Shaft Vibration due to Gear Rattle and Bearing Misalignment

In a gearbox, vibrations due to gear rattling and bearing misalignment are well known sources of noise. In this example, two shafts connected through a pair of gears are considered. The shafts are supported on roller bearings at their ends. Initially, the driven shaft is unloaded and the driver shaft rotates with a varying speed. Due to backlash, intermittent tooth meshing causes vibrations in the shafts. After some time, a resisting torque is applied to the driven shaft, making the tooth meshing smooth. In order to analyze the effect of bearing misalignment on rotor vibrations, a time-dependent analysis is performed for two cases. In the first case, all bearings are aligned with the shafts, and in the second, one of the bearings (number 2) has a small angular misalignment.

This model requires the Multibody Dynamics Module and the Rotordynamics Module.

Model Definition

The model consists of two shafts connected through a pair of spur gears. The spur gear of the first (driver) shaft transfers rotation to the larger spur gear of the second (driven) shaft. Both shafts are supported at their ends using roller bearings.

The geometry is shown in Figure 1 below.

Figure 1: System geometry.

Shafts

The shafts are made of structural steel, with diameter and length of 20 mm and 300 mm, respectively. The mean angular speed of the driver shaft is assumed to be ω0 = 20 rad/s. It fluctuates about this mean speed as:

Bearings

Each shaft is supported by deep groove ball bearings at the ends. The bearings have the same dimensions and material properties. The properties are given in Table 1.

Table 1: Bearing Properties.
Property	Value
Number of balls	20
Ball diameter	1.33 mm
Pitch diameter	21.33 mm
Contour radius, inner race	0.7049 mm
Contour radius, outer race	0.7049 mm
Young’s modulus	200 GPa
Poisson’s ratio	0.3

Gears

The properties of the spur gears are given in Table 2.

Table 2: gear properties.
Property	value
Number of teeth (Wheel)	20
Pitch diameter (Wheel)	100 mm
Number of teeth (Pinion)	30
Pitch diameter (Pinion)	150 mm
Pressure angle	25º
Gear ratio	1.5
Backlash	1 mm

The density of the gears, used to compute inertial properties, is equal to the shaft density.

constraints and loads

•

The axial rotation is prescribed at one end of the driver shaft.

•

A resisting load torque of 100 Nm is applied at the opposite end of the driven shaft. The torque is activated only after the driver shaft has completed a 45° rotation.

Results and Discussion

Figure 2 shows the axial stress variation in the shafts. In addition to the torque, there are mainly two forces acting on the shafts. One, a gear mesh force acting in the pressure angle direction which bends the shafts in opposite direction and other the reaction forces from the bearing. Reaction forces in the bearing mainly support the shaft against the gear mesh force. Moreover, due to the angular misalignment in one of the bearings, an additional reaction moment is also present to overcome the misalignment in that bearing. The net axial stress in the shaft will be a combination of the bending of the shaft about two axes, one perpendicular to the pressure angle direction, and the other parallel to the misalignment axis. From the stress distribution in the shafts it is clear that the bending in the shaft due to gear mesh force is larger as compared to the bending due to the bearing misalignment. The bearing force direction on the shafts confirms that gear meshing forces are directly transmitted to the bearings in the pressure angle direction.

Figure 2: Stresses and bearing forces.

Figure 3 shows a comparison of angular speeds of the wheel and the pinion, for the cases of an aligned and a misaligned bearing. There is a slightly higher torsional vibration for the case of a misaligned bearing. The rattling vibrations in the misaligned case also persist for longer duration.

Figure 3: Angular velocity of the shafts for aligned bearings (left), and for a misaligned bearing (right).

The axial vibration at the wheel, after the rattling subsides, is shown in Figure 4. In the case of a misaligned bearing, axial vibrations (accelerations) are mainly due to the misalignment in the bearing. As can be seen in Figure 4 that the accelerations for the case of a misaligned bearing are significantly higher than that with aligned bearings.

Due to the angular misalignment in the bearings the force transmitted through the roller to the respective races has an axial component. This is the reason for the significant axial vibration of the shafts with misaligned bearing.

Figure 4: Axial acceleration at the wheel, for aligned bearings (left), and for a misaligned bearing (right).

Frequency spectra for axial vibrations are shown in Figure 5. Compared to the case of aligned bearings, the case of a misaligned bearing shows participation from a broader range of frequencies. Vibration amplitude at higher frequencies are significantly higher in misaligned case.

Figure 5: Frequency spectrum of the axial acceleration at the wheel, for the aligned (left) and misaligned (right) bearing.

Figure 6 shows the bearing force components for bearing 2. The case where the bearing is aligned is shown on the left, and the case where it is misaligned, on the right. In the misaligned case the force variations are slightly larger and contain higher frequencies.

Figure 6: Force in bearing 2, for the aligned (left) and misaligned (right) bearing.

Figure 7 compares the moments in bearing 2. The result for the aligned bearings is presented on the left, and for the misaligned bearing on the right. One can clearly see a large moment about the x-axis in the misaligned case. The moment about z-axis also has slightly higher amplitude variation in misaligned case.

Figure 7: Moments in bearing 2, for the aligned (left) and misaligned (right) bearing.

A comparison of the rotor tilting in bearing 2 is shown in Figure 8 with results for the aligned case on the left and for the misaligned case on the right. Since the misalignment is quite small, the mean titling and amplitude do not differ significantly in both cases. However, in the misaligned case, there is a high frequency variation in the tilting.

Figure 8: Rotor tilting in bearing 2, for the aligned (left) and misaligned (right) bearing.

The gear mesh contact force for the cases of aligned and misaligned bearings are shown in Figure 9. During the rattling vibration, intermittent contact in the gear meshing is clearly visible. The contact force variation is only lightly influenced by the misalignment in the bearing, however, after the shaft loading, rattling persists for longer in the case when bearing is misaligned.

Figure 9: Gear mesh contact forces, for aligned bearings (left), and for a misaligned bearing (right).

Rotordynamics_Module/Tutorials/gear_rattle_with_bearing_misalignment

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

Create a list of parameters for the speed and the loading torque.

Global Definitions

Parameters: General

In the window, under click .

In the window for , type Parameters: General in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Value	Description
omega0	20[rad/s]	20 rad/s	Mean angular speed
T0	100[N*m]	100 N·m	Loading torque
T	2*pi/omega0	0.31416 s	Time period
isMisaligned	0	0	Is misaligned

Create a list of parameters for the gears.

Parameters: Gears

In the toolbar, click

and choose .

In the window for , type Parameters: Gears in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Value	Description
N1	20	20	No of teeth on gear 1
N2	30	30	No of teeth on gear 2
dp1	100[mm]	0.1 m	Pitch diameter of gear 1
dp2	150[mm]	0.15 m	Pitch diameter of gear 2
gr	N2/N1	1.5	Gear ratio
rc	0.5*(dp1+dp2)	0.125 m	Center to center distance
bl	1e-3[m]	0.001 m	Backlash

Create a list of parameters for the bearings.

Parameters: Roller Bearings

In the toolbar, click

and choose .

In the window for , type Parameters: Roller Bearings in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Value	Description
db	1.33[mm]	0.00133 m	Ball diameter
dp	21.33[mm]	0.02133 m	Pitch diameter
rin	0.53*db	7.049E-4 m	Inner race radius
rout	0.53*db	7.049E-4 m	Outer race radius

Define a variable for the varying angular speed.

Definitions

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
Omega	omega0(1+0.05sin(10omega0t))	rad/s	Angular speed

Start by creating the gear system geometry using the geometry from the Part Libraries.

Part Libraries

In the toolbar, click

and choose .

In the window, under click .

In the window, select in the tree.

Click

Geometry 1

Spur Gear 1 (pi1)

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	20	Number of teeth
dp	dp1	0.1 m	Pitch diameter
lsr	3	3	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

Click

Spur 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	30	Number of teeth
dp	dp2	0.15 m	Pitch diameter
dhr	0.2/gr	0.13333	Hole diameter to pitch diameter ratio (Set 0 for no hole)
wgr	0.2/gr	0.13333	Gear width to pitch diameter ratio
lsr	3/gr	2	Shaft length to pitch diameter ratio (Set 0 for no shaft)
xc	rc	0.125 m	Gear center, x-coordinate
egy	1	1	Gear axis, y-component
egz	0	0	Gear axis, z-component
th	360[deg]/2/N2	6 °	Mesh alignment angle

Click

Create the work planes to partition the shafts at the bearing locations.

Work Plane 1 (wp1)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

In the text field, type -0.13.

Click

Work Plane 2 (wp2)

Right-click and choose .

In the window for , locate the section.

In the text field, type 0.13.

Work Plane 3 (wp3)

Right-click and choose .

In the window for , locate the section.

In the text field, type -0.11.

Work Plane 4 (wp4)

Right-click and choose .

In the window for , locate the section.

In the text field, type 0.11.

Partition Objects 1 (par1)

In the toolbar, click

and choose .

Click in the window and then press Ctrl+A to select both objects.

In the window for , locate the section.

From the list, choose .

Partition Objects 2 (par2)

Right-click and choose .

Select the object only.

In the window for , locate the section.

Find the subsection. Click to select the

toggle button.

Click in the window and then press Ctrl+A to select both objects.

From the list, choose .

Click

Partition Objects 3 (par3)

Right-click and choose .

Click in the window and then press Ctrl+A to select both objects.

In the window for , locate the section.

From the list, choose .

Click

Partition Objects 4 (par4)

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Click in the window and then press Ctrl+A to select both objects.

Click

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Create a to specify the angular speed for the driver shaft.

Multibody Dynamics (mbd)

Attachment 1

In the window, under right-click and choose .

Select Boundary 86 only.

Hinge Joint 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Specify the e0 vector as


0	x
1	y
0	z

Locate the section. From the list, choose .

You create the elastic joint to allow the lateral and tilting motion of the shaft. You will constrain this motion later by adding the support to the shafts.

Prescribed Motion 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the ωp text field, type Omega.

Radial Roller Bearing 1

In the toolbar, click

and choose .

Select Boundaries 87, 88, 118, and 119 only.

In the window for , locate the section.

From the list, choose .

Specify the vector as


1	x
0	y
0	z

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

In the dp text field, type dp.

In the rin text field, type rin.

In the rout text field, type rout.

Radial Roller Bearing 2

Right-click and choose .

In the window for , locate the section.

Click

Select Boundaries 97, 98, 134, and 135 only.

Radial Roller Bearing 3

Right-click and choose .

In the window for , locate the section.

Click

Select Boundaries 343, 344, 378, and 385 only.

Radial Roller Bearing 4

Right-click and choose .

In the window for , locate the section.

Click

Select Boundaries 353, 354, 382, and 389 only.

Add angular misalignment in the bearing located opposite to the prescribed end of the driver shaft. Use a parameter isMisaligned to enable/disable the misalignment in the bearing.

Radial Roller Bearing 2

In the window, click .

Misalignment 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the θ0y text field, type 0.1[deg]*isMisaligned.

Spur Gear 1

In the toolbar, click

and choose .

Select Domain 1 only.

In the window for , locate the section.

In the n text field, type N1.

In the dp text field, type dp1.

In the α text field, type 25[deg].

Locate the section. Specify the eg vector as


0	x
1	y
0	z

Locate the section. From the list, choose .

Spur Gear 2

Right-click and choose .

In the window for , locate the section.

Click

Select Domain 7 only.

Locate the section. In the n text field, type N2.

In the dp text field, type dp2.

Locate the section. Specify the Xc vector as


rc	x
0	y
0	z

Gear Pair 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Select the check box.

Locate the section. From the list, choose .

In the pc text field, type 1e8.

Backlash 1

In the window, click .

In the window for , locate the section.

In the bl text field, type bl.

Increase the penalty factor 200 times to reduce the error in the backlash.

In the pb text field, type ((1[1/ms])^2)*mbd.grp1.Ie*200.

Rigid Connector 1

In the toolbar, click

and choose .

Select Boundary 359 only.

Add a loading torque on the driven shaft. Activate the torque only after the driver shaft has completed a 45 degree rotation.

Applied Moment 1

In the toolbar, click

and choose .

In the window for , locate the section.

Specify the M vector as


0	x
T0*(t>T/8)	y
0	z

Use a swept mesh for the shafts.

Mesh 1

Free Triangular 1

In the toolbar, click

and choose .

Select Boundaries 86 and 342 only.

Size 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Swept 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Select Domains 2–6 and 8–12 only.

Free Tetrahedral 1

In the toolbar, click

In the window for , click

Study 1

Step 1: Time Dependent

Use a small time step in the beginning to resolve the rattling vibration.

In the window, under click .

In the window for , locate the section.

In the text field, type range(0,T/20000,T/4) range(T/4+T/10000,T/10000,T).

Click the

button in the toolbar.

In the dialog box, select in the tree.

In the tree, select the check box for the node .

Click .

Add a batch sweep to solve for two cases: first, all bearings aligned, and second, one of the bearings misaligned.

Batch Sweep

In the toolbar, click

and choose .

In the window for , locate the section.

Clear the check box.

Locate the section. Find the subsection. Clear the check box.

Clear the check box.

Select the check box.

Locate the section. In the text field, type 2.

Locate the section. Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
isMisaligned (Is misaligned)	0 1

Change the settings for the time stepping and maximum number of iterations in the solver to reduce the computation time.

Solution 1 (sol1)

In the toolbar, click

In the window, expand the node, then click .

In the window for , click to expand the section.

From the list, choose .

In the window, expand the node, then click .

In the window for , click to expand the section.

From the list, choose .

In the text field, type 10.

Batch Data

In the toolbar, click

Results

Displacement (mbd)

Displacement is the default plot. Duplicate it and follow the instructions below to create the stress plot as shown in Figure 2.

Stress (mbd)

In the window, expand the node.

Right-click and choose .

In the window for , type Stress (mbd) in the text field.

Surface

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type mbd.SYY.

Click to expand the section. Select the check box.

In the text field, type -3e7.

In the text field, type 3e7.

Gears use the model. Therefore, it is not possible to plot the stress in the gears. Duplicate the existing dataset and restrict the new dataset selection to gears only. You will use this dataset to display the gears in a stress plot.

Study 1/Parametric Solutions 1 (3) (sol2)

In the window, expand the node.