2010:020

M A S T E R ' S T H E S I S

Evaluation of 3D Rotordynamics Capabilitieswithin NX Nastran

Florian Thiery

Luleå University of Technology

Master Thesis, Continuation Courses Engineering mechanics

Department of Applied Physics and Mechanical EngineeringDivision of Solid Mechanics

2010:020 - ISSN: 1653-0187 - ISRN: LTU-PB-EX--10/020--SE

Acknowledgements

First of all, I would like to thank my supervisor Jan-Olov Aidanpää as wellas my co-supervisor Marcus Sandberg for giving me the opportunity to carryout my Master’s thesis at the Solid Mechanics Department and Division ofFunctional Product Development in Luleå University of Technology. Theyhave always been very kind and friendly when answering questions or havingnew ideas to carry through this project.

Then I would like to thank Christina Hamsch, from the International of-fice, for allowing me to stay one more semester to complete the Master’s de-gree. On the other side I would also like to thank the "Scolarité Office" fromENSEM (engineering school) who allowed students like me to stay longer inLTU to finish their Master’s thesis.

I would also like to express my sincere gratitude to my family, parents andgrand-parents from France, who always supported me with my studies andagreed with the way I decided to pursue my education whatever the choicesI made in the last 5 years.

Finally, I am very grateful to Amir Jourak, who helped me giving litera-ture and always shared happiness during coffee breaks; Matthias Wissemberg,friend of mine who made a lot to obtain the authorization to stay in Sweden;I also send a warmly appreciation to Luisa Sa Vitorino, Martin Johanssonwithout forgetting all the other students (exchange or Swedish) for their loveand support during my whole studies.

Luleå, April 2010

Florian Thiery

1

Abstract

In today’s industry, companies within the jet engine industry are trying tomake their product development more effective. One way is to use productmodels to enable upfront evaluation through simulation.

To model the dynamic behaviour of rotors, mainly beam and shell ele-ments are used in the common rotor-dynamics software. As the geometrybecomes more complex, 3D rotor-dynamics can be used to solve vibrationproblems as well as the interactions with bearings and the non-rotating struc-ture. This 3D-modelling could also give a more realistic post-processing thanthe current software in the market and the dynamics of the whole engine couldbe assessed.

To try to solve these rotor-dynamics problems, NX7 was used, a commer-cial CAD/CAM/CAE software developed by Siemens PLM software. Thesolver used is NX Nastran Rotor-Dynamics. The main interest in this choice(with the 3D capabilities) is that both the CAD and CAE will be performedon the same platform and adjustments can be done faster if some problemsin meshing or modelling occur.

The aim of this Master’s thesis is thus to develop a whole engine with theComputer Aided Design capabilities of NX7 and investigate the possibilitiesof doing rotor-dynamics analyses with NX Nastran Rotor Dynamics. Firstof all, simpler models were studied to validate the analyses and more com-plicated elements were added piece by piece until all the problems are solvedto build the whole engine and analyse it.

The results presented in this Master’s thesis reveal several aspects ofrotordynamics analysis. First of all, a section is dedicated to the mode shapesof the rotor for several common models. Then plots of Campbell diagramare shown and compared with theoretical results. Finally, a mass unbalanceanalysis is performed for all types of rotor and the displacement and rotationresponse are plotted for some particular nodes. These displacement curves

2

are also compared with the Campbell diagrams to check if the simulation arerun properly.

The main problem in this Master’s thesis is that the program is today tooundeveloped for efficient 3D modelling of rotors. Indeed, it was found thatrotating and stationary parts could not be coupled in the Campbell diagram.Also a lot of commands and execution needs to be done manually and thepossibilities of post-processing have to be developed to facilitate an effectiveevaluation of the results.

The future of this project will be to try to integrate optimization tools tothese analyses, in order to have a better design and better dynamics results.That is why the 3D rotordynamics simulation within NX are important: itcould enable to design a whole engine, study the dynamics of it and optimizethe whole system to come to a more efficient engine, all of this by using thesame software. This could result in time-savings and more efficient design ofengines in years to come.

3

Table of symbols

Symbol Description Unit

m Mass kgd,C Viscous damping kg/sK Stiffness N/m (kg/s2)Ω Rotational speed rad/sF Force Newton (kg.m/s)E Young’s modulus GPaρ Density kg/m3

ν Poisson’s ratio no unitI Second moment of area m4

α Flexibility coefficient m/NH Moment of Momentum N.s/mM Moment N/mJp Polar moment of inertia kg.m2

Jd Diametric moment of inertia kg.m2

ξ Damping ratio no unitx,y Position mx,y Velocity m/sx,y Acceleration m/s2

e Eccentricity mθ,ϕ Angle rad

θ,ϕ Angular velocity rad/s

θ,ϕ Angular acceleration rad/s2

Nota Bene: The notation used here are general. They can be com-pleted with a subscript, which could mean "rotation", "translation", "x-direction",...

4

Contents

1 Introduction 10

2 Generalities 13

2.1 Rotor dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Linear Jeffcott rotor . . . . . . . . . . . . . . . . . . . 13

2.1.2 The gyroscopic effect . . . . . . . . . . . . . . . . . . . 15

2.1.3 Critical speeds and Campbell diagram . . . . . . . . . 21

2.1.4 Response to unbalance . . . . . . . . . . . . . . . . . . 22

2.2 Finite element method . . . . . . . . . . . . . . . . . . . . . . 23

3 The NX Nastran analysis 27

3.1 Theoretical foundation of rotor dynamics within Nastran . . . 27

3.1.1 Additional terms in the equations of motion . . . . . . 27

3.1.2 Equation of motion for the fixed reference system . . . 28

3.1.3 The Steiner’s inertia terms in the analysis . . . . . . . 30

3.1.4 Real eigenvalue analysis . . . . . . . . . . . . . . . . . 31

3.1.5 The fixed system eigenvalue problem . . . . . . . . . . 31

3.1.6 Solution interpretation . . . . . . . . . . . . . . . . . . 32

3.2 Equation of motion for modal frequency response . . . . . . . 32

4 Modelling of the 2D-3D rotor in NX7 34

4.1 The middle span rotor (Jeffcott rotor) . . . . . . . . . . . . . 34

4.1.1 The geometric modelling of the rotor . . . . . . . . . . 34

4.1.2 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5

4.1.3 The boundary conditions . . . . . . . . . . . . . . . . . 42

4.2 The Overhung Rotor . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 The geometric modelling of the rotor . . . . . . . . . . 42

4.2.2 1D-Meshing . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Defining NX Nastran input for rotor dynamics 45

5.1 Modifications of the .dat file . . . . . . . . . . . . . . . . . . . 45

5.1.1 The file management section . . . . . . . . . . . . . . . 45

5.1.2 The executive control section . . . . . . . . . . . . . . 46

5.1.3 The case control section . . . . . . . . . . . . . . . . . 46

5.1.4 The bulk data section . . . . . . . . . . . . . . . . . . 47

5.2 Interpretation of rotor dynamics output . . . . . . . . . . . . . 48

5.2.1 The .f06 file . . . . . . . . . . . . . . . . . . . . . . . . 49

5.2.2 The .op2 file . . . . . . . . . . . . . . . . . . . . . . . . 49

5.2.3 The .csv file . . . . . . . . . . . . . . . . . . . . . . . . 50

5.3 Procedure to perform an entire rotor dynamics simulation . . 51

6 Analysis of the results 53

6.1 The middle span rotor . . . . . . . . . . . . . . . . . . . . . . 53

6.1.1 Quality of mesh and accuracy . . . . . . . . . . . . . . 53

6.1.2 Campbell diagram . . . . . . . . . . . . . . . . . . . . 54

6.1.3 Parameters influencing the results . . . . . . . . . . . . 59

6.2 The Overhung rotor . . . . . . . . . . . . . . . . . . . . . . . . 60

6.2.1 Mode shapes and Campbell diagram . . . . . . . . . . 60

7 Solving the mass unbalance problem 63

7.1 Mass unbalance cards and modal frequency analysis . . . . . . 63

7.1.1 Additional cards in the bulk data section . . . . . . . . 63

7.1.2 Dynamic load . . . . . . . . . . . . . . . . . . . . . . . 64

7.2 Mass unbalance response of the Jeffcott rotor . . . . . . . . . 66

7.2.1 Synchronous analysis with internal damping . . . . . . 66

7.2.2 Synchronous analysis with external damping . . . . . . 70

6

7.2.3 Asynchronous analysis . . . . . . . . . . . . . . . . . . 72

7.3 Mass unbalance response of the overhung rotor . . . . . . . . . 74

8 Modelling and analyses of a more realistic rotor 77

8.1 Modelling of the rotating part . . . . . . . . . . . . . . . . . . 77

8.1.1 Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . 78

8.1.2 Boundary conditions . . . . . . . . . . . . . . . . . . . 81

8.2 Analysis of the modal complex eigenvalue solution . . . . . . . 82

8.2.1 Shaft without blades and no non-rotating structure . . 82

8.2.2 Shaft without blades and with the non-rotating structure 84

8.2.3 Shaft with blades and no non-rotating structure . . . . 85

8.3 Frequency response analysis of the rotor alone . . . . . . . . . 87

9 Conclusion and discussion 90

7

List of Figures

2.1 The mid-span rotor . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 The unbalance disc on a shaft . . . . . . . . . . . . . . . . . . 14

2.3 Simple view of the overhung rotor . . . . . . . . . . . . . . . . 15

2.4 Forces and moments acting on the system in 3D . . . . . . . . 16

2.5 Forces and moments acting on the system in 2D . . . . . . . . 17

2.6 Geometric representation of z’ . . . . . . . . . . . . . . . . . . 18

2.7 Example of Campbell diagram . . . . . . . . . . . . . . . . . . 21

2.8 Response of an unbalance rotor as a function of the frequency 23

4.1 Revolution of the sketch of the rotor . . . . . . . . . . . . . . 35

4.2 Type of solid elements . . . . . . . . . . . . . . . . . . . . . . 37

4.3 Properties of the shell elements . . . . . . . . . . . . . . . . . 38

4.4 Connections of 1D elements at the end of the shaft (ref [2]) . 40

4.5 The "spider" rigid connection (in blue) . . . . . . . . . . . . . 41

4.6 The overhung rotor model . . . . . . . . . . . . . . . . . . . . 43

4.7 General View of the 3D-rotor . . . . . . . . . . . . . . . . . . 43

4.8 Cage surrounding the shaft with bearings . . . . . . . . . . . . 44

5.1 Data entries of the rotor dynamic analysis . . . . . . . . . . . 49

5.2 Diagram of the entire rotor-dynamics procedure . . . . . . . . 50

6.1 Translation and tilting mode of the mid-span rotor . . . . . . 54

6.2 Example of a CHEXA(20) mesh and the eigenfrequencies . . . 55

6.3 Campbell diagram for the midspan rotor . . . . . . . . . . . . 57

6.4 Damping values as function of the speed . . . . . . . . . . . . 58

6.5 1st and 2nd modes shapes of the overhung rotor . . . . . . . . 60

6.6 Campbell diagram for the overhung rotor . . . . . . . . . . . . 61

8

7.1 Bulk data entries for the dynamic load . . . . . . . . . . . . . 65

7.2 Visualization of the dynamic load . . . . . . . . . . . . . . . . 65

7.3 Displacement for the backward whirl . . . . . . . . . . . . . . 67

7.4 Rotation for the backward whirl . . . . . . . . . . . . . . . . . 68

7.5 Theoretical displacement calculated with Matlab for the back-

ward whirl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.6 One and a half degree of freedom system . . . . . . . . . . . . 70

7.7 Displacement response for several damping ratios . . . . . . . 71

7.8 Displacement for the backward whirl of the asynchronous anal-

ysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

7.9 Rotation for the backward whirl of the asynchronous analysis . 73

7.10 Displacement of node 909 for the backward simulation . . . . 75

7.11 Displacement of node 909 for the forward simulation . . . . . 76

8.1 Rotor modelling with blades . . . . . . . . . . . . . . . . . . . 78

8.2 Real rotor to be modelled . . . . . . . . . . . . . . . . . . . . 79

8.3 Meshing of the blades . . . . . . . . . . . . . . . . . . . . . . . 80

8.4 Connection of the bearings to the non-rotating structure . . . 81

8.5 Mode shapes of the rotor . . . . . . . . . . . . . . . . . . . . . 83

8.6 Unrealistic modes of the rotor . . . . . . . . . . . . . . . . . . 84

8.7 First two modes of the rotor with the structure . . . . . . . . 85

8.8 Mode shape of the blades . . . . . . . . . . . . . . . . . . . . . 86

8.9 Campbell diagram of the realistic rotor . . . . . . . . . . . . . 87

8.10 Backward and forward displacement of the realistic rotor . . . 88

9

Chapter 1

Introduction

The engineering design challenge presented by aerodynamic and hydrody-

namic flows, design of blading, can sometimes cause the rotordynamics re-

quirements of turbo-machinery (e.g. jet engines, gas turbines, compressors)

design to be overlooked. It has happened that expensive turbomachines have

been built and found to be incapable of producing their rated performance

due to assumptions in the model.

Thus in designing, operating and troubleshooting turbo-machinery, rotor

dynamics analysis can help accomplish the following objectives [3] :

1. Predict critical speeds (speeds at which vibration due to rotor imbal-

ance is a maximum)

2. Determine design modification to change critical speeds

3. Predict amplitudes of synchronous vibration caused by rotor imbalance

4. Predict threshold speeds and vibration frequencies for dynamic insta-

bilities

5. Determine design modification to suppress dynamic instabilities

So basically, this project is considering three parts: designing, performing

rotor-dynamic analysis, upgrading the engine (optimizing), but in reality

10

more complex aspects to model have to be taken in account. The following

diagram shows how they are related.

The part which will be developed in this project is the rotor dynamics

analysis in 3D. Normally, these studies are done by using beam elements in

the common rotor dynamics software (see Appendix A). Currently, only a few

software are able to perform 3D rotor dynamics analyses. They usually are

non-specific codes for rotor dynamics (i.e. they were not created to perform

only rotordynamics analyses). These are:

• Ansys, the version 11 workbench and classic is capable of solving the

rotordynamic equations (3-D/2-D and beam element)

• SAMCEF, a finite element based code (3-D/2-D and beam element)

11

• Nastran

Even if these software are not made to study the rotor-dynamics as a

priority, the most interesting part is that compared with the 1D beam element

solvers, all types of element can be used (beam, shell, solid elements) and

connected all together with springs and dampers to a big structure when the

whole designed model is very complicated. But still investigation on these

elements has to be done to be sure that the 3D rotordynamics analyses are

done properly with simple structures. Once this is secured, the 3D meshing

could be develop in a good way to build more accurate models.

12

Chapter 2

Generalities

2.1 Rotor dynamics

The aim of this chapter is to introduce the general concept of rotordynamics

analyses. This is done by deriving the equations from a well-known model

called the Jeffcott Rotor [4], also called the "De Laval rotor" in Europe.

2.1.1 Linear Jeffcott rotor

Figure 2.1: The mid-span rotor

13

The figure 2.1 shows a view of the general Jeffcott rotor. This model

includes unbalance, viscous damping and whirl-instability (see section 2.3).

To understand the behaviour of the Jeffcott rotor, figure 2.2 shows the

view of the disc with a mass unbalance. The disc is mounted on a shaft and

is unbalanced, i.e. differs by the distance e from the geometric center (GC).

Figure 2.2: The unbalance disc on a shaft

The equations of motion can be derived from the previous figures. A

position vector ~r is used from the origin O till the center of gravity of the

disc. Thus:

~r = (x + e cos(Ωt))~i + (y + e sin(Ωt))~j (2.1)

~r = (x − eΩ2 cos(Ωt))~i + (y − eΩ2 sin(Ωt))~j (2.2)

The stiffness and damping of the shaft are called k and c respectively,

and also m the mass of the disc. The Newton’s second law applied to the

system gives:

m~r = −(kx + cx)~i − (ky + cy)~j (2.3)

14

Equation 2.2 into equation 2.3 gives the equations of motion:

mx + cx + kx = meΩ2 cos(Ωt) (2.4)

my + cy + ky = meΩ2 sin(Ωt) (2.5)

When Φ = Ω, this gives the phenomena of synchronous whirl. This occurs

when the angular velocity of the bended shape is equal to the angular velocity

of the rotating speed. The whirl is a two dimensional orbit motion made by

the geometric center. The orbit can have the same or the opposite direction

as the spin direction of the rotor (respectively called forward whirl and

backward whirl).

2.1.2 The gyroscopic effect

The gyroscopic effect happens when a spinning object is subjected to a

rotational perturbation when a restoring moment appears. At high spin

speeds or high moments of inertia, this phenomena becomes much significant.

This is analysed by determining the equations of motion of an overhung rotor.

Figure 2.3: Simple view of the overhung rotor

The stiffness of the beam can be obtained with the beam theory. The

following expressions can be found in an elementary case (cantilever beam

submitted to an end force and end moment).

15

For a beam submitted to a end force F, both displacement and phase are

given by:

δ =FL3

3EI= α1F (2.6)

φ =FL2

2EI= α2F (2.7)

For a beam submitted to a end moment M, both displacement and phase

are given by:

δ =ML2

2EI= α2M (2.8)

φ =ML

EI= α3F (2.9)

The associated moments and forced can be derived from a free body

diagram. A free body diagram is a pictorial representation of the forces

acting on a free body, showing both contact and non-contact forces acting

on this body (see figure 2.4).

Figure 2.4: Forces and moments acting on the system in 3D

16

If the deformations of the beam are studied in 2D with the equations

(2.6)-(2.9), the flexibility matrix for the beam can be derived with the help

of figure 2.5

Figure 2.5: Forces and moments acting on the system in 2D

Thus the following equations are obtained:

x = α1Fx + α2My (2.10)

y = α1Fy + α2(−Mx) (2.11)

−ϕ = α2Fy + α3(−Mx) (2.12)

θ = α2Fx + α3My (2.13)

In a generalized form, this gives:

q = [α]F =⇒ F = [α]−1q = [K]q (2.14)

The α matrix contains the flexibility coefficients like this:

[α] =

α1 0 0 α2

0 α1 −α2 0

0 −α2 α3 0

α2 0 0 α3

(2.15)

17

Thus the stiffness matrix ([K] = [α]−1 in equation 2.14) is given by:

[K] =

K1 0 0 K2

0 K1 −K2 0

0 −K2 K3 0

K2 0 0 K3

(2.16)

In this way the forces on the beam can be solved. The forces on the disc

are the same but with opposite signs.

The gyroscopic effect appears when the beam bends. Indeed, the disc will

rotate in a new direction Z’ which deviates from Z. The change in rotational

direction results in a gyroscopic restoring moment of the disc. To describe and

derive these moments, the moment of momentum H has to be determined.

The moment is equal to the time derivative of the momentum, i.e. H = M .

Thus it follows that:

Hz′ = ΩJp (2.17)

Figure 2.6: Geometric representation of z’

From the geometry of figure 2.6, it follows that:

H2

zz′ + H2

xz′ + H2

yz′ = H2

z′ (2.18)

18

H2

xz′ = H2

zz′ tan θ (2.19)

H2

yz′ = H2

zz′ tan(−ϕ) (2.20)

As the angles are assumed to be small, the equations 2.17, 2.19 and 2.20

into equation 2.18 give:

H2

zz′(1 + θ2 + ϕ2) ≃ H2

z′ (2.21)

H2

zz′ ≃ ΩJp (2.22)

The equation 2.22 into equation 2.17 and 2.18 gives the components in x

and y. According to figure 2.5, the moment of momentum from the rotations

around ϕ and θ should be added. Thus one get:

Hx = Jdϕ + ΩJpθ (2.23)

Hy = Jdθ − ΩJpϕ (2.24)

Hz = ΩJp (2.25)

The time derivative of the functions gives the moments caused by the

gyroscopic effect:

MHx = Jdϕ + ΩJpθ (2.26)

MHy = Jdθ − ΩJpϕ (2.27)

MHz = 0 (2.28)

The equations 2.15, 2.26 and 2.27 give together with the result of rotating

unbalance the following equations of motion:

19

mx = −K1x − K2θ + meΩ2 cos(Ωt) (2.29)

my = −K1y + K2ϕ + meΩ2 sin(Ωt) (2.30)

Jdϕ + ΩJpθ = K2y − K3ϕ (2.31)

Jdθ − ΩJpϕ = −K2x − K3θ (2.32)

Equations 2.29 to 2.32 can be written into a matrix form:

m 0 0 0

0 m 0 0

0 0 Jd 0

0 0 0 Jd

x

y

ϕ

θ

+ Ω

0 0 0 0

0 0 0 0

0 0 0 Jp

0 0 −Jp 0

x

y

ϕ

θ

+

K1 0 0 K2

0 K1 −K2 0

0 −K2 K3 0

K2 0 0 K3

x

y

ϕ

θ

= meΩ2

cos(Ωt)

sin(Ωt)

0

0

(2.33)

In a shorter way, it can also be written:

[M ]r + Ω[G]r + [K]r = [F ] (2.34)

The equation can be used to simulate the dynamics of the system, for ex-

ample by using Matlab. The property of the gyroscopic matrix is dependent

of Ω, that means the eigen-frequencies will depend on the rotating speed of

the rotor. This is a specific result of the rotating systems.

20

2.1.3 Critical speeds and Campbell diagram

Every rotor-bearing system has a number of discrete natural frequencies of

lateral vibration. When one of the natural frequencies is excited by rotor im-

balance rotating at shaft speed, the shaft speed which coincides with natural

frequency is called a critical speed.

In mathematical terms, the natural frequencies are called eigenvalues

and the mode shapes eigenvectors. The mode shape describes the expected

curvature (or displacement) of the rotor at a particular mode. Theoretically,

a distributed mass-elastic system has an infinite number of eigenvalues and

associated eigenvectors. But in practice, only the lowest critical speeds and

associated whirl modes are excited in the speed range of a typical turbo-

machine.

Figure 2.7: Example of Campbell diagram

21

As the natural frequencies vary with the spin speed (equation 2.34), the

best way to find these critical speeds is to plot the Campbell diagram [5].

It consists of plotting the natural frequencies as a function of the rotational

speed. Theoretically, these damped natural frequencies are the imaginary

part of the eigenvalues. The critical speeds are found where the natural

frequencies coincide with the speed spin (ω = Ω). In figure 2.7, frequencies

that increase (blue line) is called forward whirl whereas the one that decrease

(red line) is named backward whirl. The straight red line shows a linear whirl,

which means that the eigen-frequency is not depending on the rotational

speed. The critical speeds can be observed when the black line (called 1P

line) crosses the other curves (red or blue).

2.1.4 Response to unbalance

As shown in equation 2.33, a sinusoidal force is applied on the system. This

is due to the unbalanced mass, which is one of the main problems in rotor-

dynamic systems. For a 1-dimensional system, the equation to solve is:

mx + cx + kx = mueΩ2 sin(Ωt) (2.35)

The excitation is due to an applied force F0 sin(Ωt) where F0 = mueΩ2.

This is the centripetal force of the mass mu towards the center of rotation.

If the rotor is balanced, no force exists, but this is very seldom the case. The

amplitude of the excitation depends on both the out-of-balance mass, and

its effective distance from the axis of rotation, as well as on the rotational

speed.

The response will be given by x = X0 sin(Ωt − φ) where:

X0 =mueΩ

2/k√

(1 − r2)2 + (2ξr2)(2.36)

tan φ =2ξr

1 − r(2.37)

Here ω is the natural frequency ω =

√

k

m, ξ =

c

2√

kmand r =

Ω

ω.

22

In a non-dimensional form, it is easier to see how in general the response

depends on the frequency ration r and on the damping ratio ξ. Therefore:

X0

e

m

mu

=r2

√

(1 − r2)2 + (2ξr2)(2.38)

Then the figure 2.8 is obtained by plottingX0

e

m

mu

against r for several

damping ratios.

Figure 2.8: Response of an unbalance rotor as a function of the frequency

2.2 Finite element method

The finite element method is a very general method to handle difficult prob-

lems that cannot be solved analytically. This method is widely used in many

23

fields (thermodynamics, electrical engineering, ...) and one that is very suit-

able for is in structural mechanics. FEM is based on the subdivision of the

structure into finite elements.

Element formulation

Many different element formulations have been developed depending on their

shape and characteristics: beam elements, shell elements, plate elements,

solid elements and many others. Each element is the model of a small de-

formable solid. Inside each element the displacement u(x, y, z) of the point

with the coordinates x,y,z is approximated by combination of the shape func-

tions N . FEM is usually developed using matrix notation to obtain formulas

that can easily be transferred into computer code. The displacements are

usually written in three dimensions:

u(x, y, z, t) = N(x, y, z)q(t) (2.39)

The displacements can contain more dimensions if rotations of the el-

ements are considered. In this case, q is a vector with the n generalized

coordinates, and N the matrix containing the shape functions. There are as

many rows in N as in u and as many columns as the number n of degrees of

freedom. For a two dimensional element, the equations can be written:

(

ux(x, y, t)

uy(x, y, t)

)

=

(

N(x, y) 0

0 N(x, y)

)(

qx(t)

qy(t)

)

(2.40)

The shape functions can be chosen arbitrarily, but they have to satisfy

several conditions such as:

• be continuous and derivable up to the required order, which depends

on the element type

• be able to describe rigid body of the element leading to vanishing elastic

potential energy

24

• lead to constant strain field when the overall deformation of the element

dictates so

• lead to deflected shape of each element that matches the shape of the

surrounding elements

The strains in each element can be expressed as functions of the deriva-

tives of the displacements with respect to space coordinates; this makes pos-

sible to write the strains as:

ε(x, y, t) = B(x, y)q(t) (2.41)

B is a matrix containing the derivatives of the shape functions. The stress

can be expressed as:

σ(x, y, t) = Eε = E(x, y)B(x, y)q(t) (2.42)

E is the stiffness matrix of the material. Since only isotropic material is

used, E is a scalar. The potential energy can then be expressed as:

U =1

2

∫

V

εTσdV =1

2qT (

∫

V

BT EBdV )q (2.43)

The integral in equation 2.43 is the stiffness matrix of the element:

K =

∫

V

BT EBdV (2.44)

The mass matrix is derived from the expression of kinetic energy of the

element and the shape functions:

T =1

2qT (

∫

V

ρNT NdV )q (2.45)

M =

∫

V

ρNT NdV (2.46)

The form of the matrix B is dependent of the shape functions N and

can hence be in different dimensions and considered in different degrees of

25

freedom. This gives different dimensions of the stiffness matrix depending of

the assumed shape functions. Then these two matrices are introduced into

Newton’s second law:

Mq + Kq = F (t) (2.47)

The Fung and Tong book [6] can be investigated to have more details

about the finite element method (shape functions for solid elements, assembly

process, ...).

26

Chapter 3

The NX Nastran analysis

This chapter has similarities with Chapter 2 as it presents theoretical foun-

dation of rotor-dynamics.

3.1 Theoretical foundation of rotor dynamics

within Nastran

A said in Chapter 1, for a rotating structure, additional terms occur in the

equations of motion depending on the chosen analysis system. The fixed

system will mainly be used, and the rotor is rotating about the z-axis (default

axis). The equations of motion are described in this system. One has to set

"GRID" points that belongs to the rotor to distinguish between the rotating

and fixed parts of the structure.

3.1.1 Additional terms in the equations of motion

• Coriolis forces and gyroscopic moments

Rotor dynamics analyses in NX Nastran include both Coriolis forces

and gyroscopic effects. In the rotating analysis system, the Coriolis

forces of the mass points are included. In the fixed analysis system,

the gyroscopic moments due to nodal rotations are included.

27

• Damping

The damping in a rotor system is divided into two parts: the internal

damping acting on the rotating part of the structure, and the external

damping acting on the fixed part of the structure and in the bearings.

The rotor can become unstable at speeds above the critical speed as the

internal damping has a destabilizing effect. On the contrary, the exter-

nal damping has a stabilizing effect (more information about damping

is given in Appendix C).

3.1.2 Equation of motion for the fixed reference system

The generalized equation of motion is as follows:

[M ]q + (Ω[C ] + [DI + DA])q + ([K] + Ω[DB])q = 0 (3.1)

The generalized matrices are given by:

• [M ] = [Φ]T [M ][Φ] = [I] the generalized mass matrix

• [C] = [Φ]T [C][Φ] the generalized antisymmetric gyroscopic matrix

• [DI ] = [Φ]T [DI ][Φ] the generalized internal viscous damping matrix

• [DA] = [Φ]T [DA][Φ] the generalized external viscous damping matrix

• [K] = [Φ]T [K][Φ] = diag[ω2

0] the generalized elastic stiffness matrix

• [DB] = [Φ]T [DB][Φ] the generalized anti symmetrical internal damping

matrix

For a rotating mass point, the terms are as it will follow.

The mass matrix is the same as for non-rotating structures. A lumped

mass approach is used here:

28

[M ] =

m

m

m

Jx

Jy

Jz

(3.2)

In the gyroscopic matrix, only the polar moment of inertia appears. With-

out any polar moments of inertia,there is no rotational effects in the fixed

system.

[C] =

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 Jz 0

0 0 0 −Jz 0 0

0 0 0 0 0 0

(3.3)

The internal rotor damping matrix is given by:

[DI ] =

dI,x

dI,y

dI,z

dI,Rx

dI,Ry

dI,Rz

(3.4)

The external damping acting in the non-rotating bearings is as follows:

[DA] =

dA,x

dA,y

0

0

dA,Rx

dA,Ry

(3.5)

29

Because the displacements in the rotating part act as a velocity in the

fixed part, an antisymmetric matrix appears in the stiffness term :

[DB] =

0 dI,T 0

−dI,T 0 0

0 0 0

0 dI,R 0

−dI,R 0 0

0 0 0

(3.6)

In the previous matrices, the following notations have been used:

m mass

Jx inertia about the x-axis

Jy inertia about the y-axis

Jz inertia about the z-axis

dI viscous damping of rotor

dA viscous damping of rotor

The subscript T refers to "translation" whereas the subscript R indicates

"rotation". Moreover, in equation 3.6, the damping is assumed to be equal

in the x- and y-direction.

3.1.3 The Steiner’s inertia terms in the analysis

The ZSTEIN term is an option which can be found on the ROTORD card

(the rotordynamic entries are defined in this card) in the data file. This

allows to include the Steiner’s in the analysis. It is important to include this

term in the fixed reference system. In this case, the polar moment of inertia

is calculated as :

Jz = Jp = Σm(dx2 + dy2) (3.7)

WARNING: to use the ZSTEIN option, the nodes must have rotations.

In a model made of solid elements, this is obtained by adding a layer of

30

shell elements around the solid elements. One should also check if the nodal

rotations are not constrained by any optional cards.

3.1.4 Real eigenvalue analysis

The first step in a rotor dynamics analysis is to perform a real eigenvalue

analysis :

(−ω2

0[M ] + [K])ϕ = 0 (3.8)

The eigenvectors ϕ are collected into the modal matrix Φ. The dis-

placement vectors can be described by a linear combination of the modes:

u = [Φ]q. This is mathematically true if all modes are used. It is reasonable

when using a sufficient number of modes. The selected modes used for the

rotor dynamic analysis must be chosen according to these criteria:

• The frequency range of the real modes must be well above the frequency

range of interest in the rotor dynamic analysis

• The real modes should be able to describe the rotor motion. The modal

displacement in x and y has to be included

• The modes must be able to represent the generalized forces

3.1.5 The fixed system eigenvalue problem

For harmonic motion q(t) = q exp(λt), the following problem is solved in the

fixed reference system:

(λ2[M ] + λ(Ω[C] + [DI + DA]) + ([K] + Ω[DB ]))q = 0 (3.9)

A loop is made over the selected rotor speeds and the results are stored

for post processing.

31

Synchronous analysis

The eigenfrequency is equal to the rotor speed for the points crossing the

1P-line, i.e. ω = Ω. If the damping is neglected, λ = jω. To obtain the

resonance points, the following eigenvalue equation is solved:

(−Ω2([M ] + j[C]) + [K])q = 0 (3.10)

The imaginary parts of the solutions are the critical rotor speeds. For the

modes that do not cross the 1P line, the imaginary part is zero.

3.1.6 Solution interpretation

The solutions at each rotor speed are the complex conjugate pairs of eigen-

values λ = δ±jω. The real part is a measure of the amplitude amplification.

Having positive values leads to an increase in amplitude with time and so

the mode is unstable. The system is considered as stable when the real part

is negative. If the damping is assumed low, it can be defined as:

ξ =δ

ω(3.11)

In the complex modal analysis within NX Nastran, the output is the

g-damping defined as g = 2ξ.

The imaginary part of the eigenvalue is the damped natural frequency

of the solution. The eigenfrequency is thus f =ω

2π. The units that can be

chosen in Nastran will be shown in Chapter 4.

3.2 Equation of motion for modal frequency re-

sponse

Fixed reference system

The governing equation of the frequency response in modal space with rotor

dynamics terms, in the fixed reference system considering the load to be

32

independent of the speed of the rotation is:

(−ω2

k[M ] + jωk(Ω[C] + [DI + DA]) + ([K] + Ω[DB]))u(ωk) = p(ωk) for

k = 1, 2, ..., m

Here, m denotes the number of excitation frequencies of dynamic load.

This is called an asynchronous solution and applicable to cases like gravity

loads. The load could still have frequency dependence as shown by the m

discrete excitation frequencies defined on the FREQ card (see Chapter 7).

The rotor speed is constant and the asynchronous analysis is working along

a vertical line in the Campbell diagram.

In the case the load is dependent on the speed of rotation (called syn-

chronous analysis) is found putting ω = Ω. The governing equation is as

follows:

(−Ω2

k([M ] − j[C]) + jΩk([DI + DA] − j[DB]) + [K])u(Ωk) = p(Ωk) for

k = 1, 2, ..., n

Here n denotes the number of Ωk rotation speeds at which the analysis

is executed. Such is the case for example with centrifugal loads due to mass

unbalance. In this case the analysis is done along the 1P excitation line.

33

Chapter 4

Modelling of the 2D-3D rotor in

NX7

In this section, two rotors will be modelled in order to validate the models

created in NX 7. The results will be then compared with the theory if

possible. The rotor which will be analysed are a mid-span rotor and an

overhung rotor. The choice of both rotor has been done to try to model a

range of rotors as different as possible.

4.1 The middle span rotor (Jeffcott rotor)

4.1.1 The geometric modelling of the rotor

The parameters of the mid-span rotor are L = 1 m, ρ = 7800 kg/m3, E = 220

GPa, D = 1 m, d = 0.05 m and t = 0.1 m.

First of all, the model needs to be created under NX7. One have to open

a new part and start to "sketch" the model. Once the sketch is completed,

the solid body should be created by revolving the sketch around the z-axis.

The result of these operations gives the rotor figure 4.1.

Nonetheless, the design has to be studied before meshing. Indeed the

rotor can be modelled in two ways: either the rotor is made in one single

34

Figure 4.1: Revolution of the sketch of the rotor

part, or the shaft and the disc are made separately and glued together. The

choice of the model has an influence on the mesh. For reasons of accuracy,

the 2-part rotor will be used in the rest of the modelling.

4.1.2 Meshing

The mesh of the rotor is done in the Advanced Simulation application of

NX7. To be more accurate, a FEM file should be created and it is in this

section that the entire mesh will be completed. This section will be split

in categories (1D, 2D and 3D meshes), but in the order used to create the

model, i.e. the 3D mesh first, then the 2D mesh and the 1D to finish.

4.1.2.a The 3D mesh

Three important points have to be considered when meshing the rotor:

• The element type

• The element size

35

• For a 2-part rotor, the mesh mating condition between the contact

surfaces

The element type

Fundamentally, there are two types of 3D mesh which could be used. There

are referenced as CTETRA and CHEXA. The first one is a tetrahedron

whereas the second one is a 6-face element. Moreover in each element one

can choose either an element with nodes just in the corners or an element

with nodes in the corners and in the middle of the edges. For the CTRETA

element, it can be CTETRA(4) or CTETRA(10), and for CHEXA one can

pick CHEXA(8) or CHEXA(20). The number in parenthesis indicates the

number of nodes for each element. It is important to choose elements with

nodes on the edges because the accuracy will be much better using them.

WARNING: The elements with the greater number of nodes should

always be used. Indeed, the accuracy of the elements with only nodes in the

corner is really poor when simulating. For instance, the strains are constant

in a 4-node tetrahedron, which is the simplest solid element and which is not

very accurate [6].

In addition to this, a CPENTA element can sometimes be found in the

data file. This is an automatic creation by the software in some parts of the

rotor when the CHEXA-mesh is used.

After completing the 3D mesh, it is important to choose a material in

the properties of the 3D elements, e.g. steel, or create his own material (see

section 4.1.2.d), otherwise the simulation will not be launched.

The element size

The size of the element chosen throughout the body has an effect on the

overall results. If one select a too large size, this can result in a rotor which

is farther from the initial model. Indeed the shaft of the rotor can look more

like a polygonal object than a cylindrical one. To make a first test, the best

36

Figure 4.2: Type of solid elements

way is to try an action called "Automatic Element Size" in Nx. This is

a good approach to start with, but then it can be modified manually to find

the best results. Then one can choose smaller elements through the shaft

and the span, but the effect will be that the fastness of the simulation will

drop down as the size of the element decreases. Therefore a compromise

should be found to have both good results and a rather good quickness of

the simulation.

The mesh mating condition

As the rotor can be divided in several parts, it is important that the meshes

should be well-connected, the nodes being the same between the contact sur-

faces. This is done by using a command in the fem file called "Mesh Mating

Condition". If this is not done, the simulation will result in displacements

of both the shaft and the span totally disconnected, being independent from

each other.

4.1.2.b The 2D mesh

For this example, the 2D mesh will be used just for one goal. Indeed, to have

the analysis in the fixed reference system, it is necessary to put the ZSTEIN

entry to YES in the rotor bulk section (cf Chapter 5) and that the nodes have

rotations. In a model with solid elements, this can be done by adding a layer

of shell elements around all the solid elements.These shell elements have a

37

negligible effect on the model’s overall stiffness. When using this option, one

must ensure that the nodal rotations of the shell elements are not constrained

by the AUTOSPC option.

To insert all these shell elements in NX in the advanced simulation win-

dow, the path to follow is: Insert->Element->Surface Coat. Then all the

solid elements should be selected. There is no need to choose the type of

element (i.e. CQUAD8 or CTRIA6) as the creation will be automatic.

Once this is done, the properties of the shell elements has to be completed.

The two most important ones are Material 1 and Default Thickness. In this

model, the material will be steel and the shell elements will be 1 mm thick.

Figure 4.3: Properties of the shell elements

4.1.2.c The 1D mesh

The 1D mesh elements will be found at both ends of the rotor. All these 1D

elements will be connected to each other to the central point of the rotor.

38

But as the software does not allow to connect for example one spring from

one node to the same, it is useful to create 3 nodes in the middle of each end

surface.

As the connection between the 3D elements and 1D may cause some

problems, it is important to insert a rigid connection between the bearings

and the end surface. Then the bearings will be attached to this connection

and a end constrained point. To sum up, there are 3 types of 1D connections

in the model:

• The rigid connection (RBE2)

• The bearings (CELAS1)

• The damping of the bearings (CDAMP1)

The rigid connections

There are two types of rigid connections inserted into the model: one between

all the points of the end surface and the middle node of this same surface and

another connection from this middle node to the bearing node (both nodes

have the same coordinates).

The first connection is called "spider connection". One have to go in

Insert->Mesh->1D Connection, to choose Node to node in the connection

type box, to select RBE2 in the type of element. Then the central node has

to be selected as a source and all the other nodes of the surface should be

selected as a target. Plus the rotational degrees of freedom have to be let

free to allow rotations in X,Y and Z. Doing that, there will be no problem of

connection while simulating.

The second connection is made in the same way as the previous one,

excepted that the source is the same but the target node is one at the same

coordinates which has been created before. Plus all degrees of freedom need

to be constrained in this RBE2 connection.

39

Figure 4.4: Connections of 1D elements at the end of the shaft (ref [2])

The most important thing to remember is that one should be careful in

which sense to connect these nodes, otherwise over-constrained nodes may

appear when putting the end constraints.

40

Figure 4.5: The "spider" rigid connection (in blue)

The bearings and damping of bearings

The bearings and damping of bearings need to be inserted between the second

node and the third node created in the middle of the end surface. As the RBE

connection, it can be found in Insert->Mesh->1D Connection by changing

the type of element. In this model, two bearings will be created as well as

damping of bearings in the X and Y-direction.

For the viscous damping (external), a small value has been chosen in

order to avoid numerical problems as a start. The value has been fixed to

107 N.m for the stiffness.

4.1.2.d Material

There is two choices concerning the choice of the material: either one pre-

defined material can be selected (such as Isotropic Steel), or a new material

can be created. It is better to create an own material as the inserted values

would be then the real ones. It is also in this case that one can put some

value for the internal damping, as the predefined material do not have it (and

the values of the predefined material cannot be changed).

41

4.1.3 The boundary conditions

As shown on figure 4.4, both end nodes will be totally constrained (i.e. the six

degrees of freedom). To do that, one have to go in User Defined Constraint

in the simulation file. Then the nodes must be selected one by one and all

degrees of freedom have to be fixed.

Then the end nodes of the rotor should be constrained as the beam is

simply supported. That means one have to proceed as before, but only the

displacements in the x,y and z-directions are fixed as well as the rotation

around the z-axis.

Once all of this done, the first step is to analyse the eigen-frequencies of

the rotor to see if the mesh is satisfying. Then the rotor dynamics entries

can be inserted into the ".dat" file.

4.2 The Overhung Rotor

The aim of studying this rotor is that unlike the mid-span rotor, the eigen-

frencies of the first modes will not be the same as the rotor speed increases,

because of the gyroscopic effects. Due to that, all types of simple rotors can

be studied whatever their geometry. Plus as the boundary conditions are

changed, one have to find another way to put the bearings in this model

without affecting the overall results.

4.2.1 The geometric modelling of the rotor

The parameters of the mid-span rotor are L = 1 m, ρ = 7800 kg/m3, E = 220

GPa, D = 1 m, d = 0.05 m and t = 0.1 m (see figure 4.6).

As the mid-span rotor, the overhung rotor will me modelled with two

parts, i.e. the shaft and the disc. Then the control of the size of the mesh is

much easier.

Once the sketch is done regarding the dimensions, one have to revolve it

to finally obtain the 3D-rotor (figure 4.7)

42

Figure 4.6: The overhung rotor model

4.2.2 1D-Meshing

As the 3D-mesh and 2D-mesh will be the same as in section 4.1.2, the 1D

mesh will be emphasized as the boundary conditions differ from the mid-span

rotor.

Indeed, it is impossible to put the bearing as in the mid-span rotor because

it will not be taken in the analysis proceeding in this way. Thus the idea

is to create a small cage around the end of the rotor in order to be able to

connect the bearings from this "cage" to the rotor.

Figure 4.7: General View of the 3D-rotor

43

But still some problems can appear by working like this. How many nodes

have to be created to be sure that the boundary condition are respected? How

many bearings should be created and to which rotor nodes should they be

connected? What is the value of each bearing as several are connected to

each other?

The answers to these questions will be analysed and discussed with the

results of the simulation. But to start with something, the idea is to create

three bearings on each side of the small outside cage and connect them to

two center points of the rotor (one for each side).

Figure 4.8: Cage surrounding the shaft with bearings

If the results are satisfying, this model will be considered as a good one,

otherwise some elements will be changed. Another important thing is that

small damping will be introduced first to avoid numerical issues, trying to

be as more time-saving as possible.

Concerning the 2D and 3D mesh, it is exactly the same for the Jeffcott

rotor.

44

Chapter 5

Defining NX Nastran input for

rotor dynamics

The aim of this chapter is to describe the main commands to properly analyse

the rotor and structure and show how to process the results. Indeed, as

described in chapter 3, the solution used is the analysis of modes also known

as SEMODES-103. But to run the rotor dynamic analysis, one need to run

solution 110 (complex modal) which is not supported by NX7. Thus once the

solution has been launched, one have to Write, Edit and Solve in the Solution

Attribute Box and save the text file as a .dat file. Then modifications can

be done manually by opening the .dat file and adding the bulk data entries

of the rotor as well as changing the file management section, the executive

control section and the case control section.

5.1 Modifications of the .dat file

5.1.1 The file management section

In the file management input file, one should assign statements to have results

files which will be used for post-processing. These files generated by the

software are called GPF and CSV. To get it, the followings statements need

to be written in this section:

45

assign output4=’name_of_file.gpf’, unit=22, form=formatted

assign output4=’name_of_file.csv’, unit=25, form=formatted

The file to be used within NX7 will be the CSV file which is an excel file.

It contains values such as rotor speeds, eigen-frequencies or damping values.

This file then will transform in an AFU file to process the Campbell diagram.

This will be done using a journal called CampbellCSVtoAFU.vb (available

in the UGS directories).

5.1.2 The executive control section

In the Executive Control Section of the input file, the following command

SOL 103 has to be replaced by the command solving the Modal Complex

Eigenvalues, i.e. SOL 110.

5.1.3 The case control section

In the case control section, the RMETHOD entry is used to invoke the rotor

dynamics capability (RMETHOD = n). The number n refers to the RO-

TORD entries containing and SID = n option. The number of rotors is

specified by the number of continuation cards following the first continua-

tion card of the ROTORD entry. But in most of the examples which will

be studied, just one rotor will be specified corresponding to one continuation

card.

In modal dynamic response and rotor dynamic analyses, one should first

perform a real eigenvalue analysis and examine the modes before proceeding

to the next step in the analysis. With the rotor dynamic analysis, it is helpful

to be able to deselect local modes or other modes such as axial modes.

The following MODSEL entry lets one select or deselect specific modes

(MODSEL = n). Here n refers to a SET card that holds the list of selected

and retained mode numbers. The default is all modes in the model. The

46

numbers omitted from the list are removed from the modal space. Moreover,

numbers larger than the number of eigenvalues computed are ignored.

An entire example of the entries can be checked in Appendix B.

5.1.4 The bulk data section

For the rotor dynamics analysis, one should define all relevant rotor dynamics

data on the ROTORD bulk data entry. With the ROTORD entry, the main

card and the first continuation card contain information that is common to

all the rotors present in the system. To specify rotor information, one or more

continuation cards have to be used. The fields assignments are as follows:

Here are the names of the field and what are the contents:

• SID: sets and identifier for all the rotors. In the case control deck, one

must selected RMETHOD = SID.

• RSTART: starting value of the rotor speed.

• RSTEP: step-size of the rotor speed.

• NUMSTEP: number of steps for the rotor speed including RSTART

• REFSYS: the reference system is chosen. It could be either the fixed

reference system (FIX) or the rotating reference system (ROT).

• CMOUT: rotor speed for complex mode output request.

• RUNIT: units used for rotor speed input (RSTART and RSTEP) and

output (units for output list and Campbell’s diagram). Can be either

RPM, CPS, HZ or RAD.

• FUNIT: units used for the frequency output (RPM, CPS, HZ or RAD)

• ZSTEIN: option to incorporate Steiner’s inertia terms (default = NO)

• ORBEPS: threshold value for detection of whirl direction (default =

1E-6)

47

• ROTPRT: controls .f06 output options

• SYNC: option to select Synchronous or Asynchronous analysis for fre-

quency response (Synchronous: =1, Asynchronous: =0

• ETYPE: excitation type (Mass unbalanced: =1, Force excitation: =0)

• EORDER: excitation order (always =1 for the fixed reference system)

• RIDi: rotor ID for output identification; must be unique with respect

to all other RIDi values.

• RSETi: set number for grid points of rotor i. By default, all grid points

if only one rotor.

• RSPEEDi: relative rotor speed for rotor i.

• RCORDi: points to CORD** entry (coordinate system entry card)

specifying rotation axis of rotor i as z. (default = 0 for global z axis)

• W3i: reference frequency for structural damping defined by PARAM

G for rotor i

• W4i: reference frequency for structural damping defined on material

card for rotor i

• RFORCEi: points to RFORCE bulk data entry for rotor i. (Default

= 0 for no rotational force applied; a rotational force is required for

differential stiffness to be calculated.)

In figure 5.1 is shown an example of the rotor dynamics entries inserted

in the .dat file:

5.2 Interpretation of rotor dynamics output

When a rotor dynamics analysis is performed with NX Nastran, the software

writes the results into a .f06 file,an .op2 file and two ASCII files.

48

Figure 5.1: Data entries of the rotor dynamic analysis

5.2.1 The .f06 file

In the first part of the .f06 file a list of the rotor speeds, complex eigenvalues,

frequency, damping and whirl direction is printed. The software prints one

table for each solution.

In the second part of this file a summary of the results from the Campbell

diagram is printed. It includes the resonance of forward whirl, the resonance

of backward whirl, instabilities (points of zero damping) and critical speeds

from the synchronous analysis (only for analyses in the fixed system).

Moreover, if a simulation does not end properly, it is in this file that

the errors will be specially determined and thus the .dat file can be changed

according to these mistakes.

5.2.2 The .op2 file

If PARAM,POST,-2 is set into the input file, NX Nastran writes the results

into a file named ".op2 file". This file contain everything, i.e. the geometry

of the rotor, the fem and the results of the simulation. The modes can be

plotted with NX7 by importing this file. The processing will be explained

more accurately in the next part.

49

5.2.3 The .csv file

In addition to write the results to the standard NX nastran .f06 and .op2

files, a specially formatted file called .csv can be chosen by the software. The

.csv file contains data concerning the Campbell diagram. It can be processed

by using excel, or using the journal described in section 5.1.1 .

Figure 5.2: Diagram of the entire rotor-dynamics procedure

50

5.3 Procedure to perform an entire rotor dy-

namics simulation

To use the rotor dynamics software properly, a step by step procedure is given

to avoid as many errors as possible while simulating. Here is the summarized

procedure:

1. The geometric model has to be created under the modelling section

(File->new and create a .prt file).

2. Once the rotor is created, go into the advanced simulation to create

both .fem and .sim files. Here the solution SEMODES 103 has to

be used. In this part the meshes has to be created as well as the

constraints.

3. Once the model is completed (make a Model Setup Check), run the

simulation as Write, Edit and Solve Input File. A text file will

appear, save it as a name.dat file and close it.

4. The simulation runs. Once is finished, the results can be checked in

the Post Processing Navigator.

5. If the results seem correct (compared with the theory for instance), the

rotor dynamics cards can be written in the .dat file as explained in

section 5.1. Once the file is saved, the simulation has to be run with

NX Nastran.

6. If the simulation succeeded, the results can be checked in NX7. The

file has to be imported as an .op2 file under the Advanced Simulation

section.

7. Once imported, the Campbell diagram can be plotted by using the

macro CampbellCsvToAfu. The right .csv file should be taken and

converted. The diagram can be seen in the Simulation Navigator.

8. If the results are not satisfying (strange modes, bad range of frequen-

cies,...), changes can be made in the .dat file and the simulation be run

again.

51

IMPORTANT: If the simulation has not run entirely, the .f06 can be

checked to find the reason of the simulation’s abortion

52

Chapter 6

Analysis of the results

In this chapter, two main results will be discussed. First of all, the natural

frequencies of the rotor are checked before proceeding to the rotor dynamic

analysis. Once the eigenfrequencies seem to be satisfactory, then the Camp-

bell diagram will be analysed (concerning eigenfrequencies, damping values,

whirl direction,..) and compared with the eigen-frequencies of the theoretical

model.

6.1 The middle span rotor

The results of the eigenvectors (mode-shapes) are well-known for this type or

rotor. Indeed, the first one is associated to a translation mode whereas the

second mode is a tilting mode. In this case, the translational and the rota-

tional modes are uncoupled. The modal simulation of the model introduced

in the chapter 3.1 gives these shape results in figure 6.1 .

The eigen-frequencies are not given here but they will be given in the

next section depending on the accuracy of the mesh.

6.1.1 Quality of mesh and accuracy

As said in section 3.1.2.a, two main parameters are important when meshing:

the mesh size and element type. To check the differences in the results

53

Figure 6.1: Translation and tilting mode of the mid-span rotor

according to the mesh, these parameters will be taken into account:

• the type of element will be chosen between CTETRA4 and CTETRA10

(plus one test with CHEXA(20))

• the size of the elements will be only changed for the disc and will remain

the same for the shaft (the size are 160, 80 and 40 mm to have several

layers over the thickness of the disk) excepted for a few examples

The results of these simulations are put in Table 6.1.

If the size of the shaft is not specified, that means by the value by default

is 52.4 mm. As the mesh number 6 (CTETRA(10), disk: 170mm, shaft:

52.4 mm) give pretty accurate results with relatively high speed computer

calculations, it will mainly be proceeded in that way for the next meshes to

come. That means that CTETRA10 will be used using an automatic creation

(the sized is automatically fit with the geometry by the program).

6.1.2 Campbell diagram

Critical speeds

The Campbell diagram is a plot of the eigenfrequencies as a function of the

rotational speed. On it can be checked the value of the eigen-frequencies at

54

Figure 6.2: Example of a CHEXA(20) mesh and the eigenfrequencies

element size (mm) freq 1 (Hz) freq 2 (Hz)

CTETRA(4) 170 12.87 27.63

CTETRA(4) 80 12.81 27.67

CTETRA(4) 40 12.75 27.80

CTETRA(4) 20 12.74 27.80

CTETRA(4) 20 (disk and shaft) 11.56 25.18

CTETRA(10) 170 10.86 23.61

CTETRA(10) 80 10.85 23.64

CTETRA(10) 80 (shaft:30) 10.77 23.47

CTETRA(10) 40 (shaft:30) 10.75 23.47

CHEXA(20) 216 (shaft:13.2) 10.68 23.17

Table 6.1: Eigenfrequencies values depending on the mesh

55

ω = 0 rad/s, the linear, backward or forward whirls and the critical speeds.

The first eigen-frequencies can be checked theoretically to see if the re-

sults are acceptable. For the translation mode, the translational stiffness

is kt = 48EIL3 = 3.106N/m and thus the frequency corresponding is ω1 =

√

kt

m=√

3.106

616= 69.78rad/s = 11.10Hz. For the tilting mode, the ro-

tational stiffness is kr = 12EIL

= 1.106N/m and the frequency associated

ω2 =√

kr

Id

=√

1.106

39.5= 159rad/s.

The first eigenfrequencie is 10.98 Hz for the green and yellow line, the

second one is 23.88 Hz (see figure 6.3). These values are a bit different

from the modal analysis done before but one has to remember that some

rigid connections and bearings have been added so that the results can be

influenced.

The green and yellow line do not vary with the rotational speed. This is a

typical result of a midspan rotor, i.e. the eigenfrequencies of the translational

modes are constant whatever the speed (there is no gyroscopic effect). Then

the dotted-red line indicates a backward motion whereas the blue line shows

a forward one. Thus all motions presented here match with the theory of the

midspan rotor.

However, the third and fourth modes are purely rotational motions of

the disk without translational displacements. These two modes split in two

direction with the rotor speed, and two planes of motion are coupled due to

the gyroscopic effect.

In figure 6.3, the critical speeds can be observed when the curves are

crossed by the 1P-line. Thus in this model three critical speeds are observ-

able, one forward and two backwards. The critical speeds are given in Table

6.2.

These results are given for some specified values that have an impact on

the critical speeds. For instance the thickness of the shell elements (added

to have nodes rotations), the internal damping can modify these results. A

study will be done according to these parameters too.

56

Figure 6.3: Campbell diagram for the midspan rotor

type of mode whirl eigenfrequency (Hz) critical speed (RPM)

bending backward 10.98 658.4

bending forward 10.98 659.0

tilting backward 23.88 1015

tilting forward 23.88 -

Table 6.2: Eigenfrequencies and critical speeds for the Jeffcott rotor

Damping

As said in Chapter 2, internal damping has a destabilizing effect. Thus the

rotor can become unstable over critical speeds.

The relation between the undamped natural frequency and the shaft

speed where the instability occurs is given by [5]:

57

Ω0 = ωn(ce + ci

ci

) (6.1)

As the external damping was considered very low in the simulation, it

can be checked that the instability occurs at the first critical speed. Indeed,

it appears at the value of 684 RPM (where the damping value is zero).

Figure 6.4: Damping values as function of the speed

In this example, it is important to notice that the first two eigenvalues

have the same imaginary part (damped natural frequencies) but two different

real parts (damping exponents). At zero speed, the two modes have the same

damping factors. As the speed increases, the cross-coupled stiffness due to

internal damping increases. Thus it shows that internal damping destabilizes

the forward mode and stabilizes the backward mode.

58

6.1.3 Parameters influencing the results

As said in the previous section, some parameters can change the results. The

fact is too know how much it differs when changing one value and see if it

can be controlled properly.

Thickness of the shell elements

In the built model, it has to be remembered that a thin layer has been added

to the 3D model to assure rotation of all the nodes. When inserting this

shell-elements, the thickness of the layer is asked otherwise the simulation

cannot run.

Several thickness’s have been tested. The results are given in Table 6.3.

thickness (mm) 0.01 0.1 0.3 0.7 1 5 10

critical speed 1 654.6 658.4 666.6 682.6 693.9 815.0 922.4

critical speed 2 655.3 659.0 667.2 682.6 693.9 815.3 922.9

critical speed 3 1010 1015 1026 1047 1062 1214 1339

critical speed 4 12866 14145 18971 - - - -

Table 6.3: Critical speeds of the Jeffcott rotor for different shell thickness

It can be seen that for an overall thickness greater than 1 mm, the results

begin to be totally wrong. But for smaller values, the change in critical

speeds is less accentuated. However, a fourth critical speed appears when

decreasing the thickness. This means that the slope of the forward tilting

eigenfrequencies drops as well as the thickness.

Material internal damping

Some tests have been made changing the internal damping (0.01, 0.05, 0.1,

0.3, 0.5 and 0.8). As the results do not change significantly and the reason

of these small changes are unknown, the table of results is not presented

here. Nonetheless internal damping will be studied more accurately in the

frequency response analysis as its influence is much bigger.

59

6.2 The Overhung rotor

As for the Jeffcott rotor, the first two modes will be studied. Both the modes

and the Campbell diagram will be totally different from the Jeffcott rotor.

Indeed, both the stiffness and gyroscopic matrices are different according to

the theory. That is why this example have been checked.

In this section, the results will been given for some typical values accord-

ing to those used in the Jeffcott simulation. No general studies will be made

according to the thickness of the shell elements, internal damping or stiffness

as the influence of them have been evaluated in the Jeffcott section. There is

no internal damping in this case, just external damping. The value of each

stiffness is 1.107 N/m.

6.2.1 Mode shapes and Campbell diagram

Figure 6.5: 1st and 2nd modes shapes of the overhung rotor

As figure 6.5 shows, both modes can be said to be tilting modes. The

first eigenfrequency is 2.44 Hz and the second one is 14.60 Hz. Since the

disk is totally located off-center, the translational and rotational motions are

coupled. Also two planes of motion are coupled by the gyroscopic effect.

60

type of mode whirl eigenfrequency (Hz) critical speed (RPM)

coupled backward 2.44 135

coupled forward 2.44 157

coupled backward 14.60 658

coupled forward 14.60 7083

Table 6.4: Eigenfrequencies and critical speeds for the overhung rotor

As a result, all the natural frequencies are affected by the disk gyroscopic

moments. The natural frequencies associated with forward whirl increase as

the speed increases, and the natural frequencies associated with backward

whirl decrease as the speed increases.

Figure 6.6: Campbell diagram for the overhung rotor

From Table 6.4, it can be seen that a fourth critical speed appears when

simulating. But as shown in the Campbell diagram, this one cannot be

61

checked as the simulation was run till 300 RPM, so this critical speed should

not appear. It can be assumed that an interpolation is made for the fourth

curve, and thus maybe this one crosses the 1P-line which naturally gives this

new critical speed. But still it cannot be sure that it is a good value.

62

Chapter 7

Solving the mass unbalance

problem

The aim of this chapter is to perform mass unbalance problems for the Jeff-

cott and the overhung rotor. With NX Nastran, both synchronous and asyn-

chronous analyses can be performed. This analysis will always be performed

after the solution 110 to check if the results match with theory.

To make this modal analysis, some additional rotor cards should be in-

serted and the solution has to be changed too.

7.1 Mass unbalance cards and modal frequency

analysis

7.1.1 Additional cards in the bulk data section

In the rotor bulk data section, three cards will be added. There are called

SYNC, EORDER and ETYPE. Each card has an accurate meaning.

• SYNC: option to select synchronous or asynchronous analysis for the

frequency response (synchronous = 1).

• EORDER is the excitation order.

63

• ETYPE is the excitation type. There is two types: either the mass

unbalance m ∗ r is specified on the DLOAD card and the program will

multiply by Ω2 (ETYPE = 1), or a force excitation mrΩ2 is defined on

DLOAD (ETYPE = 0)

For calculating the modal frequency response using the synchronous, the

rotation speeds are defined by RSART, RSTEP and NUMSTEP in the RO-

TORD entry. On the contrary, for asynchronous analysis, the unique rotation

speed is defined by RSTART. The RSTEP and NUMSTEP entries are ig-

nored. The frequency increments will be put into a card called FREQ in that

case.

7.1.2 Dynamic load

To perform the analysis, it is important to define the dynamic load, for

both the mass unbalance or the force excitation. According to NX Nastran,

the DLOAD is associated with an RLOAD1 entry which specify the load

properties. The RLOAD1 card defines a load as follows:

P (f) = A[C(f) + iD(f)] exp(i(θ − 2πfτ) (7.1)

τ is defined by the card DELAY, θ by DPHASE, C(f) and D(f) by two

table entries and A by DAREA. The type of the dynamic excitation is a

load, defined by type = 0 (see figure 7.1). The nodes and directions where

the force is applied are put into the DAREA card.

In our case, the following values are created: τ = 0, A = 1, D(f) = 0,

C(f) = 1 (no frequency dependence). Concerning θ, it can have several

values. Two RLOAD1 will be created to have a rotating force. For the

first one, θ = 0. For the second one, it will be either +90 or −90 degrees.

Proceeding like this allows to first check the forward whirl and then the

backward one. This is more considered as a trick to create a rotating force.

The dynamic load can be put wherever on the structure. But the nodes

have to be checked previously if one want to put the force correctly. For

64

Figure 7.1: Bulk data entries for the dynamic load

instance, one rotating force (two RLOAD1) can be set in the middle of the

disc to check the bending response, whereas one force can be put on the edge

of the disc to check the tilting response.

Figure 7.2: Visualization of the dynamic load

Once all these cards are inserted, the .dat file has to be run in NX Nastran.

65

7.2 Mass unbalance response of the Jeffcott ro-

tor

7.2.1 Synchronous analysis with internal damping

For this analysis, the force excitation magnitude will be fixed to m ∗ e =

193kg.mm . As the results for the bending response can be seen with the

tilting response, only the second one will be simulated, it means that one

dynamic force will be created on the edge of the shaft at the outer surface

of the disc to excite both modes at the same time (exactly at the node 2561,

see figure 7.1). The analysis is perform from 0 to 30 Hz with a step of 0.1

Hz.

Concerning the modelling, only internal damping is inserted, with a

damping ratio ξint = 0.1 . Concerning the external damping, the value is

c = 1kg/s which can be disregarded, so that ξext = 0.

WARNING: It is important to write the suitable units of the magnitude

excitation in the .dat file in order to have good results. Indeed, the units used

are not part of the International System of Units. The unit is here kg.mm

As a synchronous analysis is performed, the card entries are as follows:

SYNC = 1, ETYPE = 1 and EORDER = 1 in the fixed reference system.

Backward whirl

First the backward simulation is performed. The results of the Campbell can

help to predict where the maximum displacements will appear. Indeed, it

should appear when the 1P-line crosses the backward solutions, i.e. at the

first and third critical speeds, respectively 658 RPM and 1015 RPM.

The magnitude response will be plotted for the displacement and the

rotation at the node 1521.

The results with NX gives a maximum displacement of 1.96 mm at the

frequency of 10.1 Hz for the bending mode. Concerning the tilting mode,

66

Figure 7.3: Displacement for the backward whirl

the maximum response can be checked by plotting the nodal rotation. The

maximum rotation is 0.10 degrees for a frequency of 15.8 Hz.

In order to partially check the results, the displacement will be plot as

a function of the rotational speed using Matlab. Theoretically ([?]), the

displacement for the Jeffcott rotor is given by:

X =1

√

(1 − r2)2 + (2ζr)2

meω2

K(7.2)

In that case, me = 193kg.mm, r =ω

ωn

with ωn =

√

kt

m, K the stiffness

of the bearing (K = 5 ∗ 107N/m), and ζ = 0.1 the damping.

67

Figure 7.4: Rotation for the backward whirl

According to these results, the maximum displacement should appear at

r=1, i.e. ω = ωn. Both the maximum amplitude and the end amplitude

match with the numerical simulation. As a result, the model can be said to

be correct and thus the other simulation could be correct too. It would be

certainly better to check analytically the results for each test but it will be

impossible to proceed in that way for the real rotor or even the overhung

one. Indeed, the theoretical results are much more complicated to obtain

these cases.

Forward whirl

The parameters are exactly the same as the backward whirl. Just the phase

is changed in the second RLOAD1 force from -90 to 90 degrees.

In this case, the Campbell diagram should show only one maximum dis-

68

Figure 7.5: Theoretical displacement calculated with Matlab for the backward whirl

placement, the same as the backward whirl (659 RPM). Thus the curve

should be the same with no second peak in this range of frequency.

When plotting the displacement, the peak is at the same frequency but the

width of the peak is less pronounced, that means the damping ratio should be

smaller. But as it was not changed in the analysis from the backward to the

forward, the curve should be exactly the same as the backward whirl. As a

result, the internal damping can effect the results, but it should be admitted

that the control of it is not easy, giving here strange damping ratios for the

forward whirl. Indeed, the material damping is not really well explained in

the NX Nastran guide, thus it is hard to know what is the influence of it.

69

7.2.2 Synchronous analysis with external damping

In that case, no internal damping will be included (ξint = 0), but the external

damping will be much higher with ξext = 0.1. The important is to find the

good value for the viscous damping C in order to get the expected damping

ratio (see equation 7.3).

This one is defined by ξ =C

2√

Kmwhere C is the total damping and K

the total stiffness (C = 2C and K = 2K in this case). So this gives:

C = ξ√

Km (7.3)

The numerical application of this formula gives C = 0.1∗√

5 ∗ 107 ∗ 616 =

1.75 ∗ 104 kg/s. But when this value is used for the damping, it seems that

the damping ratio is lower that expected. Thus the viscous damping has

to be increased compared with the theoretical value to have ξ = 0.1. This

phenomena could be due to the fact that at the end points of the rotor,

the damping is not only connected in parallel with a spring but connected

in parallel with a spring and in series with another spring. This system is

called the "one and a half degree of freedom" (because there is 3 degrees of

freedom in the eigenvalue problem instead of 4).

Figure 7.6: One and a half degree of freedom system

To see the influence of the external damping, several curves have been

plotted to see the evolution of the peaks according to the damping ratio (see

70

figure 7.7). As expected, the width of the peaks really depend on the value

of the damping ratio. If the value of ξ is small, thus the peak will be more

narrow and the magnitude greater. As ξ increases, the peak becomes wider

and the magnitude decreases too. For a value of ξ ≥ 0.707, no more peak

can be observed. One can notice that the displacement value when f → ∞is the same for all curves. This value is exactly the same as found with the

model with internal damping ( d(f → ∞) = 0.37 mm).

Figure 7.7: Displacement response for several damping ratios

71

7.2.3 Asynchronous analysis

For the asynchronous analysis, the response is calculated along a vertical

line in the Campbell diagram. The rotor speed is constant in this analysis

and is defined by RSTART. NUMSTEP and RSTEP are both equal to 1.

The FREQ1 card defines the frequency range of the analysis. The value of

EORDER has no effect in this analysis (as the response is calculated along

a vertical line). A mass-unbalance excitation is chosen here.

To have results in the frequency range that is studied and to see the

differences of this analysis and the synchronous one, the value of the rotor

speed is fixed to 1500 RPM (25 Hz).

Figure 7.8: Displacement for the backward whirl of the asynchronous analysis

72

Backward whirl

As the maximum displacement should occur where the vertical line at 1500

RPM crosses with the backward curve, that means that the maximum dis-

placement should appear at the same location as the synchronous analysis,

i.e. at 10.9 Hz more or less. The second peak for the tilting mode should

appear at 14.1 Hz. Both the displacement and rotation confirm these results

(see figure 7.8 and 7.9)

Figure 7.9: Rotation for the backward whirl of the asynchronous analysis

Forward whirl

As the aim of the asynchronous whirl was to check if the analysis work, the

results will not be developed for the forward whirl, assuming that they are

accurate enough and seem to work for the backward whirl. Even if this analy-

sis seem not to be interesting in the mass unbalance problem (compared with

73

the synchronous analysis), it is still recommended to perform this analysis

according to the NX Nastran guide.

7.3 Mass unbalance response of the overhung

rotor

The way to perform the mass-unbalance problem is exactly the same as the

Jeffcott rotor. The only difference is that the node where the dynamic load

will be applied has no importance as the first two modes will be excited at

the same time (wherever the force is put).

Concerning damping, only external viscous damping has been inserted

(c = 5 ∗ 107kg/s), as the control of the internal one is not domesticated. The

other parameters of the rotors are exactly the same as the one used when

checking the Campbell diagram.

Only the synchronous analysis will be studied in this section, the asyn-

chronous analysis has been used mostly to explore the NX Nastran capabil-

ities and see if it worked.

Backward whirl

According to the Campbell diagram, two maximum displacements should

occur when doing the backward simulation: the first one at 136 RPM (2.26

Hz) and the second one at 658 RPM (10.96 Hz) according to the results in

the .f06 file. The plot of the displacement of node 909 shows that the values

are quite the same ( 1.4 Hz and 10.1 Hz), even if the frequencies are both

under-evaluated.

Concerning the first peak, the matters is that it is more a "double-peak"

appearing at the frequency 1.4 Hz and 1.7 Hz, as if both the backward whirl

and the forward whirl have been excited. The reason of these two peaks is

still not determined.

74

Figure 7.10: Displacement of node 909 for the backward simulation

Another important remark is that the width of the peaks do not change

even if the external damping is increased, unlike the results obtained with

the Jeffcott rotor. It is as if it was not considered in the entire analysis.

Maybe this could be due to the specific "1 and a half degree of freedom" of

the system, the stiffness can be too high and affect the effect of damping.

The interesting thing would be to try to make some other tests by reducing

the stiffness and check if the damping have an influence. Otherwise, the mesh

of the end of the shaft should be done in another way to try to get better

results.

Forward whirl

The phase is again changed from −90 to +90 degrees. All the other param-

eters are the same.

A short look at the Campbell diagram tells that a first peak will appear

75

at 158 RPM ( 2.6 Hz). A second peak should appear but far away, at 7 083

RPM ( 118.5 Hz). As the simulation is run from 0 to 50 Hz, the second

peak cannot be seen entirely on the plot of the displacement. Indeed, the

displacement curve matches with these expectations, even if the first value

is still under-evaluated, but the beginning of the second peak can be seen.

Figure 7.11: Displacement of node 909 for the forward simulation

For this model, it can be checked that some improvements have to be

made to have better results. But for the time being, the study of the realistic

model will be done considering that the first models are good enough.

76

Chapter 8

Modelling and analyses of a more

realistic rotor

The aim of this chapter is to model a more realistic rotor as well as the the

non-rotating structure and the interactions components (mainly bearings).

The model created will be simpler but as realistic as possible for the time

being (see Appendix D for a view of the entire structure). For further exper-

iments the model can be updated and some new parts can be integrated too.

One has to remember that the first aim is to see what are the possibilities of

analysis for such big structures and check if they are done in a proper way.

Once the simplest models are validated, step by step improvements can be

done to reach a "perfect model" in the near future.

In order to do things properly, the rotating part and the bearings will be

analysed to check if the results are correct. Then the non-rotating structure

will be added and the influence of this one will be observed.

8.1 Modelling of the rotating part

As in previous chapters, the geometric model has been created under the

Modelling section of NX7. This part is made up of a shaft, several disks and

blades. The blades will be first disregarded as it can introduce additional

problems.

77

The shaft modelling has been decomposed in several parts (it is cut at

each bearing support) so that some mesh points can be inserted and make

the general mesh easier.

Figure 8.1: Rotor modelling with blades

8.1.1 Meshing

As an accurate description of the meshes has been done in Chapter 3, a

general view of how the mesh has been proceeded will be given part by part.

Mesh with solid elements

Only the shaft will be meshed with solid elements. The type of element used

is CXEHA(10). For the size of the mesh, Automatic Element Size has been

chosen. As said before, some mesh points have been inserted in the middle

line of the shaft to connect the bearings stiffness.

As there are several parts in this rotor, each part should be meshed inde-

pendently. But as the rotor should be analysed in one whole part, the meshes

between all these parts should match. So before making the 3D-mesh, it is

important to insert several mesh mating conditions between the connected

faces. Doing that, all the different meshes should be connected with each

other.

78

Figure 8.2: Real rotor to be modelled

Mesh of the blades

To start with simple simulations, the blades are not meshed yet. But then

they will be modelled either with shell elements or solid elements. As stiffness

problems may happen with the shells elements, the rotor will be first meshed

into solid elements. Indeed, if the blades are meshed with shell-elements,

the NX Nastran documentation [2] indicates that a RFORCE entry has to

be created to calculate the "geometric (differential) stiffness matrix. As this

RFORCE card is difficult to use, the choice of solid elements is made even if

the simulations could take a longer time.

To get the end of the rotor and the blades meshed correctly together, the

best thing is to use the command Unite in the modelling part. Then the

connection points will automatically coincide.

79

Figure 8.3: Meshing of the blades

Shell elements will be used for the non-rotating structures surrounding

the bearings. This will only be used to attach the bearings (it is the simplest

way to create nodes) but it will be deleted then, excepted the connection

nodes (see figure 8.3).

Mesh with 1D-connections

As for the overhung rotor, the stiffness’s of the bearing will be connected

from the center of the rotor to the non-rotating structure in several parts.

To make it symmetric, they will connected in a "star" shape with an angle

ofπ

3rad between each. The equivalent stiffness is thus:

Ktotal = Ks ∗ (2 + 4(cos θ)2) (8.1)

Exactly at the same emplacement of the bearings are added some external

viscous damping, also in a "star-shape". The value of this damping will be

first arbitrary and then chosen according to the damping ratio wanted.

WARNING: As the rotor points into the x-direction, it is important to

put both damping and stiffness’s into the y and z direction (DOF 2 and 3 in

the .dat file), otherwise the results will be totally wrong.

80

Figure 8.4: Connection of the bearings to the non-rotating structure

8.1.2 Boundary conditions

The constraints is mainly divide into two parts: one will be put on the rotor

whereas the other will concern the non-rotating structure.

Rotating structure

The constraints are put in one or several nodes on the rotor central line. To

avoid local modes, the best thing is to constrain one of the nodes (the central

one) in the x-direction (axis of rotation). Fundamentally, constraining 1 or

3 nodes should not change the results significantly in this case.

Non-rotating structure

The constraints in this case will depend on the non-rotating structure:

• If there is no non-rotating structure, the end points of the stiffness will

be all constrained in all directions

81

• If there is a non-rotating structure (like a tube), 3 aligned nodes in

both ends of the tube will be constrained in all directions.

The second case will be simulated to check the displacement of the non-

rotating structure when the rotor is submitted to a mass unbalance.

8.2 Analysis of the modal complex eigenvalue

solution

As well as the Jeffcott rotor and the overhung rotor, the eigenvalue solution

will be performed to find the eigenvectors and eigenfrequencies. But with the

previous rotors, the mode shapes and Campbell diagram were known for these

particular rotors, so it was not to difficult to deselect the bad modes (local

modes or axial modes). But for this particular rotor, no studies have been

made before and no data is available for the real engine, thus the selection

of the modes has to be made more carefully.

As explained before, several simulations will be performed on several

models. Basically, the first one will be the rotor without blades and no

non-rotating structure, the second one will have the blades added on the

first model, and the third one will be a model with a simple non-rotating

structure.

8.2.1 Shaft without blades and no non-rotating struc-

ture

The first step is thus to solve the modal analysis and check if the modes are

realistic. Unrealistic modes will be unselected using the rotor card MODSEL

when performing the frequency response analysis.

When natural frequencies are analysed, the realistic modes can be found

by animating the modes (see figure 8.5). At the same time, the bad modes

82

Figure 8.5: Mode shapes of the rotor

can be checked as well. The figure 8.6 shows an unrealistic mode with a

translation in the x-direction (even if the central node was constrained in

that direction) and a "expansion" mode (the rotor becomes bigger in all

directions). It is typically the kind of modes that should be unselected when

performing the frequency response analysis.

Then the problem is to try to find which modes are right and which modes

are wrong. If this is not done properly, then the mass-unbalance simulation

83

Figure 8.6: Unrealistic modes of the rotor

will sometimes give wrong results, as it will be seen when performing it.

8.2.2 Shaft without blades and with the non-rotating

structure

In that case, the analysis is made with the outside structure. When perform-

ing Solution 110, the first two modes show what problems have to be solved

when adding the outside structure (see figure 8.7).

Indeed, the first mode seem to be similar to the first one observed on

figure 8.5, which is quite normal. But the second one is a mode of the outside

structure, so the modal complex eigenvalue solution will also be performed

for this mode, which will result in wrong results when plotting the Campbell

diagram (only straight lines should appear) or solving the mass unbalance

simulation.

To try to make a good analysis of this system and be able to see the first 3

or 4 modes (the most important ones), the external structure could be made

much stiffer to increase the natural frequency of the tube. Proceeding like

this, the good modes could be seen and the results should be more realistic.

So introducing this structure indicates that some new problems have to be

solved accurately to improve the model.

But it is important in any case to specify these few words coming from the

NX Nastran 6 guide :"NX Nastran (6) only supports the analysis of uncoupled

84

Figure 8.7: First two modes of the rotor with the structure

systems. In other words, when you model the rotor, any interaction between

the support structure and the rotor is not considered in the analysis. You can

use either fixed or spring stiffness boundary conditions to model the rotor’s

attachment to the support structure." [2]

These few lines have recently been skipped in the NX Nastran 7 version,

in which one can perform analyses with coupled rotors. So probably it is

possible to analyse the entire system using the new NX Nastran version, the

non-rotating structure has to be tested to check if this is right or not.

8.2.3 Shaft with blades and no non-rotating structure

As it is quite certain that the outer structure is not taken in account in the

analysis, the analysis of the blades is only done with the rotor and bearings.

When making the simulation, it can be seen that the problem will be

similar to the external structure problem. Indeed, the first 10 modes (and

probably more) shows that just the modes of the blades are found (around

250 Hz for all of them), which is below of the value of the first mode when

performing the analysis without blades.

85

Figure 8.8: Mode shape of the blades

Thus, because of these first modes, the mode shapes of the rotor will not

be seen neither. The ideas to solve this problem are quite the same as in the

previous section:

1. Change the material of the blades to get them stiffer (increasing the

natural frequency)

2. Start the extraction mode after a given value (such as 300 Hz) to sup-

press the modes of the blades in the analysis

But in both cases it is some tricks to try to get the good modes. The first

idea is not so perfect as it does not take into account the real properties of

the blades. The second idea could be bad if the frequency of the first mode

of the rotor is near the frequency of the blades (which is actually the case),

then the starting value has to be chosen carefully.

Once the problems of the structure and the blades are solved in a good

way, it should be easier to study the realistic model. Indeed, for the structure,

it will only be a problem of meshing. There is fundamentally no differences

at all to complete the mesh.

86

8.3 Frequency response analysis of the rotor

alone

The Campbell diagram of the rotor is just plotted for the model created in

section 8.2.1. Only the first 3 modes (so 6 curves considering the backward

and forward whirl) have been plotted considering the range of frequency

already large for a standard rotor.

Figure 8.9: Campbell diagram of the realistic rotor

WARNING: Only the right curves have been plotted on the diagram.

As the MODSEL has NOT been activated, that means 20 curves (!) should be

contained in this diagram, among them 14 unrealistic curves (i.e. 7 unrealistic

modes).

The frequency simulation is done to see what happen when you have a

more complicated rotor; no accurate results will be shown in this simulation

as the results in the previous section were not good enough.

87

The mass unbalance has been inserted at the edge of one of the disk.

As there is more than one disk and a significant distributed mass of the

shaft, it could produce a multiplicity of critical speeds, which may require

multi-plane balancing [4].

The rotor entries for "solution 111" have to be changed carefully, such as

the range of the study or the number of grid points for the rotor. Then the

backward and forward simulation can be performed as done in the previous

chapters.

Figure 8.10: Backward and forward displacement of the realistic rotor

On the figure 8.10, two maximum displacements can be seen for the back-

ward whirl and four for the forward. From the Campbell diagram (figure 8.9)

it is found that 3 backward and 2 forward modes are expected. The extra

peaks in the response analysis cannot be explained.

Thus it would be important to try to update the model and check the

results with some real experimental values. Indeed, it would be easier to

88

deselect the bad modes to perform a good mass unbalance analysis. For

now, it is impossible to say what is right or wrong with this model, more

studies have to be done concerning the mesh, or performing the frequency

response with a multi-plane balancing.

89

Chapter 9

Conclusion and discussion

The results of these 3 types of rotor, i.e. the Jeffcott, the overhung and the

more realistic model show both good and bad results. Indeed, the Jeffcott

one has proper results compared with the theory. Concerning the overhung

model, the Campbell diagram is quite correct but the effect of damping has

to be improved. For the whole engine, a more accurate study should be made

and compared with some experimental results.

As there is no 3D rotordynamics numerical analysis available in the lit-

erature, the tricky thing is to know what is the influence of each parameter

on the model, such as the stiffness, damping, shell thickness and plenty of

others. As the modelling is in 3D, some errors of discretization can occur,

and the modelling of the constraints is more difficult since a lot of points

have to be constrained.

The other aim of this project was to have some better post-processing

capabilities. As the solution 110 is not supported by NX 7, it seemed first

to be complicated. At least it was possible to visualize the 3D mode shapes

and the Campbell diagram in NX, but it is still impossible to visualize the

rotor rotation with this software. The possibilities of post-processing have

to be developed to facilitate an effective evaluation of the results.

In order to check what parts of the project that could be improved, the

"Further work and improvement" section will partially give these answers.

90

Further work and improvements

Further improvements can be made in several ways. One of the most impor-

tant improvements is that Nastran 6 has evolved in Nastran 7 recently, that

means some new capabilities are available on this new version.

Meshing improvements

The mesh improvements include the different 1D, 2D and 3D mesh. For

example:

• The 3D mesh should be created more symmetric (the discretization can

lead to a non-symmetrci rotor)

• 1D springs and dampers should be added to model the bearings in a

better way

It could be a good idea to try to use some other dampers or springs,

or non-linear elements when meshing the rotor (for instance frequency de-

pending bearings). For example, a damper can be modelled in 5 different

ways (from CELAS1 to CELAS5, see the NX online manual for detailed

definition). The difference between each one would be interesting to check.

Insertion of the rotor dynamic entries

As told before, the solution 110 is not supported. As a result, the rotor

dynamic entries have to be added by hand when proceeding the "Write, Edit

and Solve" box, which is quite long to write. A good thing would be to create

a macro which allows NX to insert directly the rotor entries in the program

without typing it manually.

91

New capabilities within Nastran 7

Several capabilities have been developed in the new version of Nastran. Now

it is possible to:

• perform a Transient Response Analysis (SOL 112) in order to study

the behaviour of the rotor when passing a critical speed (see Appendix

E for an example)

• perform rotor dynamics analyses on structures with one or more spin-

ning rotors (at different speeds) and include several cards for the bear-

ings

One main improvement is that it is no longer noted that the support

structure is not taken into consideration compared with the documentation

of Nastran 6. So it could be possible to perform a transient analysis due to

mass unbalance and see the effect of this simulation on the entire structure,

which is one way of solving the blade-off problem.

With these new capabilities, it could be interesting to perform the tran-

sient response analysis on the model created in this project, and try to de-

velop a more complete model. Then, one have to wait for specific 3D rotor

dynamics codes to be developed in order to solve three dimensional rotor

problems without false nodes and visualization problems.

92

Bibliography

[1] Rachid Rahouadj. Dynamique des structures 2. Cours de L’Ecole Na-

tionale Superieure d’Electricité et de Mécanique, 2003.

[2] NX Nastran 7. Rotor Dynamics User’s Guide. Siemens, 2009.

[3] Arvid Andersson and Henrik Wennerstrand. A finite element program for

rotordynamical applications. Master’s thesis, Lulea Tekniska Universitet,

2003.

[4] John M. Vance. Rotordynamics of Turbomachinery. Wiley Interscience,

1988.

[5] Wen Jeng Chen and Edgar J. Gunter. Introduction to Dynamics of Rotor-

Bearing Systems. Eigen Technologies, 2001.

[6] Y.C. Fung and Pin Tong. Classical and Computational Solid Mechanics.

World Scientific, 2001.

[7] Rolf Gustavsson. Rotor Dynamical Modelling and Analysis of Hydropower

Units. PhD thesis, Lulea Tekniska Universitet, November 2008.

[8] Giancarlo Genta. Dynamics of Rotating Systems. Springer New York,

2005.

[9] Michel Lalanne and Guy Ferraris. Rotordynamics Prediction in Engineer-

ing. Wiley, 1998.

93

Appendices

A-Software Capable of Solving the Rotordynam-

ics

There are many software packages that are capable of solving the rotordy-

namic system of equations. These codes make it easy to add bearing co-

efficients, side loads, and many other items needed in rotordynamics. The

non-rotordynamic specific codes are full featured FEA solvers, and have many

years of development in the solving techniques. These non-rotordynamic spe-

cific codes can also be used to calibrate a code designed for rotordynamics.

Rotordynamic Specific Code

• MADYN 2000: Commercial combined finite element (3D Timoshenko

beam) lateral, torsional, axial and coupled solver for multiple rotors

and gears, various bearings (fluid film, spring damper, magnetic)

• Dyrobes: Commercial 1D beam element solver

• XLRotor: Commercial 1D beam element solver

• iSTRDYN: Commercial 2D Axis-symmetric finite element solver

• ARMD: Commercial 1D beam element solver

• XLTRC2 : Academic 1D element solver

• ComboRotor: Combined finite element lateral, torsional, axial solver

for multiple rotors evaluating critical speeds, stability and unbalance

response

94

• Dynamics R4: Commercial software developed for design and analysis

of spatial systems

• MESWIR: academic computer code package for analysis of rotor-bearing

systems within the linear and non-linear range

Non-Rotordynamic Specific Codes

• Ansys: Version 11 workbench capable of solving the rotordynamic equa-

tion (3D-2D and beam element)

• Nastran : Finite element based (3D-2D and beam element)

• SAMCEF: Finite element based (3D-2D and beam element)

95

B-Example of bulk data entries in the .dat file

FILE MANAGEMENT

ASSIGN OUTPUT4=’jeffcott_perfect.gpf’,UNIT=22, FORM=FORMATTED

ASSIGN OUTPUT4=’jeffcott_perfect.csv’,UNIT=25, FORM=FORMATTED

EXECUTIVE CONTROL

ID,NASTRAN,truerotor_sim1-solution_1

SOL 110

TIME 999

CEND

CASE CONTROL

ECHO = NONE

RMETHOD = 99

SPC = 100

METHOD = 1

CMETHOD = 2

OUTPUT

DISPLACEMENT(PLOT,REAL) = ALL

BULK DATA

BEGIN BULK

PARAM ROTGPF 22

PARAM ROTCSV 25

PARAM K6ROT 100.0

PARAM POST -2

PARAM POSTEXT YES

PARAM,AUTOSPC,YES

PARAM,PRGPST,NO

PARAM,GRDPNT,0

ROTORG 11 1 THRU 10001

SID RSTART RSTEP NUMSTEP REFSYS CMOUT RUNIT FUNIT

ROTORD 99 0.0 10. 200 FIX -1.0 RPM HZ +ROT0

ZSTEIN ORBEPS ROTPRT

+ROT0 YES 1.0E-5 3 +ROT1

RID1 RSET1 RSPEED1 RCORD1 W3-1 W4-1 RFORCE1

+ROT1 1 11 1.0 1 0. 0.

EIGRL 1 10. 200. 5 2

EIGC 2 CLAN 4

CORD2R 1 0. 0. 0. 0. 0. 1. +XCRD001

96

C-Information complement for damping

Damping is a mathematical approximation used to represent the energy dis-

sipation observed in structures. Damping is difficult to model accurately

since it is caused by many mechanisms, including viscous effects, external

friction, internal friction and structural non-linearities. Because these effects

are difficult to quantify, damping values are often computed based on the

results of a dynamic test.

Viscous and Structural Damping

Two types of damping are generally used for linear-elastic materials: viscous

and structural. The viscous damping force is proportional to velocity, and

the structural damping force is proportional to displacement. Which type to

use depends on the physics of the energy dissipation mechanisms.

The viscous damping force fv is proportional to velocity and is given by:

fv = cu, where c is the viscous damping coefficient and u the velocity.

The structural damping force fs is proportional to displacement and is

given by: fs = iGku, where G is the structural damping coefficient, k the

stiffness, u the displacement and i the complex number (i2 = −1)

For a sinusoidal displacement response of constant amplitude, the struc-

tural damping force is constant and the viscous damping force is propor-

tional to the forcing frequency. The following figure shows it as well as the

two damping forces are equal at a single frequency for constant amplitude

sinusoidal motion.

At this frequency, Gk = cω∗ or c =Gk

ω∗

where ω∗ is the frequency at

which the structural and viscous damping forces are equal for a constant

amplitude of sinusoidal motion

If the frequency ω∗ is the natural frequency ωn, the previous equation

becomes c =Gk

ωn

= Gωnm. If the critical damping is inserted as ccr =

2√

km = 2mωn, the damping ratio is thus:

97

Structural Damping and Viscous Damping Forces for Constant Amplitude Sinu-

soidal Displacement

ξ =c

ccr

=G

2

The quality or dynamic magnification factor is defined by Q =1

2ξ=

1

G,

which is inversely proportional to the energy dissipated per cycle of vibration.

The Effect of Damping

Damping is the result of many complicated mechanisms. The effect of damp-

ing on computed response depends on the type and loading duration of the

dynamic analysis. For short duration loadings, damping can be ignored as

the structure reaches its peak response before significant energy has had time

to dissipate. Damping is important for long duration loadings, and is criti-

cal for loadings (such as rotating machinery) that continually add energy to

the structure. The proper specification of the damping coefficients can be

obtained from structural tests or from published literature if possible.

98

D-Conversion of the csv file with the CampbellCsv-

ToAfu journal (Virtual Basic)

’ NX 6.0.2

’ CampbellCsvToAfu.vb - Import Campbell .csv file to .afu file

’

’

’

’------------------------------------------------------------

Option Strict Off

Imports System

Imports System.IO

Imports System.Collections

Imports System.Windows.Forms

Imports NXOpen

Imports NXOpenUI

Module Module1

Dim theSession As Session = Session.GetSession()

Dim lw As ListingWindow = theSession.ListingWindow

Sub Main()

lw.Open()

’ Get .csv file to process

Dim openFileDialog As New OpenFileDialog

openFileDialog.Filter = "csv files (*.csv)|*.csv|All Files (*.*)|*.*"

openFileDialog.FilterIndex = 1

openFileDialog.Title = "Select NX Nastran Rotordynamics .csv Output File"

openFileDialog.InitialDirectory = theSession.Parts.BaseDisplay.FullPath

openFileDialog.CheckFileExists = True

If openFileDialog.ShowDialog = DialogResult.OK Then

Dim csvFile As String = openFileDialog.FileName

’ Generate afu filename

Dim afuFile As String = Left(csvFile, InStrRev(csvFile, ".")) + "afu"

’ Check to see if file exists. If it does, prompt to delete it

If (File.Exists(afuFile)) Then

Dim title As String = "File Exists"

Dim message As String = "AFU File " + afuFile + " already exists." + vbNewLine +

Dim result As MsgBoxResult = MsgBox(message, MsgBoxStyle.OkCancel, title)

If (result = MsgBoxResult.Ok) Then

99

theSession.AfuManager.DeleteAfuFile(afuFile)

Else

Return

End If

End If

’ Create the afu file

theSession.AfuManager.CreateNewAfuFile(afuFile)

lw.WriteLine("Creating AFU File: " + afuFile)

parseData(csvFile, afuFile)

End If

End Sub

Sub parseData(ByVal csvFile As String, ByVal afuFile As String)

Dim csvData() As String = System.IO.File.ReadAllLines(csvFile)

Dim csvIdx As Integer = 0

Dim values() As String

’ Find start of RPM data

Do

values = Split(csvData(csvIdx), ",")

csvIdx += 1

Loop Until values(0) = " 10001"

’ Get RPM Data

Dim rpmList As New ArrayList()

Do

values = Split(csvData(csvIdx), ",")

csvIdx += 1

rpmList.Add(Trim(values(0)))

Loop Until (Left(values(0), 6) = " ")

csvIdx -= 1

rpmList.RemoveAt(rpmList.Count - 1)

Dim rpmCount As Integer = rpmList.Count

Dim rpmData(rpmCount - 1) As Double

For rpmIdx As Integer = 0 To rpmCount - 1

rpmData(rpmIdx) = rpmList.Item(rpmIdx) * 1.0

Next

lw.WriteLine(vbNewLine + "Data stored for " + rpmCount.ToString + " RPM values" + vbNewLine

createPerRevLine(afuFile, rpmData, 1.0)

createPerRevLine(afuFile, rpmData, 2.0)

100

lw.WriteLine(vbNewLine)

’ Loop through the rest of the data

Dim data(rpmCount - 1)() As String

Do

Dim labels() As String = Split(csvData(csvIdx), ",")

csvIdx += 1

Dim Idx As Integer = 0

For i As Integer = csvIdx To csvIdx + rpmCount - 1

Dim temp() As String = Split(csvData(i), ",")

data(Idx) = Split(csvData(i), ",")

Idx += 1

Next

Dim yValues(rpmCount) As Double

For j As Integer = 0 To labels.Length - 2

For i As Integer = 0 To rpmCount - 1

yValues(i) = data(i)(j)

Next

lw.WriteLine("Creating record " + Trim(labels(j)))

createAfuRecord(afuFile, Trim(labels(j)), rpmData, yValues)

Next

csvIdx += rpmCount

Loop Until csvIdx >= csvData.Length

End Sub

Sub createPerRevLine(ByVal afuFile As String, ByVal rpm() As Double, ByVal scale As Double)

Dim ordData(rpm.Length - 1) As Double

’ Generate 1P and 2P lines

For rpmIdx As Integer = 0 To rpm.Length - 1

’ordData(rpmIdx) = rpm(rpmIdx) * (Math.PI / 30) * scale

ordData(rpmIdx) = rpm(rpmIdx) / 60 * scale

Next

’ Create 1P and 2P records in AFU

Dim recordName As String = scale.ToString + "P"

createAfuRecord(afuFile, recordName, rpm, ordData)

lw.WriteLine(recordName + " line generated")

End Sub

Sub createAfuRecord(ByVal afuFile As String, ByVal recordName As String, ByVal abscissaData()

Dim afuData1 As CAE.AfuData

101

afuData1 = theSession.AfuManager.CreateAfuData()

afuData1.FileName = afuFile

afuData1.RecordName = recordName

afuData1.SetIdInformation("x+", 1, "x+", 1)

afuData1.SetAxisDefinition(CAE.AfuData.AbscissaType.Uneven, CAE.XyFunctionUnit.RpmRpm, CAE.AfuData.OrdinateType.Real,

afuData1.SetRealData(abscissaData, ordinateData)

Dim createTime1 As String

createTime1 = theSession.AfuManager.CreateRecord(CAE.XyFunctionMacroType.General, CAE.XyFunctionGeneralType.CampbellDiagram,

afuData1.Dispose()

End Sub

Public Function GetUnloadOption(ByVal dummy As String) As Integer

’Unloads the image immediately after execution within NX

GetUnloadOption = NXOpen.Session.LibraryUnloadOption.Immediately

’----Other unload options-------

’Unloads the image when the NX session terminates

’GetUnloadOption = NXOpen.Session.LibraryUnloadOption.AtTermination

’Unloads the image explicitly, via an unload dialog

’GetUnloadOption = NXOpen.Session.LibraryUnloadOption.Explicitly

’-------------------------------

End Function

End Module

102

E-Modelling of the whole engine

103

F- Help for the transient response analysis

Equations of Motion

The equations of motion are shown for the modal method. The following

equation is solved in the fixed reference system:

[M ] ¨u(t) + (Ω[C] + [DI + DA]) ˙u(t) + ([K] + Ω[DB]) ¯u(t) = ¯p(t)

The generalised force is similar to the case of frequency response: ¯p(t) =

[Φ]T p(t). The physical displacement method is found from u(t) = [Φ] ¯u(t).

The standard NX Nastran methods are used for the recovery of element forces

and stresses.

The equations are solved numerically with the standard numerical meth-

ods. The initial conditions are equal to zero: ¯u(0) = 0 and ˙u(0) = 0.

The forcing function must be defined by the user with the NX Nastran

cards. For the transient analysis, the user must provide an excitation function

which is compatible with the time step (TSTEP) used and with the rotor

speed values on the ROTORD card. A time function with linearly varying

frequency must be defined for the synchronous and asynchronous analysis.

Example of simulation

The forcing frequency is defined by:

p = p0 sin(ωt)t (9.1)

For a linearly varying frequency one have ω(t) = 2πat. The slope of

the curve is a. The angle is thus ϕ(t) = ω(t)t = 2πat2. The instantaneous

frequency is the time derivative of the angle: ω =dϕ

dt= 4πat. In order to

obtain a frequency F at time T, one have F = ωend/(2π) = 2aT The slope is

then given by a = F/(2T ). For example, to simulate from 0 to 500 Hz in 10

seconds the constant is a = 500/(2 ∗ 10) = 25. At the end of the simulation,

the frequency is F = 2.25.10 = 500 Hz and the period Tend = 1/500 = 0.002.

104

The time step is now dependent of the wanted number of integration

points per cycle. A reasonable value is 10 points and hence, a time step of

0.0002 can be used. This means 50000 time steps for the simulation of 10

seconds.

Similar to the frequency domain when a rotating force was defined by

real and imaginary parts in x and y direction, a rotating force can be defined

in the time domain by taking sine and cosine functions for the x and y axis.

These data must be entered into NX Nastran cards by means of TABLED1

cards.

The following graph shows the result of the simulation applied on the

Jeffcott rotor.

It is recommended to start the simulation with the frequency response

analysis as the way to perform the transient simulation is close to it.

105