The influence of an internal resonance on non-linear structural vibrations under subharmonic resonance conditions

Journal of Sound and Vibration (1985) 102(4), 473-492

THE INFLUENCE OF AN INTERNAL RESONANCE o N NON-LINEAR STRUCTURAL VIBRATIONS UNDER SUBHARMONIC RESONANCE CONDITIONS D. T. MOOK

Department of Engineering Science and Mechanics R. H. PLAUT

Department of Civil Engineering AND N. HAQUANG

Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, U.S.A. (Received 28 February 1984, and in revised form 5 September 1984) The response of structural elements to a simple harmonic, transverse excitation is considered. The effects of both initial curvature and midsurface stretching are included; thus, the governing equations contain both quadratic and cubic terms. A perturbation technique', the method of multiple scales, is used to determine the response. Attention is focused on the subharmonir resonance n ~ 2~Oz, where ~ is the forcing frequency and w2 is a natural frequency. If the system possesses an internal resonance of the form oJ2-~2or, energy may be transferred from the second mode to the first mode. The structural response is investigated in the presence, and in the absence, of this internal resonance. A comparison of the results reveals that the amplitude of the response can be significantly reduced by the presence of such an internal resonance. This suggests a means of passive vibration control. Also, the internal resonance causes a saturation phenomenon and a role reversal between the directly and indirectly excited modes.

1. INTRODUCTION In studying vibrations o f structural elements (such as beams, arches, plates, and shells), one typically expresses the deflection as an expansion, using the linear, free-vibration m o d e s as the basis functions:

w(r, t)= Z u.Ct),~.(r).

(1)

n=l

Here w represents the deflection, r the position o f a material point in the u n d e f o r m e d state, t the time, a n d 4~, the free-vibration modes. The time-dependent coefficients in expansion (1), u,(t), m a y be viewed as the generalized co-ordinates. Typically, the equations governing the u, have the following form:

j

+h,(t;

1 k=l

e)=0, n= 1,2,.... 473 0022-460x/85/200473 + 20 $03.00]0

j=l

k = ! I=1

(2) O 1985 Academic Press Inc. (London) Limited

474

D.T.

MOOK, R. H. P L A U T A N D N. H A Q U A N G

The dots represent derivatives with respect to time, the to, are the natural frequencies (i.e., the frequencies of infinitesimal amplitude, unforced, undamped vibrations), e is a small, constant parameter, the A~kn, ~ , and F~kz, are constants, and the h, represent the forcing terms. Here the convention is adopted that ( o , < ( o 2 < ( o 3 < . - - . The quadratic terms are due to curvature, which may be present in the unloaded shape (e.g., arches and shells) and/or the result of a static load. The cubic terms are due to stretching of the neutral axis, or midplane, or midsurface. Here the following form is considered for the forcing function:

h,(t; e) = - 2 K , ( e ) cos/2t.

(3)

/2 and the K, are independent of t. Two cases are to be considered: (i) s neat 2co2 and 2co~ not near oJ2; (ii)/2 near 2(o2 and 2co~ near OJz. For treatments of some other cases, the reader is referred to the book by Nayfeh and Mook [1]. Uniformly valid (as t~oo), approximate (for small, but finite, e) solutions of equation (2) are obtained, by using the method of multiple scales [2]. Also some solutions obtained by numerical integration are presented. The general form of the approximate solution of equations (2) and (3) is the sum of the solution of the homogeneous part of equations (2) plus a particular solution, even when non-linear terms are taken into account. However, unlike the solution of the linearized form of equation (2), the homogeneous solution does not always decay. Under some conditions, it approaches a steady state (i.e., a condition in which amplitude and phase are constants). Under other conditions, a steady state never develops and energy is continuously exchanged between some of the modes. The present analysis reveals how the modes begin to interact when the amplitude of the response grows beyond the range described by the solution of the linearized problem.

2. GENERAL NOTIONS OF THE ANALYSIS Following the method of multiple scales, one assumes that an approximation to the solution can be expressed in the form

u.(t; e)~u.o(To, 7"1. . . . )+eu.l(To, T I , . . . ) + " " , T . = e"t.

(4~5)

Each function u.~ is independent of e. The original independent variable (or scale) is replaced with multiple scales; consequently, the derivatives with respect to t become expansions in terms of the partial derivatives with respect to the 7".:

d/ dt = Do+ eD~ + e2D2 +" " . , d2 dt----~ = D~ + e2DoDt + e2(2DoD2 + D~) + . . . , D,, = O/OT,.

(6a) (6b, c)

The number of scales needed to form the approximation is determined by the humber of terms in, or the order of, the approximation. A one-term expansion requires two scales, a two-term expansion requires three, etc. Substituting equations (4)-(6) into equation (2) and then equating the coefficients of each power of e to zero, one obtains z _ 2K. cos O T o , Dou.o+ (o.U.o-

Do",,,+ a,,u.,-- -2(DoD,+ ~,,Do)u.o- ~ ~. A~k.UjoUko, 2

2

m

j ~ 1 k=1

(7)

(8)

INTERNAL 2

RESONANCE

NON-LINEAR

475

INFLUENCE

2

Dou.2 + to.u.2 - -2( DoDl + #.Do) u,i - (2DoD2 + D2 + 2##D~)u.o

where A~. = Ajk. + Akj,.

(10)

The solution of equation (7) can be written in the familiar form U~o= a. cos (to,,To+fl.)+{2K,,/(to2.-a2)} cos/2To,

(11)

where a. and ft. are arbitrary functions of the T. for n ~>1. Instead of equation (11), it is convenient to express the solution of equation (7) in complex notation: [i(w. To+ ft.)] + exp [-i(w.To + ft.)]} + {KJ(w~ -/22)}

U.o = 89

x[exp (i/2To)+exp (-i/2To)] = A. exp (ito.To) + E. exp (i/2To) + co,

(12)

where A.(T1, T2,...) =89 exp (iB.), E. = K./(~,~ -/22).

(13, 14)

cc represents the complex conjugate of the preceding terms. At this point, the complex functions, A., of the so-called slow scales, 7". for n > 0, are arbitrary. Subsequently, they will be chosen in such a way that renders the expansion uniformly valid. The E~ are real constants. I f / 2 is near one of the to. (i.e., there is the familiar resonance that one is used to treating in linear analyses), then E. becomes very large; it is called a small divisor term. With the method of multiple scales, small divisor terms are a manifestation of resonance. Because the small divisor appears in the first term of the expansion w h e n / 2 is near w., this is called a primary resonance. In this paper primary resonances are not considered; instead a resonance in which the small divisor appears in the second term of the expansion--called a secondary resonance is considered. Substituting equation (12) into equation (8) leads to Do2u.i + to2 u.i = -2ito.(D1A. +/~.A.) exp (ito.To) - 2i/2/~.E. exp (i/2 To)

- ~ ~ Ajk.{AjAk exp [i(%+tOk)To]+AjAk exp [i(%--Wk)To] .i=1 k = l

+AjEk exp [i(%+/2)To]+AjEk exp [i(%-/'/)To] +E~Ak exp [i(tok +/2) To] +EjAk exp [i(tOk --/2) To] + ~Ek[ 1 + exp (i2/2To)]1 + co,

(15)

where an overbar denotes complex conjugate. It follows that

u.~ = - ( DIA,, + I.t,,A.) To exp (ito.To) +

2i/2/.t.E. exp (i/2To) 2

2

2 --0.) n

-- ~,, ~ Ajkn{R.jkAjAk exp [i(%+Wk)Tol j=l k=l

+S~jkAj,4k exp [i(% -- Wk)To] + EjEk[S, an+ R,an exp (i2/2To)]} - ~," ~. A~.Ek{Ajg.ni exp [i(/2+%)To] ./=1 k = l

+,4jS~aj exp [i(/2-- w/) To]} + co,

(16)

476

D. T. M O O K , R. H. P L A U T A N D N. H A Q U A N G

R.Sk= I/[O,2--(O,j+tOk)2],

S.sk = I/[O,2--(O,j--O,k)2],

R~as=l/[o,2--(/2 +wj)2], (17.a-c)

Rnaa= 1 / ( o , 2 - 4 0 2 ) ,

s.aj = 1/[o,2-(o-o,y],

S.~,a = 1/o,2~. ( 1 7 d - f )

It can be shown (see, e.g., the books by Nayfeh and Mook [1] and Nayfeh [2]) that the solution of the homogeneous problem needs to be included in only the first term of the expansion. Equation (16) contains a secular term in the scale To, namely -(D~An+ I-t,,An)Toexp(ito,,To). It follows from equation (17d) that, when /2 is n e a r 2o,2, $2/~ 2 contains a small divisor, and it follows from equations (17a, b) that, when 2o,, is near o,2, R2~I, S~21, and Sty2 contain small divisors. Small divisors are a manifestation of resonance. Because R2H, S~21, and S,~2 do not involve the frequency of the external excitation, the resonance associated with 2o,t being near o2 is called an internal resonance. As the functions unt appear in equation (16), they are not part of a uniformly valid expansion. To render the expansion uniformly valid, one must choose the An in such a way that eliminates the troublesome (i.e., secular and small divisor) terms. An essential step in determining the equations that govern the An is the introduction of detuning parameters. They serve two purposes: first, they quantitatively express how close the resonant combinations of frequencies are to being satisfied exactly; second, they convert small divisor terms to secular terms in the scale To, thereby making it possible to eliminate all the troublesome terms by simply eliminating all the secular terms. These points will be demonstrated in the next sections when the details of treating the specific resonances mentioned above are considered. ,i

3. /2 NEAR 202 AND 2toI AWAY FROM 02 3.1. GENERAL In this section, it is supposed that /2 is near 2o,2 and no other resonances exist. A detuning parameter o- is introduced: J'2 = 2 o , 2 + CO'.

(18)

(/2 - o,2) To = o,2 To+ treTo = o,2 To+ o'Tl.

(19)

Then

It follows from equation (15) that the secular and small divisor terms will be eliminated from equation (16) if

D,A,,+/.tnAn=O

forn~2

and

i(DiA2+/.t2A2)+F,'~2exp(itrTa)=O, (20,21)

where

F= 1 ~ A*22Ek.

(22)

2~2 k=l

The solution of equation (20) has the form An = Mn exp (-ttnTi),

(23)

where Mn is a constant. Thus, the An for n ~ 2 decay. To obtain the solution of equation (21), one first puts 32 = [B,(7"!) + iB,(7"l)] exp (89

(24)

INTERNAL RESONANCE NON-LINEAR INFLUENCE

477

where B, and Bi are real. Then, substituting equation (24) into equation (21) and separating the result into real and imaginary parts, one obtains

B'+Ix2Br-( 89

(~o'-F)B,+B~+#2B, =0,

(25a, b)

where the prime denotes differentiation with respect to T1. Equations (25) are linear equations having constant coefficients; thus, the solution has the form

(B, B,) = (M,, M~) exp (AT1), h = - p 2 + Fx/~-~o. 2

(26, 27)

and Mr and Mj are constants. A, M,, and M~ may be complex. I f 4 F : < o.2, B oscillates and decays. If o.2 ~<4F 2 < o.2+ 4/t2 ' B decays without oscillating. If 4F 2= o . 2 + 4 ~ 2 , B approaches a constant value. If 4F2> o.2+4/x2, B grows without oscillating. Thus, in the parameter space defining the excitation, the boundary between regions of growth and decay is given by 4F 2= o'2+4//. 2.

(28)

Note that when 4F2> o.2+4/x22 growth occurs for all non-trivial initial conditions; thus, equation (28) is the boundary of the region where the trivial solution is unstable. At this level of approximation, the predicted growth is exponential, i.e., unbounded. Of course, the actual growth is bounded; non-linear interactions limit the growth. So far the analysis reveals where the growth occurs, but provides no information regarding the final amplitude of the motion. Next the analysis is modified to obtain this information. 3.2. STEADY-STATE SOLUTIONS When F is greater than the critical value given by equation (28), the results obtained above show ttiat initially the amplitude of the response grows exponentially. After IAnl has become large compared to IEnl, the non-linearity can limit this growth and a steady state can develop. In this section, approximations to these steady-state solutions are obtained~ Recognizing that IA~I is large compared to IE.I, one lets

u~ = e-~vn,

(29)

where vn = O(1) as e ~ 0 and t, is positive but arbitrary at this point. Substituting equations (3) and (29) into equation (2) leads to

j=l k=l

jffil k f l

t=l

(30) The choice v = 89makes the order of the quadratic terms equal to the order of the excitation and leads to the earliest interaction of excitation, nonlinearity, and damping; hence, the analysis in this section is based on this choice. Instead of equation (4), one now assumes an expansion of the form

vn(t; e)--V,o(To, 7"1, T2)+el/~v,t(To, 7"1, T2)+ev,2(To, Ti, T2)+" 9 ",

(31a)

where, instead of equation (5),

T,

= t;n/Et.

(31b)

Then, following the general procedure described in section 2, one obtains

V,,o=A,(TI, T2) exp (ico,,To)+ cc.

(32)

478

D.T.

IMOOK, R. H. P L A U T A N D N. H A Q U A N G

The equation governing v., has the form

D2ov.t+to2v., =-2itonD,A.

exp (itonTo)- ~ ~ jffil kffil

+Aj,Tikexp [i(toj -- Wk) To]} + K,

Ajkn{AjAkexp [i(toj+tOk)To]

exp (i~To) + cc.

(33)

To eliminate secular terms from the vnt, one must make the A, independent of T~. Then v., = E~ exp ( i a T o ) -

X Z 4~{R,k4Ak exp [i(toj+tOk)To]

jr1 k=t

+ S.ikAjAk exp [i(toi - Wk) To]} + co, S.jk are as defined in equations (17a, b).

(34)

where the Rnjk and The equation governing the v~2 has the form 2

2

Doon2+ tOnOn2 -2ito. (D2A~ + # . A , ) exp (ito. To) =

- ~ ~. ~ (Fjk,o-- ~.. A*,~.Ajk,~R,~jk){AjAkA,exp[i(toj+Wk+W,)To] j=l

kffil I=1

rrl=l

+AjAk,4: exp [i(coj + tOk-- tot) To]}

- ~j = t k~= l I=1 ~ (Fjk,.-- m~.= l A*=.Ajk~S~jk){AiAkA,exp[i(%-tOk+tO,)To] +AjAkAa exp [i(toj -

tok - tol) To]}

-2ton ~ Hk~{Akexp [i(12 + t0k) To]+.4k

exp [i(I2 --tOk)To]}+CC,

(35)

k=l

1

oo

Hk.-=-f'~w~j~=lA~knEj.

(36)

Again one introduces a detuning parameter defined by equation (18). Then one finds that secular terms in the scale To are eliminated from the v.2 if i(D2An + p~A~) -

4A. ~ c%AjAj+ ~.2F,'~2 exp (itrT2) = 0,

(37)

1=1

kffit[.4 k~..4~kSk~+.4jk..4jnk(Rki.+ Skj~)]

%" - 8(1 + 8j~)~o.

F~t. = l'~k,.+ Fkj,.+ Fj,k.+ FkO.+ Fok.+ F,ky~.

,

(38a)

(38b)

with F as defined earlier in equation (22). F is a special case of Hkn. At this point, it is convenient to re-introduce the amplitude and phase, equation (13): A.(T2) = 89

exp [ifln(T2)].

(39)

Substituting equation (39) into equation (37) and separating the result into real and im~iginary parts leads to

a" +p.a.+~,,2Fa2sin

y=0,

a.fl',,+a,,

j=l

Otjnaj2_~n2Fa2cosy = 0 ,

(40,41)

where y = trT2- 2/32.

(42)

INTERNAL

RESONANCE

NON-LINEAR

INFLUENCE

479

It follows from equation (40) that all the an for n ~ 2 decay exponentially. Thus, in the steady state (a" = 0, 5"= 0), the solution is given by a,=0

for n ~ 2 ,

a2(/-t2+Fsin 5')=0,

(43,44)

a2(89 + a22a~- F cos 5") = 0,

(45)

5" = 0 = o-- 2fl;

(46)

where

has been used. One possible solution is a2 = 0.

(47)

However, it was seen above that this solution is not always stable; the boundary of the region of growth is defined by equation (28). Assuming a2 # 0, one can square and then add equations (44) and (45) in order to eliminate 5". The result is a22= (1/a22)[-(o'/2) + x/F- 5 - bt~]. There are two cases to consider: a22o'~> 0 and c~22cr< 0. If a22o'~ 0, then one (real, positive) solution for a2 exists when F 2 _ /z2-rzcr 2- I 2 = F22.

(48)

(49)

If a22cr < 0, then two solutions for a2 exist when FI2 = / ~ 22 < rr 2 J

2-- I 2 ~-/2,2"1-~0r =

F~

(50)

and one solution for a2 exists when inequality (49) is satisfied. It follows from equations (39) and (42) that A2 = 89 exp [i89

5")]

(51)

and hence that V2o= 89 exp [i(to2 To + 89

89

+ co.

(52)

Then by using equation (18), one can reduce equation (52) to V20 = a 2 COS

[89

- 5")].

(53)

In a similar way, one finds u,, = e-l/282na2 cos [ 89

5")]-~A22na2[R,22 ' 2 cos (.Ot - 5") + S,22] - 2 E , cos Ot

+ O(el/2),

(54a)

i.e., u,, = e-~/282na 2 cos [89

- T ) ] - [89

B 2 =zA22nRn22a2+4En 1 2 2 4 2_2E,,A22,,Rn22a~cos 5",

2+ B cos (.Ot- q~)] + O(el/2),

~b=tan

(54b)

-ll A,22Rn22a2 sin y q /IAn22R,,22a2cos 2 5"-4EnJ/ .

(54c) The one-half subharmonic may appear in the second mode at first order, depending on the excitation parameters/(2 and or, while at second order, the fundamental frequency may appear and a drift term may also appear. The amplitude a2 and the phase 5' are give~ by the steady-slate solutions above. When a2 is non-zero, the first approximation to the response of the structure is the second mode, which contains the one-half subharmonic of the frequency of the excitation.

480

D, T. M O O K , R. H. P L A U T A N D

N. t l A Q U A N G

The one-half subharmonic nearly equals the second natural frequency. The other modes first appear in the second approximation to the response; each mode contains the frequency o f the excitation, not the natural frequencies associated with the modes. When a2 is zero, the solution is essentially that of the linearized problem. To investigate the stability of the non-trivial steady-state solutions, one introduces a small disturbance and then determines its character (growing or decaying). Accordingly, one lets

a,, =~,, + 8a,,,

fl,, = fin + 8fin,

(55a, b)

where the tilde denotes the steady-state (constant) values. Substituting equations (55) into equations (40)-(42) and dropping non-linear terms in the disturbances, one finds the 8a,, decay for n ~ 2. If n = 2 and a2 ~ 0, one can use equations (43)-(46) to obtain ~a~+ (89 + a22a2)a2 ~y = 0,

4ot22a2(~a2- 2/~2 ~y - ~y' = 0.

(56a, b)

Equations (56) are linear and have constant coefficients; thus the solution has the form

8a~ = ASa2,

By'= A 8%

(57)

where 2

-2

-2

A = -p.2 +.,//z2- a22a2(2tr+4a22a2).

(58)

The disturbances grow, and hence the steady state solution is unstable, when the real part o f A is greater than zero. Thus the solutions are unstable when /x2 - a22ti~(2tr + 4a22ti 2) >/z22,

(59)

-a22cr > 2a22a 2.

(60)

which reduces to

Inequality (60) can only be satisfied if a22o- < 0, and then only for the smaller of the two values of ~2 given by equation (48). 4. ~ NEAR 2co2 AND 2~0~ NEAR toa In this section, simultaneous external and internal resonances, associated with .(2 being near 2co2 and 2cot being near co2 simultaneously, are considered. No other resonances exist. One now introduces two detuning parameters oh and or2: /2 = 2co2+ ecrl,

oJ2 = 2oJn + co'2.

(61a, b)

After substituting equations (61) into equation (15), one finds that secular terms in the scale To are eliminated from the unt if 2ira, (DtAn + ~,,An) + 8nnA *2tlA2.~n exp (itr2Tn)

+ 8,,2[AnI2A 2 exp (-itr2Tn) + 2to2Ffi~2exp (itrt Tt)] = 0.

(62)

It follows that the An decay for n > 2, while At and A2 are governed by 2itot(D~An + ~tAn) + A*nt A2,4n exp (icr2T1) = 0,

(63a)

2ito2(DtA2+lz2A2)+A~t2A 2 exp (-icr2T0 + 2to2Ffi,2 exp (icrt Tt) =0.

(63b)

One solution is A~(7"1) --- A2(Tt) -= 0.

(64)

INTERNAL RESONANCE N O N L I N E A R INFLUENCE

481

One can investigate the stability of this solution by superposing small disturbances. It follows from equation (62) that those superposed on A,, for n # 2, decay. For n = 2, one lets (65)

A s --) t~A2.

Substituting equation (65) into equation (62) leads to 6,42 = M exp [(A +i~o-~)TI],

(66)

where M is a complex constant that depends on the initial conditions and A =-P-2 +',/-F-5-~o-l.l2

(67)

4 F 2 > o.12-]-4/-t22.

(68)

Thus, 6As grows when

This is the same condition obtained in section 3 where there was no internal resonance. The initial instability is not influenced by the presence of an internal resonance, and the second mode is the first to grow if growth occurs. A second solution has the form

A~(TO--O

and

A2(T0#0.

(69)

As long as AI =0, A 2 has the form given by equations (65) and (66). To determine stability, one considers small disturbances 6A,. From equation (62), it follows that the 6An decay for n > 2 and that 2iwx(D~ 6A~+tx~ 8A~)+A*nAs-6-A~ exp (io-2T1)=0,

(70)

where A2 is given by equations (65) and (66) and acts as a parametric excitation. If As decays after some initial disturbance, then 6A~ will also decay and the trivial solution results. If A2 grows, then eventually 6A~ will grow. This is confirmed by numerical integration and is discussed further in section 5. A third solution has the form that neither A~ nor A2 is zero. It is convenient to introduce the amplitudes and phases, equation (13), into equations (63). This yields

tol(a~+lxlal)+88

n y2=0,

t

!

, I . tolalfll-~A211ala2cos 72=0, (71,72)

2

to2(a2 q-/.t2a2) --gAllEa I sin ")/2+ tOEFa2 sin 71 = O, t

!

2

t o 2 a 2 f l 2 - ~ A l l E a I c o s " ) 2 - to2Fa2 c o s Yl = 0,

(73)

(74)

where

yl=o'iTi--2fl2

and

y2=o-zTl+fl2-2fli.

'(75)

In the steady state, al, a2, y~ and 3'2 are constants. Then it follows from equations (71), (72) and (75) that

a2 = ( ah/lA *Hl)ql 6t~~+ (o', + 2o-s)2

(76)

and from equations (73), (74) and (75) that

a2= p +,f-P~ Q,

(77)

where 4to2to~(4F 2 - o.~-4/z~)[(oq +20"2)2+ 16/z~]

P= A211All2

Q--

* 2 2 (A2,) Am

(78, 79)

482

D. T. MOOK, R. H. P L A U T ' A N D N. HAQUANG

Also, one gets sin 3'1=

t~2 t~1w~Anza~ F , z, w2FA 211a2

o'1 cos3'~=-2F

(ol + 20-2)o.qAma~ 4w2FA *lla~ '

(80a, b)

sin 3,2= --41.qlA*nl(A*ll)-l[16t.~+(0-1+20-2)2] -I/2,

(80c)

cos 3,2= (0"1+ 20-2)IA*~lI(A*,)-l[16~ + (o'1 +20-2)2] -1/2.

(80d)

Note that the principal value of 72 is independent of A*lh the coefficient of the non-linear term, but the quadrant in which 3,2 falls depends on the sign of A*n. One can define values Ft and F2 with meanings similar to those in section 3.2: i.e., two non-trivial solutions exist if F~ < F2< F~ and one non-trivial solution exists if F2> F :2. They are given by

F~=[21.t10-,+(0"1+20"2)lx212/[(0..,+20"2)2+16tz~],

F 2,=/x2-v~0"1. 2--, z

(81)

In the steady state,

u, = 6,,:al cos [-~(g2t- 3,1-23,2)]+ 6,,zaz cos [89

- 3,t)]+2E, cos .Ot + O(e),

(82)

where ah a2, Yl and 3'2 are given directly above and the E, are defined by equation (14). The results show a saturation phenomenon for the second mode: the amplitude of the second mode, a2, is independent of the amplitude of the excitation (see equation (76)). The amplitude a2 depends on the damping coefficient/x~ of the first mode, a combination of the detuning parameters, and the absolute value of the coefficient of the non-linear term in the equation governing the first mode. This occurs in spite of the fact that only the second mode is directly excited. The amplitude of the first mode, which is only excited through the internal resonance, does depend on the amplitude of the excitation. When there is no damping and ~r1+2o'2 is zero, both al and a2 are zero regardless of the value of F. 5. NUMERICAL RESULTS The numerical results presented in this section correspond to systems which exhibit "softening" behavior. The particular values of the parameters used to obtain the figures are described in reference [4], and are associated with the motion of a shallow, elastic arch subjected to a harmonically varying, vertical load. Damping is present. The values are given in the Appendix. Figure 1 shows plots of the regions where the different steady-state solutions determined in sections 3.2 and 4 exist. The ordinate, F, is defined by equation (22) and is a measure of the amplitude of the excitation. The abscissa, either 0- or 0-1, is the detuning parameter for the subharmonic resonance. For excitations in Region I, there is one solution, the trivial one, and it is stable. For excitations in Region II, there are two solutions: the trivial one, which is unstable, and a stable non-trivial solution. For excitations in Region III, there are three solutions: the trivial solution, which is stab!e, and two non-trivial solutions, one stable and one unstable. Figure l(a) is applicable when no internal resonance is present. The boundary of Region II is the same as the boundary given by equation (28) Which separates regions of growth and decay in the analysis of section 3.1. When the internal resonance exists, typical regions of existence are illustrated in Figures l(b)-(d). The detuning parameter {or the internal regonance, 0"2,is zero in Figure l(b), positive in Figure l(c), and negative in Figure l(d). If one compares these figures with Figure l(a), one sees that the boundary between Regions I and III in the range o'1<0 is distorted by the internal resonance,

483

I N T E R N A L RESONANCE N O N - L I N E A R I N F L U E N C E F"

F

I 0

~

0

(o)

Vl

(b)

F

F

IT

TTI

I

0

(c)

~t

0

~t

(d)

Figure 1. Regions in the parameter plane defining the excitation where the various possible steady state solutions occur: Region I, only the trivial solution exists and it is stable. Region II, the trivial solution and one non-trivial solution exist; the former is unstable and the latter is stable. Region III, the trivial solution and two non-trivial solutions exist. The trivial solution and the high-amplitude non-trivial solution are stable; the low amplitude non-trivial solution is unstable. (a) No internal resonance; (b) internal resonance with tr2 = 0;-(e) internal resonance with trz>0; (d) internal resonance with ~2<0. - - , Stable solutions; . . . . , unstable solutions.

reducing the set of forces capable of producing a non-trivial solution; however, in the range oh > 0 a second part of Region III appears when the internal resonance is present. Figure 2 shows plots of the steady-state-response amplitudes a~ and a 2 as functions of the measure o f excitation amplitude F. Stable solutions are designated by solid lines and unstable solutions by dashed lines. The negative values of a~ and a2, which can be obtained by symmetry with respect to the axis ai = 0, are not shown. Figures 2(a) and (b) correspond to the case of no internal resonance, so that only the directly excited mode is involved. The detuning parameter o" is negative in Figure 2(a) and positive in Figure 2(b). In terms of Figure l(a), one is proceeding upward along a vertical line, first on the left of tr = 0 and then on the right o f tr = 0. In Figure 2(a), one sees the possibility o f spontaneous jumps occurring when the amplitude of the excitation is varied slowly. When F is increasing from a low value, an upward jump occurs as F reaches the value of F2 (F1 and F 2 a r e defined in equation (50)). When F is decreasing, a downward jump occurs as F reaches the value o f F~. When F is in the range Fa < F < F2, two stable solutions exist, and hence there are two possible responses. The initial conditions determine which response develops. Both amplitudes a~ and a2 are involved in Figures 2(e) and (d), where the internal resonance is present. F~ and F2 are defined in equation (81). In Figure 2(c), P > 0 in eqffation (77) and hrnce two non-trivial solutions are possible for a~, while P < 0 for the parameters used in Figure 2(d) and hence at most one non-trivial solution is possible. In terms o f Figure l(b), one is proceeding upward along a vertical line, first at the right

484

D. T. MOOK, R. H. P L A U T /~ND N. H A Q U A N G

~

o2

k B

CX

E

E \

\

OI

I

I

~

F2

I F

F

(b)

1o) .~

(21

c

I

:,I<

I I ~

\

E

E ",

. . . . . .

\

O2

~f,'"

.%

"O I

I

........... -

I

02 --(21

I

Fz

F

F2

(c)

F

(d)

Figure 2. Modal amplitudes as functions of F. (a) N o internal resonance with tr < 0; (b) no internal resonance with tr > 0; (c) internal resonance with P (equation (77)) greater than zero; (d) internal resonance with P < 0.

edge and then at oq = 0. Increases in F leave the amplitude ofthe second mode unchanged (saturation), but cause the amplitude of the first mode to increase. Comparing Figures 2(a) and (b) with 2(c) and (d), one sees the influence of the internal resonance on the amplitudes ofthe responding modes. When there is an internal resonance, there is a role reversal. The behavior of the first mode resembles that of the second mode when there is no internal resonance, while the second mode becomes saturated. In Figures 3 and 4, the response amplitudes a~ and a2 are plotted as a function of the detuning parameter for the excitation frequency. The value of F is fixed in each figure, corresponding to moving along a horizontal line in Figure 1.

E 0

'\

_~_

i .... I 0

er

Figure 3. Modal amplitude as a function of o- when there is no internal resonance.

INTERNAL

RESONANCE

NON-LINEAR

485

INFLUENCE

n

o.

E

t-)

0I

>tm x->

Ii~,,~. ~

E

.

I

02

I O

-

-

I 0

o.I

0

(o)

O.I

Ib)

1II

/ OI ,,/

E

o

i I. . . . . . . . .

11

01 / ' ,,

I

E

! i

,,----0

t

/

/

ii

i:/o , !

I

I

o (c}

I/I

CL

O.I

O

o-I

(d)

Figure 4. Modal amplitudes as functions of o.t when there is internal resonance for fixed values of F. (a) and (b) o'2=0; F in (a) is less than F in (b); (c) and (d) o'2>0; F in (c) is less than F in (d). The case of no internal resonance is shown in Figure 3. In addition to the trivial solution, one sees two branches extending indefinitely as o. becomes more negative. However, if the analysis were extended to cover a wider range of detuning, the upper and lower branches would close. The subharmonic phenomenon is possible in only a limited frequency range around perfect tuning. Internal resonance is present in Figure 4. In Figures 4(a) and (b), o.2 = 0 (i.e., the internal resonance is perfectly tuned), and there is symmetry about o.1 = 0. The value of F used in Figure 4(a) is such that the corresponding horizontal line in Figure l(b) passes through Region II but not Region III, while a higher value o f F is used in Figure 4(b) and the excitation is in Region III for two ranges of 0-1. The curves in Figures 4(c) and (d) correspond to Figure 1(c), with tr2> 0, and the symmetry is destroyed. Had o-2 been given a negative value, the curves would have leaned to the left instead o f the right. Note that the minimum (non-trivial) amplitudes in Figure 4 occur at or near perfect tuning o f the external resonance. In the range of frequency between the zero values of al, the trivial solution is unstable. The multivalued regions indicate that spontaneous jumps will occur if the excitation frequency is slowly swept up or down. Some results o f numerical integration are presented in Figure 5. No internal resonance exists in Figures 5(a) and (b), and the responses are obtained from equations (40) and (41). with n = 2 and a j - O , j # 2 . In Figure 5(a), the excitation corresponds to a point in Region II of Figure'l(a), and for two different sets o f initial conditions one finds the response approaching the same steady state. (In fact, the same steady state is reached for all initial conditions.) In Figure 5(b), the excitation corresponds to a point in Region

486

t). T. M O O K , R. H, P L A U T A N D N . H A Q U A N G

02

02

2

E
(o)

(b)

2 E

E -

02

Figure 5. Modal amplitudes as functions of time. (a) The excitation corresponds to a point in Region II of Figure l(a); two different sets of initial conditions: (b) the excitation corresponds to a point in Region III of Figure l(a); two different sets of initial conditions; (c) and (d) the same excitation, corresponding to a point in Region II of Figures l(b)-(d); different initial conditions. III. One sees that different initial conditions may lead to different steady states--the large-amplitude, non-trivial solution and the trivial solution. In Figures 5(c) and (d), the internal resonance is present and the responses are determined from equations (71)-(74). The force is described by a point in Region II of Figure l(b), so the trivial solution is unstable. The parameters in the governing equations are the same, but the initial conditions are different. Clearly, the second mode experiences growth before the first mode begins to grow in Figure 5(c) where both initial values are near zero. One can notice in the numerical data, but not in the figure, that at begins to grow about the time a2 reaches its saturation value. In Figure 5(d), the initial value of at is noticeably away from zero. After the motion is started, the value of at actually declines until a2 approximately reaches its saturation value, and then a t begins to grow. To illustrate further some of the features of the response discussed above, the equations governing u~ and u2, obtained from equation (2), were numerically integrated. All the quadratic and cubic terms were included. Again the shallow elastic arch was used as the source of the numerical values of the various coefficients. For the numerical examples, all the K,, except K2, are zero, and the critical value of K2 is approximately 4740. In Figure 6, K2 is 3790; thus, the subharmonic resonance is not activated. At the left-hand sides of the records, one can see the initial conditions and the first few cycles of the transient motion. Because the system is lightly damped, the solutions approach the steady state slowly; hence, there is a break in the record to allow time to pass before the steady-state solution is shown at the right-hand side. In part (a) there is no internal resonance, while in part (b) there is. The natural frequencies may be obtained from the left-hand portions of the records. In both parts the solutions approach the same steady state. The frequency of the respons'e in the second mode is exactly that of the excitation

INTERNAL RESONANCE NON-LINEAR INFLUENCE

~

487

~AAAf^.

v

{oi

(D) Time Figure 6. u~ a n d u 2 as functions o f time when the amplitude o f the force is below the critical value, results o f numerical integration. (a) Without internal resonance; ( b ) with internal resonance.

and twice the second natural frequency (here the external detuning tr l = 0 ) , and the amplitude agrees closely with 2E, given in equation (14). There is no discernible response in the first mode. These results are nearly the exact solution of the linear problem. Clearly the internal resonance, which is a non.linear phenomenon, has no significant effect on the solution. In Figure 7, K2 is 5685. This value is 40% above that used to make Figure 6, and it is above the critical value. The frequency of the excitation is the same as the one used to

hAA~ ~ VV 1 UZ

hA A A / ^ ~ _ ~ ~

v vv v

v

(b }

lq

.

.

.

.

.

~ ~ ~

.

Time

Figure 7. u I and u2 as functions of time when the amplitude of the excitation is above the critical value, results of numerical integration. (a) Without iniernal resonance; (b) with internal resonance.

488

D . T . MOOK, R. H. PLAUT AND N. HAQUANG

make Figure 6. If the system were linear, then, when comparing Figures 6 and 7, one would see a 40% increase in the amplitude and no change in the frequency of the response of the second mode, and the first mode would remain dormant. Instead, one sees something profoundly different. In Figure 7(a), where there is no internal resonance, the amplitude of the second mode has increased approximately 40 times and the frequency has decreased to exactly half that o f the excitation. The first mode remains dormant. To the eye, the response appears to consist o f a single harmonic, b u t as will be seen below, it actually contains one strong component at half the frequency of the excitation and one weak component at the frequency o f the excitation. These numerical results and the analytic predictions are in close agreement. In Figure 7(b) there is an internal resonance, which is perfectly tuned (the excitation and second natural frequency are the same as the ones used in part (a)). Here the solution does not approach a steady state. The record is divided into three segments in an attempt to illustrate this behavior. The left-hand portion shows the initial conditions and a few cycles of the motion. The natural frequencies can be obtained from this portion. The middle portion shows the results after a rather long time. In this segment the first mode is noticeably excited. The right-hand portion shows the results a long time after the middle segment. In this portion, the first mode does not appear to be excited and the amplitude of the second mode has increased noticeably. If one had continued to show segments for longer times, one would have seen that the pattern of the second and third segments is repeated. There is a continual exchange between the first and second modes, and a steady state (in which the modal amplitudes are constant) never develops. This occurs in spite of the fact that the excitation is a simple harmonic function and there is damping. Such 1

I

I

u2

x uz

Frequency Figure 8. Fast Fourier transforms.of the numerical results in Figure 7, semilogarithmic plot. x, Without internal resonance; ~ with internal resonance.

INTERNAL RESONANCE NON-LINEAR INFLUENCE

489

behavior is in sharp contrast with the behavior o f a linear system. The obvious effect of the internal resonance is to restrict the amplitude of the response. This suggests that an internal resonance could be used to provide a passive control of the large-amplitude vibrations that can occur when a subharmonic resonance is activated. To aid in interpreting the numerical results, the fast Fourier transforms of the numerical output shown in Figure 7 were computed. These transforms can be readily compared with the analytical results given in equations (54) and (82). In Figure 8, the transforms o f u, and u2 are shown on a semilog plot. When there is no internal resonance, the first m o d e is not excited, and the second exhibits two p e a k s - - o n e at the frequency of the excitation .O and the other a t / 2 / 2 . The ratio o f the former peak to the latter is approximately 75. These results are in close agreement with the analytical predictions given by

U!

1 U2

t

1

1

(o)

1 (b)

UI

A

U2

II

1

J (c)

(d)

Ut

1

]L U2

(f)

(e) Frequency

Figure 9. Fast Fourier transforms of the numerical results showing the continual exchange between the first and second modes. Each part represents the solution at a different time.

490

D . T . M O O K , R. H. P L A U T A N D N. H A Q U A N G

equation (54) (for the numerical results, e = 0.001). When there is an internal resonance, al, a2, and E , are all O(1), as indicated by equation (82). According to the analytical predictions, the peak at -Q is independent of the internal resonance, as these transforms show. When there is an internal resonance, there is a slow continual exchange between the modes; hence, the transforms of ut and u2 change slowly with time. The results in Figure 8 represent conditions at one time. The response can be understood better if the sequence of transforms of Ul and u2 shown in Figure 9 is considered. The parts of Figure 9 are equally spaced in time, and the scales are conventional. Part (a) is a repeat of the results for internal resonance shown in Figure 8. The exchange between the modes is clearly evident. If the sequence were continued, the next figure would be part (a) again. The time required to produce an FFT is very short compared to the time between parts in the figure. Equations (71)-(75) can explain the continual exchange between the modes. For the case considered above (o'~=o'2=0), as time increases, it can happen that y~-->-zr/2, 2'2-> ~r/2, a~ -->0, and a2 grows exponentially. Clearly, at --- 0 is sufficient to satisfy equations (71) and (72), and 2"1--- - 7 r / 2 , 2"2~ rr/2, and fl~-- 0 are sufficient to satisfy equations (74) and (75). This leaves equation (73), which is now a linear equation in a 2. If the condition defined by equation (28) is met (i.e., IFI >/x2), then a2 grows exponentially. But exponential growth does not continue indefinitely. Had the analysis been carried to higher order, in a manner similar to the one used in section 3.2, a limited amplitude would have been found, as the numerical solution shows.

I

I

u

b --

~

l

l

l

I

(o)

(b)

(c}

(d) /

U2

c2 '--

I

U2

U2

_~o

(e)

I

I

I

(f)

(g)

(h)

U2

2

U2

"

o

1 o (i)

I o (])

I

o (k)

I

o (t)

Amplitude Figure 10. Phase planes for various segments o f the records o f u~ and u 2 s h o w n in Figure 7.

2

INTERNAL

RESONANCE

NON-LINEAR

INFLUENCE

491

This type of behavior has recently been observed in an experiment. H a d d o w [5] studied the response o f a two-degree-of-freedom structure to harmonic motion of its suppo.rt. Though the quadratic non-linearities in that case are due to inertial effects, Haddow's equations governing the amplitude and the phase have the same form as equations ( 7 1 ) - ( 7 5 ) . In his experiment, he observed a "beating" phenomenon in which there was a continual exchange between the modes. To the eye, such motion appears chaotic. The FFT's o f the experimental time series given by Haddow show that during the beating the peak at the frequency o f the excitation remains constant, as the analysis predicts. In Figure 10, phase planes made from various segments o f the complete record are shown for the case when the motion appears to be chaotic. In each part o f the figure, the points representing the motion make only one or two loops around the trajectories. The trajectories are experiencing a relentless transition through the various stages shown, and a phase plane covering a much larger period appears nearly solid. To arrive at the steady-state solution for the parameters used above, one must choose initial conditions very close to the steady state. When the excitation is detuned (cq = 80), the region of initial conditions leading to a steady state is not so narrow. In Figure 11,

/Jl

...VV V V,o l./I

I/2

/~%

~'--'-

VVV

v'v'v'-

,L

UI

'V V V Vj"""v"

, [

Frequency

Time

Figure 11. u! and u, as functions of time and the fast Fourier transforms of these records when there is external detuning (o't = 80) and a steady'state develops. Results for three different amplitudes of excitations, showing the saturation phenomenon. Amplitude increases in equal increments from (c) through (a).

492

D.T.

M O O K , R. H. P L A U T A N D N. H A Q U A N G

ul and u2 are shown as functions of time for three different levels of excitation. Transforms of steady states are also shown. Here a steady state is reached. These transforms clearly show the saturation phenomenon. In the transform o f u2 the peak at one half the frequency of the excitation, which is proportional to a2, remains constant, and the peak at the frequency o f the excitation grows linearly with the amplitude o f the excitation. In the transform o f u~, the peak at one quarter the frequency of the excitation, which is proportional to a~, increases as the amplitude of the excitation increases.

ACKNOWLEDGMENT This research was supported by the U.S. National Science Foundation under Grant No. MEA-8300507.

REFERENCES 1. A. H. NAYFEH and D. T. MOOK 1979 Nonlinear Oscillations. New York: Wiley-Interscience. 2. A. n. NAYFEH 1981 Introduction to Perturbation Techniques. New York: Wiley-Interscience. 3. A. n. NAYFEH 1983 Journal of Sound and Vibration 89, 457-470. The response of single-degree-offreedom systems with quadratic and cubic nonlinearities to a subharmonic excitation. 4. D. T. MOOK, R. H. PLAUT and N. HAQUANG 1983 Virginia Polytechnic Institute and State University, College of Engineering Technical Report VPI-E-83-39. The response of multidegree-offreedom systems with quadratic and cubic nonlinearities to harmonic excitation, with application to a shallow arch, I" single-frequency excitation. 5. A. G. HADDOW 1983 Ph.D. Thesis, University of Dundee. Theoretical and experimental study of modal interaction in a two-degree-of-freedom nonlinear structure. APPENDIX The values of the coefficients used in the numerical examples correspond to a shallow arch[4]. When there is no internal resonance, ~ol = 24.7, co2= 65.2, ~ = 130.4, A m = 5235, A211 = 0, A l l : = 2617, all ----23"4, a12 = --51"4, or21= --136"0, and a22 = 4.67. When there is internal resonance, co~= 34.2, ~o2= 68.4, ~ = 136.8, A m = 5399, A2H = 0, and All2 = 2699.