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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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Contents 1
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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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Contents 4
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Part 4. Lubricated bearings4.1 Program LUBINPUT4.2 Program LUBSHORT
4.3 Program LUBPLOT
4.4 Program ALIGN
55
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56
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
63
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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
7172
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7475
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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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Contents5
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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
99
100101
102
102
Part 13. Solution programs: coupled torsional-axial natural
frequencies13.1 Function DYNTA
13.2 Function CAMPBTA
103104
105
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
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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 one end (it is immaterial which end)
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);
r = ratio between the spin speed of the element and thenominal speed.
Data to be introduced in matrix GEOMColumn 1
Column 2
Column 3
Column 4
Column 5Column 6
Column 7
number of the geometry (reference to matrix ELEM)
code of the element type (2) (reference to matrix ELEM)
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
Data to be introduced in matrix MATER
Column 1
Column 2Column 3
Column 4Column 5
number of the material (reference to matrix ELEM)
Young's modulusPoisson's ratio
densityloss factor
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Part 3: Elements24
24
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.
The data to be introduced in matrix ELEM are
Type
N1
N2
N3N4
Geometry
Material
Rotation index
3
number of the node at one end (it is immaterial which end)
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.
0 if the element belongs to the stator;
1 if the element belongs to the rotor (spinning at the nominalspeed);
r = ratio between the spin speed of the element and thenominal speed.
Data to be introduced in matrix GEOM
Column 1
Column 2
Column 3Column 4
Column 5
Column 6
number of the geometry (reference to matrix ELEM)
code of the element type (3) (reference to matrix ELEM)
stiffness inx(y) directionrotational stiffness aboutx(y) axis
torsional stiffness
stiffness inzdirection
Data to be introduced in matrix MATER
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.
If two nodes are present, their co-ordinates must be the same.
The data to be introduced in matrix ELEM are
Type
N1
N2
N3
N4Geometry
Material
Rotation index
4
number of the node at one end (it is immaterial which end)
number of the node at the other end (0 if grounded)
0
0number of the type of geometry (reference to matrix GEOM)
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.
0 if the element belongs to the stator;
1 if the element belongs to the rotor (spinning at the nominalspeed);
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.
Data to be introduced in matrix GEOM
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.
The data to be introduced in matrix ELEM are
Type
N1N2
N3
N4
Geometry
Material
Rotation index
5
number of the node0
0
0
number of the type of geometry (reference to matrix GEOM)
0
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 2Column 3Column 4
Column 5
number of the geometry (reference to matrix ELEM)
code of the element type (5) (reference to matrix ELEM)masstransversal moment of inertiaJt (Jx=Jy)
polar moment of inertiaJp (Jz)
No data need to be introduced in matrix MATER
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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.
The data to be introduced in matrix ELEM are
Type
N1
N2
N3
N4Geometry
Material
Rotation index
6
number of the node at one end (it is immaterial which end)
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);
No rotation index other than 0 or 1 is possible forasymmetrical elements.
Data to be introduced in matrix GEOMColumn 1
Column 2
Column 3
Column 4
Column 5Column 6
Column 7
Column 8Column 9
Column 10
Column 11
number of the geometry (reference to matrix ELEM)
code of the element type (6) (reference to matrix ELEM)
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
number of the material (reference to matrix ELEM)
Young's modulus
Poisson's ratiodensity
loss factor
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Part 3: Elements28
28
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).
If two nodes are present, their co-ordinates must be the same.
The data to be introduced in matrix ELEM are
Type
N1
N2
N3N4
Geometry
Material
Rotation index
7
number of the node at one end
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.
0 if the element belongs to the stator;
1 if the element belongs to the rotor (spinning at the nominalspeed);
No rotation index other than 0 or 1 is possible for asymmetricalelements.
Data to be introduced in matrix GEOM
Column 1
Column 2
Column 3Column 4
Column 5
Column 6Column 7
Column 8
Column 9
number of the geometry (reference to matrix ELEM)
code of the element type (7) (reference to matrix ELEM)
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
number of the material (reference to matrix ELEM)
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
29
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.
If two nodes are present, their co-ordinates must be the same.
The data to be introduced in matrix ELEM are
Type
N1
N2
N3
N4Geometry
Material
Rotation index
8
number of the node at one end
number of the node at the other end (0 if grounded)
0
0number of the type of geometry (reference to matrix GEOM)
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.
0 if the element belongs to the stator;
1 if the element belongs to the rotor (spinning at the nominalspeed);
No rotation index other than 0 or 1 is possible forasymmetrical elements.
Data to be introduced in matrix GEOM
Column 1
Column 2
Column 3Column 4
Column 5
Column 6Column 7
Column 8
Column 9
number of the geometry (reference to matrix ELEM)
code of the element type (8) (reference to matrix ELEM)
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)
No data need to be introduced in matrix MATER
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Part 3: Elements30
30
3.9 Element 9: nonisotropic (asymmetrical) concentrated mass
Element 9 is a mass element with non-isotropic characteristics. The element has
only one node.
The data to be introduced in matrix ELEM are
Type
N1N2
N3
N4
Geometry
Material
Rotation index
9
number of the node0
0
0
number of the type of geometry (reference to matrix GEOM)
0
0 if the element belongs to the stator;1 if the element belongs to the rotor (spinning at the nominal
speed);
No rotation index other than 0 or 1 is possible for asymmetricalelements.
Data to be introduced in matrix GEOMColumn 1
Column 2Column 3Column 4
Column 5
Column 6
Column 7
number of the geometry (reference to matrix ELEM)
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)
No data need to be introduced in matrix MATER
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Part 3: Elements 31
31
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.
If two nodes are present, their co-ordinates must be the same.
The degrees of freedom connected with a nonlinear element must be defined asmaster degrees of freedom.
The data to be introduced in matrix ELEM are
Type
N1
N2N3
N4Geometry
Material
Rotation index
10
number of the node at one end
number of the node at the other end (0 if grounded)0
0number of the type of geometry (reference to matrix GEOM)
not needed (0)
not needed (0)
Data to be introduced in matrix GEOM
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
No data need to be introduced in matrix MATER
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Part 3: Elements32
32
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
( )
( )
Recommended