mesh having varying stiffness.
Figure 4: The variation of pinion angular velocity with gear rotation.
Figure 5: The variation of contact force with gear rotation.
Figure 6: Reaction forces at the center of driver gear (case-3).
Figure 7: Reaction moments at the center of driver gear (case-3).
Eigenfrequency Analysis
The results of the eigenfrequency analysis, for both the gear meshes, are given below:
Table 2: natural frequencies and mode shapes
mode number
natural frequency (rigid mesh) (Hz)
natural frequency (Elastic mesh) (HZ)
Mode type
First
 0*
 0*
Rigid body rotation
Second
-
 293.7+583.2i
Gear mesh twist
Third
 601.3
 590.3+71.7i
Elastic joint: rotation
Fourth
 728.1
 728.1
Elastic joint: translation
Fifth
 728.1
 728.1
Elastic joint: translation
Sixth
 2417
 2669+143.5i
Elastic joint: rotation
* Ideally the frequency of rigid body rotation should be zero. However, due to small numerical errors, it is not exactly zero but very close to zero.
 
It can be seen in Table 2 that the second mode is present only when the gear mesh is elastic. This mode corresponds to the twisting of the gear mesh. The rest of the modes that correspond to the displacement and rotation of the elastic joint are present for both gear meshes. It can be seen that the elasticity of the gear mesh affects only the rotational modes. The translational modes are the same in both the cases.
Figure 8: Second mode of the helical gear pair with an elastic mesh.
Figure 9: Sixth mode of the helical gear pair with an elastic mesh.
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 in the 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 the motion of a gear, you can use Prescribed Displacement/Rotation or Fixed Constraint subnodes. Alternatively, you can mount the 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 preferably opt for the penalty method for large rigid body systems.
Application Library path: Multibody_Dynamics_Module/Tutorials,_Transmission/helical_gear_pair
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 helical_gear_pair_parameters.txt.
Part Libraries
1
In the Home toolbar, click  Windows and choose Part Libraries.
2
In the Part Libraries window, select Multibody Dynamics Module > 3D > External Gears > helical_gear in the tree.
3
Click  Add to Geometry.
Geometry 1
Helical Gear 1 (pi1)
1
In the Geometry toolbar, click  Build All.
To customize the gear geometry, enter the gear parameters in the input parameters of the part.
2
In the Model Builder window, under Component 1 (comp1) > Geometry 1 click Helical Gear 1 (pi1).
3
In the Settings window for Part Instance, locate the Input Parameters section.
4
In the table, enter the following settings:
Name
Expression
Value
Description
n
n
20
Number of teeth
dp
dp
0.1 m
Pitch diameter
alpha
alpha
25 °
Pressure angle
beta
beta
30 °
Helix angle
lsr
0
0
Shaft length to pitch diameter ratio (Set 0 for no shaft)
egy
1
1
Gear axis, y-component
egz
0
0
Gear axis, z-component
Helical Gear 2 (pi2)
1
In the Geometry toolbar, click  Part Instance and choose Helical Gear.
2
In the Settings window for Part Instance, locate the Input Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
n
n
20
Number of teeth
dp
dp
0.1 m
Pitch diameter
alpha
alpha
25 °
Pressure angle
beta
-beta
-30 °
Helix angle
lsr
0
0
Shaft length to pitch diameter ratio (Set 0 for no shaft)
xc
dp
0.1 m
Gear center, x-coordinate
egy
1
1
Gear axis, y-component
egz
0
0
Gear axis, z-component
th
360/(2*n)[deg]
9 °
Mesh alignment angle
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.
Define Rectangle functions to use them in the expressions of the transmission error.
Definitions
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 0.2.
4
In the Upper limit text field, type 0.8.
5
Click to expand the Smoothing section. In the Size of transition zone text field, type 0.4.
Rectangle 2 (rect2)
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 0.6.
4
In the Upper limit text field, type 0.9.
5
Locate the Smoothing section. In the Size of transition zone text field, type 0.2.
Add Material
1
In the Home 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 Home toolbar, click  Add Material to close the Add Material window.
Multibody Dynamics (mbd)
Add Helical Gear nodes and specify the gear properties. For the automated creation of Helical Gear nodes from geometry parts, use Automated Model Setup section of Multibody Dynamics node.
In the Settings window for Multibody Dynamics, click Physics Node Generation in the upper-right corner of the Automated Model Setup section. From the menu, choose Create Gears.
Helical Gear 1
1
In the Model Builder window, expand the Gears node, then click Helical Gear 1.
2
In the Settings window for Helical Gear, locate the Initial Values section.
3
From the list, choose Locally defined.
Initial Values 1
1
In the Model Builder window, click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values: Rotational section.
3
Specify the ω vector as
0
x
omega
y
0
z
Helical Gear 2
1
In the Model Builder window, expand the Component 1 (comp1) > Multibody Dynamics (mbd) > Gears > Helical Gear 2 node, then click Helical Gear 2.
2
In the Settings window for Helical Gear, locate the Center of Rotation section.
3
Specify the Xc vector as
dp
x
0
y
0
z
4
Locate the Initial Values section. From the list, choose Locally defined.
Initial Values 1
1
In the Model Builder window, click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values: Rotational section.
3
Specify the ω vector as
0
x
-omega
y
0
z
Add Hinge Joints to connect the two gears with the ground.
Hinge Joint: Fixed-Gear 1
1
In the Physics toolbar, click  Global and choose Hinge Joint.
2
In the Settings window for Hinge Joint, type Hinge Joint: Fixed-Gear 1 in the Label text field.
3
Locate the Attachment Selection section. From the Source list, choose Fixed.
4
From the Destination list, choose Helical Gear 1.
5
Locate the Axis of Joint section. Specify the e0 vector as
0
x
1
y
0
z
Hinge Joint: Fixed-Gear 2
1
Right-click Hinge Joint: Fixed-Gear 1 and choose Duplicate.
2
In the Settings window for Hinge Joint, type Hinge Joint: Fixed-Gear 2 in the Label text field.
3
Locate the Attachment Selection section. From the Destination list, choose Helical Gear 2.
Hinge Joint: Fixed-Gear 2 (Elastic)
1
Right-click Hinge Joint: Fixed-Gear 2 and choose Duplicate.
2
In the Settings window for Hinge Joint, type Hinge Joint: Fixed-Gear 2 (Elastic) in the Label text field.
3
Locate the Joint Elasticity section. From the list, choose Elastic joint.
Joint Elasticity 1
1
In the Model Builder window, click Joint Elasticity 1.
2
In the Settings window for Joint Elasticity, locate the Spring section.
3
In the ku text field, type ku.
4
In the kθ text field, type kth.
Hinge Joint: Fixed-Gear 1
Prescribe the motion of the wheel.
1
In the Model Builder window, under Component 1 (comp1) > Multibody Dynamics (mbd) click Hinge Joint: Fixed-Gear 1.
Prescribed Motion 1
1
In the Physics toolbar, click  Attributes and choose Prescribed Motion.
2
In the Settings window for Prescribed Motion, locate the Prescribed Rotational Motion section.
3
From the Prescribed motion through list, choose Angular velocity.
4
In the ωp text field, type omega.
Hinge Joint: Fixed-Gear 2
In the Model Builder window, under Component 1 (comp1) > Multibody Dynamics (mbd) click Hinge Joint: Fixed-Gear 2.
Applied Force and Moment 1
1
In the Physics toolbar, click  Attributes and choose Applied Force and Moment.
Apply the resisting torque on the pinion.
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 T_ext.
Hinge Joint: Fixed-Gear 1, Hinge Joint: Fixed-Gear 2, Hinge Joint: Fixed-Gear 2 (Elastic)
1
In the Model Builder window, under Component 1 (comp1) > Multibody Dynamics (mbd), Ctrl-click to select Hinge Joint: Fixed-Gear 1, Hinge Joint: Fixed-Gear 2, and Hinge Joint: Fixed-Gear 2 (Elastic).
2
Right-click and choose Group.
Joints
In the Settings window for Group, type Joints in the Label text field.
Use multiple Gear Pair nodes, with different properties, to connect the two gears in different cases.
Gear Pair: Rigid
1
In the Physics toolbar, click  Global and choose Gear Pair.
2
In the Settings window for Gear Pair, type Gear Pair: Rigid in the Label text field.
3
Locate the Gear Selection section. From the Wheel list, choose Helical Gear 1.
4
From the Pinion list, choose Helical Gear 2.
5
Locate the Contact Force Computation section. From the list, choose Computed using weak constraints.
Gear Pair: Constant Stiffness
1
Right-click Gear Pair: Rigid and choose Duplicate.
2
In the Settings window for Gear Pair, type Gear Pair: Constant Stiffness in the Label text field.
3
Locate the Gear Pair Properties section. Select the Include gear elasticity checkbox.
Gear Pair: Varying Stiffness
1
Right-click Gear Pair: Constant Stiffness and choose Duplicate.
2
In the Settings window for Gear Pair, type Gear Pair: Varying Stiffness in the Label text field.
Gear Pair: Transmission Error
1
Right-click Gear Pair: Varying Stiffness and choose Duplicate.
2
In the Settings window for Gear Pair, type Gear Pair: Transmission Error in the Label text field.
3
Locate the Gear Pair Properties section. Select the Include transmission error checkbox.
Gear Elasticity 1
1
In the Model Builder window, under Component 1 (comp1) > Multibody Dynamics (mbd) > Gear Pair: Constant Stiffness click Gear Elasticity 1.
2
In the Settings window for Gear Elasticity, locate the Mesh Stiffness section.
3
In the kt,wh text field, type kt.
4
In the kt,pn text field, type kt.
5
Locate the Mesh Damping section. In the cg text field, type (ct[s]/100)*mbd.grp2.kg.
Gear Elasticity 1
1
In the Model Builder window, expand the Component 1 (comp1) > Multibody Dynamics (mbd) > Gear Pair: Varying Stiffness node, then click Gear Elasticity 1.
2
In the Settings window for Gear Elasticity, locate the Mesh Stiffness section.
3
In the kt,wh text field, type kt.
4
In the kt,pn text field, type kt.
5
From the Contact ratio in mesh cycle list, choose Varying.
6
In the ζ text field, type 0.8.
7
Locate the Mesh Damping section. In the cg text field, type (ct[s]/100)*mbd.grp3.kg.
1
In the Model Builder window, under Component 1 (comp1) > Multibody Dynamics (mbd) > Gear Pair: Transmission Error click Gear Elasticity 1.
2
In the Settings window for Gear Elasticity, locate the Mesh Stiffness section.
3
In the kt,wh text field, type kt.
4
In the kt,pn text field, type kt.
5
Locate the Mesh Damping section. In the cg text field, type (ct[s]/100)*mbd.grp4.kg.
Transmission Error 1
1
In the Model Builder window, click Transmission Error 1.
2
In the Settings window for Transmission Error, locate the Transmission Error section.
3
In the ewh text field, type et*rect1(mbd.grp4.thm_wh/(2*pi/n)).
4
In the epn text field, type et*rect2(mbd.grp4.thm_pn/(2*pi/n)).
Gear Pair: Constant Stiffness, Gear Pair: Rigid, Gear Pair: Transmission Error, Gear Pair: Varying Stiffness
1
In the Model Builder window, under Component 1 (comp1) > Multibody Dynamics (mbd), Ctrl-click to select Gear Pair: Rigid, Gear Pair: Constant Stiffness, Gear Pair: Varying Stiffness, and Gear Pair: Transmission Error.
2
Right-click and choose Group.
Gear Pairs
In the Settings window for Group, type Gear Pairs in the Label text field.
Mesh 1
In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Build All.
Compute the solution for a rigid pair.
Study 1: Transient (Rigid)
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, type Study 1: in the Label text field.
3
In the Label text field, type Study 1: Transient (Rigid).
Step 1: Time Dependent
1
In the Model Builder window, under Study 1: Transient (Rigid) 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,dT,T).
4
From the Tolerance list, choose User controlled.
5
In the Relative tolerance text field, type 1e-6.
6
Locate the Physics and Variables Selection section. Select the Modify model configuration for study step checkbox.
7
In the tree, select Component 1 (comp1) > Multibody Dynamics (mbd), Controls spatial frame > Joints > Hinge Joint: Fixed-Gear 2 (Elastic), Component 1 (comp1) > Multibody Dynamics (mbd), Controls spatial frame > Gear Pairs > Gear Pair: Constant Stiffness, Component 1 (comp1) > Multibody Dynamics (mbd), Controls spatial frame > Gear Pairs > Gear Pair: Varying Stiffness, and Component 1 (comp1) > Multibody Dynamics (mbd), Controls spatial frame > Gear Pairs > Gear Pair: Transmission Error.
8
Click  Disable.
Solution 1 (sol1)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 1 (sol1) node, then click Time-Dependent Solver 1.
3
In the Settings window for Time-Dependent Solver, click to expand the Absolute Tolerance section.
4
From the Tolerance method list, choose Manual.
5
In the Study toolbar, click  Compute.
Results
Displacement (mbd)
Click the  Zoom Extents button in the Graphics toolbar.
Add another Time dependent study and compute the solution for a gear pair with constant stiffness.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies > Time Dependent.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2: Transient (Constant Stiffness)
1
In the Settings window for Study, type Study 2: Transient (Constant Stiffness) in the Label text field.
2
Locate the Study Settings section. Clear the Generate default plots checkbox.
Step 1: Time Dependent
1
In the Model Builder window, under Study 2: Transient (Constant Stiffness) 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,dT,T).
4
From the Tolerance list, choose User controlled.
5
In the Relative tolerance text field, type 1e-6.
6
Locate the Physics and Variables Selection section. Select the Modify model configuration for study step checkbox.
7
In the tree, select Component 1 (comp1) > Multibody Dynamics (mbd), Controls spatial frame > Joints > Hinge Joint: Fixed-Gear 2 (Elastic), Component 1 (comp1) > Multibody Dynamics (mbd), Controls spatial frame > Gear Pairs > Gear Pair: Rigid, Component 1 (comp1) > Multibody Dynamics (mbd), Controls spatial frame > Gear Pairs > Gear Pair: Varying Stiffness, and Component 1 (comp1) > Multibody Dynamics (mbd), Controls spatial frame > Gear Pairs > Gear Pair: Transmission Error.
8
Click  Disable.
Solution 2 (sol2)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 2 (sol2) node, then click Time-Dependent Solver 1.
3
In the Settings window for Time-Dependent Solver, locate the Absolute Tolerance section.
4
From the Tolerance method list, choose Manual.
5
In the Study toolbar, click  Compute.
Add another Time dependent study and compute the solution for a gear pair with varying stiffness.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies > Time Dependent.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 3: Transient (Varying Stiffness)
1
In the Settings window for Study, type Study 3: Transient (Varying Stiffness) in the Label text field.
2
Locate the Study Settings section. Clear the Generate default plots checkbox.
Step 1: Time Dependent
1
In the Model Builder window, under Study 3: Transient (Varying Stiffness) 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,dT,T).
4
From the Tolerance list, choose User controlled.
5
In the Relative tolerance text field, type 1e-6.
6
Locate the Physics and Variables Selection section. Select the Modify model configuration for study step checkbox.
7
In the tree, select Component 1 (comp1) > Multibody Dynamics (mbd), Controls spatial frame > Joints > Hinge Joint: Fixed-Gear 2 (Elastic), Component 1 (comp1) > Multibody Dynamics (mbd), Controls spatial frame > Gear Pairs > Gear Pair: Rigid, Component 1 (comp1) > Multibody Dynamics (mbd), Controls spatial frame > Gear Pairs > Gear Pair: Constant Stiffness, and Component 1 (comp1) > Multibody Dynamics (mbd), Controls spatial frame > Gear Pairs > Gear Pair: Transmission Error.
8
Click  Disable.
Solution 3 (sol3)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 3 (sol3) node, then click Time-Dependent Solver 1.
3
In the Settings window for Time-Dependent Solver, locate the Absolute Tolerance section.
4
From the Tolerance method list, choose Manual.
5
In the Study toolbar, click  Compute.
Add another Time dependent study and compute the solution for a gear pair with a transmission error.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies > Time Dependent.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 4: Transient (Transmission Error)
1
In the Settings window for Study, type Study 4: Transient (Transmission Error) in the Label text field.
2
Locate the Study Settings section. Clear the Generate default plots checkbox.
Step 1: Time Dependent
1
In the Model Builder window, under Study 4: Transient (Transmission Error) 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,dT,T).
4
From the Tolerance list, choose User controlled.
5
In the Relative tolerance text field, type 1e-6.
6
Locate the Physics and Variables Selection section. Select the Modify model configuration for study step checkbox.
7
In the tree, select Component 1 (comp1) > Multibody Dynamics (mbd), Controls spatial frame > Joints > Hinge Joint: Fixed-Gear 2 (Elastic), Component 1 (comp1) > Multibody Dynamics (mbd), Controls spatial frame > Gear Pairs > Gear Pair: Rigid, Component 1 (comp1) > Multibody Dynamics (mbd), Controls spatial frame > Gear Pairs > Gear Pair: Constant Stiffness, and Component 1 (comp1) > Multibody Dynamics (mbd), Controls spatial frame > Gear Pairs > Gear Pair: Varying Stiffness.
8
Click  Disable.
Solution 4 (sol4)
1
In the Study toolbar, click  Show Default Solver.
2
In the Model Builder window, expand the Solution 4 (sol4) node, then click Time-Dependent Solver 1.
3
In the Settings window for Time-Dependent Solver, locate the Absolute Tolerance section.
4
From the Tolerance method list, choose Manual.
5
In the Study toolbar, click  Compute.
Results
Use the following instructions to plot the pinion angular velocity and the contact force for the gear pairs as shown in Figure 4 and Figure 5 respectively.
Pinion angular velocity
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Pinion angular velocity in the Label text field.
3
Click to expand the Title section. From the Title type list, choose Label.
4
Locate the Plot Settings section.
5
Select the y-axis label checkbox. In the associated text field, type Angular velocity (rad/s).
Global 1
1
Right-click Pinion angular velocity 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 > Gear pairs > Gear Pair: Rigid > Pinion > mbd.grp1.tht_pn - Pinion angular velocity - rad/s.
3
Locate the x-Axis Data section. From the Parameter list, choose Expression.
4
Click Replace Expression in the upper-right corner of the x-Axis Data section. From the menu, choose Component 1 (comp1) > Multibody Dynamics > Hinge joints > Hinge Joint: Fixed-Gear 1 > mbd.hgj1.th - Relative rotation - rad.
5
Locate the x-Axis Data section. From the Unit list, choose °.
6
Select the Description checkbox. In the associated text field, type Gear rotation.
7
Click to expand the Coloring and Style section. From the Width list, choose 2.
8
Click to expand the Legends section. From the Legends list, choose Manual.
9
In the table, enter the following settings:
Legends
Rigid
Global 2
1
Right-click Global 1 and choose Duplicate.
2
In the Settings window for Global, locate the Data section.
3
From the Dataset list, choose None.
4
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
mbd.grp2.tht_pn
rad/s
Pinion angular velocity
5
Locate the Data section. From the Dataset list, choose Study 2: Transient (Constant Stiffness)/Solution 2 (sol2).
6
Locate the Legends section. In the table, enter the following settings:
Legends
Constant stiffness
Global 3
1
Right-click Global 2 and choose Duplicate.
2
In the Settings window for Global, locate the Data section.
3
From the Dataset list, choose None.
4
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
mbd.grp3.tht_pn
rad/s
Pinion angular velocity
5
Locate the Data section. From the Dataset list, choose Study 3: Transient (Varying Stiffness)/Solution 3 (sol3).
6
Locate the Legends section. In the table, enter the following settings:
Legends
Varying stiffness
Global 4
1
Right-click Global 3 and choose Duplicate.
2
In the Settings window for Global, locate the Data section.
3
From the Dataset list, choose None.
4
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
mbd.grp4.tht_pn
rad/s
Pinion angular velocity
5
Locate the Data section. From the Dataset list, choose Study 4: Transient (Transmission Error)/Solution 4 (sol4).
6
Locate the Legends section. In the table, enter the following settings:
Legends
Transmission error
Pinion angular velocity
1
In the Model Builder window, click Pinion angular velocity.
2
In the Pinion angular velocity toolbar, click  Plot.
3
Click the  Zoom Extents button in the Graphics toolbar.
Contact force
1
Right-click Pinion angular velocity and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Contact force in the Label text field.
3
Locate the Plot Settings section. In the y-axis label text field, type Force (N).
4
Locate the Legend section. From the Position list, choose Lower right.
Global 1
1
In the Model Builder window, expand the Contact force node, then click Global 1.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
mbd.grp1.Fc
N
Force at contact point
Global 2
1
In the Model Builder window, click Global 2.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
mbd.grp2.Fc
N
Force at contact point
Global 3
1
In the Model Builder window, click Global 3.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
mbd.grp3.Fc
N
Force at contact point
Global 4
1
In the Model Builder window, click Global 4.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
mbd.grp4.Fc
N
Force at contact point
4
In the Contact force toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Use the following instructions to plot the reaction forces and moments at the wheel center when the gear pair has varying stiffness as shown in Figure 6 and Figure 7 respectively.
Reaction forces (Varying stiffness)
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Reaction forces (Varying stiffness) in the Label text field.
3
Locate the Data section. From the Dataset list, choose Study 3: Transient (Varying Stiffness)/Solution 3 (sol3).
4
Locate the Title section. From the Title type list, choose Label.
5
Locate the Plot Settings section.
6
Select the y-axis label checkbox. In the associated text field, type Force (N).
Global 1
1
Right-click Reaction forces (Varying stiffness) 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 > Gear pairs > Gear Pair: Varying Stiffness > Wheel > Force at wheel center - N > mbd.grp3.F_whx - Force at wheel center, x-component.
3
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
mbd.grp3.F_whx
N
Force at wheel center, x-component
mbd.grp3.F_why
N
Force at wheel center, y-component
mbd.grp3.F_whz
N
Force at wheel center, z-component
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type mbd.hgj1.th.
6
From the Unit list, choose °.
7
Select the Description checkbox. In the associated text field, type Gear rotation.
8
Locate the Coloring and Style section. From the Width list, choose 2.
9
In the Reaction forces (Varying stiffness) toolbar, click  Plot.
Reaction forces (Varying stiffness)
1
In the Model Builder window, click Reaction forces (Varying stiffness).
2
In the Settings window for 1D Plot Group, locate the Axis section.
3
Select the Manual axis limits checkbox.
4
In the x minimum text field, type 0.
5
In the x maximum text field, type 55.
6
In the y minimum text field, type -200.
7
In the y maximum text field, type 500.
8
In the Reaction forces (Varying stiffness) toolbar, click  Plot.
Reaction moments (Varying stiffness)
1
Right-click Reaction forces (Varying stiffness) and choose Duplicate.
2
In the Settings window for 1D Plot Group, type Reaction moments (Varying stiffness) in the Label text field.
3
Locate the Plot Settings section. In the y-axis label text field, type Moment (N*m).
Global 1
1
In the Model Builder window, expand the Reaction moments (Varying stiffness) node, then click Global 1.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
mbd.grp3.M_whx
N*m
Moment at wheel center, x-component
mbd.grp3.M_why
N*m
Moment at wheel center, y-component
mbd.grp3.M_whz
N*m
Moment at wheel center, z-component
Reaction moments (Varying stiffness)
1
In the Model Builder window, click Reaction moments (Varying stiffness).
2
In the Settings window for 1D Plot Group, locate the Axis section.
3
In the y minimum text field, type -18.
4
In the y maximum text field, type 17.
5
In the Reaction moments (Varying stiffness) toolbar, click  Plot.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies > Eigenfrequency.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Add an Eigenfrequency study and compute the solution for a rigid gear pair.
Study 5: Eigenfrequency (Rigid)
In the Settings window for Study, type Study 5: Eigenfrequency (Rigid) in the Label text field.
Step 1: Eigenfrequency
1
In the Model Builder window, under Study 5: Eigenfrequency (Rigid) click Step 1: Eigenfrequency.
2
In the Settings window for Eigenfrequency, locate the Study Settings section.
3
Select the Desired number of eigenfrequencies checkbox. In the associated text field, type 5.
4
Locate the Physics and Variables Selection section. Select the Modify model configuration for study step checkbox.
5
In the tree, select Component 1 (comp1) > Multibody Dynamics (mbd) > Joints > Hinge Joint: Fixed-Gear 1 > Prescribed Motion 1, Component 1 (comp1) > Multibody Dynamics (mbd) > Joints > Hinge Joint: Fixed-Gear 2, Component 1 (comp1) > Multibody Dynamics (mbd) > Gear Pairs > Gear Pair: Constant Stiffness, Component 1 (comp1) > Multibody Dynamics (mbd) > Gear Pairs > Gear Pair: Varying Stiffness, and Component 1 (comp1) > Multibody Dynamics (mbd) > Gear Pairs > Gear Pair: Transmission Error.
6
Click  Disable.
7
In the Study toolbar, click  Compute.
Results
Mode Shape (mbd)
1
In the Settings window for 3D Plot Group, locate the Data section.
2
From the Eigenfrequency (Hz) list, choose 24229.
Add another Eigenfrequency study and compute the solution for a gear pair with constant stiffness.
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select General Studies > Eigenfrequency.
4
Click the Add Study button in the window toolbar.
5
In the Home toolbar, click  Add Study to close the Add Study window.
Study 6: Eigenfrequency (Constant Stiffness)
In the Settings window for Study, type Study 6: Eigenfrequency (Constant Stiffness) in the Label text field.
Step 1: Eigenfrequency
1
In the Model Builder window, under Study 6: Eigenfrequency (Constant Stiffness) click Step 1: Eigenfrequency.
2
In the Settings window for Eigenfrequency, locate the Physics and Variables Selection section.
3
Select the Modify model configuration for study step checkbox.
4
In the tree, select Component 1 (comp1) > Multibody Dynamics (mbd) > Joints > Hinge Joint: Fixed-Gear 1 > Prescribed Motion 1, Component 1 (comp1) > Multibody Dynamics (mbd) > Joints > Hinge Joint: Fixed-Gear 2, Component 1 (comp1) > Multibody Dynamics (mbd) > Gear Pairs > Gear Pair: Rigid, Component 1 (comp1) > Multibody Dynamics (mbd) > Gear Pairs > Gear Pair: Varying Stiffness, and Component 1 (comp1) > Multibody Dynamics (mbd) > Gear Pairs > Gear Pair: Transmission Error.
5
Click  Disable.
6
In the Study toolbar, click  Compute.
Results
Mode Shape (mbd) 1
1
In the Settings window for 3D Plot Group, locate the Data section.
2
From the Eigenfrequency (Hz) list, choose 290.59+586.19i.
3
In the Mode Shape (mbd) 1 toolbar, click  Plot.
4
Click the  Zoom Extents button in the Graphics toolbar.
5
From the Eigenfrequency (Hz) list, choose 26751+144.17i.
6
In the Mode Shape (mbd) 1 toolbar, click  Plot.
7
Click the  Zoom Extents button in the Graphics toolbar.
Animation 1
In the Results toolbar, click  Animation and choose Player.