734
ARCHIEF
h.
Noinen cia Loire
amplitude of the oath mode
E = Young's modulus
F = trasversa force
F0 = generalized force
h = beam t hickness
¡ = second moment of area of the crosssection
K. = axial spring factor
A = axial spring constant
k1 = rotational spring constant
L = beani length.
PO i iitjal axial tension
Po,,, = nondimensional amplitude of the generalized
harmonic force = time w = transverse (lisplacement z = axial coordinate = modal constants G r, r, ,G
n = nondimensional axial coordinate
X = ratio of axial load to fundamental Euler
buck-ling load
= generalized coordinate
p = mass density
nondimensional time
= assumed spatial function or linearmode shape
40 = nondimensional linear mode shape
eigenvalue of the linear free problem
w nondimensional frequency
e,0 linear natural frequency
1.
Introduction
THE
are restrained from axial displacement has received muchnonlinear transverse vibrations of a beam whose ends attention. The common approach is to assume some form for the spatial solution, usuali a linear mode shape. and then solve the nonlinear orthnarv differential equation that results for the time variable. Most of this work has been concerned with simply supported end conditions only. The majority of authors have used only a single assumed spatial function to accomplish this; however, i\feDonalcl'has solved tlw freeReceived April 14, 196iJ, revision received November:3, 1909
This work wa substantially supported by NSF Grant GR-l2.51 and this support is gratefully ackwoovledted. This work forons
part of a thesis submit ted in tart ial fultillinent of the Phi). requirement at tue University of Gichigan by J. A. Bennett.
*Assistant Professor, Department of Aeronautical arid A5t
ro-nautical Engineering. Associate AIAA.
Professor, l)epartment of Aerospace Engineering. Associate Fellow AIAA.
AIAA JOtJItNAL
Technische Hog
Deift
A Mu!
tipj e Degree-of-Freed orn Approach to Noiii mear
Beam Vibrations
JAMES A. BENNETT*
University of illinois, Urbana. ill. AND
JOE G. EIsLEYt
University of Michigan, :Inn Aibor, iIich.
The steady-state free an.! forced response and stability for large am plitimile im,tiomi of a
la-am with elaimiped ends is investigated. Elastic restraint of tIme ends is included in order to relate theory wit li exJ,erimnen t. A loi nl t ¡mode ana1 tieni amid ii omit-rica! ten nique is used to obtain theorel kai sohom Lions for!oth rcpoflse and stability. Experimnezitool results largely confirni the iestmits of the analysis. lt iseonchimdemi that, wlmfle single mode amiahst-s ame aile-(hllate iii sone ca-es, there are circumstances where a no,iltiniode analysisisessential to pre(lict the observed results.
vibìatioii l)roblcni for an arbitrary initial condition using Otii expansion of ellipt iC functions vhich are the exact solutions to the foce single mode probltnì. Srinivasant has applitd a general ¡nodal t1>i)10ttChi in discussing a siiiiply SUpl)Olte1 beam and has Ol)tZOifleCl response solutions including seveoal modes.
rin stability uf the steuztly-state solution of forced motion
has received nitoch less attention. although EL-icy3 pcintcd out that there vcre legions iii which the single mode solutin vas not stable. rhe l)1l)leni (0f the parametric exeitzct u ou
of a niode which is initially at lest has been discusse:1 in tile
r4
There has been very little exueriinental verification ui tie theoretical results whit-li have Ix-coi oblaincri itt this pioili;u Sonic eai-l- w(I)rk w:is ilone by 13uigreen. Expeìiotuuitu1
work on the itool ooeai rt-sI)cmIse nf beams to raoidono it11sot
has hecto reviewed by Lyon.°
In the investigation reported here the response and si i-hilitv of multiple mode, large amplitude, transverse vibra-tions of a beam wet-c ))relictei by approximate aooalyt icel
and nuiiierical tech io iques it oid lIon results 'vere t hei o eeuu a io I
to experimentidiv determined values. General ecouations fr the response and stability of a tioultiple dogi-ce-of-fo-en lai beam oyere derived notti results then obtained for tIte par-ticular case which descoihes the experiment. The exilen-mental beam was. ideally, clauoupcd oui both ends; however because not all tite elasticity f the support could be removed. a term to exlocss the axial elasticity of the suI)port :ouuI
another term to express elastic rotational enti restraint oyere included in the analysis.
2.
Equations of Motion
The partial clifferciotial equal ion describing the traoisve:e vibration of a beam which is axially restrained Out1 in which large deflectiojos are pertilitt ed is
ph
+ EI
-
+ K,
f'
()2
(l.c] =F(x.t) ti) where K. is a factor expressing the amount of c'id restraint
K, = L/ltE(1,"k ± L//uE) °
Thus K, = 1.0 is fully restrained, and K, = 0.0 represents no c-nd restraint,
The only rioniiuoear effect that has been included is duc to the effect of the transverse deflection on the axial foie'
,i'(I (lite to
í'iiil (esti:niil The ('(1rvature are still..tricte(l
tt
l,í izilJ nui! hotu 1iear (1 foirant ions andrnlitudin:tl iraitia have been iteilected. )ssurlie tini t
w(x,t) =
t»(x)
(2)i I pecrfy that f,(x) satisfies the asociateti linear problem
ò2w ò4w - ò2w
p/i
+ LI4 P0---=O
iii! the appropriate houralarv cOn(litiOIis. It is therefore pi:aiiil to include :inv elastic rotational end restraint in t-lie
irr ear problem. Then, applying Galerkin's method, the
i hawing set of nonlinear ordiiiarv differential equations are
li!I n irted (IT
+
+
/:=1 =1
= Fo(r)
S
=
-m = 1.2, .1
F1,, = (IL2//l)'J!,, ÇI cWe d '1 d24,.j(117
j
- 4),,d(7 2 f01 ,,2dr Lf
F(17,r),.d17 FO Elif
2d17lrer the following itondirnensiorial quantities have been
loblIed:
r = (Ejp)"t/L, i = x/L,
ç5,,, =3.
Re.ponse
SilIce there is no known exact solution to Eq. (3), it will be 0cU5rlrV to t run to an a pIoxuaat e solut ion. The method
1 harmonic hialarice will he used. The method involves
Sinnig
traed S((lulroiI into the ciittoi't'niiai equations, and thena solution of sunirned liarnionics, substituting thisqlIitirg the cuofficients of the harmonics to zero. Aflv
l:trrr(oniCS which arise in the sulc.t itution which aie not in-i) il It') iiii t h t 115 li 1110(1 soli it 1(11) a re OCirjeet 0(1. The
effective-res ((f t his root h od is de1 )eltder0 on clIm)siiig only t hose
bar-001005 which will he irijirortalit. It has been found, for lie Ditffing equation obtained for a single assumed mode,
lait a single harmoiue terni iii the tuile expansion gives rorirate restilts over a wide range of interest. It is siso ai ((VO that when clari ping is r reglected, only in pitase and
rit of phase iilOTiOtls alise.
Thus for hair tonic forcing. assume a solution of the form
A,,, coswr (4)
Ñu (Stil uting Eq. (4) inI o Eq. (3) and neglecting the terms rt contain cos3wr, the following is obtained:
-.1 ,,,w ± F1,,J,, ± K
.ìf,,tAk.-l1A, = P0(5)
'h i5 set uf
j
noi titi tea r algol 0:00 equations tel: (till g the s and w may be solved oit i digital computer by the\egst ciii iteration teiluiiqrue. 't lie sülritiuliS \Vet'e staited r' a given initial w sin tire nrlitinuprinite single mode equa-ion. The Sirlul uris tvere til101 cunitinucd by inirieroentig (3)
Fig. i Sv
nl-metric forcing.
lint
and thirdmodes, O W/WO 2.0, X = 0.0. 6 4OL o .( 20
.\I'UiL 1970 i.; ltEE-OE'-FIF:EDo[ AFI'].tOÂC'Il Ti) HEA\I \'Iit1t.-T1ONS 735
4.
Response for a Particular Example
For the case unclei' consideration, tile equations for the first tiuee iitodes eau lie eXi)ress('(i in the following form. The coefficients may be evalu:ited liv uiunieiical integration :utd die fart th:it sever:ti of tite coefficients are zero has been used to simplify tite equations.
dt/dr1 + F111 + K,,[G11113 + Grir2Eu +
± GrriEt2r + G333E33] = P01 eoswr (6)
d22/dr2 + FrEr + K,,[G21212 + Gnn32i +
Gr.'ni3 + GriEi] = P02 coswr
d2E(,'c/r2 + -I- I' [G21113 +
-i-Gri2t + G112123 + (/n:ri3] = P03 coswr
= JI,,,,, -1-
±
G,,,0,, = -+- 1l[,,,. + ill,,,,1, + .II,,,,0 ± + i1I,,,,,,
Tri general, two types of problems arise: 1) symmetric forcing, arid 2) asymntetric forcing.
.A. Svnimetric Forcing
If tlie force is al(i)lieci at L/2 or if it is svinrrietric about L2, I'02 is identically zero as are all tite Po,,'s for the even
nnlinbele(l niodes.
1'lte respoirse for the single mude has been discussed ex-tensivclv ii-i the hiteratuu'e and is tite typical respoilse foi' the liard siing oscillattu. Since the geneialized forces fou' the even numbered nrodes ar'e zero, tite s((Iut ion ovili indicate that these modes will riot respond. The uesponse for the first nui! third modes is shown in Fig. i fra' O < w/wo < 2.0, and ii Fig. 2 fou' 5.4 < w/wo < 7.0, where W is the hncai'
mmatur:ul freqctcrtev of the first mude, The expected funda-meritai resonances of A1 in tire neighborhood of the first lir(ear
ln:(tUl'rtl frequency occur. however, there is also ait A3
resorirutce in the neighiboriroocl nf t ire first lineau' nat ural fre-queniev and ari A1 resonance in the uteighborhood of tite third
lirinir ui:iiur'al frequency. This type of resta ance cicle tc)
itou-linear' coupling will be calici! ''coupling resonance." At. w- w0 = 1.3 the couplin g resonance c:uses appi'uxiniatelv a 2% distortion iii the spatial respomise as shown iii Fig. 3. The elIcit ii tile first lutent' mintirunal freqruenicv region is to
li:iti eri the u'esponse shape r(S the frequency increases. The
0 0.4 08 ii 6 20
w arid using the solutions for the previous w as initial guesses. Since (lily teal values uf the A ,,'s oyere important, no attempt was iliade to find cornplex values oh the .1 ,,'s.
0 04
- (N
PHASE7:36 o 20 BO 40 30 20 - Io Ui o o -J aIo -08 0.6
-::
-
IN PHASE OUT 0F PHASE 5.4 58 62 6 6 70 3 54 58 62 66 70 w/Ujratio of A3/Á increases willi increasing w/w3 in the region
of the first mode resonance, I lius, the contribution of :13 to the response shape increases for increasing w/w0 in thisregion. B. Asyrniietric Forcing
When the beam is forced asyJ3lmetricaIlv, the second mode enters into the response. This response will he generated by forcing the beam at L/4.
The response for modes i and 2 is shown in Fig. 4. Of
particular interest is the absence of a coupling resonance.
Although there are three separate branches of the curve in
this area, there is no significant increase in amplitude of either mode in the resonance region of the otherniode. This can be explained by examining the coupling terms of the three modes. n the equations for modes i and 3, there is a term of the type in the former and ' in the latter. If,
say, becomes large, tue tenu in the third mode equation
vill also become relatively large and will significantly influence
the response. However, the coefficients of the and 33
terms in the second niode equation arezero; thus, there is no significant coupling between the resonant 0(1(1 nnsle.s. In fact, the coupling between the first and second,and the third and second modes is so weak that foi' all practical purposes the nonlinear coupling termsnmv be dropped and the SeCofld-)iìode response calculated from
F13 + K,G2A22 - P/A2
(7)Comparing this response with the complete thì'ee mode response, differences of less than 1% were noted. If this simplification is made, the multiple solutions for corre-sponchng to the multiple solutions for and do not occur.
EXPERIMENTAL POINTS
O W,6)0 13
Q - I_O
(4'(.j-.3
(THEORETICAL)-FIRST AND 1,0 (THEORETICAL) THIRD MODES
Fig. 2. Syniniclric forcing, first and
third modes, 5.1
ca/wo < 7.0, X = 0.0.
FIRST MODE
01 02 03 04 05
Fig. 3 Response shapes, uirt modercsomla'lee, X = (LO
J. A. BENNETT AND J. G. EISLEV AIAA JOLIJINAL
60-40 b
4
C 20 3 O 0.5 1.0 .5 20- IN
PHASE OUT OF PHASE 0.5 IO 1.5 20If higlier-omder lilodles were included, this pattern of coil-Plimig vould COlItÌIÌmmeexcept that the even modes would also
be coupled through the cubic term. It is interesting to note that the type, of coupling and thus the type and number of coupling resonances are dependent. on the boundary conlli-tions. Reference to Eq. (3) indicates that the coefficient of this term in lime 'ììOh equal ion for /:th mode coupling is
governed by ami integral of the form
fi d,,
Jo mi
If time beani is simply supported at both ends, this term is identically zero foi' uil choices ofin k, silice the linear nmude
shapes are ,, sin(nlri7). Thus, theie will be rio coupling
resonances in a benin which is simply sui)ported at both ends. On the ol hei- hand, if one end is clamped amici one is simply supported, the evaluation of this integral indicate- that all miìodes are coupled through the cubic termn.
A common approximation to the clamped-clamped beam is
= i - cos(2nlri7) j
Evaluation of time integral in Eq. (3) for this mode shape indicates that none of the mundes are coupled through the cubic term. This indicates that. one must he extremely careful in the choice of mode shapes and the number of modes that are retained.
5.
Stability
Although a solution lias been found to the steady-state
problem there is neither assurance that these solutions are unique nor that they ame stable solutions because of the non-linearity of the equations. Tlnms, it. will be necessary to
check the stability of these solutions and to investigate tl'ie possibility of further steady-state solutions.
The stability question will he invest igateli by studying the behavior of a small perturbation of time steady-stateresponse.
Let
E =
+ ô where is ihe steady state solution forEq. (3), and 6,. is a small perturbation of the oith moIld.
Substituting tins into Eq.(3), and 1-etainingonly first-om-der teims, the following is obtained.
+
+ K,
?
+
+
= O (s)For the st ability pi-0h11-mn only the svmmnet nc case will iIC
discussed. In time case of a single assumed niocic this
equa-Fig. 4 Asyinnietric
forcing, first amid
sec-ond modes, O w/o
r.
APIUL P)70 1JEG1lEE-OF-I'1EElU_\1 ÀPPIì.OACH TU BEÀ\l VIBUATIUNS 737
Skull
reduces to a Maihleu type eqiation whose stability is weil known. 1f just two tundes are included the following p:ln of _\iflni'ii eqilat Ions occur:+
3KA
+
cot1
El = O(9)
+
+
Ki12
+
K,,Gn:1i2cos11ö = O
where T = 2wr. These two \1athieu type equations aie
un-coni tieti ai al the first one is ii lei t ical to t lie asc tor 011e itiode.
It i known that the solutions to the Mathicu equation ill the 1cgioiIs of instahility are of the form c° [ß(i) J. If 3(Ï) is expanded in powers of a small caralnettr, the leading terni is periodic with period 47r/u, where n is the integer number of the stability regiohl. This is discussed in detail by Havashi.9 Thus, the instabilities that arise must be of the form
A,,, Cùsflair
or
n > I this respomise is called ultraharmonic. Thus, the ñrst instability realoil iii the '\l:tthieu diagram mal)s into tite junii iiistahilitv region in tite _-!, plane. 'l'ue subse-quent legions are identified with tite integer orderultra-liarnionics. If tIare is sonic damping in the system, only the lower circler ultraharnioitics eau he expected to arise. Two distinct types of instability arise:
first niocle instability,
= .ti coso icr, = O, n = 1,2,3
second mode instability,
= O ö2 = .12 COST1WT, ci = 1,2,3
flie a 111 ezlra ncc of a secI cii cl in cale is of part i culai ii iterest
because it is itormttallv at rest. 1f only the linear termos are coiiiilcred, the rest mode must stay at rest; but, due to the
ni ciili n e: ir ecciti cli 11g, t he rest molle imiv be excited. This
type (cf beli:ìvior has been discussed by the authors iO a
previous paper4 in sonic detail. The stability regions for moles I antI 2 only tie lreemtted iii Fig. 5.
lor the case witcit all three inccdes are considered, the hlrrtitrhatioil equations vi1l take tile formii
+
+
K(.3.h1
+
(132
+
+ ('o1)+
[K.(i3o4iA3±
3C1.13;l3\ - 1+
»1.0 + cost) = 0 (10)-7+Ï_2[P2+
K.(ju1±
X (1.0 + cosT)]2 = O+
ji,[i:c
+
1(3c3..1.i2+
+
G:ccoliA:i) XI(l +
eot)}2
42[' .(( 3I13
+
G1A32
+
+
cosI)}i
=X nl e t hat the seem id (ql tat ii cmi is in dei ien dci it of t lie fi ist.
and third. It is ccf the Miii neu type and uutv be si>1yd in a
lii:iitner :tcl:ilogous ti) tlt:ct uecl before. When tIce third no cile i ii icluded, the iii t crcept of the second imistahi ht y
re-STABILITY BOUNOAPIES
- FIRST MODE SECOND MODE
1"ig. 5
Svniinetric forcing, iiitabilitv regions for first
and second modes, X 0.0. Flags are on tite stable side of I)mIfl(Iary.
gion of the second 111011e with the respomise curve shifts
about lower in w!w0. The effect oit the third and
higher regions is iitfinitesimal.
The coupled Matliicoc type equations of the first and third
10(11 les may he ahi:tivze I Icy a cl irecit numerical applicaI ion of Flcw1imet thieorv»° This IicVcdl\TS integrating the equivalent first order differential eqciatiomis over a period to assemtihle a
moat mix, c:illecl the mt iccnoc horny mat mix, amid then obtaining its eigenvalues. l'ue eigeitv:clues directly determine the
sta-bilitv of t he point under eomisideratiomì. The method only analyzes one point on the respomise curve at a time so it is
necessary to determicte tite iiìstability POimits of several re-sponse curves iii order to cccnstruct regions of instability. Stability boimmìdaries fur specific cases are given later in the
repoi't 11h11 are discussed :ihciig vithì the experimental results
6.
Experiimteiital Eiuipuient
An experiment \as designed to pm'oyide data to augment
the theorci ical results 101(1 to iim'ovide insight itito the accuracy of the theoretical llreclictic)mts. The primai'y information
de-sued waS 15 i'(Sl.)d)its(' curve inclinI i ng regions of instability for a beatii. The experiìmieittal edluihciliehit is discussed muore fully
in Ref. 7.
The heino was cccmistriic'ted of tool steel, 10 in. X i in. X
0.0313 iii. It. ivas clamtcpcd in steel blocks G iii. X 6 in. X 4), in. Tice claniciiIg surface was 2 in. X i in. The beam
was hcd'aterl by three steel in. diaiuctci' pins. The corre-spomichmng holes in the bc:uic were individually reamed to fit
the ]ciilS. The jaw vois belch with three - n. diameter steel
bolts. The rciocthitutig blocks ivere placed in the ways of a
l:tthe bed 1111(1 held ht pcIit ion hn a c.l:imittiing plate undermìeath
the ways. Omce end remtuin ed attached, but the other end
t) })p&'ltIiI('ri ial -ellen Id i i(: .-1ioi'selìoe magnet,
I 1li(('(' Id' I'(WI C t i,, er a mii p J i tier, fl.fm ii ict mii
geii-4'I,lì Loi. Eil i,.1ilaeeiiieii t iisoi', Fli I ter, Gfret1tieitcy coil ii ter, i 1-osci I locope.
r-s J. A. BlNNETT AND J. G. E[ALEY
80 - THEORETICAL RESPONSE
o EXPERiMENTAL RESPONSE
ig. 7 S'ininctric forcing, (li..pIaceliIent at L/2, X 0.5.
av lie loosened to set tite juil ial tension. Two i in. X 3 in. eel bars were bolt ed to the lop urfaees of theblocks to in-ease the bend ng rigidity ut i lie setup. I)espite all attempts restrain the ends of the beam troni moving toward each ther. it ivas discovered that tite elasticity of lite test rig was significant factor. This suggested that improperly ev:ilu-ted boundary conditions nitty ben prime source of error in revious experinielils of this ty e. The positioii of the
iiiov-block was recilated by a screw artaiigenient atid then )lock was clanipcd into poii)u1t. Ihe initial axial load
irthc beam was measured by two foil-type strain gages laced on opposite sides of the benno 1.5 iii. from tile end.
The beam vas forced by an electi'oniagnet asshown iii the
rheinatic diagram of Fig. 6. The magnet was designed
itch that the motion of the henni would not significantly ffect the force applied to the benin. The magnet was
sus-ei ided on tina wires so that the siiiuoiid:ti torce applied to
he beni u could be i neti sil red (h reel i by phi cing a i i a e
icier-meter on the magnet. The niaguhl udc of the sitiusou.lal
arce could be kept COli st [liii by (01 it iitioiiIv monitoring the
ccelerometer unti at.
The nmgnet actually applied a forcing function of lIte type F(r) (1 ± COsCeT)
Ifa single mode response is sought and a solution is assumed 1 the form = d1 + A1 coso,r whered1 is a constant, after alanciug tite harmonics the tollownig equations result: w2 4 + F11A + G111K(d12A1 + -_1) - P01 = O
F11d1 + KG111(d13 ± iAl2d1) - P01 = O
Pear
the first-mode resonance region J1 becomes large it is possible to neglect d2 compared to J12. Or
w2,l1 + FiAi + -G111KA13 - = O
(12)
F11d1 + K,G111(Al2d1) - Pot O
'his essentially uncoupies tite equatiulis in tite resonance egion, tinci also iidicaies that the sinusoidal amplitude is lentical to the (ase for llore simtusoh ial foreiig. Outside of le resonance region the respomise is determni ted by tite linear ither litait the nonlinear terni.
- 4 o 60 01
-- 20
0.4 00 6 20w/w-Wig. 8 Svniiiictric foiing, il i-placenicì t at L/2. X= 0.0.
8.0- - THEORETICAL RESPONSE 80 6,0 40 O EXPERIMENTAL RESPONSE THE ORET(CAL STABILITY BOUNDARIES FIRST MODE SECOND MODE THIRD MODE WIW'
'l'bemefore the siiiusoicltil part of tite iesi)t)ll-e Thotol(l oc-cutately indicate the leshiOlise to pille sinUS(ii(ltl forcing. l(1t1L0t iolis (1 i ) wele solyei.l a u Ii teilen liv tiid iiidicat ed that. tuis itl)i(li)XiiiiLitiotl \vtIs ttt curate to iviiiiiii
i'lii disitlacememit was uieasured by a Beiitiv-Nevada etidv
clirm.eltt poxiineter. fiit output ivtis liiiearly r(ltte(l tu
lis-ihtcenl( I i over the ra i i ge of ci is dto ce t t ient s t I i a t w ere t et u h.
Ihe iiistztiiilities of the first itiode treme erisilv detecteth by
tite tiisiti:iitintnt mneasurillg (sluihimlient. Flie instabilities of t lie iiigltel mii()des were lI(.(t tiiu':tvs iel1 (lefilic(1 111 the dis-i(iacelio(it t ITI(C. Often it iia i)(sshi)le to see a disturbaiìce.
luit its frequen cv a ud sua PC w(re (Ii iscule. To handle t his. the outituit was filtered by a baini pass filter. The pass band ctuld l)c set. in the neigliborhootl of the litiiuu:il frequency of the n iode whose stai ditty was u nier quest it tui. The filtered
lesilonse would COlIthilul itliluttlitY tite COltiht(tl(ellts tit titis frequemicv. The order of t he ltarmttunic cotib 1 he Casi lv
deter-in med 1 tv Co mìs ci rim ig lit e flit ere I :111(1 un fi lt crei i resi (((Ilse.
Tliu. lit e i tart iculti r i i it tue that was linstal tie ut tuid i e t let
er-tIll ted I ireciseiv by 1nov ii g the t lis1 ditcetnei it noxtitiel ir along
tite henni amid observing the phn.e chamiges of tise filtered
rispo list..
'l'he :txial spring (((listant k was deterntined by measuring tite axial simili itt tite beam with strain g:tges aun! coruittarilie the actnttl stretchiutg of the iw:tiii with tite stretching which wouhi have beets devchtped if tite cutis were eolnpleteiv re-strained. For the ease at hauni a value of K, = 0.9 ivas
(thtained. The rotational spring ittmistant k1 was deterniitied
by comitparing the experi nieltttd static deflectioit curve ivitli
the niirtivl ical solution for elastic rotatittn:tl restraints.
7.
Experimental Results
Three (lifferent values of X, tite ittititti tension rttt io, were
used:
X = 0.0, X = 0.5, X = +0.5.
The theoretical andexperimental resitt)mlses for X = 0.5, 0.0, 0.5 tite plot teti in
Figs. 7, S, and 9, respectiyelv. 'l'be experiutietttal forcing corresiuomided to a value of Pg = 5.0 X 1O° atol this volite
EXPERIMENTAL STABILITY
AIAA JOEI1NAL
Fig. 9
Syin-iSle Iii e foie i 11g. (I i .})I Lire111 ei i t a t
L/2, X = 0.5.
MODE UL TR AH ARMONIO
Fig. lO 'I'Iìeoret irai litai eeriIfleii tIti illstIlbilit ie. symmetric forcing. X = 0.0. i']ag are oit tite REable ide
of Itomiriilarv.
A.
- THEORETICAL RESPONSE
o EXPERIMENTAL RESPONSE 80
Wa u.e(i j it ti) e t liet ut't i (t1 1 (ti lit I t011. 11)1' t li t't )ret lea I iii it I
('Xieiii]ieIItal tiiliiIitV i)'tl1OklijOSfat X O.() :tit' PJ'eseIttC(I in Fig. IO. IIie re-ui1t at X = 0.0iVili lie tuiti.sd i)riliarily
jitee tite rt-ult far ill iaIue uf X ivere similar.
.\. t.JIIÌI)(IiL = 0.0
The íxiti'titiieiittl )itits a2t('(' it1i tite tittatt'iittl Ct1iVtS
(pute ii-t'I1 vjth sottie tIis(lt'i)jl)(\ lit tIR' higitet ìtlltitlltll(leS. .I'hi i-tII 1)1 Iiscusse(I iii turtits of the iuistahiIitieu.
1'jlst ititait iilsta})iiiti(S \\-t't'e Ol)erve(1 expeiitiieiìtallv in t\Vt, tìetìs. fhe Jililiji t)ItCltOltt('lìOfl was tuliseiveti both for
i i Cieli0t t g t ti (I u I ec icasi ii g c wc,. }ùi i n cleasing w"o, fit e
'I :'- i'-itit'li exelut les dì ln)i1ig (loes not pitliet an inst ability
1_)oint. so there citi he tat coitflittiatioti of this. 1he 1)Oiflt of iilst:tl)ility fc)!' (leireItsilig Wí(_i1 is the point ctf ver1iitI
tan-getì(y. l'bis point is tattfittited liv the expetinteittal iesuiis.
The first ttiode Was ¿uso tuistahie as a seconul-order Ultra-hIU'inolu i('.
'Ihu' su'uuuittl rituulu shuuwi'd uiti'tharmoiucs (uf the second aiid
tu ¿ru i tut_u 1er. 'Ib e tb tu l-u titi ('r i,ilt rti au rmuul t iu_' accu n'veci il_i ait 0X1 reiuiey lOt)'tOW i';uitgt' t) /úo. Ait example .s shoivit in
Fig. I i . lite sepu uti-tu ru i et' uit ridai imam e (ut'eurre(i at a lower
frequency thiit was tiuearetieallv predicteul. Titis disci'ep-atuev vas nuute(l tat ¿ti! values of X. The itstthilitv of
siantuieaiut ituiigiuiluaie attui Ivas visible in the respinse trace.
It tru ut litceti a tut ut le sha t hat ;v: t. nuit v in met rie, as Fig.
1 2 shows. Since energy mut be gu ti tug it to tite asti lintion of lite sceotirl niocle, it titlist bu' taken frani the first or titirul
ituode. Thus, a (Ieei'ease in first tutode aunulituuIe froto that u'('ulicted frctrn the stu.'utIy-state theory ivaulul be 'x1uecteui. 'l'bis effect wits observed ex}tet'imeit1tIlv as was tiateti eiti'lier,
.nupe the expei'i uteiutuI iuiut ttil below the theoretical curve
in the legialt where tite seetuuui lituu'au' tuuuude is uttustable. This uiu.taIuiity pt'i'sisleui ututil the juli11) at_purred, ittu[ieatiuig tinut
i
5l»1\' del eri i iii uc t h e j um p i h enomeru on, it 'vu tu lui be cessaty to iia'Iuule the secotid uttocie instabilities in the analysis.'l'ue third mode huiwetl fourth- and fifth-artier uiti'ahar-tu untie bust thilit. 'flue utuaguu it udc of tite ii ustiuhil it ics vei'e tuoi as great ¿SS t itt' secautd nuuucle second order uiItl'aharmOu lic,
hut. the fourth order third niode uIt i'aharttuonic was notice-able ill the l'espanse over a ratIgo tuf ahatut Otte C5 (Fig. 13). The ittuver-order ultru1ucriuuottics of the third nuode occurreti
1ìEG1EE-oF-FiI:EI)oM AI1Pi(JAC'll 'It) BE.'s.\I VIBRATIONS 739
DIS PL4CEMENT T O.25L D SPLA CE MENT T 075L I:) I Secoiìil-order ultra-i la rulITt It nf Lhe .te()ll(I ttitttle, X = 0.0. ¿SPL AC E MEN T .7 O25i. SPL ACE MEN T AT O75L WILt,-'
after the j nui lt:u nul were not u.lt't.eci able ¿uit huuugh tlueoret ucu]ly
tue instability regions exist. Lainlela = +0.5
'lite cOcci (uf positive X is tu) (iCi'i't'itSi' the iifleii' natural
fru'qu.uetucv nuid tut increase the hurdeuuing effect. The saune iuistzihi!jties (iisCU5Se(1 iii tite X = 00 section ivcre observed for X = +0.5.
Lainiula = 0.5
The effet_t of negative X is to increase the linear ivatuiu'al frequenc''iuud to clect'cae the nonlinear harcleiuing. 'flic
u lecrease itu the nu uiuline:u r effect ¿tIsa u hc'crc'aseul theampht udes ut tite fifth uuu-ulei' ultraiutriuuuuuuic of the third moule ¿md the tu iì'ui-order uit rthari u it_uit ic of t lie second nucu(ie to the point that they weu'e not observeti.
8.
Conclusions
Tite u'estilts of the experiuiuental and theoretical studies ii udicat e dutt tite single- t cita harmonic b:tlauuce t euh tu ique gives excellent. aphuroxinu:ute solutions far tlue auutpiitituies ciucountercul iii tite beam pi'tti)letui (tuice the henni thiu'kness).
Hou'ever, I lue sui gie sptt ial ii toile ai upraach uutt_ist be used
with soute e:uut io n. In uru ti dei ut s in w hit_lu the liait h tu ea r
cOlil)li tug is weak, it is P tssii)le t o abt ¿ti iu quit e au'cun'ate
l'espanse u-itt-ves by (lt-utili tug tite nonli tuent' coupling terms
atid cousit lent ig each nut tilt' iuuuliv i(lulailv tu n' response pin'poses. However, itu titutse cases ivluei'e the euuefficient of the cubic coupling tet w is ]ion zeu'o, cautu thug i'esoivaiiee will ill)! tear
which ittay significantly iuuuudif' tite t'espoilse shapes.
lt is also necesstu'v to u'uuiusiíler the PosSibilitY of modal
itustabilities. A matie iiu:uv i-esiuuunul itu an uultrahai'nuouuic type of response nt sottie iututlt i1 tie at tite fou-eiuug frcqtieuicy. It is
further utecessat'y to i'atusiu lei' tise stability of the rest runucies
because thiu'y uiva\' uuscillat u' lite tu) ltuu'anuetnie excitation.
References
I \ [e L) ut u titi, P, I I-,' Nuu u ii u uc'ai i )vi tLt.Ifli(' Couupliiug in a Bc'uiitt
\'ibi'itioui,'' .Jouur,u,uul of .tppliuul iIuc/ucinics, \'ol, 22, Nut, 4, l)ee. 1955, pp. 573-37-S,
i Srinivasau, A. \',, 'Nouil'utear \'ittrat.iouus of Beams ¿uttd Plate5,'' ¡,uluu-ru,ational .Iouuu-n(u/ of Von/un-ar 11cc/tunics, \ol. I, Nu, 3, Nov, 1960, pii, i 71)-191.
'J Eislev, J, (l,, 'Nuuuutiuuc-ur \'ibrations of Beams ¿nid
lice-t,Aitgiuliui' Phaies, Z' Ouu/uuu/'t tutu' ttugu uu'u.tuuul(u 5fat/tuuuuotit tutu']
I'/uuisu/,-, \'al. I 5, Nut. 2, \ i uuulu i 964, i)J' i 67-174.
4 F:isle', J. ( 'tuuul Itc'uuiut'II ,.1. X,, ''Staluiiitv (uf Large
Atuu1uli-liude 1"onu'eul Sluuthuuu uf u iiiuuiy Suiiupuui't.cul Beato,'' In/duri uil tonal
Io ut'ri uit of .\'ouu ¡¡ru tar .1 (u.u_/tui uit rs, t tu be pu ti uti,-lted.
Buiguu'cuu, I),, ''Fi-ce \'ibr:ut iuuius uuf a Piti Euudu'd Cairumn vith
Cuuttstuuttt I )itaiuu'c Bet weert Euuuis,'' .Jotuu'nutl ni tjtp/uu/ ,tiutiuuin-t'cs, Voi. iS, No. 2, Jutai' 1)1St. up 133-139.
6 Lvuuut, lt. ti., ''Ulusevvuuiuuus tutu Outs [hule tuf Noutlitseanit ¿us
ltanduuiuu Viluniui ions u uf Si ruait ires,'' TN D-1X72, \ [ aix-h 1963, NASA.
Beuutuett, .1, A., ''A ,\iuuir iplu' \luuule Appi'oaeh to Nu,nliuueat'
Beam Vi! ur:tu iuuuus,' ' Pit. I ) , i.lueis, 1)161), U utiv, u uf ?uliu'ltigan, Ann Au'tuuur, Mich,
iruuve, W. E,, ¡lu-lui Xtuuutuu'icl 5[ulhocl,, Preuutii'e-lI:uti, Euugtt.'i-uuui (htfs, N.J., 1966.
it t\'isshti , C' .,V out t itt ut u' O,'uriUutt ¡unis tri P/u tus irs! Sijstu nus,
Mt'Gr:uw-ihil, Nu'w Yuurk. I'.t64, pp. 59-92,
° Pa rs, L.,-\,., ,1'I'ruat u-su u ru .1 ivalìl ucti/ Dytuuuuit iCS, Heiuìeman
Eduicat it uit ti Bu uuuks, Lou uuiu uit' 11)63, pp. 461--463.
-I II
tIItItIl1I1tt1i( Uf t h C CC4)fl ti in oti C,
RESPOiSE Fig. 13
F'oiirtli-oreler
liltraliar-iiìi)flc f tite
thirti moite, X =
ES PONS E
X = 0M. F ILTE RED 0.0. LT ERE D
