Methodology of modelling the flexural-torsional vibrations in transient states of the rotating

power transmission systems

Tomasz Matyja

Faculty of Transport

Department of Logistics and Aviation Technologies

11:55

Rotating power transmission systems

2

Dynamic behaviors investigated by methods of Rotordynamics

Turbochargers

Ship shaft line

Helicopter drive system

Much more …

Motivation for research and work

Model of rigid rotor with six degrees of freedom

Decoupling equations of the rotor motion

Proposed decomposition method of the typical rotating system

Simulink library for modeling and simulation of dynamic phenomena in rotating power transmission systems

Examples of simulation studies

Conclusions

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Plan of presentation

3

Due:

- the use of new materials (eg composites),

- lightest and less stiffness shafts,

- higher speeds.

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Motivation for research and work

4

More important are:

- non-linear nature of dynamic phenomena in rotating

systems,

- couplings between different forms of vibration:

flexular, torsional and longitudinal,

- transient analysis.

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Methods used in the Rotordynamics

5

Transfer Matrix Method [Prohl]

Finite Element Method

1D – Timoshenko beam elements and rigid rotors

2D – axisymmetric elements

3D – volume elements

Typicallyflexuralvibrationonly

Old method, flexural vibration of linear systems

Allows to considering couplings

The dynamics of rotating systems is so different from the dynamics of the structure that specialized software is necessary.

DYNROT (FEM, MATLAB) [Genta: Dynamics of RotatingSystems. 2005]

MESWIR [Kiciński: Rotor Dynamics. 2005]

ANSYS [Rotordynamics with ANSYS MechanicalSolutions.2009]

MSC NASTRAN [Rotordynamics user’s guide,2016] DyRoBeS [Chen W.J.: Introduction to Dynamics of Rotor-

Bearing Systems] XLRotor [Vance J., Zeidan F., Murphy B.: Machinery

Vibration and Rotordynamics]

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Software used in Rotordynamics

6

academic

There has now been a lot of rotodynamicscomputer software, both academic and commercial.

commercial

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Motivation for research and work

7

Existing methods of modelling the rotating system and available software not always allow to study coupled vibrations in transient states.

OrSimulations of this type are very time-consuming, due to numerical integration of the equations of motion and a large number of degrees of freedom.

Hence the idea of building own tools for modeling rotating systems

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Model of rigid rotor with six degrees of freedom

8

three translational coordinates 𝑥𝑐 , 𝑦𝑐 , 𝑧𝑐

three rotational coordinates {𝜓,𝜃,𝜑}Angles {ψ,θ} describe inclination of a rotor’s plane and must be small. Rotation angle φ connected with the axis ζ may have arbitrary values.

C is a geometrical center of a rotor

(𝜉, 휂, 휁) movable coordinate system

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Selection of Euler angles (Rot 1-2-3) allows to avoid the „gimbal lock”

9

𝑅 =

=

𝑐𝑜𝑠휃𝑐𝑜𝑠𝜑 𝑠𝑖𝑛𝜓𝑠𝑖𝑛휃𝑐𝑜𝑠𝜑 + 𝑐𝑜𝑠𝜓𝑠𝑖𝑛𝜑 −𝑐𝑜𝑠𝜓𝑠𝑖𝑛휃𝑐𝑜𝑠𝜑 + 𝑠𝑖𝑛𝜓𝑠𝑖𝑛𝜑−𝑐𝑜𝑠휃𝑠𝑖𝑛𝜑 −𝑠𝑖𝑛𝜓𝑠𝑖𝑛휃𝑠𝑖𝑛𝜑 + 𝑐𝑜𝑠𝜓𝑐𝑜𝑠𝜑 𝑐𝑜𝑠𝜓𝑠𝑖𝑛휃𝑠𝑖𝑛𝜑 + 𝑠𝑖𝑛𝜓𝑐𝑜𝑠𝜑

𝑠𝑖𝑛휃 −𝑠𝑖𝑛𝜓𝑐𝑜𝑠휃 𝑐𝑜𝑠𝜓𝑐𝑜𝑠휃.

Transformation matrix

𝑅𝜃=

𝜋2

=0 sin 𝜓 + 𝜑 −cos 𝜓 + 𝜑

0 cos 𝜓 + 𝜑 sin 𝜓 + 𝜑1 0 0

.휃 →

𝜋

2lock if

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Static and dynamic unbalance

10

C - geometrical center

P - rotor’s mass center

𝛾- precede angle

of principal axes,

𝛿 - inclining angle

of principal axes

eccentricity

𝑒 , 𝛽

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Inertia matrix of the rotor

11

𝕁123 ≡ 𝕁 =

𝐽𝑡 0 00 𝐽𝑡 00 0 𝐽𝑝

𝕁𝜉𝜂𝜁

=

𝐽𝑡 + 𝐽𝑝 − 𝐽𝑡 cos2 𝛾 sin2 𝛿 𝐽𝑝 − 𝐽𝑡 𝑠𝑖𝑛𝛾𝑐𝑜𝑠𝛾 sin2 𝛿 − 𝐽𝑝 − 𝐽𝑡 𝑐𝑜𝑠𝛾𝑠𝑖𝑛𝛿𝑐𝑜𝑠𝛿

𝐽𝑝 − 𝐽𝑡 𝑠𝑖𝑛𝛾𝑐𝑜𝑠𝛾 sin2 𝛿 𝐽𝑡 + 𝐽𝑝 − 𝐽𝑡 sin2 𝛾 sin2 𝛿 − 𝐽𝑝 − 𝐽𝑡 𝑠𝑖𝑛𝛾𝑠𝑖𝑛𝛿𝑐𝑜𝑠𝛿

− 𝐽𝑝 − 𝐽𝑡 𝑐𝑜𝑠𝛾𝑠𝑖𝑛𝛿𝑐𝑜𝑠𝛿 − 𝐽𝑝 − 𝐽𝑡 𝑠𝑖𝑛𝛾𝑠𝑖𝑛𝛿𝑐𝑜𝑠𝛿 𝐽𝑝 − 𝐽𝑝 − 𝐽𝑡 sin2 𝛿

in the system of principal axes

in local moving frame

Axisymmetric rotor

More general case

𝕁𝜉𝜂𝜁 =

𝐽11 𝐽12 𝐽13

𝐽21 𝐽22 𝐽23

𝐽31 𝐽32 𝐽33

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Lagrange equations of rotor motion

12

𝕄 =

𝑚 0 00 𝑚 00 0 𝑚

0 0 00 0 00 0 0

0 0 00 0 00 0 0

𝐽11 𝐽12 𝐽13

𝐽21 𝐽22 𝐽23

𝐽31 𝐽32 𝐽33

, 𝕢 =

𝑥𝐶

𝑦𝐶

𝑧𝐶

𝜓휃𝜑

,ℚ =

𝑄𝑥𝐶

𝑄𝑦𝐶

𝑄𝑧𝐶

𝑄𝜓

𝑄𝜃

𝑄𝜑

, 𝕍 =𝑉𝑃

Ω𝜉𝜂𝜁.

𝐸𝑘 =1

2𝕍𝑇𝕄𝕍.

𝜕𝕍

𝜕 𝑞

𝑇

𝕄 𝕍 +𝑑

𝑑𝑡

𝜕𝕍

𝜕 𝑞

𝑇

−𝜕𝕍

𝜕𝑞

𝑇

𝕄𝕍 = 𝑄𝑞 . 𝑞 ∈ {𝑥𝐶 , 𝑦𝐶 , 𝑧𝐶 , 𝜓, 휃, 𝜑

Mass matrix, generalized coordinates, generalized forces, velocities vector

total kinetic energy of the rotor

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Inertia coupling equations

13

𝜓[ 𝐽11 𝑐𝑜𝑠2 𝜑 − 𝐽12𝑠𝑖𝑛2𝜑 + 𝐽22 𝑠𝑖𝑛2 𝜑 + 2휃(𝐽13𝑐𝑜𝑠𝜑 − 𝐽23𝑠𝑖𝑛𝜑) + 휃2𝐽33]+

휃 (𝐽11 − 𝐽22)𝑠𝑖𝑛𝜑𝑐𝑜𝑠𝜑 + 𝐽12(𝑐𝑜𝑠2 𝜑 − 𝑠𝑖𝑛2 𝜑) + 휃(𝐽13𝑠𝑖𝑛𝜑 + 𝐽23𝑐𝑜𝑠𝜑) + 𝜑 [𝐽31𝑐𝑜𝑠𝜑 − 𝐽32𝑠𝑖𝑛𝜑 + 휃𝐽33] + 𝜑 휃 (𝐽11 − 𝐽22)𝑐𝑜𝑠2𝜑 − 2𝐽12𝑠𝑖𝑛2𝜑 + 휃(𝐽13𝑐𝑜𝑠𝜑 − 𝐽23𝑠𝑖𝑛𝜑) + 𝐽33 + 𝜑 𝜓 (𝐽22 − 𝐽11)𝑠𝑖𝑛2𝜑 − 2𝐽12𝑐𝑜𝑠2𝜑 − 2휃(𝐽13𝑠𝑖𝑛𝜑 + 𝐽23𝑐𝑜𝑠𝜑) + 2𝜓 휃 [𝐽13𝑐𝑜𝑠𝜑 − 𝐽23𝑠𝑖𝑛𝜑 + 휃𝐽33] + (휃 2 − 𝜑 2)(𝐽13𝑠𝑖𝑛𝜑 + 𝐽23𝑐𝑜𝑠𝜑) = 𝑄𝜓 − 𝑄𝑧𝐶

𝑒 ∙ 𝑠𝑖𝑛(𝛽 + 𝜑 ) .

For exampleequation of rotational motion due to the first rotational coordinate.Linearization: sin 𝛼 ≈ 𝛼 cos 𝛼 ≈ 1 𝛼 ∈ 𝜓, 휃 - small angles .

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Decoupling the equationsof the rotor motion

14

Φ1 𝑞, 𝑞, 𝑞 = 0

𝑞 = Φ2( 𝑞, 𝑞)

𝑞1 = 𝑞, 𝑞 𝑞1 = Φ3(𝑞1)

a very difficult taskImplicit form:

Explicit form:

Standard form:

The decoupling simplifies numerical integration.

ModelForm

of the equations

Inertia matrixLevel of the linearization

𝜶 ∈ {𝝍, 𝜽 Numeric efficiency

1 Explicit

form

Simplifying

assumptions

𝐉𝟏𝟐 = 𝐉𝟐𝟑 = 𝟎

𝐬𝐢𝐧𝛂 ≈ 𝛂

𝐜𝐨𝐬𝛂 ≈ 𝟏

Faster, the smallest number of

arithmetic operations

2 Implicit

matrix

form

Full without

simplifying

assumptions

𝐬𝐢𝐧𝛂 ≈ 𝛂

𝐜𝐨𝐬𝛂 ≈ 𝟏

Matrix multiplication required

3 Implicit

matrix

form

Full without

simplifying

assumptions

𝐬𝐢𝐧𝛂 ≈ 𝛂

𝐜𝐨𝐬𝛂 ≈ 𝟏 −𝟏

𝟐𝛂𝟐

Additional cost of calculating

the correction matrix

4 Implicit

matrix

form

Full without

simplifying

assumptions

Without linearization Additional cost of calculating

sines and cosines of angles ψ, θ

11:5515

Four models of the rotor

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First method of decuplingModel n0. 1 of the rotor

16

𝐽11 cos2 𝜑 + 𝐽22 sin2 𝜑 𝐽11 − 𝐽22 𝑠𝑖𝑛𝜑𝑐𝑜𝑠𝜑 𝐽31𝑐𝑜𝑠𝜑 + 휃𝐽33

𝐽11 − 𝐽22 𝑠𝑖𝑛𝜑𝑐𝑜𝑠𝜑 𝐽11 sin2 𝜑 + 𝐽22 cos2 𝜑 𝐽31𝑠𝑖𝑛𝜑𝐽31𝑐𝑜𝑠𝜑 + 휃𝐽33 𝐽31𝑠𝑖𝑛𝜑 𝐽33

𝜓 휃 𝜑

=

𝐹𝜓(𝜓, 휃, 𝜑, 𝜓, 휃, 𝜑, 𝑄𝑥𝐶, 𝑄𝑦𝐶

, 𝑄𝑧𝐶, 𝑄𝜓, 𝑄𝜃 , 𝑄𝜑)

𝐹𝜃(𝜓, 휃, 𝜑, 𝜓, 휃, 𝜑, 𝑄𝑥𝐶, 𝑄𝑦𝐶

, 𝑄𝑧𝐶, 𝑄𝜓, 𝑄𝜃 , 𝑄𝜑

𝐹𝜑(𝜓, 휃, 𝜑, 𝜓, 휃, 𝜑, 𝑄𝑥𝐶, 𝑄𝑦𝐶

, 𝑄𝑧𝐶, 𝑄𝜓, 𝑄𝜃 , 𝑄𝜑

,

ℍ

𝜓 휃 𝜑

=

𝐹𝜓

𝐹𝜃

𝐹𝜑

.

𝜓 휃 𝜑

= ℍ−1 𝜑 ∙

𝐹𝜓

𝐹𝜃

𝐹𝜑

.

𝑥𝐶

𝑦𝐶

𝑧𝐶

=1

𝑚∙

𝐹𝑥𝐶

𝐹𝑦𝐶

𝐹𝑧𝐶

,

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Other method of decuplingModel n0. 2 of the rotor

17

𝕍 = 𝔸(𝕢) 𝕢

𝔸𝑇𝕄𝔸 𝕢 + (𝔸𝑇𝕄 𝔸 + 𝔹𝕄𝔸) 𝕢 = ℚ .

𝕢 = ℂ ℚ − (𝔸𝑇𝕄 𝔸 + 𝔹𝕄𝔸) 𝕢 .

𝔸 =

1 0 0 0 0 −𝑒 sin 𝛽 + 𝜑

0 1 0 0 0 𝑒 cos 𝛽 + 𝜑

0 0 1 𝑒 sin 𝛽 + 𝜑 −𝑒 cos(𝛽 + 𝜑) 𝑒 𝜓 cos 𝛽 + 𝜑 + 휃 sin 𝛽 + 𝜑 0 0 0 𝑐𝑜𝑠𝜑 𝑠𝑖𝑛𝜑 00 0 0 −𝑠𝑖𝑛𝜑 𝑐𝑜𝑠𝜑 00 0 0 휃 0 1

𝔹 =

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 − 𝜑 𝑠𝑖𝑛𝜑 − 𝜑 𝑐𝑜𝑠𝜑 휃

0 0 0 𝜑 𝑐𝑜𝑠𝜑 − 𝜑 𝑠𝑖𝑛𝜑 − 𝜓

0 0 0 − 휃𝑐𝑜𝑠𝜑 + 𝜓𝑠𝑖𝑛𝜑 휃𝑠𝑖𝑛𝜑 + 𝜓𝑐𝑜𝑠𝜑 0

.

mathematical verification - comparison of equations

verification by simulation methods -comparison of vibration signals from simulations

experimental verification- comparison of measured vibration signals with simulation results

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Three methods used to verify the model

18

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Mathematical verification -comparison of equations

19

Proposed6 DOF model with

static and dynamic imbalance Simplified

6 DOF model v. 1

4 DOF model from

literature

linearization

switching off torsional and longitudinal vibrations

6 DOFideal rotor

turning off imbalance

From Euler-Newtonequations

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Verification by simulation methods -comparison of vibration signals from

simulations

20

Jetcrofft’s rotor6 DOF

Jetcrofft’s rotor4 DOF

Developed by the author

11:55

Verification by simulation methods -comparison of vibration signals from

simulations

21

Slower startup - less torsional vibrations -greater compatibility

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Experimental verification-comparison of measured vibration

signals with simulation results

22

Frequency o the flexural vibrations measured on the bearings were compared. More comprehensive comparison is planned.

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Interactions between the rigid rotor and the shaft

23

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Proposed decomposition method of typical rotating system

24

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Interactions between the inertial and compliance elements

25

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Simulink library for modeling and simulation of dynamic phenomena in rotating power transmission systems

26

Model of thin rigid rotorFrom:

To:Forces kinematic

output

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Thin vs. Long Rotor

27

Presented model can be easily generalized to the case of a long rotor.

thin rotor long rotor

Two additional outputs are necessary

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Block modelling the shaft

2828

kinematic signal from left inertia element

kinematic signal from right inertia element

kinematic signal internal DOF

force acting to left inertia element

force acting to right inertia element

Block modeling shaft must have 12 external DOF (6 on each end).

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FEM model of the shaft

29

FEM - Timoshenko beam elements – better !Rigid Finite Element Method (beam elements) – faster !

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Rigid Finite Element Method

30

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A special beam element

31

𝑵 =

𝑁𝑢𝑢

1 𝑁𝑢𝑣1 0 𝑁𝑢𝜓

1 𝑁𝑢휃1 𝑁𝑢𝜙

1 𝑁𝑢𝑢2 𝑁𝑢𝑣

2 0 𝑁𝑢𝜓2 𝑁𝑢휃

2 𝑁𝑢𝜙2

𝑁𝑣𝑢1 𝑁𝑣𝑣

1 0 𝑁𝑣𝜓1 𝑁𝑣휃

1 𝑁𝑣𝜙1 𝑁𝑣𝑢

2 𝑁𝑣𝑣2 0 𝑁𝑣𝜓

2 𝑁𝑣휃2 𝑁𝑣𝜙

2

𝑁𝑤𝑢1 𝑁𝑤𝑣

1 𝑁𝑤𝑤1 𝑁𝑤𝜓

1 𝑁𝑤휃1 𝑁𝑤𝜙

1 𝑁𝑤𝑢2 𝑁𝑤𝑣

2 𝑁𝑤𝑤2 𝑁𝑤𝜓

2 𝑁𝑤휃2 𝑁𝑤𝜙

2

𝑁𝜓𝑢1 𝑁𝜓𝑣

1 0 𝑁𝜓𝜓1 𝑁𝜓휃

1 𝑁𝜓𝜙1 𝑁𝜓𝑢

2 𝑁𝜓𝑣2 0 𝑁𝜓𝜓

2 𝑁𝜓휃2 𝑁𝜓𝜙

2

𝑁휃𝑢1 𝑁휃𝑣

1 0 𝑁휃𝜓1 𝑁휃휃

1 𝑁휃𝜙1 𝑁휃𝑢

2 𝑁휃𝑣2 0 𝑁휃𝜓

2 𝑁휃휃2 𝑁휃𝜙

2

0 0 0 0 0 𝑁𝜙𝜙1 0 0 0 0 0 𝑁𝜙𝜙

2

.

𝑁𝑀𝑥

𝑀𝑦

𝑇𝑦

𝑇𝑥

𝑀𝑧

=

𝐸𝐴 𝐸𝑆𝑥 −𝐸𝑆𝑦 0 0 0

𝐸𝑆𝑥 𝐸𝐼𝑥 −𝐸𝐷𝑥𝑦 0 0 0

−𝐸𝑆𝑦 −𝐸𝐷𝑥𝑦 𝐸𝐼𝑦 0 0 0

0 0 0 𝜒1𝑦𝐺𝐴 0 𝜒1𝑦𝐺𝑆𝑦

0 0 0 0 𝜒1𝑥𝐺𝐴 −𝜒1𝑥𝐺𝑆𝑥

0 0 0 𝜒1𝑦𝐺𝑆𝑦 −𝜒1𝑥𝐺𝑆𝑥 𝐺(𝜒1𝑥𝐼𝑥 + 𝜒1𝑦 𝐼𝑦 )

휀𝜅𝑥

𝜅𝑦

𝛽𝑥

𝛽𝑦

𝜅𝑧

.

𝕄𝕢 + 휂𝑉𝕂 + Ω𝔾 𝕢 + 𝕂 + 휂𝑉Ω𝕂𝐶 + Ω 𝔾 𝕢 = ℚ + ℚ𝑢 − Ω 𝕗𝜙 ,

Coordinate system axes do not have to be the principal axes of inertia

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Universal dynamic model of the bearing

32

𝑭𝐵 = 𝕂(𝒒𝑏 − 𝒒ℎ) + 𝕂𝑛(𝒒𝑏 − 𝒒ℎ)𝑛 + 𝔻(𝒒 𝑏 − 𝒒 ℎ),

11:55

Universal dynamic model of the clutch

33

𝑭𝑊 = 𝑭𝑊(𝒒𝐿 , 𝒒 𝐿 , 𝒒𝑅 , 𝒒 𝑅),

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Broadband torsional vibration damper

34

Viscous damper

Rubber damper

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Influence of dynamic imbalance on the vibrations

35

Machine startup simulation

𝜔𝑛 = 1000 𝑟𝑎𝑑/𝑠

𝑡𝑛 = 2.5 𝑠

Four cases of the main rotor dynamic imbalance:00; 50; 100; 150.

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Simulink model

36

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Trajectory of the mass center – main rotor

37

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Trajectory of the mass center –external rotor

38

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„X” component of transverse vibration

39

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Gyroscopic effect - inclination angle of the rotor

40

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Torsional vibration velocity

410 0.5 1 1.5 2 2.5-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

d

/dt

[rad/s

]

t [s]

difference between the main rotor speed and drive speed

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Torsional vibration acceleration

42

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Gyroscopic effect – dynamicimbalance 50

43

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Distance from the geometric axis of rotation

44

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RMS of accelerations of torsional vibrations

45

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Scheme of the rotating system with transverse crack

46

𝕂′ = 𝕂 − 𝑓 𝜓 ∙ Δ𝕂

𝑓 𝜓 =1 + cos 𝜓

2

1- crak open0- crak closed

RFEM elements

11:5547

Selected results of simulations

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Spectrogram of the gyroscopic acceleration of the central rotor without crak

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Spectrogram of the gyroscopic acceleration of the central rotor with crak

• Presented method of modeling the rotating systems is an alternative to existing systems based on FEM.

• Its advantage is the relatively small number of degrees of freedom which reduces simulation time and allows to study non-steady states with low cost of computer hardware.

• The authors library of Simulink blocks can be freely expanded with new elements.

• Another advantage is ability to easily integrate with tools and models available in Simulink and Matlab.

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Conclusions

Thank you for your attention

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