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USAAMRDL-TR- 75-39C
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DESIGN CRITERIA FOR ELASTOMERIC BEARINGS
Volum« III - Program User's Manual
Thiokol/Wosofch Division CO
* A Division of Thiokol Corporation i> Brigham City, Utah 84302
I© i < § March 1976
Final Raport
D D C rpprpnr -
MAY 26 »T6
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Approved for public release; distribution unlimited.
Proporod for
EUSTIS DIRECTORATE U. S. ARMY AIR MOBILITY RESEARCH AND DEVELOPMENT LABORATORY Fort Eustis, Vo. 23604
EUSTIS DIRECTORATE POSITION STATEMENT :
The data contained In this report are the results of an effort designed to improve the state of the art of elastomeric bearing design for helicopter rotor head applications. The products of this effort are a design manual and a computer program based on finite- element techniques. The results of this program are contained in the following four volumes:
Volume I - Final Report Volume II • Design Manual Volume III • Program User's Manual Volume IV - Programmer's Manual
Volume I contains the development and background information used in producing the design manual.
Volume II presents design considerations and procedures, bearing applications, methods of analysis, and techniques for predicting bearing performance.
Volumes III and IV contain the computer code and examples of problems showing sample inputs and outputs.
The products of this effort provide a good foundation for building a comprehensive manual and computer code for the design and analysis of elastomeric bearings for helicopter rotor head applications. It was recognized at the onset of this program that both the manual and the code would be first editions. The results of this effort were expected to define areas requiring further development. Further investigations coupled with feedback from users and/or evaluators are expected to provide material for upgrading the content and format of the manual and codes.
Mr. John Sobczak of the Military Operations Technology Division served as project engineer for this effort.
DISCLAIMERS
TIM finding* in this report art not to be comtrued at an official Department of the Army position unlen to designated by other authorised documents.
When Government drawings, specifications, or other data are used for any purpose other than in connection with a definitely related Government procurement operation, the United States Government thereby incurs no responsibility nor any obligation whatsoever; and the fact that the Government may have formulated, furnished, or in any way supplied the said drawings, specifications, or other data is not to be regarded by implication or otherwise as in any manner licensing the holder or any other person or corporction, or conveying any rights or permission, to manufacture, use, or sell any patented invention that may in any way be related thereto.
Trade names cited in this report do not constitute an official endorsement or approval of the use of such commercial hardware or software.
DISPOSITION INSTRUCTIONS
Destroy this report when no longer needed. Do not return it to the originator.
UNCLASSIFIED ■ CATION OF THIS PAGE (WhKl 0«( Knfnd)
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USAAMRDLffTR-75-39C
RT DOCUMENTATION PAGE 2. OOVT ACCESSION NO
RESIGN j^JllTpilA FOR |_LASTOMFRir gFARlNOS^ Volume XÖ.,-^Program User's Manual* / -
7. AUTNORf*;
READ INSTRUCTIONS BEFORE COMPLETING FORM
3. RECIPIENT'S CATALOG NUMBEN
». TVPI OF NCPORT » PERIOD COVERED
PROGRAM USER'S MANUAL
•■ PERFORMING ORO. REPORT NUMBER
B. CONTRACT OR GRANT NUMBERS
/a I. PERFORMING OROANIZATION NAME AND ADDRESS
Thiokol/Wasatch Division A Division of Thlokol Corporation BrIgham City, Utah 8*1302
tl. CONTROLLING OFFICE NAME AND ADDRESS
Eustls Directorate, U.S. Army Air Mobility Research and Development Laboratory Fort Eustls, Vlrqlnla 2360't
!«■ MONITORIN9 AGENCY NAME I ADDRESV" Mlnmt Inm Canlralllnt OHIe»)
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DAAJ^2-73-C-i)JB91 */* IV
10. PROGRAM ELEMENT. PROJECTTT
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UNCLASSIFIED IB«. DECLASSIFICATION/DOWNGRADING
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!•• SUPPLEMENTARY NOTE«
Volume III of a '(-volume report
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5
I». KEY WORDS (Conllnu* on rmvoroo oU» II noeofty and Httilllr by block ntmbor)
Elastomerlc Bearings Stability Helicopter Rotor Natural Rubber Finite Element Analysis
Service Life
ABSTRACT (Cmtllnu* on toiotto tldo II Htcmomr md Idtnillr »r »loc* numbor)
This document describes the usage and Input of a program that performs a stress analysis on an axlsymmetrlc body with Isotropie materials. The body may have asymmetric loads, in which case several passes through the program are required to obtain the complete solution. The accumulation routine will accumulate all the harmonics of the Fourier expansion. The proRram Is a compressible formulation; that Is, Poisson's ratio must be less
00 FORM I JAN 71 1473 COITION OF I NOV B* IS OBSOLETE UNCLASSIFIED V
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than .5, but It may be as close to It as desired.
An extensive Input module has been Included In the program to make It as user oriented as possible. This Includes routines that will automatically generate bearing geometry based on basic Input parameters.
The basic program Is written In FORTRAN IV with some support routines written In IBM 370 Assembler Languaue.
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TABLE OF CONTENTS
LIST OF ILLUSTRATIONS k METHOD 5 LOAD DEFINITIONS 7 PROGRAM COORDINATES 8 THE LINE GENERATOR 9 THE GRID GENERATOR 10 INPUTTING THE PROGRAM 11 TIMING 16 RESTRICTIONS 17 VARIABLE DIMENSIONS 18 FREFRM INPUT 21 FLAG AND TITLE RECORDS 23 INPUT FOR PROGRAM 2k
Title 2k General Data - The flag is GENE 2i» Run Parameters - The Flag is RUN 25 Output Options - The Flag is OUTP 26 Special Points - The Flag is SPEC 27 Nodes - The flag is NODE 27 Line Generators - The flag is LINE 28 Bearing Generator - The flag is BEAR 29 Grid Generator - The flag is GRID 32 Type of Material - The flag is TYPE 33 Boundary Conditions - The flag Is BOUN 3«» Pressures - The flag is PRES 35 Body Force Loads - The Flag is BODY 38 Temperature Distribution - The Flag is TEMP 38 Material Properties - The flag is MATE 39 Selective Print Option - The Flag is SELE kO Incremental Loading - The Flag is INCR 1*0 Large Deformation - The Flag Is LARG kl Geometry Plots - The Flag is GEOM k2 Accumulation Control - The Flag is ACCU i»3 Stability - The flag is STAB l»i» End of Harmonic - The flag is END kk
SAMPLE INPUT U5 SAMPLE PROBLEM 1 i»6 SAMPLE PROBLEM 2 i»8 SAMPLE OUTPUT 51 APPENDIXES
A. Service Life Program ••• 72 B. S3359F Fourier Coefficient Generator 73 C. 370 Pperatloo Instructions 77
LIST OF ILLUSTRATIONS
Fifiure
1 Bearing Example
2 Sliding Boundary Condition
3 Right-Handed Pressure Convention
k Left-Handed Pressure Convention
5 Sample Problem 1
6 Sample Problem 2
7 Deformed Sample 2
Ea&Si /
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HETHQP
The theoretical basis of the program and is not repeated in this manual.
is contained in Volume I
Variable dimensions of the grid have been used throughout so that any size problem can be run If sufficient computer size and time are available.
The solution scheme is a Gaussian reduction, answers with reasonably short run times.
This gives good
The program may be run as a standard axlsymmetric program giving either one or both of the axisymmetrlc or torsion solutions. Multiple passes may be made through the program Inputting the proper load coefficients to gain an asymmetric loading solution. The accumulation routine will accumulate the displacements and strains for all harmonics to give the final solution.
The program is a finite-element program. To solve for the displacements« stresses and strains throughout a body« it is divided into many small pieces called elements by defining a grid network of nodes over a cross section of the body. For each element« a stiffness matrix is defined according to the basic assumptions for that element. A force vector Is created by applying the pressures and loads to the nodes. The element stiffness matrices and force vectors are then assembled Into a master set of equations/ and the equations are solved using a Gaussian reduction technique. This method of solution has been extensively used at Thlokol for many years and has yielded good answers in reasonably short run times.
There are two basic elements available In this program/ and one or the other must be selected In the RUN Input section. The first Is the linear displacement element. The assumption for It is that the displacements are a linear function through the element. This element Is used for the basic axisymmetrlc runs, the asymmetric loading runs and the torsion runs.
The second element Is the isoparametric element. It is a quadratic element where the displacements are assumed to be a quadratic function through the element. This element will In general give better results for the same size or a smaller grid. It has not, however/ been as extensively used as the 11near element.
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In the Type of Material section, there are five element types given. Types I, 2, k and 5 are all linear elements and differ only In the number of degrees of freedom they can handle.
LOAD. DEFINITIONS
For the single-pass, axlsymmetric run, loads are Input directly as they apply to the entire body of revolution.
For asymmetric loading, a Fourier expansion of the form
M M f(e) ■ A0 ♦ L An cos n0 + Z R_ sin n0
n»! ' n»l n
must be generated, where N Is the number of terms needed to accurately represent the load. All loads are defined In terms of this function of theta. The loads are then Input with the constant term Ap In the first pass and either An or On being entered for the n+lst pass.
The assumption Is Implicit In the program that the loads are symmetric about the 0 - 180 plane through the body. Because of this symmetry, the radial and axial loads can go in only as an even function and the tangential loads only as an odd function. That Is, only the A terns can be used for R or Z loads, and only the B terms can be used for the e loads.
The units of the function f and the coefficients A and B must be In pounds per square Inch for pressure, or pounds for nodal point load, over the entire surface affected.
Pressure, shear and traction loads act on the entire surface of revolution on which they are applied. That is, the total load is equal to n DLP where D Is the diameter, L the lenr.th, and P the pressure or shear.
Nodal point forces apply a load directly to a point and are not distributed over any surface. They are a load on a ring as the loaded point is revolved about the axis.
See Appendix B for the description of a program to generate the Fourier coefficients.
PROGRAM CQQRPINATES
The normal orientation of the coordinate system Is with the R-axis horizontal and the Z-axis vertical. This is referred to here as a right-handed system; that is, R rotated into Z gives a vector out of the paper.
The Indexing system of the program can also be layed out as a coordinate system. 3y definition, a right-handed system is one in which the I axis rotated into the J axis gives a vector out of the papor.
The two systems, coordinate and Indexing, must always agree. That Is, they must both be either right-handed or left-handed. If one Is right and the other left, the program v/ill give negative area errors on the grid and stop execution.
All angles in the system are measured from the positive R axis counterclockwise in the right-handed system. If the system is transformed, this relative orientation must be maintained.
There are "local origins" referred to In the Input section. These are simply temporary origins to aid In the designating of that one node or line. The point will be translated back into the global coordinate system by adding the coordinates of the local origin to the coordinates of the point.
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Hü Uil£ GENERATOR
By means of the Line Generator Record, line and arc segments may be generated Internally to connect any* two points with a set of nodes. The following options are available:
A. Equal Interval - In the straight line all Intervals between nodes are equal. In an arc the angular Intervals are equal.
B. Square root of r - Where N Is the total number of intervals and I goes from 1 to fJ-1, each Interval Is (2i-l)/N2 of the total length. Fora line, this is an Increment of length; for an arc. It is an angular interval.
C. Geometric progression - The increments of length or angle will be in a geometric progression. Any of r (the common ratio), a (the first Interval), or I (the last Interval) may be specified. In addition, the second node in the line may be specified and r will be calculated using the interval from the first node to the second node of the line ns a.
Any of these options may be used to generate any line in the figure. The lines may go in either ascending or descending order of I or J.
«Either I or J must be constant for all the nodes in a line or arc.
Uli QMH GENERATOR
The grid generator used in this program is one developed at Thiokol by Dr. William Cook.* This Is a linear Interpolation method which yields a good grid under most conditions regardless of the curvature of the boundaries.
A two-dimensional space Is defined In £ and q, and the grid boundary nodes are mapped Into this space. A linear Interpolation scheme Is then used to fill In the remainder of the £ / ^ space. Two functions, f and g, are then defined that map the ?, T\ space Into the X^Y space of the grid. These functions are then used to define the remainder of the grid.
There may be any number of major partitions In the grid and any number of subpartitions within each of these.
A major partition Is defined as a closed rectangular section of matrix (the figure may be of any shape, but the section of matrix must be capable of being defined by a minimum and maximum of I and J), all of which will be generated by one and only one grid generation record. The subpartitions are closed rectangular sections of matrix lying within the major partition.
All boundaries of each partition must be completely specified, and there must be no nodes specified that do not lie on the boundaries of a partition.
Each major partition must be specified on a Grid Generator Record, but the program will pick up all subpartitions lying within the specified major partition.
Due to the nature of the Cook grid generator, the following restrictions must be observed when using It:
1. All sections of the matrix that are to be generated by It must be rectangular and the boundaries must be completely defined.
2. If Interior nodes are specified, they must form rectangular subsections of the section of matrix being generated. There Is no limit to the number of such subsections allowed.
*Cook, William A., BODY ORIENTED (NATURAL) CO-ORDIMATES FOR GENERATING THREE-DlMEriSIOMAL MESHES, international Journal for Numerical Methods in Engineering, Vol 3, 197U, PP. 27-U3.
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INPUTTING THE PROGRAM
There are several things about the program that you should be aware of as you set up your Input. The sections below define both general things to keep In mind and the specific methods of Input required for the various options. For additional information, see Vol II of this report.
i General
The RUN section must always be used to specify both the geometry type and the element type. in this version of the program, the geometry type Is alv/ays ax'symmetric (option 3). The element type can be either linear displacement (option k) or Isoparametric (option 5).
Each node defines an element, except for Isoparameterlc, where it is every other node. The elements are referenced by the node with the smallest Indices of the nodes comprising the element. Those nodes v/hich lie on the boundary of the geometry and do not reference a real element are given a type code of 9, which signifies a missing element. This type code of 9 is also used to code an internal element which has no material In It. Type 9 elements require no material code, and this can be set to 0 or left blank.
Plots can be produced quite simply by specifying the orientation desired, the range of Indices of the nodes to be plotted, and the Y paper size. The data will then be scaled to fit the paper height, and the same scale factor will be used for the X direction. The actual scale values used, the minimum values and the maximum values, if calculated, will be printed when each plot is produced.
Several Input sections use index Increments referred to as II and Jl . The value 11 Is an I Index Increment, that Is, we go from II to 12 In Increments of II. The value Jl Is a J Index Increment, and we go from Jl to J2 In Increments of Jl. For example. If we have a grid where ve have material 1 In the elements with an even J Index and material 2 In the elements with an odd J Index, we can apply these In only 2 records by using a Jl of 2. Another example of its use Is In the Isoparametric Input. This Is explained more fully in section VII below.
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I I Ib£ ßflslc Axisvmmetric Problem
This is the case for which there are no iterations and the basic linear displacement elenent is being used. The RUN section must show this by inputting options 3 and i».
I I I Asymmetric Loadinti
The asymmetric loading of an axisymmetric body is accomplished by making multiple passes through the program v/ithin one run. Each pass has the harmonic coefficients for one term in the Fourier expansion entered as load conditions. Run option 9/ asymmetric loading, must be specified in each pass.
The node number, Lll in the General Data, and the highest harmonic allowed, L12 in the General Data, must both be specified in each pass, and the highest harmonic allowed must be nonzero.
The accumulation of the output is necessary for proper interpretation of the results. Fach pass will ilive results that are the maximum for each coordinate direction. These values have meaning only when multiplied by the proper sine and cosine terms nnd added to the other terms for that node or element.
The undeformed geometry can be plotted at any time In the run. Accumulation and plotting of the deformed geometry must be In the last pass only. Plots can be made of the displaced geometry at any angle 0 or axial location 1 by accumulating the data where desired and then Inputting a plot section immediately after. The plot routines will always plot the last data accumulated. See the sample input for an example of this.
IV Incremental LflfldllLS
When this option Is used, the program automatically cycles through as many ttnos as there are load Increments. The cumulative totals of the displacements and strains may be printed out for each Increment under the control of the Selective print option. The print suppress option (output option 1) and the Selective print section control the output of the final results, and the Incremental loading print flag with the Selective print option controls all others.
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All loads and boundary conditions may be increments. The fraction of the load to in each step may be specified, or on increments will be calculated in an progression.
applied in be applied a flajj the ari thmetic
A set of flags is provided so that specified loads may not be applied incrementally.
Each iteration is handled essentially as a separate run with the program looping through the entire solution scheme. The displaced geometry reflecting all the previous loading is used in each iteration so that stiffness due to bending will be taken into cons ideration.
LaLM Deformation
the The use of this option takes into consideration nonlinear terms In the strain displacement relationship and iterates for the final solution. You can also enter strain dependent material properties to take Into consideration the nonlinear material behavior.
A Convergence criteria and a series of underrelaxation factors must be entered. The underrelaxation factors must be positive values no greater than 1.0, and at least the last two values must be 1.0, The program will iterate until convergence or the maximum number of iterations is sat Isfled.
As in the incremental loading option, the print suppress flag controls the final output and the large deformation print flag controls all others.
The energy calculation, reaction check are not valid for this allowed.
forces and accuracy option and are not
vi Incremental Load I M aM Larzs. Deformation
These two options may be combined to allow a large deformation iteration Inside each incremental loading loop. The controls on each remain the same, and the same set of underrelaxation factors will be used for each increment of load.
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VI I Isoparametric Elements
The Isoparametric element in this program Is a quadratic element v/lth three nodes on each side. The geometry Is Input and generated just as If a linear displacement element were being used. Ilote, however, that all element boundaries and material boundaries must fall on nodes with odd Indices.
The Internal handling of this element requires some special consideration in the input. The element type codes and material numbers must only be on the nodes which designate element corners. To achieve this, 11 and Jl In the TYPE section must be Input as 2^.
VIM Torsion
The torsion solution is part of the zero harmonic In the asymmetric loading case, because It is the only clement at present that has a theta degree of freedom. A RUN parameter of 9 must then be sped fled.
There are to different ways that torsion can be run: (1) with the full asymmetric calculation j»ivin<: all three degrees of freedom, and using a type k element: (2) with torsion only. In which case a type 5 element will be used. The type 5 element Is faster than the type i*. Accumulation can be made for both elemonts, or for any combination of 2, h or 5 elements, since they are all asymmetric. Note that the torsion loads act on a moment arm equal to the radial coordinate of the point where they are applied.
IX Stabl1 Itv
This option Is used to determine the stability of a flat, vertically stacked bearing. The only Input sections required are the General Input, the Run Input, and the Stability Input. The values In the General section will not be used, but due to program flow the section must be Included. Run option 20 must be specified.
X Service Life
To determine the service life of a bearing, this program must be used In conjunction with program S3359SL. When a run Is made where data needs to be saved for the service life calculations, output option 20 must be specified. The stresses and
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strains wl SERVLIFE. identified output: th is the ti used to ide These sets they ran be proper Job
11 then be output on a file whose name is Each set of data put on this file will be by two integers which are printed on the
e first is the Julian date and the second me of day in centiseconds. These will be ntify the set of data for prosram S3359SL.
can be put on separate reels of tape, or stacked together on one tape by using the
Control Language.
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TIMING
Timing on this program Is a problem because so many factors affect the execution speed.
In general the time required for a run is a function of I2 and J.
The asymmetric loading problem requires about tv/o times as long for each pass as a simple axisymmetric problem.
Each iteration on iterative runs must be counted as a full execution.
From our experience/ the best way to estimate the time Is by experience with the guidelines given above. However, the following equation will give a starting point:
Time (In minutes) ■ F lz J ♦ 1.2
where F ■ .0012 for an asymmetric run and .0006 for an axisymmetric run, I Is the l-dlmenslon of the grid, and J Is the J-dlmenslon of the r.rld. This equation Is for the IBM 370/155; the factor F and the constant 1.2 may change for other machines.
16
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RFSTRICTIOHS
1. The Isotropie Poisson's ratio is restricted to 0 < v < .5. At least under some circumstances a zero Poisson's ratio will make the stiffness matrix singular.
2. If a node Is on the axis of revolution of an axlsymmetrically loaded body« the radial displacement of that node must be fixed at zero.
3. At least one axial component of displacement must be specified to avoid rigid-body axial motion. A sliding boundary condition may also fulfill this requirement if it restrains axial motion.
k.
5.
6.
In any through Type 9 "voids"
row of the each value elements may
in the grid.
grid, I must run consecutively IM IN i II I MAX for that row. be used where needed to provide
In the input, a maximum of 9 digits Is allowed on each side of the decimal point In a number.
There Is a maximum of 32/767 materials allowed in the program. However/ due to the special internal handling/ the number of materials isoparametric elements is limited to 50.
used with
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VARIAßLE PIMENSIQNS
To facilitate running of both lar^e and small problems, the working arrays in the program have been variably dimensioned. Two things should be noted here about the program. First, the program will take as much core storage as It is given; that is, on the step resource usage given at the end of the run, the amount of core used and requested will always be the same. Second, the actual amount of working storage used in each phase will be printed out as part of the output.
The table on the following pages gives the maximum core required for a run. There are many factors which affect the core required for any given run, so the first guess should be taken from the table and then adjusted after a complete run has been made. The excess core available for each part of the program is given with the output.
To find the region required for a run, a look in the table for the I and J size of the grid will give the region in K bytes. If an asymmetric loading case, or an isoparametric case is not being run, the region may be reduced by up to one-third.
For a stability calculation, only 190K will be needed for the run.
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19
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20
FREFRM INPUT
Input to the prosram is via subroutine FREFRM. This Is a record-oriented free-form Input routine. It allows data to be entered without regard to card columns and yet be Input as individual unique records.
This routine has the following characteristics:
1. Numbers are not restricted to any particular columns on the card.
2. The numbers may be separated with any of the followins:
a. A comma is a separator.
b. The sign of the number is a separator.
c. The single quotation mark (') v/hen defining an alpha string Is also a separator.
d. The end of a card (column 72) Is a separator; that is, if no other separator Is encountered before the end of a card, that end will terminate the number. Note that an end of card In an alpha string Is an error,
e. The L of an L-number Is a separator.
f. A record terminator (;) will terminate the number as well as the record. A record terminator In an alpha string Is an error. A string of dissimilar separators Is only one separator. For example, a comma followed by an end of card, followed by an L or quote, is still only one separator.
3. L-Numbers may be used to specify the relative location of the value In the record. An L-Number Is an Integer preceded by an L which sets the counter to a specific location In the record, the first location being 1. Thus the third location is given by L3. The L-Number must be senarated from the following value by a valid separator. The number must always follow the L immediately. The presence of anything other than a digit Immediately following the L will
21
/
cause an error.
k» A logical record must be terminated by a semicolon
5. Alphanumeric data can be entered anywhere in the string but must be set off with single quotation marks ('). The quotation mark is a separator. Two consecutive quotation marks will cause a quotation mark to be entered in the string. There are 8 characters in each location as defined by the L-Mumbers.
6. If two commas appear consecutively/ no value is input between them. The value previously put in that location will remain there.
7. If two signs appear consecutively/ a zero is assumed between them.
8. If a record begins with an alpha character/ it will be assumed to be a flag record or a title/ and no other data may be entered In It. These records may begin with 'L'/ which usually signifies an L-Mumber/ if and only if the second character in the string Is alphabetic.
9. Comments may asterisk (*) asterisk and be ignored.
be punched in any card followed by any desired
all data following it in
by punch I nj; an comments. The that card wi11
10. Only the first 72 columns of each card will be used, so columns 73-80 nay be used for sequence numbers If des I red.
11. Data entered in a location in a record location in subsequent records until It entering a new value in that location.
stays in that i s changed by
12. A maximum of J Hi decimal point in a
{Its is allowed on each side of the number.
22
^\
^•s»»*^Tw^*WW*W(5!'|SMfiW^ffiH
FLAG AND TITLE RECORDS
Any record which begins with alphabetic data is considered to be a flag or title record. Flag records are used to flag the beginning of each group of records. If the first four columns of the record contain a flag sequence^ the record is considered to be a flag record. If not, it is considered to be a title record and its use is determined from its location in the input stream. The record terminator must not appear in the string. A title record may span several cards, but each flag or title record must end with a record terminator. A flag record cannot exceed a single card.
In the following pages, only four characters are given for each flag since this is all that is checked. Additional characters may be added to make a full statement if desired.
23
INPUT FOR PROGRAM
As many of the following groups as needed may be entered. Some of the groups are required; others are optional.
The Title records and General Data must be the first two groups Input. All other groups may be Input In any order, with the only restriction being that of logical sequence of events. Obviously you can not use an Item of data until It Is Input; therefore. Line Generators must follow the Node records that define their end points, etc. In general, any group may be Input more than once; however, th?s practice should be restricted to avoid overly complex Input sets.
Each case run through the program Is considered to be a separate case, and all Information must be entered for each case. This Is also true of the various parts of an asymmetric loading run. Except for the accumulation, each case Is considered to be completely separate.
Note that only columns 1 through 72 may be used for data.
Title
As many cards as desired may be entered horn, with the last card containing a record terminator. Each title card must start with an alphabetic character.
General QalA ' The flag Is GENE
LI Minimum I value
L2 Minimum J value
L3 Maximum I value
Ik Maximum J val ue
L3-L9 Not used at present
L10 Base temperature
Lll Mode number If asymmetric loading
L12 Highest mode allowed If asymmetric loading
2U
Ma Parameters - The Flag Is RUN
Run options will be selected from the following table by entering the option numbers as a sequence of Integers. The blank options represent options not available In this version of the program.
1.
2.
3.
I».
5.
6.
7.
8.
9.
10.
11.
12.
13.
111.
15.
16.
17.
18.
19.
20.
Axlsymmetric geometry
Linear displacement element
IsoparameterIc element
Asymmetric loading of an axlsymmetrlc geometry
Mesh only - check Input and test the grid. Geometry plots v/lll be produced If requested.
StabilIty calculation
25
■
Output Potions - The Flag Is OUTP
The options are selected by entering the option numbers from the following table. These options apply to the entire output set; If output Is desired for specified sets of nodes or elements only, then use the Selective Print option.
If no output options are specif led, the output will be the displacements for each node and the stresses and strains for each element.
1. Print Suppress. The displacements and stresses will not print except as specified by the Selective Print option.
2. Punch the deformed grid.
3. Print the deformed grid.
1*.
5.
6.
7.
8. Accuracy check. Calculate (F-KU) for each element and print the 10 elements with the highest error.
9. Reaction forces. Print the reaction forces for the nodes with displacement boundary conditions.
10.
11.
12.
13.
1U. Print the strain energy.
15. Print the element material properties.
16. Print the loads for each element with nonzero loads and the boundary conditions for each node with nonzero boundary conditions.
26
. A
17.
18.
19.
20. Output data on tape for service life calculation.
Special Points - The Flag Is SPEC
This section allows for a maximum of 100 special points that are not part of the grid. Their purpose Is to provide convenient points to be used as local origins for nodes, arcs, etc.
LI K The special point number. It will be referenced by this number whenever It is used.
L2 X or R coordinate
L3 ¥,6 or Z coordinate
Lk If nonzero the values In 12 and L3 are considered to be R and 0, in decrees. If zero, they are considered to be X and Y.
L5 The number of a previously input special point may be Input here, and It will be used as a local origin for point K.
Modes - The flag is NODE
Each of these records defines one geometry point in the grid.
LI 1
L2 J
L3 R(I,J)
LU Z( 1,J) or e in degrees
L5 Spherical flag - In input instead of R and Z.
27
L6 Local origin flag
0 - no local origin
1 - special point whose number Is In L7 Is the local origin
2 - the node whose Indices are given In L7 and L8 will be used as the local origin
3 - the coordinates of the local origin are given In L7 and L8
L7 K, IL or RL depending on L6
L8 JL or ZL depending on LC
Line Generators - The flag is LIME
Each record generates a sequence of node points between the existing end points. Either 11 - 12 or Jl * J2 must hold.
LI 11
L2 Jl
L3 12
Li* J2
:] Beginning node of the line.
j Ending node of the line.
L3 Rotation code
♦1 If rotating from the positive R to the posi 11vc Z axls.
-1 If rotating from the positive Z to the posi 11ve R axls.
L6 Nodal spacing option code
•0' equal interval option
'1' the square root of r option
^ geometric progression with "r* specified
28
SS.
'S* geometric progression with 'a' specified
' I»' geometric progression with ' l1
specified
'S' geometric progression with the second node In the line defined previously and used to define 'a'
L7 This field will contain a, r or Z depending on which option was specified In L6.
L3 GUIN -The smallest Interval allowed for a geometric progression.
L9 GMAX - The largest Interval allowed for a geometric progression.
L10 If the sequence of nodes Is to fo.-"i an arc, the way In which the arc center Is given Is specified here.
0 - not arc - generate a straight line
1 - the special point whose number Is given In Lll Is the arc center
2 - the node whose Indices are given In Lll and L12 Is the arc center.
3 - the center of the arc Is at the coordinates given In Lll and L12
Lll K, IC or RC depending on L10
L12 JC or ZC depending on L10
Bearinfi Generator - The flag is BEAR
This section contains parameters necessary to generate a spherical or conical bearing model only.
The node points will be generated and element type and material codes will be set as specified for the elements generated. Type of Material records do not need to be Input for the elements of the bearing.
29
See Figure 1 for the relative location of the indices.
LI The minimum I value In the bearinn. This may be any value. Default of 1. IMMB
L2 The maximum I value in the bearing IMXB
L3 The minimum J value in the bearing. Default of 1. JMMB
LU-6 Not used at present.
L7 IEQU If this is nonzero the nodes will be equally spaced across the bearing In the I direction Instead of having the first and last interval half the others.
L8 Element type - see Type of Material section for valid types.
I L9 +0 if a spherical bearing.
♦1 if a conical bearing.
L10 The inside radius of the bearing for spherical bearings only.
Lll Radial coordinate for node IMMB, JI1NB.
L12 Radial coordinate for node IMIIB, JMXB*.
L13 Radial coordinate for node IMXB, JMIIB.
Lll» Radial coordinate for node IMXB, JMXB*.
L15-18 Contain the axial coordinates of the corner nodes. For a spherical bearing their only function is to place the node In the proper quadrant; they need not be accurate since they will be recalculated to be sure that the node lies on the proper circle. For a conical bearing they wl11 be used as input.
L15 Axial coordinate for node IMflB, JMNB.
»JMXB Is JMAX for the bearing and Is Internally calculated.
L16 Axial coordinate for node IMNB, JMXB*.
L17 Axial coordinate for node IMXB, JMNB.
L18 Axial coordinate for node IMXB, JMXB*.
IMAX#,IMAX
"•|MIH#Jf'AX
Figure 1. Bearing Example
The complete geometry goes from IMIN#JMIN to IMAX^JMAX. Parts A and C are attachment appliances on the ends of the bearing B. The bearing is defined between indices IMMB/ JMNB and IMXB, JMXB.
L19 TS thickness of the shim.
L20 NS number of columns in one shim.
L21 ISN number of shims.
L22 TE thickness of the elastomer.
L23 NE number of columns in one elastomer.
L2U I EN number of elastomers.
L25-29 Not used at present.
«JMXB is JMAX for the bearing and is internally calculated.
31
•A
L30
L31
L32-U9
L5Ü-99
L100-1U9
L150-199
L200-2U9
MS material number for shim.
Default of 1
ME material number for elastomer.
Default of 2
Mot used at present.
**SHIMT shim thickness for each of the shims If not constant. Any value left blank or zero will be replaced by the constant value In L19.
**ELAST elastomer thickness for each elastomer If not constant. Any value left blank or zero will be replaced by the constant value In L22.
**MATS material number for the shims If not constant. Any value left blank or zero will be replaced by the constant value In L30.
**MATE material number for the elastomers If not constant. Any value left blank or zero will be replaced by the constant value In L31.
Grid Generator - The flag Is GRID
LI 11
L2
L3
U
dl
12
J2
These specify the reclon to be generated. The generator used Is the Cook grid generator, and the region specified must be a closed rectangular system In the I, J plane.
♦♦Starting with JMNB and proceeding In order out to JMXB
32
Tvoe af Material - The flag Is TYPE
Each of these records assigns a material number and an element type to a block of elements. A material property table must be input for each number assigned here. Note that these are element indices.
LI II
L2 Jl
L3 12
LI» J2
L5 Element type
The valid element types are:
1 Axisymmetric with R and Z degrees of freedom
2 Axi symmetric v/ith R# 6 and Z degrees of freedom. Used for asymmetric loading.
3 Isoparametric quadratic element with three nodes per side and R and Z degrees of freedom.
k Element to be used on mode 0 of an asymmetric loading run when theta loads are used. Torsion in connection with radial and axial loads.
5 Element to be used on mode 0 of an asymmetric loadin;; run when torsion Is the only load. This element Is much faster then type h.
9 Hull or void element. This type must be assigned to the last node In each row and column. It may also be used to define holes In the material. It is automatically assigned to the nodes on I MAX and JMAX.
L6 Material number
L7 II - If Input, I will go from II to 12 In Increments of 11
L8 Jl - If Input, J will go from Jl to J2 In Increments of Jl
33
s-\
Boundary Conditions - The flag is BOUN
Each record assigns boundary conditions to a set of nodes.
LI II
L2 Jl
L3 12
U J2
L5 R boundary condition code*
LG R boundary condition value or harmonic coefficient
L7 0 boundary condition code*
L8 e boundary condition value or harmonic coefficient
L9 Z boundary condition code*
L10 Z boundary condition value or harmonic coefficient
Lll-12 not used at present
L13 Sliding boundary condition code, 0 or 1 only.
LIU Angle in degrees of the line the nodo must slide on. See Figure 2.
L15 II if input, I will go from II to 12 in Increments of I I
L1G Jl if input, J will go from Jl to J2 in increments of Jl
*The valid codes are: ü - no boundary condition 1 - displacement boundary condition 2 - A nodal force is to be applied. Modal forces are the
forces on the entire circumference.
3U
> z
Figure 2. Sliding Boundary Condition
Pressures - The flag is PRES
Each record puts pressure along a line of nodes. P is normal to the element surface and S is parallel to it. The values PR/ PTH and PZ are traction loads in the coordinate system and will decompose into pressure and shear components internally.
LI II
L2 Jl
L3 12
Ll» J2
L5 Di Direction code:
0 if the pressure is to be applied to the elements to the left of the line moving from point 11/ Jl to point 12/ J2.
1 if the pressure is to be applied to the elements on the right.
See Figures 3 and i».
Note; This code is correct for a right-handed indexing system. For a left-handed indexing system/ the codes must be interchanged. That Is, 0 Is pressure to the right and 1 to the left.
is
L6 PI
L7 P2
L8 SI
L9 S2
L10 PR1
Lll PR2
L12 PTH1
L13 PTH2
Lid PZ1
L15 PZ2
Pressure at node II, Jl
Pressure at node 12, J2
Shear at node II, Jl
Shear at node 12, J2
Radial traction at node II, Jl
Radial traction at node 12, J2
Theta traction at node II, Jl
Theta traction at node 12, J2
Axial traction at node II, Jl
Axial traction at node 12, J2
36
z,J Z,J
12^2
ll'J1 Code - 1 ■M
Figure 3. Right-handed Pressure Convention
RJ R,l
I2,J2
Z,J ll.Jl Code ■ 0
Z,J
Figure I». Left-handed Pressure Convention
57
• A
fijUly Force Loads - The Flag Is BODY
LI
L2
L3
Ik
IS
L6
L7
II
Jl
12
J2
Radial (pw2)
body force harmonic coefficient
Theta body force harmonic coefficient (F)
Axial body force harmonic coefficient (F/L8)
Temperature Distribution - The Flag Is TEMP
There are two ways in which a temperature distribution may be applied in this program. Only one form may be used In any one run.
A. Constant temperature over the whole body.
The first record must be the word CONSTANT enclosed In single quotation marks. This record must be followed by one record of the following form:
LI The constant temperature
3. Radius vs temperature tables
The first record must be the character string R vs T enclosed In single quotation marks. This record will be followed by as many records of the following form as necessary to completely define the temperature of each element:
These are node point
Indices which define
the area over which
LI 11
L2 Jl
L3 12
33
•
u L5
L6
L7
La
J2
Rl
Tl
R2
T2
J thls table applies.
Temperature at radius Rl
Temperature at radius R2
Material Properties - The flag is MATE
This program accepts isotropic materials only.
For nonstrain or temperature-dependent materials/ only one record is entered for each material.
For materials that are strain or temperature dependent, several records will be entered for each material. One record must be entered for each entry in the table. The material number and name need not be entered on each record of the table. If the table is strain dependent/ the strain value in the first record must be nonzero and the table must be so input that a zero strain value can be found. If the table is temperature dependent/ the temperature value in the first record must be nonzero.
LI
L2
L3
Ik
IS
16
L7-9
Material number
Young's modulus (E) or large deformation Kl*
Poisson's ratio (v) or large deformation K2*
Alpha for thermal loading
Temperature for which these material properties apply If they are temperature dependent.
Strain at which these material properties apply If they are strain dependent (large deformation).
A 2U-character material name enclosed In quotation marks.
I «See Vol II for definition.
39
x^V
NOTE: Due to special Internal handllnc of material properties, da not use L-Numbers in the input. Always specify missing parameters by commas.
Selective Print Option - The Flag is SELE
This section may be used to ask for output of displacements/ stresses and material properties if the overall output of those I terns has been suppressed in the Output section. As many records ns desired may be entered.
LI II
L2 Jl
L3 12
Ik J2
Incremental Loading - The Flag is I NCR
This section will cause the program to iterate, applying a portion of the load each tine and accumulating the results.
LI The number of increments (M) over which the loads will be applied. If this is positive, then the second record for this set must be entered with N fractions on It. If negative, the loads will be applied In an arithmetic progression.
L2 If a nonzero value is input, printout of material properties, displacements and stress/strains will be made each Iteration for those nodes specified by the selective print. The complete output will be printed on the final increment unless the print suppress flag, output option 1, is turned on.
The following set of flags tells which loads will be applied incrementally. If the flag is zero, the load will be applied in increments; if nonzero the full load. If any, will be applied In each Iteration.
L3 Pressure and traction loads PR, PTH and PZ
/
1*0
Li»
L5
L6
L7
L8
L9
L10
Shear
Temperature
R body force
Z body force
R nodal load
Z nodal load
Nodal displacements
Incremental loading fraction record
This record must be entered If N Is positive and may be entered to override calculated values If N Is negative.
LI Fraction for 1st Increment
L2 Fraction for 2nd Increment
etc.
for N (max. of 100) Increments
Large Deformation - The Flag Is LARG
This section controls the large deformation iteration. One record of the following form must be entered.
LI
L2
L3
LU-103
The number of the last Iteration to be allowed.
If nonzero, printout of displacements will be made each Iteration for those nodes specified by the Selective print option.
Convergence factor. Convergence is atcained when the absolute value of the largest change in displacements Is less than this value.
The underrelaxation factors for Iterations 1-99. The default value Is 1.0.
kl
-■
Geometry Plots - The Flag Is GEOM
This section Is used to request the plotting of the geometry, either deformed or undeformed. Plots for which the displacement multiplier Is zero will be produced during the Input phase at the time the plot section is encountered. It Is possible to plot the geometry at any point in its development by putting geometry plot sections In at the desired points, such as after the node records/ etc.
Note that data will not carry over from one record to the next in this section.
LI
L2 11
L3 Jl
LI» 12
L5 J2
L6 Th
Axis parameter option
1 If the R-axis is to be horizontal to the right
2 if the Z-axls is to be horizontal to the right
These Indices define the region of
the geometry to be plotted.
L7
The displacement multiplier. If zero, the original geometry will be plotted. If nonzero, the displacements will be multiplied by this before being added to the node coordinates for plotting.
X-axis scale factor*
*The X and geometry. maximum an not specif size need /(Max - Ml factor wl1 are not In the paper, entered, t
Y axes here refer to the paper rather than the Not all of these parameters need be entered. The
d minimum values will be obtained from the data if led. Also, both the scale factors and the paper not be specified. The relation Scale ■ paper size n) will be used to find any missing data. The scale 1 always be used if entered. If the scale factors put, they will be calculated to keep the plot on
If the X paper size and scale factor are not he Y scale factor will be used for both axes.
l»2
•-v
L8 Y-axis scale factor*
L9 Minimum value on the X-axis*
L10 Maximum value on the X-axis*
Lll Minimum value on the Y-axis*
L12 Maximum value on the Y-axis*
L13 X-axis paper size*
LIU Y-axis paper size*
L15 The X location in inches of the beginning of the title.
L16 The Y location in Inches of the beginning of the title.
L17 The letter size for the title In inches.
L18-19 Not used at present
L20-25 Up to 48 characters of plot title information as an alpha string enclosed in quotation marks.
Accumulation Control - The Flag is ACCU
This section must be entered only on the last case of an asymmetric loading run. Under control of these records the displacements and strains will be read In for each mode and accumulated. The accumulated output will be printed. It may also be plotted by putting a geometry plot request after the desired displacements have been calculated; thus there may be several Accumulation and Plot sections alternating In the input.
L
*The X and Y axes here refer to the paper rather than the geometry. Not all of these parameters need be entered. The maximum and minimum values will be obtained from the data If not specified. Also^ both the scale factors and the paper size need not be specified. The relation Scale ■ paper size /(Max - Min) will be used to find any missing data. The scale factor will always be used if entered, if the scale factors are not input/ they will be calculated to keep the plot on the paper, if the X paper size and scale factor are not entered, the Y scale factor will be used for both axes.
U3
Note that to plot a transverse cut, t goes from 1 to the maximum number of nodes In the Radial direction and J goes from 1 to 19.
LI Option Flag
1 Output will be an axial cut at angled.
2 Output will be a transverse cut at the axial distance Z.
L2 e or Z depending on LI.
Stability - The flag is STAB
This section is used to input the data for the stability calculation. Run option 20 must be specified for the calculations to be made. The stack is assumed to be a vertical stack beginning and ending with a rubber pad; hence the number of shims is assumed to be one less than the number of rubber pads.
LI The thickness of each rubber pad.
L2 The thickness of each shim.
L3 The Inside diameter of the stack If it is cylIndrlcal.
Li» The outside diameter.
L5 The number of rubber pads.
L6 The Young's modulus for the rubber pads.
L7 The convergence accuracy factor. When (1 - old buckling load/new buckling load) is less than this value, the Iteration has converged.
L8 The maximum number of Iterations to be allowed.
End of HarmonIc - The flag Is END
An end record must terminate the Input for each portion of the run. This signals the program to stop reading and solve the problem as defined.
iti*
SAMPLE INPUT
Two cases are shown here. They are Illustrations only and do not necessarily reflect real problems.
Sample case 1 Is a flat bearing. It Is set up In a right-handed system. Figure 5 Is a plot of the geometry, and arrows have been drawn In to show the R, Z, I and J axes. This Is purely a bearing with no additional parts added. The output for this case Is not given In this document since Its form Is similar to that of case 2.
Sample case 2 Is a spherical bearing with a steel attachment on one end. It Is oriented In a left-handed system. Figure 6 Is a plot of the geometry, and arrows have been drawn In to show the axes. Note how the input sections are ordered to achieve the desired results. The output from this case Is given here to show Its form.
kS
■
SAMPLE PROBLEM 1
SlfPLE PROBLEM 1 ; GENERAL CAT« ;
1» 1* 6» Ik i RUN i
2» * ; OUTPUT :
6* 9» 14 : BEARING ;
1> 6* it L8# 1» U» Cß C« 3.C* 3.0» 0* 2.0* 0# 2.C» .55» 2» 2» .3» 3» 3 t
BCLNOARY CONO S 1» i» i» 14» i» o ; 1» 1» 6» i» o* G» o» c» i» o ;
BOCY FORCE ; i» i» 6» i» o» c» ic. ; i» t» 6» 9 : 1» ii* 6» i« ; 1» 4» 6» 6» 0» 0* 30. ; 1» 9» 6* ii ;
MATERIALS ; 1» 1.0E6» .3»»»» • STEEL SHIM* ; 2» 3C0.» .*<;<;<;9**** «sueef« P/C ;
GEOMETRY PLOT ; 1» 1» 1* 6» U» 0« 3.» 3.» -.333»» -.333*»*» l.C* <3.5» .12*
L20 *SAf«PLE 1* .* 1» 1* 1* 6» l«i* 5.0* 4.» 4.« C»* C»*»» 1.0* 8.5* .12* L20» 'OEFORMEO SAMPLE 1* :
ENC ;
dB
E 0)
43 0 i. a.
c E IC V)
0) L. 3
UJ _l a. a: CO
M
SAMPLE PROBLEM 2
SAPPLE PR061EP 2* ZEftCTH HAKHCNIC i GENERAL CATA ;
1» l» b» It» III» C» l ; RLK S
3» 4* 92 NCCES ;
i» i» o.o» 9.o ; 6» 1« 3.0» 9.0 ;
BEARING ; 1» 6» 3* Ld* 2» 0» ICC» C» C» 3.0» 3.C» 1G.C» 12.C* 9.0» 11.0»
.3» 2» 2* .24» 3» 3; LIKES :
1» l» fc» l J 1* 1» 1» 3 J 6» 1» 6» 3 ;
GRIO : 1» i» 6» 3 ;
TYPE ; l» 1* 9» 2» 2» I :
BCLNOARIES S i» 2» i» it» i» o ; i» i» 6» i» i» o» i»c» i» o ;
PRESSURE ; 1» 16» 6» lö» I» 50.» tO, :
MATERIALS ; I» 1.CE6» .3»*»» * STEEL • ; 2» 300.» .49S9<J,,,» •RLaecK« ;
GECHETRY PLCT .* 2» i» l» 6» 16» C» 2.0» 2.C» d.C»» -l.C»»#» 2.C» d.3» .42 »
L2C» • SACPLE 2 INCtFCkMtC»; EftC OF HARHCNIC ;
SAMPLE PROBLEM 2» FIRST MRKNIC : GENERAL DATA ;
1» 1» 6* 16» LU» 1» 1 : RtN ;
3» %» 9; NCCES ;
i» i» o.o» 4.c : 6» 1» 3.0» 9.0 :
BEARINC ; 1» 6» 3» LB» 2» 0» ICC» C» C» 3.C» 3.0» 10.C» 12.C» 9.0» 11.0»
.3« 2» 2» .24» 3# 3; LINES ;
i» i» 6» i ; I» 1» 1» 3 I 6» l» 6» 3 ;
<»8
«mo i 1» 1* 6« 3 3
TYPE ; 1« 1« 9* 2» 2» i ;
BCUNOARIES i I» I» 6* 1» I» 0* I'O» 1* 0 3
PRESSURE 3 6» 1« 6» 16» 0* 100.» 100.;
MATERIALS ; I» 1.0E6» .3*«»* * STEEL • 3 2» 300.» •499««»»»» «RLBBER* i
ACCUNULATION ; 1» 0 I
GEOMETRY PLOT I 2» 1» 1» 6» 16» 5» 2.0» 2.C» fl.C»» -1.0»»»» 2.C» 8.3» .12 »
L20i ENC OF HARMONIC 3
SARRiE 2 OEFORfEO 5)r*3
i»9
S>L
SRMPLE 2 UNDEFORMED
R.l
"^7.,.!
Figure C. Sample Problon 2
50
SAMPLE OUTPUT
The output shown Is for Sample Input Case 2. Some of the pages are combined to save space, and in some cases the full output Is not shown. Note that the input for the second pass has been heavily abbreviated since most of it is the same as for the first pass.
Figure 7 Is a plot of the deformed geometry with displacements multiplied by 5.
the
„■'-
SAMPLE 2 DEFORMED 5X
Figure 7. Deformed Sample 2
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71
APPENDIX A
Service Life Program
This program ts designated as S3359SL. It is designed to accept a tape or tapes created by S3359/ when output option 20 has been specified, and accumulate a selected set of stresses and strains.
The energy is calculated for each element from the principal stresses and strains as the accumulation is done.
The accumulated energy, stresses and strains and the principal stresses and strains calculated from the accumulated stresses and strains are printed out at the end of the run. The locations of the highest energy and the highest value of the principal stresses are printed out also.
Input
When the S3359 runs were made, a pair of Integers giving the date and time for each set of data put on the tape was written out. To select the desired sets of data from the tape for accumulation, the date and time of each of the desired sets must be Input exactlv as they were written out and In the same order as they appear on the tape.
The input Is via FREFRM as follows:
Ti tie - A title record that will be printed on the top of each page of the output. This must be only one card.
Control Records
As many records as desired to select the data sets.
LI The data as printed by S3359.
L2 The time as printed by S3359.
L3 The factor by which the stresses and strains will be multiplied. The default value Is 1.0.
An end of file stops the reading and accumulating and causes the final output to be produced.
72
APPENDIX B
S33S9F Fourier Coefficient Generator
This program computes the Fourier series coefficients by use of Trapezoidal Integration Formula« then fits the function f(0) with a Fourier series curve fit. The output Includes a set of cards containing the Fourier series coefficients In a form compatible with the Input requirements of program S3359.
The function to be fit with a Fourier series curve fit must be either odd or even having period 2«. The Integration for the coefficients Is performed over the Interval [o,«] and then mult I pi led by two.
Preparation sd. Input The Input Is read In In free form. (The rules of free-form Input are given In the S3359 document In detail.) The first card Is a title card. The second card contains data specifying the symmetry of the function, the error tolerance which Is compared with the least-squares error approximation, the maximum harmonic which Is to be calculated, and a plot flag specifying what Is to be plotted. The next set of cards contains values of e and f{9) where 6 Is In degrees.
Title Record
May contain any desired alphanumeric Information
Control Record
I0EF IOEF-0 If f(e) is even, 1 If f(0) is odd
ISYM Specifies the symmetry of f(e) as shown In Tables B-l and B-2
E Real number error tolerance allowed for Fourier series curve fit
N Integer number specifying the maximum harmonic to be computed. Maximum of fifteen.
I PLOT IPLOT-0, no plot
73
IPLOT-1, plot all harmonics
IPLOT'2, plot final harmonic only
IPL0T"3, plot specified harmonics
IH(15) Integer number specifying harmonic numbers to be used In plotting. Maximum of fifteen. Required only for IPLOT-3.
Input Data Cards
Ö Real value of e In degrees
f(9) Real value of f{e)
The title card must be ended with a semicolon. All other data Is separated by commas and ended with a semicolon. In choosing N and IH, the user must keep In mind what type of symmetry (I0EF and ISYM) he Is working with. For example. If I0EF - 0, and ISYM ■ I, only even harmonics are calculated since the odd harmonics are zero. Hence, N and the values of IH should be even.
If the Fourier coefficients are to be computed for more than one function, the data for each function Is preceded by a title record.
Many functions of Interest contain some type of symmetry. In these cases certain terms In the Fourier series will be zero and therefore are not calculated. The function f. In this case. Is always either odd or even. The following rules of symmetry apply.
71»
■ ■
'
Code
ISYM=0
ISYM«1 or 2
ISYM-3
EVEN FUNCTION Table B-l
Function
f(e)-f(-e)
f(e)-f(i80+e)-f(i80-e)
f(e)-f(i80+e)»-f(i80-e)
Fgurler Terms
no sine terms
no sine terns; no odd cosine terms
no constant term
no even cosine terms
no sine terms
.
ODD FUNCTION Table B-2
Code Function
1SYM-0 fC-e)—f(e)
ISYM-l
ISYM-2or3
f(0)-f(18O*e)—f(180-e)
Fourier Terms
no constant term
no cosine terms
no constant terms
no cosine terms
no odd sine terms
no constant term
no cosine terms
no even sine terms
Note, for example. In the case where the function Is odd and ISYM«!/ that though only the even sin terms are being calculated, some of these might also be zero. These numbers most likely will not be zero In the output but only small
f(0)-f(18O-e) —f(180+e)
75
s-\
numbers. The user must decide whether or not they are zero.
QutPUt
The entire input data is listed giving the FORTRAN name. The computed Fourier coefficients are written with a least-squares error for the curve fit up to that particular harmonic. A deck of cards containing the Fourier series coefficients is punched. The cards are punched in a form compatible with the input requirements of Program S3359.
If a plot Is specified, a plot of the Fourier series curve fit Is given on 12-Inch grid paper. The actual curve Is marked by X's, and the Fourier series curve fit Is drawn with a solid line.
Sample Input
PRESSURE LOAD 1;
1,1,.001,12,3,2,12;
0.^2.;
«♦5.,2.;
90.,2.;
90,-2;
135.,-2;
180.,-2;
V
76
APPENDIX C
370 OPERATING INSTRUCTIONS
PROGRAM: STRESS ANALYSIS OF AN AX I SYMMETRIC BODY WITH ASYMMETRIC LOADS
TAPES
Density 2 s a 16 I2 s a le 2 s 8 16 2 5 e 16 2 5 6 10
Data Sei
\ Name SERVLIFE
Input/
Output '1(5) l|0 l| 0 i | 0 I 1 0 1 Reserve
List |Scratcl ifTlLlscI R 1 L IsC R 1 L ISC R 1 L ISC R 1L BC 1
CARD OUT
PLOTTER
YES
YES
ERROR PROCEDURE: Flush
ESTIMATED MAXIMUM RUNNING TIME! See user request
SPECIAL INSTRUCTIONS: None
77 J703-76