The stability of nontrivial stationary solutions a Ỷ 0 of the equation 1.12 when UJ is constant e studied by using the corresponding variational equations, which lead to the condition:
Trang 1lí C ơ học Journal of Mechanics, N C N S T of Vietnam T XIX, 1997, No 4 (11 - 20)
I N T E R A C T I O N O F T H E E L E M E N T S
C H A R A C T E R I Z I N G T H E Q U A D R A T I C
N O N L I N E A R I T Y A N D F O R C E D
E X C IT A T IO N W IT H T H E O T H E R E X C IT A T IO N S
N g u y e n V a n D ao
V i e t n a m N a t i o n a l U n i v e r s i t y , H a n o i
oduction
In nonlinear system s, the first order of smallness terms of quadratic nonlinearity and the
id excitation with nonresonance frequency and the second order of smallness terms of linear ion, cubic nonlinearity, forced and parametric excitation with resonance frequencies have no :t on the oscillation in the first approximation However, they do interact one with another in second approximation and new nonlinear phenomena occur The study of these phenomena,
ig the asym ptotic method of nonlinear mechanics [ 1Ị with a digital computer, is our aim.
Interaction between the elements of quadratic nonlinearity and forced excitation imselves
Let us consider a nonlinear system governed by the differential equation
X + X = e \ c t x 2 + q COS 2< p ( t )] — £ 2 ( 2 h x + f i x 3 ), ( 1 * 1 )
.ere t h e d o t s in d ic a t e d i f f e r e n t ia t i o n w i t h r e s p e c t t o t im e , a , q, h a n d fi a re c o n s t a n t s , T = e t
d e is a sm all dimensionless parameter characterizing the smallness of the terms behind it The ram eter £ is introduced artificially and used as a book-keeping device and will be set equal unity in the final solution The quadratic term may be due to curvature or and asymmetric aterial nonlinearity The function < p (r) is supposed to be a form
~ ^ = " (r )> r = e i > ( L 2 ) here u ( t ) is close to the natural frequency i.e to unity:
'he equation (1.1) can be rewritten as:
X + i / 2 ( r ) x = e [ a x 2 + q COS 2 < p (r)] — S2( - A z + 2 h i + f i x 3 ) ( 1 4 )
A Solution of this equation is sought by using the asym ptotic method of nonlinear oscillation
X = acos Ớ- f - t / > , ớ) + e2u2(a, t/;, Ớ) + 53 , 9 = ip + rj)} (1.5)
1]
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Trang 2= ~ ~ z i ~ + ( f ~ 7 a 2 ) + /l2< j2' e2A = w2 - 1 « 2(w - 1), 7 = £ 0 - • (1.16)
Vom equation (1.15) it follows that:
dependence of the amplitude a on the external frequency UJ is presented in figure 1 for the neters: e 2a q = 0.063, e 2h = 0.01, £27 = 0.08.
The stability of nontrivial stationary solutions ( a Ỷ 0) of the equation (1.12) when UJ is constant )e studied by using the corresponding variational equations, which lead to the condition: [l|
d W
d a 0
.use function w (1.16) is positive outside and negative inside the resonance curve, the stable
ch of the resonance curve is the upper branch, which corresponds to the upper sign before the :al in ( i 17) Thus, between the two forms of oscillations corresponding to definite values of fie form with large amplitude is stable and the form with small amplitude is unstable.
Fig 1
Following Chapter 4 of [l], the trivial solution a = 0 of the equation (1.12) is stable if the lue L 0 does not lie in that interval of the axis U/, from which the resonance curve is rising In
;ure 1 the stable branches are shown by heavy lines, while the unstable ones are shown by dotted
The passage of the system under consideration through resonance when v ( t ) is not a constant,
i t changes by the law: z/(r) = Uo+Sfit, can be examined by integration of the differential equations 12) T he parameters are chosen as to = 0, d o = 0.009, t/>0 = 0, e 2h = 0.001, e27 = 0.01,
:ctq = —0.024, V0 = 1, /1 = 10- 5 (curve 1, Fig 2); /1 = 2 - 1Q~5 (curve 2, Fig 2); /1 = - 1 0 -5 :urve 1, Fig 3); /1 = - 2 10~6 (curve 2, Fig 3).
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Trang 3a a
0.020
-0.010
0 0 0 0
From the expression (1.12) and (1.13) one can see that the quadratic nonlinearity ( a) is always
to aoftenize the system under consideration regardless of the sign of Q Moreover, two elements characterizing quadratic nonlinearity a i 2 and forced excitation qcos2<p(r) combine together and act just like a parametric excitation with an intensity aq.
The system of equations (1.12) has a trivial solution a = 0, which corresponds to a pure forced oscillation under the action of an external excitation eqcos2<p:
2 Interaction between the elements of quadratic nonlinearity and forced excita tions
The system under consideration in this paragraph is governed by d.e.
X -f 0 J2x = f ( a i 2 + q cos 2ujt) + f 2 [ A i — 2 h i — /?x3 + r cos(u;i — r?)], CJ2 = 1 + e2A (2.1) Here, the nonresonance forced excitation (<?) is of the first order of smallness, while the resonance forced excitation (r) is of second order of smallness These excitations have no effect on the oscil lation in the first approxim ation, but they interact one with another in the second approximation Similarly to the previous paragraph, the solution of the equation (2.1) is found in the series (1.5) The equations (1.11)-(1.13) now take the form:
(1.19)
X = a cos
0 + f f | -(ocá2 -I- 2a cos 2t/>i COS 26 — - sin 2rp sin 201, 6 = 0Jt + \p, (2-2)
(2.3)
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Trang 48 12
The stationary solutions of equations (2.3) are determined by the relations:
f o = 0, go = 0,
<ỵq
/ o = 2 h u a -1 asin 2 0 + r sin(i/> + r j ) ,
3
<7o = A a — 2 7 a 3 -- a c o s 2 t / > 4- r cos(rp -f 77), ỉquivalently:
m here we obtain
/0 cos rị)— (70 sin 0 = 0 , /0 sin Tp + <70 c°s 0 = 0.
2/iu>a cos tp — (A a — 27a3 + — a) sin t/> -f r sin rj = 0,
3 2/itưa sin rp + (A a — 27a3 - -a ) COS rp + r CO S rj = 0.
Ổ
)te : T h e equations (2.4) belong to the form
Y sin rịỉ + z cos t — c
l e functions sin \ị) and COS yịỉ satisfy the relationship
sin21p + cos20 = 1.
om equation (1) we have
Y 2 sin2rp = c 2 + z 2 COS2rp — 2z c COS rp
lim inating sint/> between last two equations we get
(Y 2 Hr z 2 ) COS2 yị) - 2 Z C COS yị> + c 2 - Y 2 = 0
b m here we obtain
= y2 l ' z2 [ z c ± V F C P - (ya + Z2)(C3 - n
COS
Therefore, the condition for reality of costịỉ is that the under radical expression should legative:
y 2 + z 2 > c 2.
A p p l y i n g t h e c o n d i t i o n ( 2 5 ) to e q u a tio n s (2 4 ) we have
a214 h 2w 2 -f - 27a2 + j > f2 sin2 *?>
a2| 4 / i2cư2 + (A - 27a2 - I > r2COS2 rj.
S y ste m w ith o u t frictio n (/i = 0)
In t h i s c a s e e q u a t i o n s ( 2 4 ) take th e form
Trang 527a2
-( * ♦ ? ) ]
27a2 —
( * - 3 ) ]
a sin = — r sin Ĩ7,
a cos \p= r cos T).
(2.8)
a) If 27a2 - ^ # 0 and 27a2 - ^ 7*^ 0 then eliminating the phase xp from (2.8)
we get the equation of the resonance curve C\ \
w (u;2 , a2) = 0.
where
= -rl ™ L l - + -t fgfl??
[ 2 V ( a + ^ ) ] [ 2 ^ - ( a - f ) ]
b) If 2 7a2 — -f ^ j = 0, i.e if we have the resonance curve c 2:
— a
then
From here we obtain:
2 a q o.a. sin yp = — r sin rj, -a COS v> = r COS r?.
Ó
± 3 r 2 9 r2
s i n T) = 0 = > ỴỊ = 0 , 7r; c o s T7 = ± 1 = > w = a r c COS - = > a > — —
If 2 7a2 — ^ = 0, i.e if we have the resonance curve C3:
2V = u,2 - 1 - ^ ,
3 then
o.acos yp = rcosrj, -7— a sin rp = r sin r?
o Prom here we obtain:
cos rj = 0 => TỊ = —7T , — ; 3tt sin r? = ± 1 => t/; = ±arc sin — 3r => a* >2 9r2
(2.9)
( 2 . 10 )
( 2 . 11 )
( 2 1 2 )
(2.13)
(2.14)
So, if rj = 0, — , 7T, — , the resonance curves c 2» C3 do not exist If rj = 0, 7T, then beside the
9 r resonance curve C l there is still semi-straight line C 2 in the (a2, cư2)-plane with a2 > ^ —2 2 If
^ _ 7T 3tt kken beside the resonance curve Cl there is still sem i-straight line C3 in the (a2
9r2
plane with a2 > ■ õ Õ ■
4org*
S y ste m w ith frictio n (/1 ^ 0)
Solving the system of equations (2.4) relatively sin \ị) and COS rp we have:
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Trang 6• / — I — P i (o 1
a sin rị) = a co s t/; = - ~ , ( 2 1 5 )
>r the case D 7^ 0:
D = 4w2h2 + ( A — 2 7 a 2 ) 2 — a — ,
D i = —r 2/icj sin rj + — 2 7 a 2 + — ^ COS ?7j , ( 2 1 6 )
2hoj cos TỊ — — 2 7 a 2 -sin r / j.
Di = -1
n in a t in g 0 f r o m ( 2 1 5 ) gives the follow in g e q u a tio n for a m p litu d e (a) and f r e q u e n c y (u>)
f r l T l
[f D - 0 w e h ave
1 sin 0, cos tịỉ exist only when D i = D 2 = 0, or equivalently
D \ cos Y] — D 2 sin rj = 0, D i sin rj + D2 cos = 0.
Dm h ere w e o b ta in :
= cư* — 1 -f — cos 2rj, CJ* = - sin 277 (2.19)
le formula (2.6) w ith taking into account D = 0 and (2.19) gives a restriction to a*:
Interaction of the elements of first d e g r e e of smallness quadratic nonlinearity
n d f o r c e d e x c i t a t i o n w i t h t h e s e l f - e x c i t a t i o n o f s e c o n d d e g r e e o f s m a l l n e s s
L e t us c o n s id e r a n on lin e ar s y s t e m d esc rib ed b y th e f ollow in g d ifferen tial eq u ation :
X -I- (Jj2 x = e ( a x 2 + qc o s 2ut) + e 2 [A x + D ( l — ỏ x 2) x — /33x] ,
Vherc ũ , Ỗ are positive constants The other parameters are the same as in the previous para-
r a p h s
T h e a p p r o x i m a t e s o lu t io n o f th e e q u a tio n (3 1 ) w ill b e fo u n d in th e form (2 2 ) w i t h the unpiitude (a) and phase (v>) satisfying the relations:
Ỳ t = è [ D a u j ( 1 ~ ì a 2 ) + f asin24
da £
dt 2(jj drp e2
a dt = 2 UJ
(3.2 )
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Trang 73/? 5 q 2
7 - 1 12
Equation (3.2) have a trivial solution a = 0 The non-trivial (a Ỷ 0) stationary amplitude do
and phase t/>0 are determined from the equations:
~ sin 2 \Ị >0 = - D u ị l - -a^J ,
— COS 2rpo= A — 27ỮQ.
3
(3.3)
Elim inating the phase \po gives:
w ( d o , w 2 ) = 0 ,
w ( a 0,w 2) = (A - 27ag)2+ D2W2 ( l - ? f
-(3.4) (3.5)
From the last two equations we obtain approximately
u>2 = 1 + 2c27a l ± e2y j - D 2 ( ĩ - (3.6)
This formula is plotted in the figure 4 for the parameters: E2D = 10 3, 6 = 40, e - Ệ = 10 3 and
£2Tr = —0.005 (curve l) , 527 = 0.01 (curve 2) and e2r) = 0.025 (curve 3)
t o o *
D enoting the right kand sides of the equations (3.2) by X and y, respectively, we have:
( a ) = - T ữ( V 3 a / 0 4 5aồ’ I T T V d\p / 0= CJ(A “ 2^ )
( * r ) o ( ^ ) o = c a o j D ( 1 4 ° ° ) ’
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(3.7)
Trang 8the subscript “0” means that the derivatives are calculated at stationary values do, y>0 (3.3) tab ility conditions of stationary oscillations are
““ ( f f ) o + ( a * ) o = ,2“oD( 1 - 2°«) < °'
/ d X \ f d Y _ \ _ / d X _ \ f d Y _ \ g 4 3 d W
V (3a / 0 V d\p/ 0 \ <9t/> / 0 V d a / 0 2 uj 2 0 d á ị
(3.8)
e ao > 0, the stability conditions take the form
à ị > T
(3W
To study the stability of the zero solution a = 0 of equations (3.2) we introduce the variable
have
2LU
d a
d t
s t 2(jj
‘re the non-written terms contain u and t; with higher degrees of smallness The origin u = V = 0
= 0) of the system of equations (3.11) is unstable, because the characteristic equation of the
ar terms of (3.11) has the roots with positive real part.
In the figure 4 the stable branches of resonance curves are shown by heavy lines, while the itable ones-by dotted lines.
Conclusion
In the nonlinear system under consideration, the elements characterizing the first degree of allness quadratic nonlinearity and nonresonance forced excitation (for brief, N-F-elem ents) have
e f f e c t on the oscillation in the first approximation However, they interact one with another in
Ỉ second approxim ation and appear as a parametric excitation with m odulation of the product of
eừ intensity (a , <7) This means that each element (a and q) standing alone has no effect on the
3tem and these elem ents have equal role The resonance curve (Fig 1) is bent to the right and
ts the frequency-axis at two points This curve is typical for a nonlinear system with parametric citation The passage of the system under consideration through resonance has been examined
’ig-2, 3).
In the second paragraph theừ interaction between these elements and the second degree of uallness resonance forced excitation has been studied Some typical results for the interaction
2tween parametric and forced excitations have been obtained The interaction between N-F-
em ents and self-excitation is given in the paragraph 3 The resonance curves have oval forms Fig 4 ) and are bent either to the left or to the right, depending on the sign of the parameter 7.
Acknowledgments
TỈDe a u t h o r is grateful to Dr Tran Kim Chi for numerical calculations on the digital computer,
w ork w a s financially supported by the Council for Natural Sciences of Vietnam.
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Trang 91 M itropolskii Yu A., Nguyen Van Dao Applied asymptotic methods in nonlinear oscillations Kluwer Academic Publishers, 1997.
2 Nguyen Van D inh Interaction between parametric and forced oscilations in fundamental resonance J of Mechanics, NCNST of Vietnam, Tom 17, No 3, 1995.
Received October 15, 1997
T Ư Ơ N G TÁC CỦA CÁC PHAN TỬ ĐẶC TRƯNG CHO PHI TUYEN CAP HAI
VÀ KÍCH ĐỘNG CƯỠNG BỨC VỚI CÁC KÍCH ĐỘNG LOẠI KHÁC
Trong các hệ phi tuyến, những số hạng phi tuyến cấp hai và kích động cưỡng bức không cộng
h ư ảng có bậc bé e và các số hạng ma sát tuyến tính, phi tuyến cấp ba, các kích động thông số
và cirỡng bức cộng hưởng có bậc bé é 2 sẽ không có tác dụng trong xẩp xỉ thử nhất, song chúng tác động qua lại với nhau trong xẩp xi th ử hai và những hiện tượng phi tuyến mới sẽ xuất hiện Việc nghiên cứu các hiện tircmg này là mục tiêu của bài báo Phưcmg pháp tiệm cận của cơ học phi tuyến kết hợp vớ i máy tính đã cho phép giải bài toán đặt ra.
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