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NASA-CR-165430 19840003447
A Reproduced Copy OF
Reproduced for NASA
by the
NASA Scientific and Technical Information Facility
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fiASA CR-i65430 BAT Report OD2536-941003
FINITE aa~E;H FOi~CED VHH1!\T!QrJ i\NALYSIS
OF !\OTfITIrlG CYCLIC STRUCTURES
by
v. ELCHURI
G. c. c. smm
BELL flEROSPl;CE TEXTROi~ P. O. Box One
Buffalo, tle"i~ Yor~ 14240
NATIONAL AERm{AUTICS AND SPt;CE .I\C;·lIHISTRATION
Contract NAS3-22533
W,SA'LeViis Research Ccntt~r Cleveland. Ohio 44135
Decembel' 1991
(dASA-CR-165430) F!UI!~ ELBtlE~1 P09CED IlraATIOU ~llALISI~ Uf HOIA1ING CYCLIC S!...'H;CTtlU':::S F.l.!l:ll 'It)(:}JUical j(>r;o.rt !:l'll.l !;.orospace Co., J3l1U:Jlo 6 :1. Y.,) UC !.04/~lF l..Oi
(Textron 73 p
CSCL 20;\ Gj/J9
NH4-11515
UncI.! !J ~;;~557
....-:
1. ne~ No. ~~nmrot A<:"..f.'~Ji"" Nc. 3. Rrcipient', Caul~ No.
NASA CR-165430 i ~~~n-,~-'e-U>--d-Su~~U-u-,----------------~-'--'------------'----------------~~~~n~~--~OI~t.----------------~
Finite Element Forced Vibration A.~alysis of l~ota::ing C!,c1i( December 1981 Structures ~-6.-r~~-'-~-m-i-~--O-t~--ru-la-t-ion--Cc--da------~
7. Author(,' V. Elchuri. G. C. C. Smith
e. rtr'orrning Or~niut;on N&mr end Addteu
Bell Aerospace Textron P. O. Box One Buffalo. New York 14240
O. Performi", Org~nization Clfpott tlo. D2536-941008 .
10. Work Unit No.
11. ContrKt or Gttnt No.
NAS3-22533 1------------------------------------------------------1 13. Type 01 Report ~nd Period eo\'t(ca
12. Sponsoring At'C>'-=V Narm cr.d Ac!dr~'
NASA Lewis Research Center 21000 Brookpark Rd. Cleveland, Ohio 44135 1 S. Supplemtnt:ry N~tn Richard E. Morris - Technical Monitor
Contractor Report
Finel Technical Report
16. Abstract -------------------------------------------4 A new capability has been added to the general purpose finite element program
NAST.RAN Level 17.7 to conduct forced vibration analysis of .tuned cyclic structures rotating about their axis of sy~netry. The effects of Coriolis and centripetal accelerations together with those due to linear acceleration of the axis of rotation have been included.
This rep0rt presents the theoretical development of this new capability. Th2 work was conducted under Contract NAS3-22533 from NASA Lewis Research Center, Cleveland. Ohio, with Mr. Richard E. Morris as the Technical Monitor.
17. Key \'Io<ds 'Su~trd try AuL'Io<h!) lB. Oiltribution SIot.ment
Forced Vibrations, Rotating Cyclic Structures, NASTR.t..:,. Fini te Elements, Publicly antilable Turbomachines, Propeller~J Base Excitation
I 19. ~urity Ousil. (01 this r(~::rtl
Unclassi fied I :/0. Scc",ity Ousif. (01 this "",..,1
Unclassified
21. No. 01 Pa~
71
• Fe: sale by the Natio:1J1 Te:hnical !niorr.'.Jlion Service, Springrjel.l, Virginia 22161 •
NASA-C.l6S (R.v. lo.iS)
22. Pric~·
"
i i
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NASA CR-165430 BAT Report #02536-941008
FINITE ELEHENT For~CED VIBRATION ANALYSIS
OF ROTATING CYCLIC STRUCTURES
by
V. ELCHUR!
G. C. c. SruTH
BELL AEROSPACE TEXTRON P. O. Box One
Buffalo, NeVI York 14240
NATIONAL AERONAUTICS AND SPACE AOmNISTRATiON
Contract NAS3-22533
NASA Lewis Research Center Cleveland, Ohio 44135
December 1981
ABSTRACT
A new capability has been added to the general purpose finite element
program NASTRM Lp.vel 17.7 to conduct forced vibration analysis of tuned cyclic
structures rotating about their axis of symmetry. The effects of Cor'iolis and
centripetal accelerations together ~lith those due to 1 inear acceleration of the
axis of rotation have been included.
This report presents the theoreticai development of this nel" capabil ity.
The work was conducted under Contract NAS3-22533 from NASA Lewis Research
Center, Cleveland, Ohio, with Hr. Richard E. Morris as the Technical Monitor.
ii
. .. It
I , , .
, . i I •
sur'~~lARY
The objective of the work described herein, was the development, documentation. demonstration and deliv~ry of a computer program for the forced vibration analysis of rotating cyclic structures. Tuned bladed discs arc an
example of specific interest.
The scope \'>'as by defi nit i on to address:
_ direct periodic loads moving with the rotating structure,
specified in the frequency or time domain;
translational acceleration of the rotating axis.
The capability is operational in the N.t..STRAN general purpose program at
Level 17.7.
NASTRAtl documentation is provided and example analysis results have been
obtailled.
Re1ationships to previous ","ork are described and further developrr:ents
are recoITrncnded.
iii
ACKNOtiLEDGENEflT
The authors take this opportunity to express their deep appreciation of the programming efforts of Nr. A. i1ichael Gallo, ~jr. Steven C. Skalski and Ms. Beverly J. Dale in implementing this theoretical development in NASTRAN. This report \'t'as typed by !-Irs. Deanna L. Kutis.
iv
TABLE OF CONTEriTS
Section Title PaQe
Abstract i i
Summary . . . . iii
Ackno't/l edgEment iv
List of Tables. . vi
List of Illustrations . . . . vii
1 Intrcduction • . . . . . 2 Equations of Motion 5
3 Solution of Equations of Motion 15
4 Examples • . . 25
5 Conclusions 57
6 Recom:nendations 58
Appendix 59
Symbols 62
References . " 64
v
.... _-
Table
1
2
3
4
LIST Or: TA3LES .
Title
Principal Features Demonstrated by E}:ample Problems
Bladed Disc Natural Frequencies . • . . . . •. • • .
Effect of Co~io1is and Centripetal Accelerations on the Displacement Response of Grid Point 18 at 600 RPS
Comparison of Response at 1814 Hz •.•.••.••.
vi
Page
28
29
52
55
i i _
[
I.
Figura
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
'22
23
24
25
LIST OF ILLUSTRATIONS
Title
j'lain ProblEm Spectrum ..
Overall Program Structure and Status
Coordinate Systems . . .•. .
Overall Fl11ivchart of Forced Vibi'ation Analysis of Rotating Cyclic Structures. . . • • . • . . . . . • . .. . ..
Directly Applied Periodic Loads Specified as Functions of Time •• • . . . . • . . . . . . . .
NASTRAN Hodel of the 12-Bladed Disc
NASTRAN Cyclic Hodel of the 12-Bladed Disc
k=2 Modes of Bladed Disc . . . . . • . . .
Bladed Disc Example 1, Displace~ent Rasponse (Magnitude)
Bladed Disc Example 1, Stress Response (Magnitude)
Bladed Disc Example 1, Stress Response (Magnitude)
Bladed Disc Example 2, Displacement Response (Magnitude)
Bladed Disc Example 2. Displacement Response (Magnitude)
Bladed Disc Example 2, Stress Respvnse (Magnitude)
Bladed Disc Example 2. Stress Response (Magnitude)
Base Acceleration Data in an Inertial Coordinate System
Bladed Disc Example 3, Displacement Response (M3gnitude)
Bladed Disc Example 3, Stress Response (Magnitude)
Bladed Disc Example 3, Displacement Response (Magnitude)
Bladed Disc Example 3, Stress Response (Magnitude)
Bladed Disc Example 3. Di~place~ent Response (Phase)
Bladed Disc Example 3, Displacement Response (Mlgnitude)
Bladed Disc Example 3, Stress Response (Magnitude)
Bladed Disc Example 3, Displacement Response (Magnitude)
Bladed Disc Example 3, Stress Response (Magnituje)
vii
Page
2
3
6
17,18
19
26
27
30
32
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36
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39
42
43
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;'.J
1 • INTRODUCTION
In Reference 1 (NASA CR159728), a gener'a 1 approach to conducting dynami cs analysis of bladed discs was discussed and a logical sequence of problems defined. Figure 1 indicates the problem spectrum cnd Figure 2 shows an overall program structure of modular nature, based on additions and modifications to the NASTRAN general purpose structural analysis program.
The general set of problems can be represented by formal equations, various terms of \'1 11 i ch are included or excluded dependi ng on the proD 1 em bei n9 stud i ed. Thus the equations
[~G {iiI + [[B] + 2)1[[\1]J iu} - [~~ {u} + [[K] - rl [f41]J {lI}
= {PI - [M2] {~o} ) (1)
\'Iith some boundary condit ions, may be taken to represent the general forced
vibration dynamic problem of a tuned bladed disc. In the previous and current \'1ork, the advantag% of cycl ic sym:nctry are incorpol'ated in this formulation.
The I'lork of Reference 1 handled, \·lithin state-of-the-art techll'J1ogy, analyses of aeroelastic, modal and flutter problems of tuned cyclic structures (Figure Z). It was documented and operated in NASTRAN at Levels 16 (Refs. 2 - 5~ and 17.7 (Ref 6).
The equations
[ria] {ill + [B] {a} - [QJ {u} + [K] {u} {OJ (2)
were treated in the context of modal and flutter problems of a tuned system, where
M represented the inertia matrix; B represented the damping matrix; Q represented the induced aerodynamic matrix (complex);
and K represented the elastic cum geJmetric stiffness matrix.
In the current work. the equations
[N] {ti} + [[5J + 212 [E3 1J]{u} + [[K] rt.2 [H1JJ {u} {P} - U'2J {Ro}
(3)
are·treated in the context of forced vibration. (Figure 2) ,
-1-
q
I rv I
t'I ~ I
;.. I
A.Z
A.J
Aerothermcelastic Probjsms
(Con~ldnt Speed)
"As Manufactured" I * Shipe dl the Deslgr. Point
Off-Oesl~n Point I * Performsnce A~.lysls
Tr'f'.!ent 1 A Aercther~oelastlc Per fornlance
t Elastic and geometric (differential) stiffness cffetts.
!!..Olli: 1. Altar design point A.I. A.2 gives
the elastic system "Kap· within Aerodyn~mic prosra~ limitations
Junr~~ystrm
Q~n.'ndI:S
(Cons t.'11 t ~11I·(·d. I'ressure Rat10 allu flew Rate)
T.l In-Vacuo Modal An~lysls
T.2 "Running" Modal A~alysis
T.3 flutter UoundArtcs
T.4 Sub-critical Eigp.nvollUt!S
T.S Forced Vibrdtlon Rc.pon,c
Oasis for P~rturbatlon and Fo~ricr Exp~nsion Technlqucs
..
* ,;,
*
,~
2. T,l. 1.2. are functions of selectud op~rating points on the Compressor Map.
3. T.J uses modes froru T.2. 4. T.~ uses harmonic ~erodyna~lc~
frem T.3.
S. T.S uses complc~ olgenvaluQs frCiil T .4.
Figure 1: r·lain Problem Spectrum
,-t tI.l
N.Z
14.3
~1.4
H.S
Hlstuned Syste~, Dynamics
(Constallt Speed. Pressure Ratio and Flow nata)
In Vacuo 140dal AnAly,,\s
-Rllnn1ng" 110dlll AnGlysls
Flutter Oound.r1e$
Sub-critic. 1 E~cenvalucs
Forced VlbrAtfon Rospon5c
6. M.l, ".2 USc tte modal lnforn.Hion from 1.1. T.2.
1. H.J. M.4 use T.3 har~Qnlt acrodyna~ics ~nd Fourier e"p'ln~iol1 technlquc~. (Cascade th~orlci 11~lt validity)
8. H.S uses coruplQX etganvalues frot.;. H. ~.
9, *. Operational o . Mcthod$ Formu14tc~
A • Approach Conc~pt~elizQQ.
Q
e
0
e i
0 1 --l
C) ' ... ~li ~ ~~ o ::~ t,:) ~~ ~,~ ~- .. .q :-;: C- ._. : ... f·:~ ,...... ,," ... -4 ". -< ~'.
CRIGiNCtL PA~:~ !S OF POUR QUALln'
-..,E"...x""':"i-st-.,...wl,...·n-g---:-:-NA:-::S~T~R-:-:P.N~-----....,B,....e~l-=-l/i'""7jA-S=rR,-A-U-Le-\-'e-o"l-17°-.-7-----------------Level 17.7 Bladed Discs/Cyclic Structures Analysis Curl'cnt Status
__ ~R_i~l~·d~F~onn~a~t~s __ ~~-~R~i~·id~F~o~r~~.a~t~s/Sol~ut~l~·o~n_P~a~c~k~a~e~s~ ____ ~~----------
I Statics with r--t' Stage Differential . I Design/Anaiysis, Stiffness, I RF 16 DISP
L:..:R.:...F ~D.:..:IS::.:..P_-: I I
r----------------+I-lDifferential!
.-----i--i Sti ffness
Stage 3-d Steady State ~erodynafllic Analysis
I I Operational I (Refs. 1-6)
I I I I I I I I Nonna1 Nodes using
cyclic syrrmetry, RF 15 DISP
r~odal Flutterf--__ Analysis,
Stage Flutter Analysis RF 9 AERO
Stage 2-d loperatior.a1 Unsteady state cascade I (Refs. 1-5) aerodynamic analysis I
RF 10 AERO • I I
Direct Frequency i [ Forced Vi-b-r-at-i'-o-n---' and Random Response r" Analysis of Rotating J
External forces 'Operationij1 and Base accelera- I (?resent \',ork and
Analysis, I Cyclic Structures RF 8 orsp I (Direct formulation),
i RF---sorSP wi th ALTER
~t_i~on _______ ~1 R2f. 8)
I I I I
Forced vibration AnalYSi
J of Rotating Cyclic Structures (Modal Iformulation)---
I Il-----. ! I I I I
Nistuned Bladed Discs Modal. Flutter and Forced Vibration Analyses
Bladed Discs :'!is:uning Specifications
Figure 2: Overall Program Structure & Status
-3-
I I I I Approach defined,
to be imp1er~nted.
I
I Mode rna th development I complete (Ref. 1).
Flutter and forced I vibrations approl'ch I formulated.
where [Ml, [B1. [Kl are as before, [81] represents Coriolis forces w.atrix {P} represents appl ied surface load vector n~21 represents base forcing mass matrix
and {Ro} represents base acceleration vector.
The specific form of the forced vibration equations incluJing th~ types of forcing functions to be incorporated were selected by NASA. These ~rc specifically
- directly appl ied loads moving Ivi tt-. tile rotating str:..cture;
- inertial loads due to translational accelerations of the ax's of rotation ("base acceleration").
The loads may be periodic and specified in the frequenc1 or time domains. Solution procedures follow generally those of the cyclic symmetry furmulation of the NASTRAN Theoretical ~lanual (Ref. 7). The capability has been dev{~10 ~j
on the IBM 370 system at Bell Aerospace Textron and documented. delivered and demonstrated on the UNIVAC 1108 at tlASA LeloJis in tlASTRAN LE:"el 17.7.
Five demons~ration examples are presented, one being a c~mp1€te strJcture example to shm'l compatibility wHh the cyc1 ic structurf' formulation. A sir:plc twelve bladed disc is modelled <.:nd forced at conditions related to its nat.ural frequencies. The response examples include:
- phys i ca 1 component ford ng (freq;.!ency doma in) , - harr.1oni c cor.lf)Onent forc i ng (frequency dOfo~a in), - harmonic component forcing of th~ rotationa~ axis, - physical periodic forcing (time uOMa;n),
harmonic component periodic fOI'cing (tir.le domain)
It would be logical to extend t~~ current ~ork to include the gen~rat;on of applied and induced oscillatory aerodynamic loads S0 hat the farced vibrati'Jns of subcritical engine stages can be addressed dlrectly. Rotational base accelerations could also be of practical interest in investigating the gyrosccpic effects on rotat1ng machinery.
-4-
, .~ ..
I . I i
[
2. EQUATIONS OF NOTION
The equations of !J1otion of a tuned cycl ic structure !""otating about its axis of syrrtiletr:i and subjected to steady sinusoidal and general per"iodic excitation are derived using Lagrange's formulation.
Figure 3 illustrates the problem by conSidering a l2-bladed disc as an example. The bladed disc consists of " tHel ve 2r)° segments--identical in their geometric, material and constraint properties. rh~ ~isc rotates about its axis of symmetry at a constant angular velocity. The axis of rotation itself is pern.itted to oscillate 1 inearly in any given inertial refel'ence. In addition, the bladed disc is allowed to be loaded with steady sinusoidal or general periodic loads moving with the structure. Under these conditions, it is desired to determine the dynamic response (displacement, acceleration, stress, etc.) of the bladed disc.
The cyclic symmetry feature of the rotating structure is utilized in deriving and solving the equations of forc~d motion. Consequently, only one of the cyclic sectors is modelled and analyzed using finite element~. yielding substantial savings in the analysis cost. Results, hOI-lever, are obta ined for the entire structure. The Coriolis and centripetal acceleration terms have been included. For Clarity of derlvation, the equations of motion are first derived for an arbitrary grid point of the cycl ic sector finite element model. and then extended for the complete model.
COORDIN/\TE SYSTEHS
These are shoHn in Figure 3. V-XYZ is an inertial coordinate SystE 1.
O-XSYSZB is a body-fixed coordinate system such that aXB coincides with the axis of rotation of the structure and is ah/ays parallel to €lX. For a NASTRAN finite element model of the bladed d;sc, O-XBYBZB also represents the ~asic coordinate system. A-xyz is a body-fixed global coordinilte systerl in which the displacements of any grid point P are desired. The unit vectol"S associated with these coordinate systems are also shown in Figure 3.
DEGREES OF FREEDOM
The rotating structure is permitted four rigid body mo~ions including three translations (a10ng OX, CJY and ElZ) and one rotation at a constant angular velocity n about its axis of rotation OXB.
-5-
".
" " " I, J, K
IB, JB, i, 3. k
ORtGH"~kL r[tG~ ~~; OF POOR QW\Ul"t
.-4... Parallel to ilJsic Y a::is
x (Parallel to Basic X axis)
U~it vectors along Inertial ~YZ axes
Re Unit vectors along Basic XBYgZg axes Unit vectors a10ng Global xyz axes
Figure 3: Coordinate Systems
-6-
y
A
/~Z
Global C.S.
!
i ,
l.
L
.-I 1
r -I ! i
All grid points of the structure are each permitted six degrees of freedom. The displacement at any grid point in any sector can be expressed in any bodyfixed coordinate system as a combination of:
l} the steady state displacement due to the steady rotation of, and the steady state loads applied to, the structure, and
2) the vibratory displacement (superposed on the steady displacement) due to the vibratory excitation provided by the directly appiied loads and the inertial loads due to the acceleration of the axis of rotation ('base' acceleration).
The purpose of the present development is to determine the vibratory response.
LAGRAtlGE FORf1ULATION
Referring to Figure 3, the complete tuned structure consists of N identical cyclic sectors. If u represents all the vibratory degrees of freedor:J of the complete structure, the equations of motion can be derived via the Lagrange formulation,
ddt ( ~Tuo ) _ aT + .£!! + uD = oW
o au au dB ou (1)
where T and U represent the kinetic and strain energies, respectively, of the complete structure; D is the Rayleigh's dissipation function representing th~ energy lest in the system due to resisting forces proportional to veloci ties ~
(e.g. viscous damping forces); and 6W represents the virtual work done on the structure by the external forces through virtual displacements 6u.
The complete set of degrees of freedom u can be subdivided into N subsets, each containing un degrees of freedom for each of the II cyclic sectors. Since any given cyclic sector is 'connected' to adjacent cyclic sectors only on its two sides, un satisfies the intersector boundal'y compatibil ity condition
n n+ 1 ( ) uside 2:: usid~ 1 n = 1,2 •... , N. 2
Equations (1), therefore, Ciln be written as N sets of equations coupled only ilS given by equations (2):
n=1,2, •.. ,Il. (3)
-7-
, ~.
~
! •
" l'
,.'
I 1 1 I I I I 1 ." .;)
OF POOR QUAUTV
For clarity of presentation, l'lithout loss of generality, equations (3) are first applied to obeain the equ<J.tions of motion of an arbitrary grid point in any cyclic sector by considering its three tr'Clnslational degrees of freedo~,l. Inclusion of the three rotational degrees of freedom at the arbitrary grid point. and extension to include the remaining grid point$ in the cyclic sector are considered subsequently.
KINETIC ENERGY
With reference to Figure 3. point P is an arbitrary grid point of the nth cyclic sector with a mass of 'm' units lumped from the adjacent fini~e elements.
The kinetic energy of the mass at P can be written as:
(4)
where (5)
(6)
o 0 A 0 A 0 A
POP = XOpIB + 'iOPJB + ZOpKB ' (7)
n = nI (8)
and
fB) [1 0 0
III l ~ J' : c s
-s c
(9)
with c ~ cos nt and s :::: sin nt (lO)
Substitution of equations (5) through (9) in equation U·) results in
T = ~-0 0
zoJ r J[.:~ LxO YO m
L
+1 " 0 ZOP{ r ~OP! LXoP Yor m \ YOP 2
In _ (Zop )
-8-
o~...::~~: ,~; .. ;._ t t :~.:.:::. ; ..... ~
r OF POOR QU;:'UT'f i 0 0 0)1 XOP
1 I
1 n2 ~OP Yop ZoPJ l~ YOP +- ,,-
!: _ 1 2
.. , 0 ZOP
::H 0
[:
0 Xo
1 liop Yap zOPJ 0
+ me Yo 0
-ms Zo
[:
0
~mn XOP ~ + n LXop y op ZOPJ 0 YoP
m ZOP
r: 0 oJ ) ~o } + n Lxop Yap zopj -ms _: L;: (11)
i..0 -me •
In order to introduce the global coordinates of point P, consider now the position vector to P written as
( 12)
i.e.
(13 )
where
[TBasiC to GiObal] (14 )
-9-
....
T J ., 1 .. ,; .
r r: I, I
!
T· ! !
: .
! , . I ,
"
Therefore equation (13) yields
C·;\;~;:i". ,~:. ~-';~J' .. : ::J OF POOR QUALITY
(15 )
(16 )
The global position vector to the point P can be further thought of as consisting of three. component vectors, i.e.,
f xAP ( [XAP l \ Ux l 1 Ux
( l :AP ~ ~ : AP) + l i } + ~y j 07}
AP AP initial Z Z L I I 'I I Initial position A Disr1acements due to vibratory forces
IDisp1acements due to n and steady loads
I I Steady part Vibratory part
Substitution of equations (17) in equations (15), and differentiation once with respect to t yields
o
(1'~)
Therefore, \'/ith the help of equations (15), (17) and (18), equati:>n (11) expresses the kinetic energy of the ~ass at P in terms of the displacements and velocities of P expressed in the global coordinate system A-xyz.
STRA HI ENERGY
The strain energy due to the displacements at point P expressed in the basic coordinate system can be written as
u = t Uxop - XOP , in.) , (Yap - Yap, in.) , (lOp - Zap, il1.11 .,. ·f KXX KXY KXZ 1 \ XOP Xop , i 11. I
* ~YX "YY ~'{Z L YOP - YOp,in. ? (19)
L KZX KZy KZZ ZOP - ZOP, in. J -10- -..
.. -t'
r J
-' " £ g.
r . '
r l.
i. 1 -! i.
l.
i
! '
1 .
I
By using equations (15) and (17), this becomes
(20)
where Global ~B Tn· GB
[K J = [TU J [KUaslCJ [T J (21)
DISSIPATED ENERGY
The energy dissipated by the damping forces acting against the motion of the point P is expressed by the Rayleigh's dissipation function,
BXX BXY
ByX Byy
Szx ,BZy
sic 0
BXZ [ :OP 1 ByZ Yap
o
BZZ Zap
Use of equation (18) transforms this to
\'lhere
D = 1 l~ ~y 2 x
Global ~zJ [B J
Global T Basic [B J = [TGB] [B] [TGB]
VIRTUAL WORK
o
The virtual work done by an oscillatory appl ied load
p = p i + P J~ + P k x' y z
(22)
(23)
(24)
(25)
moving with the point of application, through individually dchieved virtual
I' displacements 6ux' 6Uy and 6uz can be stated as
(26)
-11-
~
I
j-, 1 .. r !
I .. r (-.
r' r -
I
I !
!
I f
.-,
i I
1,
I .. - , ; I
i' I
EQUATIONS OF FORCED i~TI.Qri
O:\ICIi~f.'\L ['f..e::: ~;; OF POOR QUAi-lTV
Substitution of the cxpressions fOI' T (equation 11), U (equation 20), 0 ~quation 23) and 6H (equation 26) in the Lagrange equations (3) results in the following equations of forced motion of point P expressed in the displacement (global) coordinate system A-xyz:
[r.1] {in + [[B] {- 2)2[B 1]J fij} + r~K] - )22[1011]] {u} = {P} [t12] dio
}
The terms appearing in equations (27) are:
r
{PI = i
mol = p: 1 Zo
[11] = [M ] = [TGB] g1 oba 1 T [m~ JO~ mOO] [TGB]
global [B] = [B J
global T [~ 0
-n [B1J = [B1 j [TGBJ 0 [TGB] m
global [KJ = [K J
-12-
(27)
(28)
(29)
(30)
(31)
(24) .
(32)
(2l)
{33}
rr, .
I T T
" l . , I , ..
, .-. ,
!
i-
I.
i 1.
..
.. j
I"
I
and t 0
~s] [r"21 = [TG3] ~ me (34)
-ms me
where c and s are given by equation (10) .
Equations(27) describe the translatory motion of an arbitrary point P in an arbitl'ary sector n of the rotating eyc1 ic structure subjected to a directly applied vibratory load {P} and base acceleration {Ro}'
These equations can be extended to include the three rotational degre~s of f,'cedom at point P by noting that:
viz. by
1) in a lumped mass model, only the translational degrees of freedom at any grid point contribute to the kinetic energy of the structure. and
2) the coupling between various degrees of freedom may exist only via the stiffness matrix. (Instances where the damping matrix is defined proportional to the stiffness Matrix also may result in coupled equations of motion.)
Accordingly. the matrices derived from kinetic energy considel'ations, [i1], [81], [N j ] and [H2] of equation (27) can be expanded as typifieci
[M] = [~M~]J ~ ~] 6x6 0 I 0
I
, (35)
where [Mtt] is the 3x3 (!rans1ationa1j mass matrix of equation (27). With subscripts t and r representing the lrans1ationa1 and ~otational degrees or freedom at point P, the stiffness and damping matrices may be expanded as
and
[K] = [~K~t]_ :_[~r~] (36)
[KrtJ I [KrrJ I
[B] = (37)
By similar reasoning, the equations of forced vibratory motion of all the cycl ic sectors of the total structure can be \'Iri tten as
-13- .-o
Fi;
H J T
,
~ .
j'
1 I
T -
1
J'
n :: 1 t 2 t •••• Pi. (38)
The intersegment boundary compatibility is specified by equation (2).
-14-
r
r r
1,.
r I ~ , ..
_. i I
j -
-1
f '<.
3. SOlUTIOii 0;= EQUATiONS OF MOTIor~
The method of solution of the equations of forced motion (equations 38 and 2. Section 2) is based upon the fom in \'/hich the excitation of the rotating structure is specified. As noted earlier. the present development considers excitation prescribed as:
1) directly appl ied loads moving \'Jith the stl'ucture and
2) inertial loads due to the translational acceleration of the axis of rotation ('base' acceleration).
These steady-state sinusoidal Qt' general pel';odie loads are specified to represent:
1) the physical loads on various segments of the complete structure, or
2) the circumferential harmonic components of the loads in (1).
The sinusoidal loads are specified as functions of frequency and the general periodic loads are specified as functions of time.
The translationai acceleration of the axis of rotation is specified as a function of frequency in an inertial coordinate system.
Because of its eventual implementation in the NASTRAN general purpose finite element structural analysis program, the following solution procedure generally follO\-Js the theoretical presentation of cyel ic symmetry given in the N.n_STRAr~
Theoretical Manual (Ref. 7).
METHOD OF SOLUTION
The method cf solution of the equations of motion consists of four principal steps:
1) Transfonnation of applied 10ads to frequency-dependent circumferential harmonic components.
2) Application of circumferential har~onic-dependent inter-segment compatibility constraints.
3} Solution of frequency-dependent circumferential harmonic components of displacem~nts.
4} Recovery of frequency-dependent response (disp11cements, stresses, loads, etc.) in various segments of the t0tal structure.
-15-
- .. ....,
~ n ... \1 U
An overall flo't:chart outlining the solution algorithm is ShOl'iO in Figure 4. Provision to include the diff\!rentiai stiffness due to the steady loads is also shown.
1. Transformation of Applied Loads
J The transfornation to frequency-dependent circumferential harmonic components depends on the form in \'Jilich the excitation is specified by the user. The follow-
-i ing options are considered in the present developrr.ent to specify the form of exci-; i~' tation due to the directly applied loads and base acceleration loads:
t .. r r l
r ..
I . . . !
t ... '
Directly applied loads specified as:
periodic functions of time on various segments. - p~riodic f,lnctions of time for various circumferential harmonic indices - functions of frequency on various segments - functions of frequency for various circumferential harn:onic indices.
Base acceleration specified as:
- fur.ction of frequency for circumferential harmonic indices 0 {axial} and 1 (latel~al).
Details of each of the above five loading conditions are as follows:
Directl.UEP.l ied loads (seament-dependent and pedodic in time)
If ?n represents a general periodic ioad on sector n specified as a function of time at M equallY spaced instances of time per period (Figure 5). the lead at mth time instant can be written as
m ·'0 £.L r- £.c pn = pn + I l:n cos(m-l£b)
l=l
-9..5 J -11/2 - m- n + pn sin(m-ltb) + (-l) lp , (1)
m = 1. 2 ••.•• M
where b = 2Ti/I·1. 9..L = (M-l )/2 for odd 11, £.L = (11-2)/2 for even M. The last tcnn -"9.." in equation (1) exists only when M is even. The coefficients pn
("9.." = 0; 9..c. 9.$. 9..=1. 2, .•• , 9..L; 11/2) in equation (l) are independent of time. and are defined by the relations
-0 Ii1 1 ~, pn = M 1 lln
m=l (9. = 0) Part of (2)
-16-
J
-j j .
1 I ~.
r:i rcumf. ; ·Iarmonic -
Dependent -. , .
Differential Stiffness t1atrix
(-- EnteC) --r Finite Eiement Hodel of one Cyclic Sector, Rotational Speed, Constraints, Loads
Generation of Stiffness, Mass and Dzmping Matrices
Application of Constraints and Partitioning to Stiffness. Mass
and Damping Matrices
General, periodic in tili'2
Segment -Dependent
Circumf. Harmonic -Dependent
Fourier decomposition Phase 1 (time)
Equation (6)
Fourier decomposition Phase 2 (circumferential), Equation (7)
. SeglT'I-?nt -Der-endent
Fouri"~r d;;;~posi' tion, Phase 1 (time), Eq. (1)
Fourier decomposition, Phase 2 (circumferential), Equation (3) I
'--_____________ .-6-_---y ___ --' ______ . ___ .. __ . _____ 4'
'j
.J \..l
Application of Constraints and Partitioning to Load Matrices
FIGURE 4: Overall Flo~lchart of Forced Vibration Analysis of Rotating Cyclic Structures
-17-
!
,<11
J
, ;
•
, i T -;
.. ! . I
i.
t
r- --I Application of inte~-segment . co~patibility constraints to
sti ffr.ess, mass, c;dmpi n9 and load matrices
~ So1ution of independent harmonic displacements
OF:iGHJM .. r:;:..t=.7~ r::; OF rOOR QtiAL1Y'1
I ......... ---No-·
Ye-:,
Recovery of segment-dependent independent displacements
(Inverse Phase 2, if necessary)
J Recovery of oependent
disr>lac~ments
! Output requests for diSPlacements,'1
stresses, loads, plots, et~
i ( Exit )
FIGUKt 4. (Concluded)
-18-
i r T
i i ..
! t:_
, .
l I , .
1
m m n --"k" P or P
Directly Applied Loads'
OF POOR QUALITY
T2
Figure 5: Directly Applied Periodic Loads Specified as Functions of Time
-19-
Time, t
..
-2c m 2 14
pn = -" L pn cos(m~f~b), 'I m=l
-£.s M m . pn = ~ ') pn sin(m-ltb),
H iii=l
O~T~~~': ~ ~ ... ~ ~ ... : .... ~c~ I l\.t\:'I. I, __ 10 r .. "'.~~ • ...... 1 OF POOR QUALlIYI
(
(9.,::.1. 2, •••• t L)
( 2 Contd. )
,I
and
-M/2 M m
pn = k I .(_l)m-l pn (M even only) (l=M/2). J m='
II nil
- tl Each of the coefficient vectors P on the left hand sides of equations
(2) can further be expanded in a circumferential (truncated) Fourier series
p~9.." :: ~£." + ~L r~I~t' cos (i1.:Tka) + ~rSin(n-lka)] + (_1)n-l~;;; k=l L-
where n ~ 1. 2 ••. ~. N • "9.,":: 0; lC. tS.l:: 1, 2 ..... lL; t-I/2
a = 2r./N kL :: (N-1)/2 for N odd kL=(N-2)/2 for N even.
The last term in equation (3) exists only when N is even. The Fourier
(3)
(4)
-"l" "k" coefficients P ("k" = 0; kc, ks. k :: 1. 2 •.•.• kL; N/2) in equation (3)
do not vary from sector to sector. and are defined by
_"9.,11
pO
_"ll1
pkc
_"9.," ps
1 N = N L
n=1
2 N = N I
n=l
2 N :: IT L
n=l
_"lil
(k = 0)
_" £."
n (-) P cos n-1ka
(k :: 1. 2, .... kL) ( 5 )
_" £."
pn sin (n-1ka), and
_"9.," "£"
~l/~ = 1 ~ (_1)n-l pn (N even only) (k - N/2) N 0=1
-20-
-..... . '. ~ ,-, ~... ""t -.
J
, ! ,.
_"R," -lIkll
The terms P ("9)' = 0; 2-c, R.s, 11. ::: 1, 2 •••• , R,L; rV2 and "kll = 0; kc, ks, k = 1, 2, •••• !<L; N/2) art; the trili1sforjri~d frequency-dependent circumferential harmonic components of the directly applied loads pW (m = 1,2, .••• M and n :: 1, 2, ••. , N).
Directly applied loads (Circumferential harmonic-dependent and periodic in time) •
Such loads can be represented as
m -0 £. ~-£c -lIkll -"k" L -"k" __ p = p + I P cos(m-iib}
9. .. 1
-£s l -H/2 "k" m 1 "k" + Psi n (m-Hb U + (-1) - P ,
where m = 1. 2 ••.•• M represent the time instances at which harmonic components "k" = 0; kc, ks, k = 1. 2, •••• kL; N/2 of directly applied loads are specified.
_" ,til
The coefficients P"k" on the right hand side of equation (6) are obtained using equations (2) with sector number n replaced by harmonic number "k".
Directly applied loads (frequency-and seament-dependent)
This type of loads can be represented uS
(6)
(7)
where "1" (=1. 2 •.•.• F) now represents the frequencies at which excitation is
specified. The transformed frequency-dependent circumferential harmonic compo;Hmts _".11." -"k" p (Ilk" = 0; kc. ks, k = 1,2 •••• , k
L; N/2) are obtained using equations (5)
with ".11. 11 as defined above.
Direct' y app, i ed loads (frequency- and ci rcumferent i a 1 harmoni c-dependent)
These loads ~re the transformed frequency-dependent circumferential ".t" ... -
components p-"k" ("k" = 0; kc. ks. k = 1.2 ••.•• kL; N/2) with "£." (=1,
representing the various frequencies at which the directly applied loads specified.
Base acceleration (fregLlen.f1- and cil'cumferential ham-onie-dependent)
harmonic 2, .•. , F)
are
In Appendix A. it is shown that the components of the translational base acceleratio~ contribute to inertial loads on the rotating structure in the following manner:
" . ~ l . • ... ~
~.,"f··~ .,'.~ .•. ~.~ .. ~ "-'
-21- CF POOR QUhLln'
T ,
f .. ;
i j.
r ! .
1.
2.
2.
-"£" "k ll
Axial component contributes to P \'/here Ilk" = 0, and "£ .. represents the specified excitation frequencies,_"
f"
"k" Lateral components contribute to ~ wher~ "k" = 1c and 1s, and "£" re-presents the effective excitetion frequencies which are shifted from the specified frequencies by ± n, the rotational frequency.
Appl ieation of Inter'-Segment Compatibil ity Constraints.
As sho\'/n in Section 4.5.1 of Reference 7, equations (2) of Section 2 are used to derive the compatibility conditions relating the circumferential harmonic component degrees of freedom on the b/o sides of a l"otationally cyc1 ic
(k = 0)
(k = 1, 2, •..• kL) (8 )
(k = H/2)
In order to apply these constraint relationships for any given harmonic k, an independent set uK consisting of the circumferential harmonic component (cosine and sine) degrees of freedom from the interior and side 1 of the cyclic sector is defined. uK is selected from the 'analysis' set degrees of freedom (i.e., the degrees of freedom retained after the application of constraints and any other reduction procedures), and is defined as
itc -f' and = Gck(k) u " (9) .
uks = ) -K Gsk(k u
itc and uks each contain all (and ~lnly) the 'analY:;1s'set degrees of freedom from the interior and both sides of the cyclic sector. Equations (8) are used to define some of the elements of the transformation matrices Gek and Gsk ' For k = 0 arid N/2, the matrix GSk is null.
-22-OF POOH QUALITY
J T { ...
..
..
r
OF POOR QUALITY 3.
OJ
For a given hal"monic !~, tile introduction of U''' in the equations of motion, equations (38), Section 2 , ~'esu1ts in the transformed equations of motion (Ref. 1)
(10)
where .-.KM' = GT Mn G ;. GT Hn G ck ck sk' sk'
(11 )
o,K _ ,.. T Kn r. -:- G T Kn G • and ~ - uck uck sk sk
~ = GT pKC + GT pks ck sk
As discussed in subsection 1 of Section 3, pkc and pks are the transformed frequency-dependent circumferential harmonic components of the directly applied and base acceleration loads.
At any excitation freqUency w". let
} (12) uK = uKeiw~t
=K =K where P a;.d u are complex quanti:ies. Equation (10) can be rewritten as
(13 )
The excitation frequency w~ is given by
(14)
= Win for lateral base acceleration loads.
w~ = w for all directly applied and axial base acceleration '} loads, and
Equation (13) is solved for uK for all excitation frequencies and all harmonics as specified by the uset'o The cosine and sin~ harmonic components of displacements are recovered using equ,}tions (9).
4. Recovery of Frequency-Dependent D~lacements in Various Segments
This step is carried out only when the applied loads are specified on the various segments of the complete structure.
-23-
Po
"
J J T
" o , 1.
· .
- .
· , "
i" · -
For loads specified (is functions of time, equation (3) is used to obtain the _II ou .
displacements un"" in various 5cgmE:nt5 v.'ith ".~" ;: 0; lc, 1s, £ ;: 1, 2, .•• , 1max.
The circumferential harmonic k is varied from kmin to "max. The use," specifies "
1max' kmin and k~ax·
For loads specified as functions of frequency, equation (7) is used to _110 11
obtain the displacements ul1- in various segments with "t" representing the
excitation frequencies. The circumferential harmonic is varied from user specified kmin to kmax '
The solution pr'ocedure discussed in this section has been implemented as a
ne\"/ capab i 1 ity in NASTRJI.N (Ref. 8) •
-24-
J T
-;
•
i ; ..
1.
!.
~,-.---. " - -.. ~-
4. EXpJ!'PLES
Five inter-related examples are presented to illustrate the theoretical development of the previous sections. The ne\'/ capabil ity added to flASTRAN to conduct forced vibt'ation analysis of rotating cyclic stl"Uctures (Ref. 8), as a result of the present development, has been used to conduct these examples. A l2-bladed disc is used for demonstration.
Example 1 ;s conducted on a finite element model of the complete structure (Figure 6). Examples 2 through 5 use a finite element model of one rotationally cyc1;c sector (Figure 7). Results of example 1 are used to verify some of the results obtained in the remaining examples. Table 1 sumnarizes the principal features demonstrated by these examples.
Steady-state frequency-dependent (sinusoidal) or time-dependent (periodic) loads are applied to selected grid point degrees of freedom. The specified loads can represent either the physical loads on various segments or their circumferential harmonic components. For illustration purposes oniy, the frequency band of excitation, 1700-1920 Hz, due to directly applied loads and base acceleration is selected to include the second bending mode of the disc for a circumferential harmonic index k = 2. The 'blade-to-blade' distribution of the directly appiied loads also corresponds to k = 2. Table 2 lists the first few natural frequencies of the bladed disc for k = 0, 1 and 2. Nodes for k = 2 are sho'fJn in Figure 8.
General In2ut
1. Parameters:
Diameter at blade tip = 19.4 in. Diameter at blade root = 14.2 in. Shaft diameter = 4.0 in. Disc thickness = 0.25 in. Blade thickness = O. i25 ; n.
Young's modulus = 30.0 x 106 lbf/in2
Poisson's ratio = 0.3
Material density = 7.4 x 10-4 lbf-sec2/in4
Uniform structural damping (g) = 0.02
2. Constraints:
All constraints are applied in body-fixed global coordinate system(s). All grid points on the shaft diameter are completely fixed. Rotational degrees
of frecdonl 9Z at re~a;ning grid points are constrained to zero.
-25-
---{"':7' ... I
;.. r .z ..
, 1.
I L
I ! .. i.
r' I. !
1
O~:G~;··H:·,L r~~\~;: !:~~ OF POOR QUAUT'l
Segment 8
Cyclic Sector Mcdeiled (Segment 1)
Disc in YZ plane
Figure 6: NASTRAfl lIodel of the 12-Bladed Disc
-26-
... . 1'
r' f~ ..,. f
" ';'
I J .'
r l '
,- .
l.
r
,--j
I
( .
. . ;
1
Figure 7: NASTRAN Cyclic Model of the 12-Bladed Disc
-27-
-, ...:
.:.-'" ,;, )
I 1') 00 I
Example No.
1
I 2
3
" 5
'-------~
Fi nHe Element r~odel of
Complete Structure
Cyclic Sector
Cyclic Sector
Cyclic Sector
Cyclic Sector
--I ---I •. ,....c-.::. ... , .;' .... cl olI.--+
TABLE 1: PRINCIPAL FEATURES DEMONSTRATED BY EXArI.PLE PROBLEHS
Applied loads specified as functlons of Frequenc Jsinusoidall TimeLQeriodic1 Base
Physical Circum.Harmonic Physical Circum.Harmonic Acceleration Components ComEonents Components Components
Yes No
Yes No
Yes Yes
I Yes No
Yes No
-------------- - .. --- ----- ---- .
~ ~"<I:'!:3 !1~
Rotational Sj:eE:d
No
No
Yes
Yes
Yes i ----- --~
00 'TI ;;j
-oR 8 ····
~.:; ::rJ I"
rO "':! c.: ~."-' )-.' (.' r.: Ii;
:1 ~)
, _ J
J J J
.. '
i ..
r ..
r j.
,-I
TABLE 2: GUiDED-DISC fiATURAL FREQUENCiES
Ol~:~~~ .. ,i-"i:' ,-;.:.,;.. : \ .••
OF POO~ Q~.jALnY
--. Frequency (f·lode NO.), Hz.
k = 0 Mode Description
k :: 1 k :: 2
~ 214 (1) 208 (i) 242 (1) I en
I ZII~
591 (2) 594 (2) 622 (2) I I ---~ I
1577 (3) 1633 (3) 1814 (3) !~ I .
I I 2468 (5)** 2460 (4)
I 2433 (4) I ....,.,... =, "" . .---<::::::l
I
* k is the circumfere.,tial hat'monic index
** Mode No. 4 for k = 0 at 1994 Hz represents an in-plane shear mode not excited by the applied forces.
-29-
t·r.) •
Mode 1 242 Hz
Hade 2 622 Hz
Hode 4 2433 Hz
Figure 8: k = 2 Modes of Bladed Disc
-30-
EXPJiPLE 1
Description
This example uses the direct frequency response capability in NASTRAN, RF8, and forms the basis to verify some of the results of examples 2 through 5.
1. Parameters:
Same as general input parameters.
2. Constraints:
Same as general input constraints.
3. Loads:
where
P(f;n} = A(f) cos (n-1 .® @ ) n is the segment number, ® represents k = 2, ~ represents the total number of segments in the bladed
disc. P is specified using RLOAUi bulk data cards.
Results
Sample plots of grid point displacem~nt and element stress response are shown in Figures 9 through li. The expected behavior about a k = 2 natural frequency of the bladed disc can be seen in all these figures.
\
All the response plots (Figures 9 through 25 except 16) have been obtained using the plotting capability of flASTRAN. On any given piot, the various curves are identified, in order, by symbols X, *, +, -, • and o. The sequence of curves is indicated ojn the first 1 ine of the pOlot description at the bottom left of each figure.
-31-
..
I'
I r
!
r
G R I 0
P a J H T
0 I S r L R C E H E II T 5
H
R C N
I T u 0 E
H
c H
lE..b3
9
0
7
G
S
IJ
s
2
lE-~
9
a 7
6
5
~
3
2
r-------~r_--------TI--------~----·----~---~------~--------~!E-3
r--------i·----------r--------+-·------~·--------~------~9 I---------+·---------r--------+----------i--------~------~o
~--------~~-------_;--------~!--------~r_--------~------~7
.----------+--------4----------1--.-----+-------+------1 G
r_-------4---------4---------+--------~--------~------~3
~------+_--------+_--------4_-------4_--------~--------~2
~ ________ ~ ________ ~. ________ ~ ________ J_· ___ • ____ L_ ______ ~lE-S
lE-5
1.68 1.72 1. 76 1. 00 1. 84 1.88 E J FRECUENCl (HEArll
lij 1T3RtlI ,16 tT3RHI,9S ITJRlll
FORCED VIBRRTI0N RNRLTS!S OF R01RTING C!(LIC STRUCTURES BlRDED DISC EXfl~f'lE 1 tfUlL M:lOEl,fREQ L-,::'05l KINDE~ 2t TIPE LORDS
Figure 9
-32-
rr~ .....
it r
1 T .,. I ~
F. L E H E N .. T
S T A E 5 S E 5
, H R G N I T U 0 E
p
5 I
J t .. 1
t
0:- PO{)~ QUAUTY
IE Q D I
7
5
S
:2
IE :3 9
7
5
'"
/ ~ /' \..
/ '\.
V ~ / ~ ,
,,/ " ...
2
IE 2 9
7
5
3
2
~
~ /~
P tA~ I ~ A~ ~ I
I /'/ / / '\. 1\ ,\,
~ ~/ / V
"" ~~ ~ V // '\
~ ~'" V V V ~) ~ L " -....:
,/ V ~~ ""tD
/" --.........
~ 1 E 1
1.68 1.72 1.75 1.80 1.84 1 ,88 E3 rBEouENCl (HERTZ)
II (3).11 (5),11 (71,11 (101.11 (121.)1 (1'1)
rORC£D VIBRATION R~ALTSIS OF R01ATING CYCLIC STRUCTURES BLRDED DISC EXRMPLE I (FULL HODEL. FRED LORDS) KINDEX 2C llPE LOADS
Figure 10
-33-
IE q 9
7
s
3
JE 3 9
7
5
3
2
JE 2 9
7
s
3
2
1 E 1
j'
t: L E H E N
T
S T R E S S E S
H
R
G N
T U
D E
P S
I
••
IE Y 9
7
5
3
2
IE 3 9
7
5
::I
2
IE 2 9
7
5
3
2
I E I
,---.
'.' "";""1 rf"'S • i .\.:..t.: ~;.,
OF PGOri QL'{iLiTr
T ,/",
"-..
7 "\
/ /'
/ v
/ A~ ./
V //% \ " k/V
;/ // 7"
or' v'" /'
~
I
~.
~ r--.....
~ ~
~ ~ "'- .....,
~
i --'
1. 68 1.72 1.76 1.80 1.84 1.88 E3 FflEDUEtlCY (HER1 Z)
109131.109 IS) .109(7) .1(19 (10) .109 (12) .109 (]Ill FORCED VIBRRIIOtl RNRLlSIS ~F ROTRIING CYCLIC STRUCTURES BLROED DISC EXRMPLE I (FULL HllOEL,fflEO LOReS) KINOEX 2C TIPE LORDS
Fi gure li
-34-
IE Y 9
7
5
3
2
IE 3 !l
7
5
::I
2
IE 2 9
7
5
3
2
I E I
i t
!~ r
~" t)
": .... /--..-
Dcscripticn
This example uses the forced vibration c~pability with cyclic symmetry. The user input/output data for loads, displacements, stresses, etc., pertain to the physical representation of the various segments of the bladed disc. The frequency-dependent applied ioads correspond to k = 2, and hence the solution loops on the circumferential harmonic index k are restricted to k = 2 only via parameters KMIN and ~1AX.
Input
1. Parameters:
In addition to general input parameters,
CVelD = +1 physical cyclic input/output data K.t.!IN = 2 minim~m cir!:ur.1ferential harmonic index KMAX = 2 maximum circumferential harmonic index NSEGS ~ 12 number of rotationally cyclic segments RPS ~ 0.0 rotational speed GKAD = FREQRESPl Spec; fy the form in whi ch the dampi n9 parameters LGKAD = +1 f are used.
2. Constraints:
Same as general input constraints.
3. Loads:
where
Results
n is the segment number, ® represents k = 2, @ represents the total number cf seqments in the bladed
disc. P is specified using RLOADi bulk data cards.
Displacement and stress output results for selected grid points and elements are presented in Figures 12. through 15. Agreement betv/een results of Figures 12-13
and Figure 9, Figure 14 and Figure 10, and Figure 15 and Figure 11 is excellent.
-35-
--
T : ;
i r
- .
1
,
I' . , :
U~-3
9
C , G
5
G ,. R J 0
I -,
I
C~i; .. ~~·;j·~·!... ,-·r·.~~c ~(:i' OF roo~ QUALln'
/\ I A\ s
p a J N 2
/1 II \\1\ T II 0 J S t' L A lE-~
C 9
E 6 11
. E 1
N e; T 5
5
/ V /
/ / ,./ /
/ //
/ ~
H
A G 3 N
I T u 0 2
E
N
C H
lE-5 1. 68 1. 76 1. 80 1. 84
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Description
This example uses the forced vibration eapabil ity I':ith cye1 ie syrrmetry. The user input/output data pertain to harmonic representation. Frequencydependent excitation is provided by both directly applied and base acceleration loads.
Input
1. Parameters:
In addition to general input parameters, CYCIO = -1 har~~nic cyclic input/output data KtUN = a minimum circumferential hat'monic index !<PAX = 2 maximum circumferential harmonic index NSEGS = 12 number of rotationally cyclic sectors RPS = 600.0 revolutions per second BXTID, BYTID, BZTID l Refer to TABLEDi bulk data cards to specify BXPTID, BYPTID, BZPiID! magnitude and phase of base acceleration
component:. GKAD = FREQRESPl Specify the form in, which damping parameters are LGKAD = +1 J used.
2. Constraints:
Same as general input constraints.
3. Loads:
a) pO,2c = A(f) specified on RLOADi bulk data cards. b) Base acceleration as shown in Figure 16 .
Results
Results are shown in Figures 17 through 25 •
Figures 17 and 18 present k = 0 results (subcase 1). The excitation consists of axial base acceleration and directly applied loads. The selected frequency band of excitation, 1700-1920 HZ, lies between the second out-of-plane disc bending mode frequency (1577 Hz, k = D, Table 2) and the first ~n-p1ane shear r.1ode fre.quency (1994 Hz, k ~ 0, Table 2). Since the excitation is parallel to the axis of rotation, only the former mode responds.
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Figures 19 through 23 present k ~ 1 results (subc~ses 2 (k :: 1c) and 3 (k :: ls)}. ihe excitation is due to lateral base acceleration only. Although the frequency band of input base acceleration is 1700-1920 Hz. the rotation of the bladed disc at: 600 Hz (pararr.etcr RPS) spl its the input bandi~idth into b~o effec t i ve band'i11 dths:
(1700 - 600) :: 1100 to (1920 - 600) :: 1320 Hz, and (1700 + 600) :: 2300 to (1920 + 600) = 2520 Hz.
The only k = 1 mode in these effective band~'iidths is the first torsional mode of the blade \'Jith the disc practically stationary (2460 Hz. k :: 1. Table 2). This is shO\>in by the out-of-plane displacement magnitudes of grid points 18 (blade) and 3 (disc) respectively (Figures 19 (k = lc) and 22 (k :: ls». The corresponding phase responses of these grid points are sh~~n in Figure 21 •
Figures 24 and 25 present k = 2 results (subcase 4 (k = 2c». The excitation consists of directly applied k = 2c loads. The out-af-plane displacement magnitude of grid point 18 (Figure 24) compares well with that obtained in example 2 (Figure l~. Table 3 lists the ollt-of-plane displacement response of grid point 18 as obtained in examples 2 and 3. The marginal difference in response in example 3 is due to the Coriolis and centripetal acceleration effects at a rotational speed of 600 revolutions per second.
No k = 2s loads are applied in this example (subcase 5).
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Figure 16: Base Acceleration Data in an Inertial Coordinate System
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TABLE 3: EFFECT OF CGHlOUS Atm CENTRIPETAL ACCELERATIONS ON THE
DISPLACEHENT RESPONSE OF GRID POINT 13 AT 600 RPS •
Example 2 Exam21e 3 Frequency Segment 1 lsuhcase 1 ~ k = 2c (subcase 4)
Hz Mao. (in)/Phase (deg) !4ag. (in)/Phase (deg)
1700 7.2655 E-5/349.4 7.6132 E-5/354.3 I I
1750 1.3071 E-4/343.1 I 1.3844 E-4/347.3 i
1778 2.1580 E-4/332.7 I 2.3252 E-4/335.8
1796 3.4139 E-4/314.6 i - 3.7252 E-4/315.2 ,
1814 4.8374 E-4/269.9 4.9177 E-4/266.8
1832 3.4146 E-4/224.9 ! 3.2555 E-4/225.5
1850 2.1451 E-4/206.6 I i
2.0742 E-4/209.3
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1920 7.6125 [-5/190.4 7.5397 E-5/194. 3
-52-
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EXA~lPLE 4
Description
This example uses the forced vibraticn capability ~'/ith cyclic s}mnetry. The user input/output pertains to physical representation. Periodic loads are specified as functions of time on the segments of the bladed disc corresponding to k = 2. For clarity of illustration only, sinusoidal loads of varying amplitudes at a frequency of i8l4 lIz are specified. The Fourier' decomposition of these sine functions obviously contains contributions from first hannonie alone (t = 1)-- the parameter Lf.1AX accordingly has been set at 1 (r. = 0, le. IS).
Iii .p.!:!!
1. Parameters;
2.
In addition to general input parameters,
CYCiO :: +1 physical cyclic input/output data KHIN :: 2 min"jmum circumferential harmonic index KHAX :: 2 maximum circumferential harmonic index L~~X = 1 maximum harmonic in the Fourier decomposition of periodic,
time-dependent loads. NSEGS :: 12 number of rotationally cyclic sectors RPS = 600.0 revolutions per second GKAD :: FREQRESPl Specify the form in which the damping parameters are
. LGKAD :: +1 fused.
Constraints:
Same as general input constraints.
3. Loads:
l'ihere n is the segment number. (g)represents k = 2. ~ represents the total number of segments in the bladed
disc. A(t) = A·sin (2n·1814·t). P is specified on TLOADi bulk data cards.
-53-
T
:i , .'
r
I' i
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Results
Results are presented in T.:ble 4 and are in good agreement \'1ith those from exampl e 3.
-54-
J J , ..
• , .'
TABLE 4: crX1P/IJUSON OF RE!i!'OIlSE AT 1814 Hz,
Grid Pt.Disp. or Ei:uITiole 3 Examole 4 " = 2c (subcase 4) Segment i "(S"Ubcasel) E1em. Stresses Mag.(in)/Phas2(deq) Haq.(in)/Pllase{deg)
8 (T3Rf1), liZ 5.4297 E-4/82.6 5.·1299 E-4/82.6
- - .... -18 (nru·!). liZ 4.9177 E-4/266.8 4.9180 E-4/266.8
11 (3), uxx ,l ..... 1.4841 E 3/84.7 1. 4842 E 3/84.7
11 (5). Uyy ,l 2.0891 E 2/83.4 2.0892 E 2/83.4
--- -
11 (7), 'xy, 1 1.0774 E 2/64.7 1.0775 E 2/64.7
11 (10). Uxx,t 1.4677 E 3/263.3 1.4678 E3/263. 3
11 (12), Uyy • 2 2.2489 E 2/260.3 2.2491 E 2/260.4
11 (14), L)'y,2 1.8510 E 2/253.0 1 .8511 E 2/253.0
* Fibre distances 1 and 2.
-55-
I Example 5 k :: 2c Tsubcase 4) l<1aQ. (in l/Phase( deq)
5.4299 E-4/82.6
.. - .- .. ----
4.9180 E-4/266.8
1. 4842 E3/84. 7
2.0892 E2/83.4 - .. - - .. -- --. _.
1.0775 E2/64.7
1. 4678 E3/263. 3
2.2491 E2/260.4 I 1.8512 E2/253.0
;/
Description
This example uses the forced vibloation capc:lbility \'lith cyclic symmetry. The user input/output pertains to han~Dnic representation. Periodic loads are specified as functions of time fOt' the circumferential harrronic index k :: 2. For clarity of illustration only, sinusoidal loads are selected.
1. Param~ters:
In addition to general input parameters.
CYCIO :: -1 hanr.onic cyclic input/output data KfHN = 2 minimum circumferential harmonic index KHAX = 2 r.aximum circumferential harmonic index LMAX = 1 maximum harmcnic in the Fourier decomposition of periodic,
time-dependent loads. NSEGS = 12 nu~er of rotationally cyclic sectors RPS = 600.0 revolutions per second GKAD = FREQRESPl Spec; fy the fOl"ID in \~hich the damping parameters LGKAD :: +1 J are used.
2. Constraints:
Same as general input constraints.
3. Loads:
p2c{t) = A.sin (2n.18l4.t)
specified on TLOAD; bulk data cards.
Results
Results are pre~ented in Table 4 and agree well with those from example 3.
-55-
5. CO::CLUSlOiiS
1. A ne~'J capabil ity has been developed and added to the general purpose finite elem~nt progrilm NASTHAtl Level 11.7 to conduct forced vibration analysis of tuned cyclic structures rotuting i)bcut their axis of symmetry.
2. The effects of Coriolis and centripetal accelerations together with those due to the translational acceleration of the axis of rotation have been included.
3. A variety cf l!SC:-' options is pr'ovlozQ to specify the loads on the rotating structure.
4. Five intel'related examples are presented to illustrate the .'arious features of this development.
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5. m:CO;"J{Ef{D.I\TIONS
1. This is a nett capsbility ar.d, therefore, the examples presented herein.
have been prili'.Jl'ily designed to il1ustl"ate the various basic features of the
developr.:ent. ApiJl icat ion to a variety of real problems would substantiCllly
contribute tC\'Jilrds determining its m~rits arid limitations \'lith regards to its
applicabil'lty, u5efull1Co;S, and savings in modelling and COiT:put~tional time.
2. The capability should be extended to conduct forced response analysis
using norrml modes with cyclic symmetiAY as the basis.
3. Inclusion of induced and applied oscillatory aerodynamic loads within
the capability i':ould be a desirable step in solving the forced vibration problems
of turbomachines.
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APPENDIX
H{f-RTIAl LOADS DUE TO BASE ACCELERATION
The acceleration of the axis of rotation generates inertial loads at all grid points of the co~p1ete structure. In this appendix, the generation of these inertial loads and their transformation to frequency-dependent circumferential harmonic components ai"e discussed.
As given by equation (27) of Section 2 • the inel·tial forces on the three translational degrees of freedom at an arbitrary point P of the modelled cyclic sector, expressed in the global (displacement) coordinate system. are
{pG} = -[M2]{~0} = [TBG]{pB} (1)
where
{pB} = F: 1 B 0 or' 0
:H~: 1 [ ~ - : m : l~ c
0 -s Pz (2)
with c = cos nt and s = sin nt.
Since ali the eycl ic sectors are identicai in all respects except for the specified loads. no general ity is lost in assu;nin':J. for s"imp1 icity, that the modelled sector is the n = 1 sector. Equation (1) can, then. be rewritten as
(3)
where c = cos (n-1 • 1 ·2n/U),- and } n (4 ) s = sin (n-1 ·1 ·21'"/11) n
substituting equation (3) in equations (5) of Section 3 , and noting that
N L cn - 0 n=l
(5)
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1
l '
I. r'
I I·
_ e~:;-.,
r cn • cos (n-T. k • 2rr/N) = lJ/2 k=l n
- 0
ISn • cos(n:T· k • 2iT/N) ::: 0 , n
ICn • sin{n:T. It. • 2rr/N) :: 0 • n
Is~ • sin(n:T. k • 2:r/1l} = N/2 n
= a
• k=l
• k~l
(5) (contd)
the circumferential ha~nic components of the base acceleration loads become
("k" = 0)
("k" ;: lc)
(6)
("k" = ls) , and
G {p1<c,ks} ;: {O}, all other "k".
In the present development, the components of ~ .se acceleraticn XO' YO and Zo are considered to be sinusoidal of frequency ~\, and are specified as
..
I Xo ;: Xo cos({j;t + 4X) , mag ..
YO = YO r:os ({j:t + 'i') , and ( (7) , mag
Zo .. ZOo cos (ut ~ ;~l) I mag ./ _ h()- ~~'\
r r ..
"
From equation (2), therefore, lte Cilfl vlr-ib~ . P B = X
P B y =
p B = Z
-mXo,mag. co~(wt + ~X}
.. -m[YO ,mag cos f··.· cos(wt + ¢'y} + Z O.mag sin nt • cos(wt ;. IPZl1 and
.. .. -m[-YO ,mag Sill., . I"os(wt + ¢y} + Zo . ,mag cos nt· cos(wt ... ¢'Z}J.
The cosine and sine products in equations (8) can be expressed in terms of individual cosine and sine terms with'freq~encies (w + n) and (w - n).
The following conclusions about base acceleration loads can, therefore, be drawn by ~ubstituting equat10ns (8) into equations (6):
1. The axial component of base acceleration, Xo(w), contributes to pO at excitation frequencies w.
.. .. 2. The lateral components of base acceletation, YO(w) and ZO(w), contribute to plc and pls at excitation frequencies (w ± n) for each w specified.
-51-
(8)
( '.
/
[
l
B
Bl
D
G ,.. ,.. ,.. I. J. K ,.. ,.. ,.. IB, JB~ KB ,.. ,.. ,.. ; , j, k
K
k
R..
!~
Ml
M2
m
N
P
Q
RO .... R -+ ,. p " , T
t
U
u
W
n
w
Damping matrix
Coriolis acceleration coefficient m3tri~
Raylei~h's dissipation function
"Symmetric Componcnts" tran5for--;nation r..atrih
unit vectors along Inertial XVZ axes
Unit vectors along Basic XB VB Zu axes
Unit vectors along Global xyz axes
Stiffness matrix
Circumferential harmonic index
Tim!! harmonic index
} (Figure 3)
Mass matrix, number of time intervals per period {Fig~re 5)
Centripetal acceleration coefficient matrix
Base acceleration coefficient matrix
Hass Nu:nber of cyclic sectors in the complete structure
Load vector
Aerodynamic coefficient matrix
B~se acceleration vector
position vectors (Figure 3)
Kinetic energy, coordinate system transformation matrix
Tim€:
Strain energy
Physical displacement degrees of freedom
Virtual \~ork
RotatioMl frequency
Forcing frequency
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B
G
K
m
n
-0
-1',C
-R.s
-kc
-ks
-W2
-N/2
StM30LS {Cootinued)
Superscripts
Basic
Giobal
Independent solution set in "syrrrnetric components"
~th tima instant
nth cyclic sector
Fouri er coeffi cients ("symmetric components")
-63-
-,
c.
I ., I. r-
1
[ . l.
2.
REFERENCES
Smith~ G. C. c., and Elchuri ~ V .• "r\l'~I'oe1astic and Dynamic Finite El€::l:ent
Analyses of ~ Bladed Shrouded Disk", NASA CR 159728, March 1980.
E1churi, V., and Smith, G. C. C., "NASTRAN Level 16 Theoretical Har.ual
Updates for Aeroelastic Analysis of G1aded Discs". l'll\SA eR 159323, !'1orch
1980.
3. E1churi, V., and Gallo, A. Hichael, "NASTRAN level 16 User's :1anual Updates for Aeroe1astic Analysis of Bladed Di:.;c::". I'IASA CR 159824. March 1980 ..
4. Gallo, A. fiichael, and Dale, B. J., "f1ASHl/{il Level 16 Prograrr.llelo 's Hanu1l1
Updates for Aeroelastic Analysis of Bladed Discs". NASA CR 159825. (·1arch
1980.
5. Elchuri, V., and Gallo. A. Nichael. "NASTRAN Level 16 Dcr.;onstration ftmuul
Updates tor Aeroelastic Analysis of Bluded Discs", W~SA cn 159826, fb.t'"ch
1980.
6. Gallo, A. Michael, Elchuri. V .• and Skalski. S. C., "Bladed-Shrouded-Disc
Aeroelastic Analyses: Computer Program Updates in NASTRAN Level 17.7".
NASA CR 165428, December 1981.
7. NASTRAN level 17.6 Theoretical Manual, W\SA SP 221 (05). October 1930~
8. Elchuri. V •• Gallo. A. Michael, and Skalski. S.C., "Forced Vibration Analysis
of Rotating Cyclic Structut'es in rlASTRAN", NASA CR 16511 29. December 19B1.
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~ ... -'
.', '
End of Document