Dynrot Handbook DYNROT HBK

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DYNROT 8.3

A FINITE ELEMENT CODE FOR

ROTORDYNAMIC ANALYSIS

Dipartimento di Meccanica

Politecnico di Torino

Torino Italy

November 2000


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Foreword

DYNROT 8.3 is a code for rotordynamics computations based on the finite

element method (FEM).

It has been evolved in various versions through more than 20 years. The presentversion is based on the MATLAB

1interactive software package and consequently

can be used on any hardware on which MATLAB has been installed. Version 4.0 or

higher of MATLAB is required.The original code was written at the end of the seventies by Giancarlo Genta and

Antonio Gugliotta, both of Department of Mechanics of Politecnico di Torino, using

HPL and then HP-BASIC language, for desktop HP 9800 computers. Subsequentversions written in Fortran and C languages were developed in the eighties and

finally the present version using MATLAB package was evolved. Its superiority lays

mainly in the ability of MATLAB of dealing with complex arithmetics, its graphictools and the possibility of easily obtaining output ASCII files.

At present the DYNROT project is coordinated by Giancarlo Genta. Its aim is tocontinue the development of the code to widen its capabilities and to make it an

even more powerful tool for the dynamic analysis of rotating machinery. A number

of persons worked and is still working in it, mostly undergraduate and post graduate

students. Among them Giacomo Brussino, Philip Miller, Domenico Bassani,Cristiana Delprete, Stefano Carabelli and Andrea Tonoli must be mentioned.

G. Genta

Torino, November 2000

1MATLAB is a trademark of The MathWorks, Inc.


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Contents3

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Contents

Foreword

Contents

Part 1. How to get started1.1 Installation1.2 Running DYNROT

1.3 Structure of the driver M-file

1.4 How to input data

Part 2. General informations

2.1 Aims of the code2.2 Theoretical background

2.3 Parametric modelling

2.4 Degrees of freedom

2.5 Units

Part 3. Elements3.1 Element 1: beam3.2 Element 2: tapered beam

3.3 Element 3: spring

3.4 Element 4: damper

3.5 Element 5: concentrated mass3.6 Element 6: nonisotropic (asymmetrical) beam

3.7 Element 7: nonisotropic (asymmetrical) spring

3.8 Element 8: nonisotropic (asymmetrical) damper3.9 Element 9: nonisotropic (asymmetrical) concentrated mass

3.10 Element 10: cubic spring

3.11 Element 11: spring with clearance

3.12 Element 12: magnetic bearing3.13 Element 13: asymmetrical magnetic bearing

3.14 Element 14: 8-coefficients bearing

3.15 Element 15: speed-dependent 8-coefficients bearing

3.16 Element 16: hydrodynamic bearing3.17 Element 17: crank with connecting rod and piston

3.18 Element 18: disc-shaft transition element

3.19 Element 19: flexible disc

3.20 Element 20: disc-blade transition element3.21 Element 21: row of blades element

3.22 Element 22: concentrated driving torque

3.23 Element 23: shaft-disc transition element3.24 Element 24: blade-disc transition element

3.25 Element 101: PID controller for magnetic bearings

3.26 Element 102: general controller for magnetic bearings

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Part 4. Lubricated bearings4.1 Program LUBINPUT4.2 Program LUBSHORT

4.3 Program LUBPLOT

4.4 Program ALIGN

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Part 5. Construction of the model5.1 Construction of the matrices of the elements

5.2 Saving matrices

5.3 Function DYNPLOT5.4 Function DYNPLOT1

5.5 Function STRPLT

5.6 Function DYNTRANS

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Part 6. Solution programs: static analysis6.1 Function DYNSTAT6.2 Function STATPLT

6.3 Function FORCES

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Part 7. Solution programs: critical speeds7.1 Function DYNCRIT

7.2 Function DYNCRITM

7.3 Function CRITPLT

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Part 8. Solution programs: unbalance response8.1 Function DYNUNBAL

8.2 Function DYNUNMOD

8.3 Function DYNUNM8.4 Function LUBUNBAL

8.5 Function UNPLT

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Part 9. Solution programs: acceleration response9.1 Function DYNACCEL

9.2 Function ACCPLT

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Part 10. Solution programs: Campbell diagram10.1 Function DYNCAMP

10.2 Function DYNDAMP10.3 Function DYNDAMOD10.4 Function DYNLUB

10.5 Function DYNMAG

10.6 Function DYNMAG110.7 Function CAMPLT

10.8 Function CAMPLT1

10.9 Function LUBPLT

10.10 Function ROOTPLT

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Part 11. Solution programs: torsional analysis11.2 Function DYNTORS

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11.3 Function FORTORS

11.4 Function TORQTORS11.5 Function CAMPBT

11.6 Function TORSPLT

11.7 Function TCPLT11.8 Function FTPLT

11.9 Function HTPLT

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Part 12. Solution programs: axial natural frequencies

12.1 Function DYNAX12.2 Function CAMPBA

12.3 Function AXPLT

12.4 Function ACPLT

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Part 13. Solution programs: coupled torsional-axial natural

frequencies13.1 Function DYNTA

13.2 Function CAMPBTA

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Part 14. Examples14.1 Example TEST1

14.2 Example TEST2

14.3 Example TEST3

14.4 Example TEST414.5 Example TEST514.6 Example TEST6

14.7 Example TEST7

14.8 Example TEST8

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References 123


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Part 1How to get started

1.1 Structure of the code

The code is based on a number of MATLAB M-files. Some of them are script

files, while others are function files. They are supplied in the directoriesPROGRAMS, MLAB4 and MLAB5 of the DYNROT 8.3 diskette.

The input data for any particular program are contained in a particular M-file,

that can be prepared by the user with any editor or generated interactively using theprograms contained in the INPUT directory (see section 1.5). Examples of these data

files are supplied in the directory DATA of the DYNROT 8.3 diskette. The same

directory contains also the data for various types of lubricated bearings.A driver called DYNROT can be run by typing DYNROT in the MATLAB

command window followed by a . A button menu then appears, from

which all parts of the code can be accessed.

The computation must start with the construction of the model. This is performed

by function DYNPREP, which calls automatically other functions when needed.

Once the model has been prepared, the solutions of the various problems areobtained by running the several M-files which solve them (e.g., DYNCRIT.M finds

the critical speeds, UNBAL.M computes the unbalance response, etc. ). They are all

function files and consequently the calling statement must contain the relevantparameters, although alternative ways of running solution routines exist. An in-line

help is provided: for example, by typing HELP DYNCRIT a short description of the

DYNCRIT function and of the calling parameters needed is supplied. The user cancall any number of solution routines, to solve the various problems related to a

particular model. If the solution is impossible, as calling the unbalance response of arotor whose unbalances have not been defined, the program stops with an errormessage (e.g. -** ERROR ** Unbalances not defined-). In other cases, a warning

message is supplied and the computation proceeds with a slightly different aim: if

the computation of the critical speeds of a nonlinear rotor is called, the computation

is performed on a linearized version of the model and the warning -** WARNING

** Nonlinear system: linearized analysis performed- is supplied. Many solutionroutines supply the results in form of messages on the screen and/or of a plot on

MATLAB graphic windows. All messages on the screen are saved in a diary-file

while each plot is drawn in a separate graphic window which can then be recalled atwill.


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After computations are performed, the results can be post processed in three

different ways. The most direct is that of calling a function M-file provided in theDYNROT diskette (e.g., function CRITPLT.M plots the mode shapes at the critical

speeds). The outputs are generally plots in various graphic windows.

The second alternative is that of loading the file containing the results fromwithin MATLAB either using the command LOAD (e.g. LOAD CRSPEED.### -

MAT, where ### is the code of a particular problem, will result in loading the scalar

quantity "neig" expressing the number of critical speeds, the one-column matrix

OMCR, containing the critical speeds, and the matrix of the eigenvectors EIGV, all

prepared by the function DYNCRIT) or through an appropriate choice of theinteractive menu. The matrices can then be saved in ASCII form and processed

using any graphical program. The numbers can be passed, through an ASCII file, to

any word processor to be included in a report.The third alternative is that of preparing a code which reads the output file to

produce a plot or a report, either directly from the outputs in MATLAB format or,

after conversion, in ASCII form.

1.2 Installation

To install DYNROT on a computer, MATLAB must be installed first.A directory for DYNROT must be created on the hard disk to contain all the

program M-files (e.g. C:\DYNROT).The directory with the M-files must be inserted in the MATLABPATH. No otherchanges in the MATLABPATH will be needed.

All M-files from PROGRAM subdirectory of the DYNROT diskette must be

copied in the chosen directory of the hard disk. Also the M-files from either MLAB4

or MLAB5 subdirectories of the DYNROT diskette must be copied in the chosen

directory of the hard disk, depending whether MATLAB 4 or MATLAB 5 is used.The lubricated bearings data contained in A:\DYNROT\DATA and all similar ones

created by the used must be stored in a directory included in the MATLABPATH.

Note that in the driver files included for the examples this directory isC:\DYNROT\DATA; if this name is changed, the driver files must be changed

accordingly. Note also that the bearing data (files starting with SH and SO) need not

to be copied, unless a particular bearing is actually used. Example TEST4 uses filesh58.m.

The data and the results of each case can be placed in different directories. They

can be subdirectories of C:\DYNROT or not. These directories must be createdbefore starting the computations and need not to be included in the

MATLABPATH. To run the test cases the data M-files must be copied on the hard

disk.

1.3 Running DYNROT

Before running the code, the M-file containing the data must be moved in a

directory which is included into the matlabpath. Alternatively, a directory for data

(e.g., C:\DYNROT\DATA) can be created and included into the matlabpath.


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The user can invoke each file from the keyboard one by one and perform all the

computation in this way. This procedure is however not convenient to start anyparticular problem as some parameters must be supplied between the various calls of

the M-files.

If the code is installed on a personal computer the most convenient way to runDYNROT 8.3 is by using the interactive driver supplied with the code. Type first

DYNROT to start the driver. A first menu will be visualized to state whether:

to run the examples in an interactive way (see section 14);

to run some utilities for lubricated bearings (see section 4);

to create a data file;

to start a new computation (in case the matrices of the particular problem havenot been computed earlier);

to continue a computation already performed.In some cases the directory in which to store the data or the results must be

supplied.

The code of the problem which must be dealt with is then requested. If the 'newproblem' option was chosen first, the problem can be the one just studied or another

one dealt with earlier. The computation then goes on using different menus, until the

user exits from the interactive driver. Note that to change problem code the user

must exit and enter again, typing DYNROT. The transformation routine

DYNTRANS causes the program to exit, as it creates a new case with a new code.To run a problem created by transforming an existing one the user must not chose

'new problem' but 'restart', as the relevant data do not exist explicitly as an M-file

separated from the original one (which has however a different code).Another way to run the code, which is best suited for computers other than

personal computers, is that of preparing a driver, i.e a M-file which acts as a batch

command file and calls all the needed script and function files. This is importantmainly to go through the first part of the computation, the building of the model. For

the subsequent parts, the solution and the post processing, the relevant function files

can easily be called by the user from the keyboard.Some drivers are included in the directory DATA of the diskette (e.g.

TEST1DRV.M). Drivers with any name can be created, the only limitation being

that the name must be different from that of the M-file containing the data. The

driver M-file must be moved to a directory included in the MATLABPATH.

Once the driver is started, all the computations included in it are performedautomatically. They can be just the construction of the model or the whole solution

of the problem, if all the required solution and post-processing functions are called.

The driver can contain any other instruction the userwants to include, and all itsoutputs are printed in the diary file.

All messages outputted by the function M-files not included in the driver but

called directly from the keyboard will not be included in the diary file, except if theuser creates explicitly a new diary file.

1.4 Structure of the driver M-file

This section describes the structure of the driver M-file, which is not needed to

run the code using the interactive driver. It can be skipped completely by the user

who plans to use DYNROT as an interactive code.


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The driver M-file is a standard script file which must contain the following

instructions:

clear

filename = 'aaaaaa';

aaaaaaa.m is the name of the m-file (maximum length of filename is 8 characters)

containing the data of the problem to be studied. Among the data, there will be a

code of three characters designating the problem to be solved. If the code is xxx, all

files containing the results will have xxx as extension (example: filename = 'test1'; infile TEST1.M there is the instruction jobcode = 'ts1';).

datapath = yyyyyyyyy;

yyyyyyyyy is a directory in which all results of the problem to be solved will be

written (example: datapath = 'c:\dynrot\test1\'; the final \ is needed). This directorymust be created in advance.

eval (['delete ' datapath 'dynrot.log']);

eval (['diary ' datapath 'dynrot.log']);

These instructions delete the old diary file and prepare a new one. The diary file

is named DYNROT.LOG and is created in the directory yyyyyyyyy.

dynprep(filename,datapath);

DYNPREP function is run to build the model. If other parameters must be

passed to function dynprep, they must have be arranged in a form of a single matrix(e.g. matrix par) and the line becomes (see section 2.3.):

dynprep(filename,datapath,par);

After this instruction all the required function M-file are called, e.g.

dynplot('xxx',0);

It calls function DYNPLOT which plots all the model. In all function M-files the

first parameter xxx is the three-characters code which designates the particularproblem and corresponds to the code entered in the data M-file aaaaaaa.M. In thepresent example the second parameter is 0.

diary off

This is the last instruction, which closes the diary file. It must not be omitted.


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1.5 How to input data

The simplest way to produce a data file is by using the interactive preprocessor

called DYNINPUT. It is an M-file which can be called directly from the MATLAB

prompt by typing DYNINPUT or by calling the interactive driver DYNROT andchosing the option create a new model from the first of the menus. The interactive

preprocessor is based on a series of menus, leading the user through the whole

process. It can be used also to modify an existing data file or to create a new file

based on an old one. However, to introduce small modifications it is far easier to editdirectly the existing file using any editor, as the preprocessor does not substitute

single data but all data of the same kind (i.e., all co-ordinates or all geometries etc.).As already stated, data can also be entered by creating a script M-file. The M-file

containing the data can be created and archived in the directory containing theresults of the particular problem, but before running the program it must be moved

in a directory included in the matlabpath.

The supplied data files and also those created by the preprocessor contain manycomment lines, so they should be self-explanatory. It is advisable to use one of them

to prepare new data files, simply substituting the data with those of the problem to

be studied.

The first two instructions must be of the type

jobcode = 'xxx';

jobtitle = 'cccccccccccccccc';

xxx is a three-characters code designating the problem whose data are specified.All files containing the results will have xxx as extension.

cccccccccccccccc is a string (maximum 256 characters) containing a description

of the problem. It is used only for description purposes, is displayed in outputs but isnever actually used in computations. Keep it as short as possible, as in some graphic

outputs there may be not much space for it.

The instructions

oscale = a;

lscale = b;

then follow. a and b are either 1, or 2 or 3. They designate the scales for spin speed

() and frequency () axes in graphical outputs:

oscale (lscale) = 1 scales in rad/s

oscale (lscale) = 2 scales in rpm

oscale (lscale) = 3 scales in HzNote that oscale and lscale may be omitted: in this case all scales will be in rad/s.The data are supplied in form of matrices, using MATLAB notation.

Matrix COORDcontains the nodal co-ordinates. It has as many rows as there

are nodes and 5 columns.

In the first column a number designating the type of node is entered. Columns 2,3 and 4 contain thex,yandzco-ordinates of the node. As the rotation axis of the

rotor is assumed to be z-axis, all nodes of type 1 must havexand yco-ordinates

equal to zero. For nodes of type 2 or 3 a nonzero value of xco-ordinate must be


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supplied, indicating the radius at which the node is located on the disc. In the

present version of the code column 3 of matrix COORD (that for y co-ordinate) isuseless: it is present just in prevision of a more complete future version of the code

in which nodes of different type will be included. A 0 in column 3 must however be

entered.Column 5 contains the number of the node (remember that the nodes must be

listed in numeric order). After column 5 a space for comments is left. A label can be

introduced to help understanding and modifying the input file.

Matrix ELEMcontains element data. It has as many rows as there are elements

and 8 columns. In column 1 a number designating the type of element is entered. Inthe present version of the code there are 24 types of elements (see Part 3). In the

following 4 columns the numbers of the nodes are listed. A zero is introduced for

nodes not needed. In columns 6 and 7 the numbers of the geometry and of thematerial are listed. Column number 8 is for the rotation index, a number linked with

the spin speed of the particular element.

Matrix MATER contains data of materials. It has as many rows as there aredifferent materials and 6 columns. In the first column the number of the material is

entered while the other columns contain the Young's modulus, Poisson's ratio,

density, loss factor and thermal expansion coefficient.Matrix GEOMcontains geometrical data of elements; the termgeometricalmust

be here intended in a generalized sense, as it contains also axial forces in case of

beam elements or gains in case of control system elements. It has as many rows as

there are different geometries and a variable number of columns. In the first column

the number of the geometries is entered, the second column contains the geometrytype (24 geometries for elements and 2 types of control systems are at presentdefined) while the other columns contain properties of the elements. Note that all

lines of matrix GEOM must contain the same number of elements: zeros must be

inserted in the empty columns of elements which require a smaller number ofparameters.

The instructions defining static forces then follow.

gxis a number defining the constant (gravitational) acceleration in direction ofx-axis.

gyis a number defining the constant (gravitational) acceleration in direction ofy-axis.

If no gravitational acceleration is accounted for, gx and gy need not to be

defined.Matrix STFORcontains data of constant concentrated generalized forces acting

at the nodes in the inflection planes. It has as many rows as there are concentratedforces and 6 columns. In the first column the number of the node on which the forceacts is entered. The following columns contain the values of the forces in xand y

directions and the values of the moments aboutx,yandzaxes.

Matrix UNFORcontains data about unbalances. It has as many rows as there areconcentrated unbalances and 5 columns. In the first column the number of the node

on which the unbalance acts is entered. The following columns contain the values

of the static unbalances in x and ydirections and the values of couple unbalancesaboutxandzaxes.

Matrix BEARINGScontains data for speed-dependent 8-coefficients bearings.

For each bearing it has a line for each speed at which the stiffness and damping

parameters are defined. If there are rbearings defined at sspeeds each, it has rs


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rows. It has 10 columns. In the first column the number of the bearing (referred to

matrix GEOM) is entered. The following column contains the values of speed atwhich the coefficients are referred. In columns from 3 to 6 the stiffness values kx, ky,

kxyand kyxare listed. Columns from 7 to 10 contain the damping coefficients cx, cy,

cxyand cyx.Matrix PHASEdefines the relative phases of the working cycles of the cranks.

It has as many rows as there are cranks and 4 columns. In the first column the

number of the node is entered. The second column contains the crank angle in

degrees. If the cylinders are not all in the same plane, the crank angle must be

substituted by the difference between the crank angle and an angle defining theplane containing the cylinder referred to a fixed direction. The third column contains

the phase angle in degrees. The fourth column contains a progressive number which

designates the characteristics of the working cycle (see matrix CYCLE).Matrix CYCLEdefines the general characteristics of the working cycles. It has

as many rows as there are different working cycles and a variable number of

columns. In the first column a 2 or a 4 is entered depending whether a 2 or 4 strokecycle is considered. Different types of cycles cannot be included in the same

machine, but a 2 stroke cycle can be modelled by a 4 stroke cycle with all odd

harmonics equal to zero. The second column contains the area of the piston. Thethird column contains coefficient k for the computation of the damping (see [1]).

The subsequent columns contain the coefficients for the computation of the mean

indicated pressure as a function of the speed (speed in rad/s) following a polynomial

expansion (pmi= C1+ C2+ C32+ C4

3).

Matrix HARFORdefines the coefficients of the harmonic terms for the drivingtorque on the pistons. It has as many rows as there are harmonics, including the

zero-order harmonic, and a variable number of columns (two for each differentworking cycle present). In the first column the coefficients for the terms in cosine of

the first cycle are entered. The second column contains the coefficients of the terms

in sine for the first cycle (the first row must contain a 0, as the zero-order harmonichas no term in sine). The subsequent columns contain the coefficients for the other

cycles, always with coefficients of the terms in cosine in odd columns and in sine in

even columns.Matrix TIMEHISTdefines the time history of the rotor. In the first column the

values of the time are entered. The second column contains the values of the speed

at the stated time. The speed is assumed to vary linearily during each time step. Thethird column contains a coefficient which multiplies the unbalance. The coefficient

is kept constant in each time step and varies abruptly at the end of the step. The

value in the last row has no importance and 0 can be entered.Matrix STATUSdefines whether the various degrees of freedom of the nodes

are master, slave or constrained. It has as many rows as there are nodes and 10

columns. In the first column the number of the node is entered.If the node is a node of type 1, the following 2 columns contain the status of the

first (complex radial displacement) and the second (complex bending rotation)degree of freedom for flexural behaviour. Zeros must be entered in columns columns

from 4 to 6. The seventh column contains the status of the torsional degree of

freedom (real torsional rotation). A zero must be entered in column 8. The ninth

column contains the status of the axial degree of freedom (real displacement) and thelast (tenth column) must contain a zero.


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In case of nodes of type 2, columns from 2 to 5 contain the status of the four

degrees of freedom for flexural behaviour in the order described in section 2.3.Column 6 must contain a 0. Column 7 contains the status of the torsional degree of

freedom, column 8 contains a 0 and columns 9 and 10 the status of the axial degrees

of freedom.In case of nodes of type 3, columns from 2 to 6 contain the status of the five

degrees of freedom for flexural behaviour in the order described in section 2.3.

Columns 7 and 8 contain the status of the two torsional degrees of freedom and

columns 9 and 10 that of the axial degrees of freedom. Status equal to 0 means that

the degree of freedom is constrained, 1 characterizes master degrees of freedom and2 slave degrees of freedom. In general, the second flexural degree of freedom of

nodes of type 1 (column 3), the last flexural and axial degree of freedom of nodes of

type 2 (columns 5 and 10) can be assumed to be slave. For nodes of type 3 also thelast flexural and torsional degrees of freedom (columns 6 and 8) can be assumed to

be slave.

A further type of degree of freedom can be defined, only for flexuraltranslational, torsional and axial degrees of freedom of nodes of type 1 (columns 2, 7

and 9). They are defined as input supermaster, output supermastersand input-output

supermasters. They are standard masterdegrees of freedom whose displacemnts areneeded by the user for further computations, as in the case of active systems in

which the displacements at the sensor and actuator locations are required. The

program generates selection matrices which, when multiplied by the displacement

vector, return the displacements in the selected nodes.

The overall characteristics of a model are listed in a vector named GLOBAL,which is automatically generated by the code. They are:

Row Content1. number of nodes

2. number of elements

3. number of materials4. number of geometries

5. number of static forces (flexural)

6. number of unbalances7. number of cranks

8. number of working cycles

9. number of harmonics (including zero-order)

10. number of entries for time histories11. nonlinearity index12. number of bladed discs

13. index for axial-torsional coupling

14. index for rotating anisotropy15. index for non rotating anisotropy

16. number of substructures

17. index for hysteretic rotating damping

18. index for hysteretic non rotating damping19. index for viscous rotating damping

20. index for viscous non rotating damping

21. number of cubic elements (flexural behaviour)


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22. number of spring-with-clearance elements (flexural behaviour)

23. number of cubic elements (torsional behaviour)24. number of spring-with-clearance elements (torsional behaviour)

25. number of cubic elements (axial behaviour)

26. number of spring-with-clearance elements (axial behaviour)27. number of isotropic magnetic bearings

28. number of anisotropic magnetic bearings

29. number of lubricated bearings

30. number of speed-dependent bearings

31. number of degrees of freedom (flexural behaviour)32. number of master degrees of freedom (flexural behaviour)

33. number of slave degrees of freedom (flexural behaviour)

34. number of degrees of freedom (torsional behaviour)35. number of master degrees of freedom (torsional behaviour)

36. number of slave degrees of freedom (torsional behaviour)

37. number of degrees of freedom (axial behaviour)38. number of master degrees of freedom (axial behaviour)

39. number of slave degrees of freedom (axial behaviour)

40. number of degrees of freedom (torsional-axial behaviour)41. number of master degrees of freedom (torsional-axial behaviour)

42. number of slave degrees of freedom (torsional-axial behaviour)

43. number of cubic elements (torsional-axial behaviour)

44. number of spring-with-clearance elements (torsional-axial behaviour)

45. number of concentrated torque elements- -51. gx

52. gy

- -61. index for omega scale

62. index for frequency scale


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Part 2General informations

2.1 Aims of the code

DYNROT is a tool for the dynamic analysis of rotating machines. Its main aim isthen to perform all the usual rotordynamics computations, such as the evaluation of

critical speeds and the plotting of the Campbell diagram.However, its ability to take

into account features as nonlinearities,deviations from axial symmetry of either the

rotor or the stator of the machine, damping and angular acceleration, allow to

simulate the dynamic behaviour of the machine in a detail much deeper than that

achieved in usual rotordynamic analysis. The effects of the flexibility of the discsand of the blades are included: when they are introduced into the model, the stress

distribution in the disc due both to centrifugal and thermal stressing are computed, totake into account also the stiffening (and then the increase of the natural frequencies)

due to the stress field.

Also the axial and the torsional behaviour of the system can be studied. In these

cases the stiffening of the discs and the blades due to the stress field are accountedfor. If the blades have their principal axes of inertia in directions other than axial and

tangential to the discs, a coupling between axial and torsional behaviour results. In

this case the code allows to study also the coupled torsional-axial dynamics. Notethat the results obtained from the uncopled torsional and axial study can be incorrect

in this case (warnings are issued if the uncoupled study is attempted).Routines for the detailed study of the equivalent system for the study of the

torsional behaviour of reciprocating machines have been included in the present

release.While building the model, the inertial properties (mass, moments of inertia and

co-ordinates of the centre of mass) of the whole model and of the various

substructures are computed, printed to the screen and in the diary file.As the whole machine is made of different parts which can rotate at different

speeds, all outputs can be obtained for the various substructures separately. A

substructure is here defined as the assembly of all the parts which rotate at the samespin speed. The spin speed is entered as rotation index: rotation index equal to zero

characterizes the nonrotating parts of the machine (stator); rotation index equal to 1

characterize all the parts of the machine rotating at the nominal speed. If the value of

the rotation index is , the spin speed is times the nominal speed.Although it is possible, to use slightly different values of the rotating index (e.g.,

1 and 1.0000001 for parts rotating at the nominal speed) to obtain separate outputs


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Part 2: General informations18

18

for parts constituting the same substructure (in the above mentioned sense), this

must be done with extreme care, particularly if hysteretic damping ornonsymmetrical elements are present.

2.2 Theoretical background

DYNROT code uses more or less a standard finite element formulation, with

some deviations aimed to allow a more straightforward analysis of rotating systems.The theoretical background on which the code is based is treated in detail in the

book by G.Genta Vibration of Structures and Machines[1]2. In particular, the details

on elements formulation, structure assembly and reduction can be found in Chapter

2; flexural behaviour in Chapter 4; torsional behaviour in Chapter 5 and magnetic

bearings in Chapter 6. As a consequence, no details on the theory on which the code

is based are given in the present guide and the reader is suggested to refer to thementioned textbook.

The most important peculiar feature of DYNROT code is the use of complex co-

ordinates to express the flexural displacements and rotations. Assume thatz-axis of afixed reference frame lays along the rotation axis of the machine and that each node

(type 1) has 6 degrees of freedom, three translational and three rotational.

The axial displacement uzis assumed to be uncoupled from the others and does

not enter flexural or torsional behaviour. In the same way, the torsional rotation zis

also assumed to be uncoupled from the others and enters only torsional behaviour.

2.3 Parametric modelling

There are cases in which the user wants to investigate the effects of the changes

in some of the data of a certain model. Obviously he can perform the relevantcomputations several times stating different values of the parameters, but this can be

time consuming. DYNROT 8.3 allows leaving some of the data in symbolic form

and then passing the relevant numerical values to the code at the time of the

execution. In this way it is possible to call the various routines within a loop, to

study the effects of the variation of the design parameters. An example, related tothe construction of a plot of the critical speeds as functions of the stiffness of the

bearings, is shown in Example TEST 8. The user can refer to this example for all

relevant informations.The parameters are passed to DYNPREP function and from it to the other

functions computing the model of the system in the form of a matrix named par. It

can have any number of rows or columns.

2.4 Degrees of freedom

To deal with the different characteristics of the different elements, three types ofnodes had to be introduced. Nodes of type 1 have two complex and two real degrees

2Numbers in braces designate references listed in last section of the present guide.


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Part 2: General informations 19

19

of freedom, corresponding to the six real degrees of freedom commonly used for the

nodes of beam elements. Only nodes of type 1 are used for all elements except fordisc and row-of-blades elements. Suitable transition elements are provided to

connect elements which require nodes of different types.

A node of type 1 has 2 complex degrees of freedom related to displacements androtations in xzand yzplanes, 1 real degree of freedom related to torsional rotation

and 1 real degree of freedom related to axial displacement. The two translational

degrees of freedom uxand uy are composed to produce the complex displacementz=

ux + i uy and the two rotational degrees of freedom x and y are composed to

produce the complex rotation=y + ix. The complex co-ordinates zand aredealt with in the way real co-ordinates usually are.

A node of type 2 has 4 complex degrees of freedom related to displacements and

rotations in xzand yzplanes, 1 real degree of freedom related to torsional rotation

and 2 real degrees of freedom related to axial displacement. For the flexural

behaviour they are a complex radial displacement (only the first harmonic of aFourier expansion of a non-axisymmetrical displacement), a complex axial

displacement (again the first harmonic of a Fourier expansion of a non-

axisymmetrical displacement); a complex circumferential displacement (firstharmonic) and the derivative of the axial displacement with respect to the radius,

which coincides with the slope of the inflected shape, as no shear deformation is

considered (first harmonic). The axial displacement (second degree of freedom) and

its derivative are divided by the radius to form a sort of rotation of the circumferenceat the location of the node about one of its diameters.

For the torsional behaviour the real degree of freedom is the circumferential

displacement divided by the radius, i.e. the torsional rotation (zero order harmonic ofa Fourier expansion). For the axial behaviour they are a real axial displacement and

its derivative with respect to the radius (zero order harmonic).

A node of type 3 has 5 complex degrees of freedom related to displacements and

rotations inxzandyzplanes, 2 real degrees of freedom related to torsional rotation

and 2 real degrees of freedom related to axial displacement. For the flexuralbehaviour they are a complex radial displacement (only the first harmonic of a

Fourier expansion of a non-axisymmetrical displacement), a complex axial

displacement (again the first harmonic of a Fourier expansion of a non-axisymmetrical displacement); a complex circumferential displacement (first

harmonic) and the derivatives of the axial and circumferential with respect to the

radius.For the torsional behaviour the real degrees of freedom are the circumferential

displacement (zero order harmonic of a Fourier expansion) and its derivative withrespect to the radius, both divided by the radius. In a similar way, for the axial

behaviour they are a real axial displacement and its derivative with respect to the

radius (zero order harmonic).

A consequence of the use of the complex co-ordinates is the need to use mean

and deviatoric matrices in case of anisotropic systems; this has however its

advantages too, as it is easy to build an averaged modelwhich can be used for first-approximation studies [2]. Another consequence is the fact that gyropscopic

matrices are always symmetrical.


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Part 2: General informations20

20

2.5 Units

All numerical quantities are expected to be introduced into the code using any

consistent system. Obviously the use of S.I. system is highly recommended and

consequently the most important units are expected to be:

length m mass kg force N

density kg/m3 stress N/m

2 angle rad

ang. vel. rad/s frequency rad/s decay rate 1/spressure Pa

There is only one exception: in few cases angles are requested to be introduced

in degrees, as it is far more immediate for the user. In this case the user is explicitlywarned to do so in the present guide.

The user is explicitly warned not to use mm for lengths and N/mm2for stresses

and Young's moduli: in this case the correct unit for densities would not be kg/mm3

but t/mm3, which could lead to misunderstandings.


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Part 3Elements

The element library of DYNROT code is particularly tailored for rotordynamicsapplications; as a consequence many standard elements are not included and

specialized elements are present. In the present version there are 24 mechanical

elements and 2 control elements. They are:

1 beam

2 tapered beam3 spring

4 damper

5 mass

6 asymmetrical beam7 asymmetrical spring8 asymmetrical damper

9 asymmetrical mass

10 cubic spring11 spring with clearance

12 magnetic bearing

13 asymmetrical magnetic bearing

14 8-coefficients bearing15 8-coefficients bearing (speed dependent)

16 lubricated bearing17 crank with piston

18 shaft-disc interface

19 flexible disc

20 disc-blade interface21 flexible blade row

22 concentrated driving torque23 disc-shaft interface

24 blade-disc interface

101 PID controller102 general controller

The data concerning the element connectivity and some general characteristicsare given in ELEM matrix while those concerning the geometrical properties of the

element are given in GEOM matrix. In this way if some elements have the same

geometrical properties the latter can be listed only once.


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Part 3: Elements22

22

3.1 Element 1: beam

Element 1 is a constant cross section beam with circular or annular cross section.

Its formulation is of the type usually referred to as simple Timoshenko beam; it is

described in detail in [1]. A consistent formulation has been used for mass andgyroscopic matrices. Two different outer diameters can be defined: an outer

diameter and a mass outer diameter. Usually the two are coincident (in this case the

mass outer diameter can be set to zero), but the second can be larger than the first ifthere is a part of the cross section of the shaft which contributes to its inertial

properties and not to its stiffness, as in the case of laminations and windings of

electric motors and generators. In this case a ratio of densities can be defined as the

ratio between the density of the part of the element not contributing to its stiffness

and the density of the inner core. The last two parameters must be entered, even if

they can be set to zero if they are useless.It has two nodes at its ends.

The data to be introduced in matrix ELEM areType

N1

N2N3

N4GeometryMaterial

Rotation index

1

number of the node at one end (it is immaterial which end)

number of the node at the other end0

0number of the type of geometry (reference to matrix GEOM)number of the type of material (reference to matrix MATER)

0 if the element belongs to the stator;

1 if the element belongs to the rotor (spinning at the nominal

speed);

r = ratio between the spin speed of the element and thenominal speed.

Data to be introduced in matrix GEOMColumn 1

Column 2

Column 3Column 4

Column 5

Column 6Column 7

number of the geometry (reference to matrix ELEM)

code of the element type (1) (reference to matrix ELEM)

inner diameterouter diameter

axial force acting on the element

outer "mass" diameter (0 if equal to the outer diameter)ratio of densities (outer part/inner core)

Data to be introduced in matrix MATER

Column 1

Column 2Column 3

Column 4

Column 5

number of the material (reference to matrix ELEM)

Young's modulusPoisson's ratio

density

loss factor


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Part 3: Elements 23

23

3.2 Element 2: tapered beam

Element 2 is a beam with circular or annular cross section with linearily varying

inner and outer radii. Its formulation is of the type usually referred to as simple

Timoshenko beam; it is described in detail in [3]. A consistent formulation has beenused for mass and gyroscopic matrices.

It has two nodes at its ends.

The data to be introduced in matrix ELEM are

Type

N1

N2

N3

N4Geometry

Material

Rotation index

2


number of the node at the other end

0

0number of the type of geometry (reference to matrix GEOM)

number of the type of material (reference to matrix MATER)

0 if the element belongs to the stator;1 if the element belongs to the rotor (spinning at the nominal

speed);



Column 2

Column 3

Column 4

Column 5Column 6

Column 7



inner diameter at node 1

outer diameter at node 1

inner diameter at node 2outer diameter at node 2

axial force acting on the element


Column 1

Column 2Column 3

Column 4Column 5


Young's modulusPoisson's ratio

densityloss factor


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Part 3: Elements24

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3.3 Element 3: spring

Element 3 is a spring element which can be connected at both ends at two nodes

of the structure (e.g. to simulate joints, bearings between two different shafts or

between a shaft and the stator) or at one node only, the other end being fixed, i.e.connected with theground(e.g. to simulate an elastic support). The characteristics

of the spring must be isotropic inxyplane.

If two nodes are present, their co-ordinates must be the same.


Type

N1

N2

N3N4

Geometry

Material

Rotation index

3


number of the node at the other end (0 if grounded)

00

number of the type of geometry (reference to matrix GEOM)

number of the type of material (reference to matrix MATER).The material must be specified to introduce the loss factor.


1 if the element belongs to the rotor (spinning at the nominalspeed);


Data to be introduced in matrix GEOM

Column 1

Column 2

Column 3Column 4

Column 5

Column 6



stiffness inx(y) directionrotational stiffness aboutx(y) axis

torsional stiffness

stiffness inzdirection


Column 1Column 2

Column 3

Column 4Column 5

number of the material (reference to matrix ELEM)Young's modulus

Poisson's ratio

densityloss factor


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Part 3: Elements 25

25

3.4 Element 4: damper

Element 4 is a viscous damper element which can be connected at both ends at

two nodes of the structure or at one node only, the other end being fixed, i.e.

connected with the ground, in a way which is similar to the spring element. Thecharacteristics of the damper must be isotropic inxyplane.



Type

N1

N2

N3

N4Geometry

Material

Rotation index

4



0



The material need not be specified as it is useless and a 0 can beentered.



r = ratio between the spin speed of the element and thenominal speed.In case of intershaft dampers, the spin speed is that of the

element in which energy is dissipated.


Column 1Column 2

Column 3

Column 4Column 5

Column 6

number of the geometry (reference to matrix ELEM)code of the element type (4) (reference to matrix ELEM)

damping coefficient inx(y) direction

rotational damping coefficient aboutx(y) axistorsional damping coefficient

damping coefficient inzdirection

No data need to be introduced in matrix MATER


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Part 3: Elements26

26

3.5 Element 5: concentrated mass

Element 5 is an isotropic mass element. Owing to isotropy, the moments of

inertiaJxandJyare equal and are indicated asJt. The element has only one node.


Type

N1N2

N3

N4

Geometry

Material

Rotation index

5

number of the node0

0

0


0


speed);



Column 2Column 3Column 4

Column 5


code of the element type (5) (reference to matrix ELEM)masstransversal moment of inertiaJt (Jx=Jy)

polar moment of inertiaJp (Jz)



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Part 3: Elements 27

27

3.6 Element 6: nonisotropic (asymmetrical) beam

Element 6 is a constant cross section beam with non-isotropic cross section. Its

formulation is of the same type of that of element 1. A consistent formulation has

been used for mass and gyroscopic matrices. Mean and deviatoric matrices aregenerated [2].

It has two nodes at its ends.


Type

N1

N2

N3

N4Geometry

Material

Rotation index

6


number of the node at the other end

0




speed);

No rotation index other than 0 or 1 is possible forasymmetrical elements.


Column 2

Column 3

Column 4

Column 5Column 6

Column 7

Column 8Column 9

Column 10

Column 11



area of the cross section

area moment of inertia aboutx-axis

area moment of inertia abouty-axisarea moment of inertia aboutz-axis

shear coefficient for bending inyzplane

shear coefficient for bending inxzplanetorsion coefficient

angle between a rotatingx-axis of the whole rotor and a similar

axis of the element (expressed in degrees)axial force acting on the element

Data to be introduced in matrix MATERColumn 1

Column 2

Column 3Column 4

Column 5


Young's modulus

Poisson's ratiodensity

loss factor


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Part 3: Elements28

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3.7 Element 7: nonisotropic (asymmetrical) spring

Element 7 is a spring element with non-isotropic characteristics which can be

connected at both ends at two nodes of the structure (e.g. to simulate joints, bearings

between two different shafts or between a shaft and the stator) or at one node only,the other end being fixed, i.e. connected with the ground(e.g. to simulate an elastic

support).



Type

N1

N2

N3N4

Geometry

Material

Rotation index

7

number of the node at one end


00


number of the type of material (reference to matrix MATER).The material must be specified to introduce the loss factor.



No rotation index other than 0 or 1 is possible for asymmetricalelements.


Column 1

Column 2

Column 3Column 4

Column 5

Column 6Column 7

Column 8

Column 9



stiffness inxdirectionrotational stiffness aboutxaxis

stiffness in y direction

rotational stiffness aboutyaxistorsional stiffness

stiffness inzdirection

angle between a rotatingx-axis of the whole rotor and a similaraxis of the element (expressed in degrees)

Data to be introduced in matrix MATERColumn 1

Column 2

Column 3Column 4

Column 5


Young's modulus (not needed: can be 0)

Poisson's ratio (not needed: can be 0)density (not needed: can be 0)

loss factor


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Part 3: Elements 29

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3.8 Element 8: nonisotropic (asymmetrical) damper

Element 8 is a viscous damper element with non-isotropic characteristics which

can be connected at both ends at two nodes of the structure or at one node only, the

other end being fixed, i.e. connected with the ground, in a way which is similar tothe spring element.



Type

N1

N2

N3

N4Geometry

Material

Rotation index

8



0


number of the type of material (reference to matrix MATER).

The material need not be specified as it is useless and a 0 can beentered.



No rotation index other than 0 or 1 is possible forasymmetrical elements.


Column 1

Column 2

Column 3Column 4

Column 5

Column 6Column 7

Column 8

Column 9



damping coefficient inxdirectionrotational damping coefficient aboutxaxis

damping coefficient inydirection

rotational damping coefficient aboutyaxistorsional damping coefficient

damping coefficient inzdirection

angle between a rotatingx-axis of the whole rotor and a similaraxis of the element (expressed in degrees)



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Part 3: Elements30

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3.9 Element 9: nonisotropic (asymmetrical) concentrated mass

Element 9 is a mass element with non-isotropic characteristics. The element has

only one node.


Type

N1N2

N3

N4

Geometry

Material

Rotation index

9

number of the node0

0

0


0


speed);

No rotation index other than 0 or 1 is possible for asymmetricalelements.


Column 2Column 3Column 4

Column 5

Column 6

Column 7


code of the element type (9) (reference to matrix ELEM)massmoment of inertiaJx

moment of inertiaJy

moment of inertiaJzangle between a rotatingx-axis of the whole rotor and a similar

axis of the element (expressed in degrees)



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Part 3: Elements 31

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3.10 Element 10: cubic spring

Element 10 is a nonlinear spring element which can be connected at both ends at

two nodes of the structure (e.g. to simulate joints, bearings between two different

shafts or between a shaft and the stator) or at one node only, the other end beingfixed, i.e. connected with the ground (e.g. to simulate an elastic support). The

characteristics of the spring must be isotropic inxyplane.

The restoring force is assumed to be of the type

( )21 xkxF +=

wherexis the relative displacement of the end nodes.

The linear part of the restoring force must be entered separately, as a linearspring element.


The degrees of freedom connected with a nonlinear element must be defined asmaster degrees of freedom.


Type

N1

N2N3

N4Geometry

Material

Rotation index

10


number of the node at the other end (0 if grounded)0


not needed (0)

not needed (0)


Column 1Column 2

Column 3

Column 4Column 5

number of the geometry (reference to matrix ELEM)code of the element type (10) (reference to matrix ELEM)

coefficient k for translations inx(y) direction

coefficient k for torsional rotations

coefficient kfor translations inz direction



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3.12 Element 11: spring with clearance

Element 11 is a nonlinear spring element which can be connected at both ends at

two nodes of the structure (e.g. to simulate joints, bearings between two different

shafts or between a shaft and the stator) or at one node only, the other end beingfixed, i.e. connected with the ground (e.g. to simulate an elastic support). The

characteristics of the spring must be isotropic inxyplane.

The spring restoring force is assumed to be of the type

( )

( )

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