Interaction between parametric and forced oscillations in multidimensional systems

Under certain conditions the oscillation of the first mode x excites parametrically the oscillation of the second one and so the two oscillations of the second mode parametric and forced

INTERACTION BETWEEN PARAM ETRIC AND FORCED OSCILLATIONS

IN M ULTIDIM ENSIONAL SYSTEM S

N G U Y E N V A N D A O (HANOI)

This paper is devoted to the investigation of the interaction between parametric and forced non-linear oscillations in multidimensional system described by two non-linear differential equations of the second order The two modes (.X, y) of the system considered are excited by sinusoidal forces The two modes are coupled non-linearly by means of the product of their coordinates Under certain conditions the oscillation of the first mode (x) excites parametrically the oscillation of the second one and so the two oscillations of the second mode (parametric and forced oscillations) may coexist and there exists some kind of interaction between them.

We shall now consider the stationary oscillations of the modes and their stability.

1 Equations of Motion Stationary Oscillations Let us consider the oscillations of the system with two degrees of freedom described

by a set of two differentia] equations of the type

x + /2^ + £ /2(/i0ir + ccc3 + cv2x) = Osinyf,

ỷ + (ư2y + eco2(hỷ + fỉy3 + bx2y) = eơ)2p cos(ví-f <5),

where h0 > 0, h > 0, a, Cy b, Q,p > 0, /?, Ổ, are constants and £ is a small parameter

We assume the following relations between the frequencies cư,r, and y:

(1.2) cư2 = Ơ 2 V2 + e ơ ) 2A , y = e v , n X # m y ,

where Ơ, e are rational numbers, A is detuming of frequencies and m, n are integers.

First, we transform the Eqs (1.1) by means of the formulae

X = < 7 s i n y /- f a x COSỠ!,

X = yqcosyt— Aai sinfli,

ỳ = -ơrasine, q = J ĩ ị p , where al , ỚJ, Ơ, 6 are the new variables which will be determined later.

in the standard form:

da ị dĩ dxp l

= E Ằ ( h 0 x + ccx3 + cy2x)sinOi ,

(1.4)

a i — J — = e XỌ ì q X + QLX3 -+■ c y ^ x ) C O S Ớ Ị, at

= eơv<Ị>(x,y,ỷ, 0 sinớ + 0 02),

at ady

~dT = eơv&(x, y , ỳ, t)cosd + 0(e2),

where

where

(1.5)

&( x9 y, ỳ , 0 = Ay + hỳ + bx2y + Py3 — pcos(vt+ Ỗ),

y)ị = ỚJ — At, rp = d — ơvt

Averaging the right-hand parts of (1.4) over the time, we receive the equations of the

first approximation for the unknowns a, a l 9 y>:

1) for ơ # e, ơ # 1

- «1.

(1.6)

n 2 eơ~vz

a = - y - h a +

a 1 = — — h^ax

à = — ~Y [vha +/?sin(y>— ($)]+

axp =

3) for e = Ơ # 1

£T

zla + ợ2ứ + — /fa3 - p cos(y — Ô)

ỏi = ” 2 ^°ứ l’

4

£ơ

4) for e = Ơ = 1

(1.9) a — — vha+ -^-q2asin2y)Jrpsin(y)—ỗ)\+

where the nonwritten terms disappear when flj = 0 By analogy with N e s s [1], the first

case is called the non-resonant case, the second case — the harmonic resonant case, the

third (1.8) — parametric resonant case and the last (1.9) — harmonic and parametric

resonant case Obviously, the most interesting is the last resonant case We shall inves­

tigate it in more detail.

The stationary solution of the system (1.9) is the one which is determined from equa­

tions = ả = ỹ) = 0 or

f l j = 0, (1.10) r / j f l - f <72 f l s i n 2 y j + / ? s i n ( Y > — Ô ) = 0 ,

Aa-\- — q2a+ pa3— Ỵ q2acos2ĩp—p c o s( y -ô ) = 0.

Eliminating \p in the two last equations of (1.10), we obtain the following relation for

the amplitude a = a0 = const of the stationary oscillation of the coordinate y:

where

M = (w2 + u2 — V 2) 2— / ? 2 [ ( w — acos2<5)2 + (w*f Z>sin2<5)2], (1.12) u = vha0, v = ~ q 2a0, IV = - a0 ỊA + q2 + ậaị

The relation (1.11) is expressed in the parameters of the initial system as

al

(1.13) - F

e + 2 q + 4 H + ^ 2- T 6 ^

- —— - h q2 + + -j-ợ2cos2<5 + I co/2 H ~ ọ sin2Ỏ

* = CO

Following this formula the resonant curves are presented in Figs 1-6.

The equations in variations for the system (1.9) are

F i g 3.

dò

it

a 0 j — = —(w' + v f cos2y)0)ôa+[2vsin2rp0+psin(yỈQ — ỏ)]ỏy>,

where T = 2t/ev and primes denote the derivatives with respect to ŨQ. The characteristic equation of the system (2.1) takes the form:

e + -ị-h0)(a0Q2 + 2vha0ọ + R) = 0,

w h e r e

(2.2) R = — (u +v' sin2^o)[2^sin2^ o+/7sin(^o~ Ổ)]

- (w ' + V ' COS 2\p o) [ 2 vC O S2 ^ 0 + p C O S (y j0 - Ô) ]

By using the relations (1.10), (1.12), the expression R may be rewritten in the form

2 ( w 2 + u 2 — V2) o a 0

As h0 > 0, h > 0, a0 > 0, the stability condition of the stationary solution is

ƠŨQ

where E = w2 + u2— V2 The resonant curves (M = 0) divide the plane (a , k y into the regions, in each of which the expression M has a definite sign ( + or —) If moving up along

the straight line parallel to the axis a0 we pass from the region M < 0 to the region M > 0

then at the point of intersection between the straight line and the resonant curve the deriv­ ative õMỊỗa0 is positive So, this point corresponds to the stable state of oscillation if

£ = vv2*f u2—v 2 > 0 and to the unstable one if £ < 0 On the contrary if we pass from the region M > 0 to the region M < 0, then the point of intersection corresponds to

the stable state of oscillation if E < 0 and to the unstable one if E > 0.

Id the limit case when h = Ổ = 0 the equation for the stationary amplitude a0 (1.13)

may be written in the form:

(2.5) + ^ r + T f i t ) [ ( V 1 + 4 ~ + 4 f t * ) - j r

From here we obtain a double root

and two other roots

Following these formulae the resonant curves are presented in Figs 1-4, where the

branch expressed by (2.6) is shown by number 1 and the branches expressed by (2.7) and (2.8) are shown by numbers 3 and 4, respectively The parameters are chosen as

(e/4)bq2 = 0.13, (3/4)e/3 = 0.1, ep = 0.1 In the shaded region (the region of parametric resonance) the expression E = w2 + u2— V1 is negative On the heavy lines, EdM/da0 is

positive, so that they correspond to the stable state of oscillations The dotted lines corres­ pond to the unstable state of oscillations The signs + and — in the figures are those of

the expression M On the branch 1 oT the resonant curves when h — 0, the stability of stationary oscillation is doubtful because on this branch E = 0.

= 0.

the form shown in Figs 5, 6, where v2h2 = 0.005 These curves are obtained by solving the Eq (1.13) on a digital computer The branch 1 in the case of h = 0 (Figs 1-4) changes

into two either stable (Fig 6) or unstable branches (Fig 5) With larger values of friction

(h) the resonant curves take the forms presented in Fig 7 (v2h2 = 0.01) and Fig 8 (v2h2

= 0.02).

From the results obtained the following conclusions may be drawn:

1) Inside the region of the parametric resonance, the parametric excitation caused

by the first mode (x) strongly influences the stability of the stationary forced oscillations

of the second mode (j>) Some branches of the resonant curves of the second mode oo which are unstable for X = 0 now become stable and * vice versa Outside the region of the parametric resonance, the mechanism of parametric excitation does not influence the stability of the forced oscillations.

2) The jump phenomenon of the amplitudes — in the case of a hardening character­

istic (Fie 1, 5), when the frequency k decreases, and in the case of a softening character­

istic (Fies 4,6), when the frequency increases — is observed quite clearly The change

of the stationary amplitudes follows the M-form.

We assume now that the amplitude of the external force acting on the second mode (v) is not small, so that the equations of motion take the form

where V ^ cư.

On the assumption that there are resonant relations (1.2), we first transform the Eqs (3.1) by means of the formulae:

3 System with a Large Amplitude of the External Force

(3 1 )

x + ?.2X + e?.2(h0x + ccx3 + c y 2x ) = Q s i n y t ,

ỷ+ơ)2y+eoj2(hỳ+fìy2 + bx2y) = Z)cos(r/+ (5),

(3.2 )

X = q s i u y t + a 1 COSỠj,

X = y qcos y t— Áũ 1 sin 6!,

y = jcos0 + c/cos(vr + <5),

(3 3 )

(X)2—V2 r2(ơ2 — 1)

The equations for new variables a 1, 6 1, a , 6 are

( 3 4 )

a i xpl = £Ẳ(/20i: + ttx*+ cy2x)cosO I,

à =* CƠT0! (*,}>, ỷ) sin 0 + ơ(£2),

axp = £ ơ r 0 i ( x , j , ỷ ) c o s ớ - f ỡ ( £ 2) ,

(x, y , ỳ) = A y + hỷ+Py3+ b x 1y ì y>ị = ỚỊ — Ằí, y = 6 — ơvt.

1) for Ơ = e = 3 (superharmonic and parametric resonance)

SẰ'

A0fli,

<3.5) à = 3r»' — I- r/ĩứ + 3 8 i/ 3 sin (y) — 3 Ổ) — 1bq2a sin 2^

ay = 3ev { — fl-f p 3 J3 + 3 <f0— cos (y* — 3 Ổ)3

4- — q a — — qzaoosl\p

2) for Ơ = 3, e = 2 (superharmonic and combination resonance)

• _ EĨ'2 u

ai — 2— °Ơ1 ’

— \ - v h a + 4 - / W 3 sin(Y> — 3 Ổ ) - <72*/sin(y>+<5)

q Q3jt - r -d 2a+ ~ ~ cos(yj-3<5)

+ ỡ(«i);

3) for Ơ = 3, e = 1 (superharmonic and combination resonance)

e?.2

— vha + — d 3 sin (v> - 3 Ổ) — q2d sin (V - Ổ)

dip = 3ei> | y

-— -3- q2dcos{xp-<5)> + 0{a^)\

4) for ơ = - ị- , e = -i- (subharmonic-parametric and combination resonance)

<3.8)  = -y -j — -^-/ỉj+-ị-^íÌ72sÌD(3v>—ổ)“ - ~ ợ2[ữsin2^ + í/sin(^— +

-5- Ữ3 + -7- í/2ữ -f -5- dtf 2cos(3y> — (5)

q2[acos2y) + dcos(y— (5)]? -f 0{al)\

1 2

5) for Ơ = - y , e = — (subharmonic and combination resonance)

• _ _ £*2 u

a\ — 2 hoai>

(3.9)

ay

- -y- [ — -^r ha + ^ - fi da 2 sin(3y>—<5) — ^ q zdún(\p+ Ô) + 0 ( a L),

- -^-q2d cos(y> + <5)1 + 0 0 !);

6) for Ơ = 3, e # 1 , 2 , 3 (superharmonic resonance)

• _ £*2 u

a\ = - —TT-noũỵ,

(3.10) à = 3ev

= 3ev I

+ 0(a x),

ơ 3 + - J í / 2ứ + ~ - c o s ( y > —3Ổ)

7) for Ơ = -J, e # 3 ’ 3 (su^ armon*c resonance)

(3.11) ả =

<2^

£T

£T'

—2- ha + — pda2 sin(3v> — Ô)

I Aa , o

p r + i ị a 3+ - ị 8 4 + 8 cos(3w — Ỏ) + ^ a j + O O J;

8) for ơ = e # ^ > 3 (parametric resonance)

( 3 12) à = e ơ r

= £ƠT

— /lứ— g"#2^ sin2y^ + 0(ứi)>

4-P Ị — a3+ — d 2a I + — — q2a cos2yj

9) for 2e — ơ — 1 = 0, e # 2, -J- (combination resonance)

SẢ2

h o ^ i»

axp = eơv Aa + /ỉ Ị -jp ữ 3 + d2a + -T- q2a - qzd COS(yi + ô)

1 0) for ơ — 1 = ± 2 e , e - Ỵ , 1 (combination resonance)

aì ^ M l ,

(3.14) ứ = £ƠT

= eơv

- ^ h a - — q2d $ m ( y - ồ ) + 0 ( a l) 9

Aa /> / 3 - 3 , \ è , Ề ,

2 + M 8 4 ) + 4 ? 8 9 C0S(V-<5)

1 1) for non-resonant case

ơ 7^ 3, ơ ^ 3 ’ Ơ # e , 2e — Ơ ± 1 ^ 0 , 2 e - f ơ — 1 = 7 * 0 ,

• - e^2 í,

0\ ~ ^ "0ữi >

(3.15) á = - -L (<Tv)2/ia + 0(ai),

a x p — e ơ v A a * 1 3 3 3 ^ 2 \ * •> ’

2 + ^ \ 8 ữ + 4 ^ ứ) + 4

The analysis of the systems (3.8) and (3.9) is rather complicated The system (3.5) has been investigated exhaustively in the previous paragraph As the study of the remain­ ing systems is not difficult we shall not consider them here.

References

1 D J N e s s , Resonance classification in a cubic system , A S M E A pplied M echanics C onference 1971.

2 N N B o g o l ĩ ƯBOV, Yu A M i t r o p o l s k y , A sym p to tic m ethods in theory o f non-linear oscillations,

M oscow 1963.

S t r e s z c z e n i e

ODDZIALYWAN1E MIẸDZY DRGANIAMI PARAMETRYCZNYMI

I WYMUSZONYMI w WIELOWYMIAROWYCH ƯKLADACH

Niniejszy arty k u ỉ zostal posw iẹcony b a d an iu oddziatyw an m iẹdzy drg an iam i param etrycznyrai i wy-

m uszonym i w w ielow ym iarow ych ukJadach, k tó re opisyw ane sạ p rzez ukỉad dw och n ie lin io w y c h rốz-

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harm oniczne D w ie w spom niane w spóỉrzẹdne sạ zw iạzane z sobạ p rzez ich iloczyn.

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d e p a r t m e n t o f m a t h e m a t i c s a n d p h y s i c s

PO LY TECH N ]cA L INSTITUTE, HANOI

Received October 14, 1974.