Forces and Moments on Bevel Gears

This model simulates a pair of straight conical bevel gears. The gears are modeled as rigid. A transient analysis is performed to compute the forces and moments at the center of the fixed gear. This model is built using the gears functionality in the Multibody Dynamics interface in COMSOL Multiphysics.

The data for this model is taken from Ref. 1. The results of the analysis are compared with the results given in the reference.

Model Definition

The model geometry is shown in Figure 1. It has three components: wheel, pinion, and a rigid bar. Both the wheel and pinion are modeled as bevel gears. The wheel is fixed on its location. The pinion is mounted on a rigid bar through a hinge joint. The rigid bar is hinged at a point lying on the axis of the wheel.

Figure 1: Model geometry. The wheel center, the hinge location of rigid bar, and the axis of both the gears are also shown.

Model parameters

The properties of the wheel and pinion are given in the table below:

Table 1: Gear properties
Properties		Wheel	pinion
Number of teeth	n	50	10
Pitch diameter	dp	100 mm	20 mm
Pressure angle	α	20°	20°
Cone angle	γ	78.7°	11.3°
Center of rotation	xc	(0, 0, 0) mm	(0, -10, 50) mm
Axis of rotation	eg	(0, -1, 0)	(0, 0, -1)

The pinion is hinged to the rigid bar about the its own axis. The rigid bar is hinged to the ground about the wheel axis. The center of this hinge joint is (0, 10 mm, 0).

Constraints and loads

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The wheel is fixed to the ground.

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The rigid bar is given a rotation of 1 rad.

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A resisting torque of 10 N·m is applied at the joint between pinion and the rigid bar.

Results and Discussion

The comparison of the computed results with the solution presented in Ref. 1 is given in the tables below:

Table 2: Contact force and reaction forces
	Fc	Fx	fy	fz
Computed values (in N)	1064.2	600.32	356.92	802.94
Reference values (in N)	1064	600.35	356.9	802.9

Table 3: reaction moments
	mx	my	mz
Computed values (in N·m)	-9.642	50	-15.017
Reference values (in N·m)	-9.641	50	-15.016

It can be seen that the computed values of the forces and moments are in the very close agreement with the reference values. The units and the coordinate system are different in the reference model. Hence reference results are adjusted appropriately before using them for the comparison.

Figure 2 shows the position of the pinion and the rigid bar after 1 rad of rotation.

Figure 2: Displacement of the pinion and the rigid bar after 1 rad of rotation.

Figure 3 and Figure 4 show the reaction forces and moments at the wheel center. The forces and moments are plotted for various positions of the pinion.

Figure 3: Reaction forces at the center of the wheel.

Figure 4: Reaction moments at the center of the wheel.

Notes About the COMSOL Implementation

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To build a gear geometry, you can import a gear part from the and customize it by changing its input parameters. Alternatively, you can also create an equivalent disc or cone to represent the gear.

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All the gears are assumed rigid. The elasticity of a gear mesh can be included on nodes using the subnode.

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All the nodes are assumed ideal and frictionless. You can add , , or subnodes when required.

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To constraint the motion of a gear, you can use or subnodes. Alternatively, you can mount the gears on a shaft or on the ground through various nodes.

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The contact force on a is computed using or . By default, the contact force computation is turned off. Use the weak constraints method for more accurate contact forces; however, you can, preferably, switch to the penalty method for large rigid body systems.

References

1. A. Cardona, “Three-Dimensional Gears Modeling in Multibody Systems Analysis,” International Journal for Numerical Methods in Engineering, vol. 40, pp. 357–381, 1997.

Multibody_Dynamics_Module/Verification_Examples/bevel_gear_pair

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file bevel_gear_pair_parameters.txt.

Part Libraries

In the toolbar, click

and choose .

In the window, select in the tree.

Click

Geometry 1

Bevel Gear 1 (pi1)

In the toolbar, click

To customize the gear geometry, enter the gear parameters in the input parameters of the part.

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
n	n1	50	Number of teeth
dp	dp1	0.1 m	Pitch diameter
alpha	alpha	20 °	Pressure angle
gamma	gamma1	78.7 °	Cone angle
adr	0.75	0.75	Addendum to module ratio
tfr	0	0	Tip fillet radius to pitch diameter ratio (Set 0 for no fillet)
rfr	0	0	Root fillet radius to pitch diameter ratio (Set 0 for no fillet)
dhr	0.6	0.6	Hole diameter to pitch diameter ratio (Set 0 for no hole)
wgr	0.03	0.03	Gear width to pitch diameter ratio
lsr	0	0	Shaft length to pitch diameter ratio (Set 0 for no shaft)
xc	xcx1	0 m	Gear center, x coordinate
yc	xcy1	0 m	Gear center, y coordinate
zc	xcz1	0 m	Gear center, z coordinate
egx	ex1	0	Gear axis, x component
egy	ey1	-1	Gear axis, y component
egz	ez1	0	Gear axis, z component

Bevel Gear 2 (pi2)

In the toolbar, click

and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
n	n2	10	Number of teeth
dp	dp2	0.02 m	Pitch diameter
alpha	alpha	20 °	Pressure angle
gamma	gamma2	11.3 °	Cone angle
adr	0.3	0.3	Addendum to module ratio
htr	3	3	Tooth height to addendum ratio
tfr	0	0	Tip fillet radius to pitch diameter ratio (Set 0 for no fillet)
rfr	0	0	Root fillet radius to pitch diameter ratio (Set 0 for no fillet)
wgr	0.4	0.4	Gear width to pitch diameter ratio
wbr	0.1	0.1	Back cone width to gear width ratio
wcr	0	0	Collar width to gear width ratio (Set 0 for no collar)
lsr	0	0	Shaft length to pitch diameter ratio (Set 0 for no shaft)
xc	xcx2	0 m	Gear center, x coordinate
yc	xcy2	-0.01 m	Gear center, y coordinate
zc	xcz2	0.05 m	Gear center, z coordinate
egx	ex2	0	Gear axis, x component
egy	ey2	0	Gear axis, y component
egz	ez2	-1	Gear axis, z component
th	360/(2*n2)[deg]	18 °	Mesh alignment angle

Cylinder 1 (cyl1)

In the toolbar, click

In the window for , locate the section.

In the text field, type dp2/8.

In the text field, type dp1/2.

Locate the section. In the text field, type xcx2.

In the text field, type xcy2.

In the text field, type xcz2.

Locate the section. From the list, choose .

In the text field, type -1.

Form Union (fin)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Clear the check box.

In the toolbar, click

Multibody Dynamics (mbd)

Add two nodes and specify all the properties of gears. For the automated creation of nodes from geometry parts, use section of Multibody Dynamics node.