COMSOL 6.4 - Rotors Connected by a Spline Coupling

This example demonstrates a method for coupling two rotors connected by a spline coupling. An eigenfrequency analysis is performed at different rotor speeds, and the results are compared with those obtained in Ref. 1.

Model Definition

The model consists an assembly of two rotors connected through a spline coupling. The rotor configuration is visualized in Figure 1. The first rotor, spanning from point A to C, has a fixed support at point A and is equipped with a disk at point B. The second rotor, spanning from point C to F, is simply supported at point D and F. A pulley is mounted at point E. The rotors are connected through point C. It is assumed that only translational motion is coupled between the rotors through the coupling, while the rotations are uncoupled.

Figure 1: Rotor geometry.

The disk mounted on the cantilevered rotor has both mass and moment of inertia. The coupling has only mass with negligible moment of inertia. In addition, the pulley mounted on the simply supported rotor has only mass but negligible moment of inertia.

The properties of the shaft are summarized in the Table 1.

Table 1: properties of the shaft.
Property	value
Density ρ	7832 kg/m3
Young’s modulus E	206820 MPa
Shear modulus G	79546 MPa
Diameter D	12.7 mm
Length l1	88.9 mm
Length l2	76.2 mm

The properties of the mountings on the shafts are given in the Table 2.

Table 2: Properties of the mountings.
Property	Value
Mass of the disk md	1.27 kg
Polar moment of inertia of the disk Ip	0.00256 kg/m2
Diametral moment of inertia of the disk Id	0.00128 kg/m2
Mass of the coupling mc	0.7 kg
Mass of the pulley mp	0.87 kg

Results and Discussion

Figure 2 below shows the whirl plots for the second forward and backward mode at
50,000 rpm. The discontinuity in the rotational degrees of freedom is clearly visible at the coupling location. The undeformed geometry of the rotor is also shown below the whirl plots.

Figure 2: Whirl plots of forward and backward whirl at 50,000 rpm.

The gyroscopic effect in the rotor stiffens the forward modes and softens the backward modes. This phenomenon is illustrated in the Campbell diagram shown in Figure 3. A comparison of the critical speeds (in rpm) with those of Ref. 1 is given in Table 3.

Table 3: Comparison of Critical Speeds with ref. 1.
Mode	COMSOL	REf. 1
First (backward)	7488	7489
First (forward)	8519	8521
Second (backward)	21690	21703
Second (forward)	21707	21720
Third (backward)	23384	23404
Third (forward)	25358	25373
Fourth (backward)	27052	27056

It is clear that the second mode is relatively unaffected by the gyroscopic effect. However, the fourth mode is very strongly influenced by the gyroscopic effect.

Figure 3: Campbell diagram.

Reference

1. W.J. Chen and E.J. Gunter, Introduction to the Dynamics of Rotor-Bearing Systems, Example 5.3, pp. 186–189, Trafford Publishing, 2007.

Rotordynamics_Module/Verification_Examples/coupled_rotors

Modeling Instructions

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New

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Model Wizard

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In the tree, select > > .

Click .

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In the tree, select > .

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Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
Ow	0[rpm]	0 1/s	Angular speed of the shaft
rho_shaft	7832[kg/m^3]	7832 kg/m³	Density of the shaft
E_shaft	206820[MPa]	2.0682E11 Pa	Young's modulus of the shaft
G_shaft	79546[MPa]	7.9546E10 Pa	Shear modulus of the shaft
d_shaft	12.7[mm]	0.0127 m	Diameter of the shaft
m_disk	1.27[kg]	1.27 kg	Mass of the disk
Ip_disk	0.00256[kg*m^2]	0.00256 kg·m²	Polar moment of inertia of the disk
Id_disk	0.00128[kg*m^2]	0.00128 kg·m²	Diametral moment of inertia of the disk
m_coupling	0.7[kg]	0.7 kg	Mass of the coupling
m_pulley	0.87[kg]	0.87 kg	Mass of the pulley
l1	88.9[mm]	0.0889 m	Distance of the disk from the clamped end
l2	76.2[mm]	0.0762 m	Distance of the coupling end from the disk
x_disk	l1	0.0889 m	Position of the disk
x_coupling	l1+l2	0.1651 m	Position of the coupling
x_support1	l1+2*l2	0.2413 m	Position of the first pinned support
x_pulley	l1+3*l2	0.3175 m	Position of the pulley
x_support2	l1+4*l2	0.3937 m	Position of the second pinned support

Geometry 1

Polygon 1 (pol1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type 0 x_disk x_coupling x_support1 x_pulley x_support2.

In the text field, type 0.

Beam Rotor (rotbm)

In the window, under click .

In the window for , locate the section.

In the text field, type Ow.

Set the to use the Young’s modulus and shear modulus as the elastic properties.

Linear Elastic Material 1

In the window, under > click .

In the window for , locate the section.

From the list, choose .

Materials

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	E_shaft	Pa	Young’s modulus and shear modulus
Shear modulus	G	G_shaft	N/m²	Young’s modulus and shear modulus
Density	rho	rho_shaft	kg/m³	Basic