Nonlinear normal modes and their superposition in a two degrees of ...

Applied Mathematics and Mechanics (English Edition, Vol. 19, No. 12, Dec. 1998)

Published by SU, Shanghai, China

NONLINEAR NORMAL MODES AND THEIR SUPERPOSITION IN A TWO DEGREES

OF FREEDOM

ASYMMETRIC SYSTEM

WITH CUBIC NONLINEARITIES* Xu Jian (~,

~)~

Lu Qishao ( g / ~ f l ) ~

H u a n g Kelei ( ~ 3 ~ , ~ ) t

(Received Sep. 2, 1996; Revised March 20, 1997; Communicated by Chen Yushu) Abstract This paper investigates nonlinear normal modes and their superposition in a two degrees ~'.freedom as)'nnnetric s)'stem, n'ith cubic nonlinearities for all nonsingular conditions,

based on the invariant subspace in nonlinear normal modes f o r the

nonlhTear equations of motion. The focus of attention is to consider relation between the validity of superposition and the static bifurcation of modal dynamics. The numerical resuhs show that the validity has something to do not only with its local restriction, but ah'o with the static bifurcation of modal d)'namics.

Key words

I.

nonlinear normal mode, asymmetric system, nonlinear vibration, nonlinear dynamics

Introduction

The modal analysis for studying the linear systems has played an important role and many scholars have been trying to apply the method for studying nonlinear vibration systems tbr many years. In a series of papers, Rosenberg It-sl introduccd the concept of nonlinear normal qaodes. According to Rosenberg, fi'ee vibrations in normal modes are vibration-inunion. In a recent work, Shaw t~I associates this view with the invariant manifold in the dynamical systems and points out that the subspace of nonlinear normal modes is also an invariant set. Meanwhile, he provides a constructive method to find nonlinear normal modes. Besides the above, many methods have been used to study nonlinear normal modes. Anand tTl analysed the free vibration of a system of the massed connected by means of nonlinear springs. He used one-term Fourier approximations for the periodic motions. In [8], group representation theory was used to investigate normal modes of symlnetric systems. Vakakis tg] introduced similar normal modes and their bifurcations by means of balancing diagrams. A numerical study of bifurcating normal modes was given in [10]. Nearly all previous work on nonlinear normal modes deals with similar ones and the chosen physical systems are symmetric ones. Although Shaw provides a new studying field for nonlinear normal modes, a basic problem has not been investigated in details, t h a t is, how many degrees the superposcd solution of nonlinear normal modes can describe solution of the * Project Supported by the National Natural Science Foundation and PSF of China t Department of Applied Mathematics and Physics, Peking University of Aeronautics and Astronautics, Beijing 100083, P. R. China 1167

Xu Jian, Lu Qishao and Huang Kelei

1168

original system if it has two or more normal modes. For it, a two degree o f . f r e e d o m system o f free vibration with cubic nonlinearities is considered as follows

:~t + xl + k ( x l - x2) + plx 3 + q ( x i -

}

-%2)3 = 0

:e2 + (1 + a)x2 + k(x2 - x l ) + p2x32 -i- q(x2 - x l ) 3 = 0 where a , k , p l , p 2

and q are parameters, and xt,x2 E ,.r

(1.1)

System (1.1) is called asymmetric

one when a ~- 0 or Pl # P2. When a = O,pl = P2, system (I.1) is called symmctric one. Taking P 2 = Pt + / 3 , then we call a and ]1 asymmetric parameters. We investigated nonlinear normal modes and their superposition of system (1.1) near the equilibrium point at origin. II.

Nonlinear

Normal

Modes and Their Singularity

An invariant set for a dynamical system is defined as a subset S of the phase space such that if ~he system is given an initial condition in S, the solution o f the governing equations o f motion remains in S all the time. According to the constructive method given by Shaw, we know they are invariant spaces for the nonlinear equation of motion and tangent to their linear counterparts, the planar eigenspace, at the equilibrium point. We rewrite system (i.I) as

:r = Yl Yl

-

xl

-

plxat

k(xl

-

-

x2)

-

q(xl

~2

Y2

Y2

- (1 + a)x2 - p2x] - k(x2 - xl)

-

1 t

x2)3

(2.1)

q(x2 - Xl) 3

or

z = [A0 +

where z = [ x l , y z , x 2 , y 2 ] r, 0 -

a 0 =

(I + k) 0 k

1

o

o

k

0

0

(2.3)

o o

- (p, + q)x~ - 3qxa~ 0

A2(z) =

(2.2)

o]

o o

&~z)]z

o

o-)

3qx~ + qx]

~]

0

0

0

- 3qx~ + qx~

0

- ( P 2 + q ) x ~ - 3qx~

0

a, k and q are not zero. Pl # P2 9 Let xl = u , y l = v, and express x2 and :)'2 functionally in terms o f u and v: X2 = 0`1 u + 0`2V + (23 tt2 + 0,4 tt/) + 0`5V2 + (2,6//,3 + a7tt2v + 0`8ttv 2 + 0,91)3 + " " l

Y2

big + b2v + b3 u2 + b4ttv + b5 v2 + b6 u3 + bTu2v + bsuv 2 + b9 v3 + "'" I

(2.4)

Substituting (2.4) into system (2.1) and gathering the coefficients o f the same power o f u and v yields a set o f identical forms, where they are obtained by M A T H E M A T I C A : u term

Nonlinear N o r m a l Modes in a Two Degrees =0

aS + bl + a 2 k - a t a s k - al-aat

1169

+ b2+ k+

]

bsk-

atk-

atbsk

= 0

(2.5)

f

v term -al+ -

b2-

a~k

a2-aa2-

= 0

bt-

a2k-

a2b2k

= 0

(2.6)

J

u z term a4 + b3 - a 2 a 3 k + a 4 k - a l a 4 k -

a3 - aa3 + b4 - a 3 k

= 0

a3 k - a 3 b 2 k + b4k - a t b 4 k

= 0

(2.7)

J

uv term - 2a3 + 2 a s + b4 - 2 a 2 a 4 k + 2 a s k

- a4 - aa4 - 263 + 265 - a 4 k -

- 2atask

= 0

a4b2k - a2b4k + 2 b s k - 2 a t b s k

= 0

(2.8)

J

v 2 term -

ct4 + b5 - 3 a 2 a n k

- a5 - aa5 - b4 -

= 0 ask -

asb2k -205a2 k = 0

I

(2.9)

u 3 term a7 + a6 - a 3 a 4 k - a 2 a 6 k + a T k - a t a T k + a2pt + a2q - 3ala2q

+ 3a]a2q - a]a2q

= 0

- a6 - aa6 + b7 - a 6 k - a 6 b 2 k - a 3 b 4 k + b7k - a t b 7 k + b2pl - a3p2 + q - 3 a l q + 3a2tq -

]

(2.

10)

a~q + b2q - 3 a l b 2 q + 3a~b2q - a31b2q = 0

u2 v term

3a6 + 2 a s + b7 - a ] k - 2 a 3 a s k . 2 a 2 a T k + 2 a s k - 2 a t a a k

-

- 3a~q + 6ata~q - 3a~a~q = 0 - a7 - aa7 - 366 + 2bs - aTk - a 7 b 2 k - a 4 b 4 k - 2 a 3 b s k - a2bTk + 2bsk - 2albsk

(2.11)

- 3a~a2P2 - 3 a 2 q + 6 a t a 2 q - 3 a ~ a 2 q

+ 3a2b2q + 6ala2b2q - 3a~a2b2q

= 0

uv2 term -

2a7 + 3a9 + b8 - 3 a 4 a s k - 3 a 2 a s k + 3 a g k - 3 a x a 9 k + 3a]q - 3ala]q

-

= 0

a8 - aa8 - 267 + 3b9 + a s k - a s b 2 k - a s b 4 k - 2 a 4 6 5 k - 2 a 2 b a k + 3bgk - 3azb9k - 3ala~p2 + 3a~q - 3ata~q + 3a]b2q - 3ala~b2q

v 3 term

= 0

(2.12)

1170

Xu Jian, Lu Qishao and H u a n g Kelei as + b9 - 2 a ] k - 4 a 2 a 9 k - a~q = O.

-

a9 - aa9

-

b8 - a g k

]

(2.13)

- a9b~k - 2 a s b s k - 3 a z b g k - a3p21

- a32q - a~baq = 0

Noticing system (2.1) and expression (2.4), we may obtain a2 = hi. Substituting it into (2.5)--(2.13), all coefficients of two normal modes can bd found by M A T H E M A T I C A . If coefficients o f model

1 is

au,a2t,a31,'",

bll,b21,b31,"',

o,12, a22, a32, " " , b12, b22, b32,"" for system approximation o f ' t w o modes are respectively Mode 1 a n = b21 = a2l

=

bl! -

8

661 -

agl

=

=

(2.1),

then

and

coefficients

ones o f mode 2 is with

cubic

order

a + ~/a2+ 4k/2k a31 =

b3t =

a41 =

k 4/61 n,61 rttl2 ~ a71 = 0,

b61 = 0 ,

b71 -

b41 =

asl

=

bs1 =

12gsl aSl. =

3h71]71, ml

bsl

(2.14)

m,1

= 0 , b91 =

as1

Mode 2 -

a

al2 = bz2 =

-

~/a

2 +

4k 2

2k

az~ = bt2 = a3", = a4:~ = a~:z = b32 = b4:z = bs:~ = 0 -

a62 --

8 k 4162.%2 12gaz Ill'2 , a72 = O, a82 -/7Z2

(2.15)

a9-2 = bc~ = 0, b n = 3hTz].72, b82 = 0, b92 = as2 m2

where 16i = ( - 62 - 3a - 6 2 k ) a 2 - (248 + 124a - 2 4 8 k ) k z + ~ / a 2 + 4k2(48 " + 48a + 32a 2 + 9 6 k +

4 8 a k + 128k z)

(2.16)

n6i = ak3pl - a k ( a 2 + 3 k 2 ) p 2 + ( a 4 + 2 a 3 k + 4 a 2 k 2 + 4 a k 3 ) q

T-'v/~a2 + 4k2( - k3Pl + a2kp2 + k3p2 - a3q - 2 a 2 k q - 2 a k 2 q )

(2.17)

gai = n61

(2.18)

hTi = 4 + 2a + 4 k -T- 4 ~ / a 2 + 4 k z

(2.19)

J7i = n'6i

(2.20)

m~ = - 16k4(1 + a + 2 k + a k ) + 4 k 4 ( 2 + a + 2k ~ 2 ~ / a 2 + 4k2) 9

(2.21)

.... j

~-..1), i = 1 expresses coefficients o f mode. 1, i = 2 ones of m o d e 2 and sign

before ~ / a 2 + 4k 2 are taken correspondingly as over and below one. Thus ' a three order approximate expression o f nonlinear normal modes is X2 =

y2

ati/./, + a6i/d, 3 + a s i u v 2 l f

at,~ + b71u2y + asiv3 J

(2.22)

Nonlinear N o r m a l Modes in a Two Degrees

1171

where i = 1, 2 are correspond to mode 1 and mode 2 respectively. Fig. 1 shows the nonlinear invariant 0.5,

P2 = 0 . 7 ,

q = 0.02,

modal surfaces when

k = 1, a = 0.05, Pz =

where the thin surface is the first nonlinear normal mode and

the thick surface the second one.

x2

~

12 -1

0

-1

0.

Y2

_.01

v 0

2

2 -2

2 ~2

7"2

T I/,

Fig. 1

0 -2

2 1

2

The invariant

u

surface

of nonlinear

Taking ct = 0 , p l = g,P2 = 0 , q = 0 coefficients of two modes.

and

normal

substituting

2

2

modes

it

into

(2.22)

(2.14)--(2.21)

yield

M o d e 1: all

= b21 = 1, a21 = b l l = a31 = b31 = a41 = b41 = a~l = bs1 = 0"1

g(k- 3), 2 k ( k - 4)

a6t

asl

b91 = asl, aT1 = b61

: 3g 3 g ( k " 1) = -'~2k(k - 4 ) ' bTl = 2 k ( k - 4)

(2.23)

ha1 = agl = 0

Mode 2: at2 --- b22 = -

(3 + 7 k ) g 2k(4 + 9k)'

a62 b92

1, tt22 = b12 = 0,32 = b32 = a 4 2 = b42 = a52 = b52 = O'l

=

asz,

aTz =

3g an = 2k(4 + 9k)' agz =

b62 =

~ 3g(1 + 3k) ,'72 = 2 k ( 4 + 9 k )

bsz = 0

F r o m s (2.23) and (2.24) have the same results as paper [6].

l

(2.24)

1172

Xu Jian, Lu Qishao and Huang Kelei

From expressions (2.14)-(2.21) of modal coefficients, it is seen that they appear to singularity when one of the following conditions is satisfied: (1) k = 0

(2.25)

(2) rai = O,

(i

= 1,2)

(2.26)

The above conditions (2.25) and (2.26) are called singularity ones of nonlinear normal modes for system (2.1). This paper only deals with nonsitagularity for nonlinear normal modes. As for singularity, we will discuss it in the other paper. III.

S u p e r p o s i t i o n in N o n l i n e a r N o r m a l Modes

It is well-known that the superposed solution of all modes for a linear system is also the solution o f the system. They can be superpose& But superposition principle for a nonlinear system may be incorrect. The nonlinear normal modes defined here are in the neighborhood at the equilibrium poinL How strong and weak nonlinearity is depends on how large or small the neighborhood is. The smaller neighborhood is, the weaker nonlinearity is. Therefore, nonlinear normal modes for superposition must be heavily dependent on the initial values. So, whether are the superposition of nonlinear normal modes affected by other factors? In the local view, one of the important distinctions between nonlinear and linear system is that there are bifurcations in nonlinear system ttq. The discussed system (1.1) is a free vibration system and we consider that effects of static bifurcations of the equilibrium solution for the modal dynamical equation on degrees of nonlinear normal modes for supcrposition describing the original solution. Letting ( u t , v l ) and ( u 2 , v 2 ) be the contribution of mode 1 and mode 2 to the displacement and velocity, then we can obtain a n o n l i n e a r relation between the physica! coordinates and modal coordinates from (2.22). Substituting (2.22) into system (1.1) yields the modal dynamical equation that two modal coordinates present: ai + (1 + k - a t ~ k ) u , + ( q + Pt - a6,k - a31iq + 3a~iq - 3atiq)a31 - aa~kui~Z~ = 0

(3.1)

Where i = 1 and i = 2 correspond to mode 1 and mode 2 respectively. Eq. (3.1) shows that when al~k

-

k

-

1

q + Pt - a6ik - aZliq + 3a~iq - 3 a l i q

> 0

(3.2)

there are static bifurcations. Specially, when denominator of the left form in (3.2) is zero, degenerate bifurcations occur and more order approximation must be considered (~1. Making data keep unity to be compared, we take 2t = 0.5, P2 = 0.7, q = 0.02 (see Fig. I). Fig. 2 shows the static bifurcation of mode 1 and mode 2 following two parameters a and k for the modal dynamical Eq. (3.1), where the thick surface is the trivial solution and the thin surface nontrivial one. If the solution of nonlinear normal mode I is noted its x~ l ) , mode x2(2) (see form (2.22)), then their superposed solution (noted as modal solution) is ~l = U1 4- l$2 "~2

g2(1)

+

]

X~2) j}'

(3.3)

Nonlinear Normal Modes in a Two Degrees

I173

u2

5

2.5

-5

0

-2.5

-"_ 2.5~0 k

2,5

0 k

(a) mode I

a

-2.5 5

5

(b) mode 2

Fig. 2 Statle bifurcation of equilibrium solution for modal dynamical equation following ~,k

~2

AAAAAA

::: -0.2

-0.4

XINld

0.2 0.I..~

~2NM

0,2 9

-0.1

-0.2

-0.2

-0.4

0.2 0 " 1 A -0. -0.2

t

~g21 '~2RM

~176 IVVVVVV'

0.2 -0.4

Fig. 3 Comparison between numerical and a =4,(ulo,vlo, u2o,v2o) = (0.1, 0, 0,1, 0)

modal

solution

when

where ul and u 2 satisfy (3.1), and x (1) and x2(2) are determined by (2.22). Next, we analyse degrees of the modal solution describing the original system (2.1). As for its accuracy, paper

1174

Xu Jian, Lu Qishao and Huang Kclci

[6] has had some good results, so we d o not repeat here. The analysis is done in three ways. First, system (2.1) is solved numerically in a four order RK method and the initial values are the same as those of modal solution (3.3). Then, the results of nonlinear n~ modes are given. Finally, two results are compared. All results of the analysis are presented in Fig. 3, Fig. 4, Fig. 5 and Fig. 61 Fixing k = I and taking Pl and P2 and q for the same value as those in Fig. 1 and Fig. 2. In Fig. 3 and Fig. 4, a = 4 and the initial values are respectively ( u t o , v m , u m , v 2 o ) = ( 0 . 1 , 0, 0 . 1 , 0) and (0.5, 0, 0.5, 0); In Fig. 5 and Fig. 6, a = - 4 and tho initial values are respectively (0.01, 0, 0.01, 0) and (0.1, 0, 0.I, 0). Thus, the initial values o f the corresponding modal solution and numerical solution may be obtained from (3.3). From Fig. 3 to Fig. 6, the thin line is the nmnerical solution, the thick line is the modal solution, and two traces are plotted together for comparison in the third row, where coefficients of the modes are given by using (2.14)--(2.21).

~_o!I

~2

-,

o

-2

-3

XlNM

:~ 2h'M

2.i

o.5

;l:

1:

-0.5 -1

7. p ~ 2/VM

~I~%INM

0.5

1 _

--

;t,

-1 -3

Fig.

4 .Comparison

between

n u m e r i c a l and a = 4, ( ulo, vlo, uao, v.~) = (0.5, O, 0.5, 0)

modal

solution

wtien

Fig. 3 presents the numerical solution keeping a good unity with the modal solution and Fig. 4 shows that their unity is affected by the initial value increasing. However, the direction

Nonlinear Normal Modes in a Two Degrees

t175

of each curve is the same. Compared with numerical solution, the modal solution has a delay or lead at time. From Fig. 3 to Fig. 4, we may see that the naethod of nonlinear normal modes is suitable for a weak nonlinear system and it may be applied for engineering. But the method pends further study for a strong nonlinear system. As for comparison between modal 9superposed solution derived from nonlinear normal modes and corresponding linear solution, paper [6] has investigated. This paper does not repeat here. The tinity is impossible in Fig. 5 and Fig. 6 although the chosen initial values are smaller at order of magnitude (see F'ig. 2). C o m p a r i n g them with Fig. 2(a), we may see that there is a static bil'urcation of the equilibrium solution for the modal dynamical equation and it has an intrinsic effect on the modal solution. To examine view what we point out, we repeat the above analysis and investigate results at a = 0 . 0 5 , a = - 0 . 0 5 , a = 2, and a = - 2 . In addition, Fixing a ' = 0 . 0 5 , we also study cases at k - ' l , k = - l , k = - 2 , k = - 4 . . It is found out that unity between the numerical and modal solution is impossible how ever small the initial values are. when parameter a or k is chosen at those values where the static bifurcations occur. These present the modal solution is not able to describe the local dynamics of the original system. When a or k is chosen at those values where the bifurcations do not appear, unity between the numerical and modal solution is only affected by the initial values, The smaller initial values are, the better t h e unity is. These illustrate that the degrees of the modal Solution 2

t

0.5 I

-0.5

.

5

~

1.5 0.5 25

30

35

40

~2NM

XINM

0.5 .i

0.3

0.5

0;1 ~

25

30

35

40

t

30

3s

~2~2NM

2.5 1.5

0.5

0.5 125

-0.~ Fig. 5 C o m p a r i s o n between n u m e r i c a l

.30 !

35

and m o d a l solution

a = - 4, ( ulo, vlo, u ~ , v2o) = ( 0 . 0 1 , O, 0 . 0 1 , O)

-t

r

when

1176

Xu Jian, Lu Qishao and Huang Kelei

describing the original system only depend on the initial values. It also confirms that the method is valid in local space. The other figures have not appeared at this paper because pages are restricted.

Zl

I

X2

^'

0.5 i u

__-

~

ZlNM

A l

U

t

3o

0.5

A l

25

35

v

I

t

,40 U

:~2NM

I

1.41 ~ 0.6 0.2

25

30

35

40

25

~ I , Y;1NM

~t

30

35

30

35

40

~2~2NM

0.5

9 F ~i5

v'

io":

~135

20'

Fig. 6 Comparisonbetween numerical and modal Solution when a,= -4,(um,vlo, u.~,v~) = (0.I, O, 0.i, O)

IV.

Conclusions

(I) On nonsingularity,stabilityof equilibrium solutions for a modal dynamical cquation that modal co-ordinates satisfy has close links with the degrees of nonlinear normal modes for superposition describing original systems. (2) Whcn parameters are chosen at those values where equilibrium solutions for a modal dynamical equation do not appear to static bifurcations, the degrees of modal solutions describing original systems depend hc:lvily on initial values of the corresponding modal dynamical equation. The smaller initial values are, the better the degrees are. Conversely, when parameters are chosen at those values where equilibrium solutions for a modal dynamical equation appear to static bifurcations, the method of nonlinear normal modes is invalid. (3) Properties of nonlinear normal modes for singularities pend further discussing.

Nonlinear Normal Modes in a Two Degrees

1177

References

[1] R. M. Rosenberg, Normal modes in nonlinear dual-mode systelns, J. Appl. Mecll., 27, 3 (1960), 263--268.

[2] R. M. Rosenberg,. On normal vibrations of a general class of nonlinear dual-mode systems, J. Appl. Mech., 28, 3 (1961), 275--283. [3] R. M. Rosenberg, The normal modes of nonlinear n-degree of freedom systems, J. Appl. Mech., 30, 1 (1962), 7--14. [41 R. M. Rosenberg, On a geometrical method in nonlinear vibrations, in Les Vibrations Forces dans systems Nonlinearies, Int. Conf. in Nonlinear Vibrations, Marseille (1964). [5] R. M. Rossenberg, On nonlinear vibrations of systems with many degrees of freedom, Adv. Appl. Met(1., 8, 2 (1966), 155--242. [ 6 1 S. W. Shaw and C. Pierre, Normal modes for nonlinear vibrations systems, J. Sottnd Vibratidn, 164, 1 (1993), 35-- 122. [7] G. V. Anand, Natural modes of a coupled nonlinear system, hTternat. J. Non-Linear Mech., 7, 1 (1972), 81~91. [81 A. K. Mishra and M. C. Singh, The normal modes of nonlinear symmetric systems by group representation theory, htternat. J. Non-Lhzear. Mech., 9, 4 (1974), 463--480. [9] T. K. Caughey, A. Vakakis and J. M. Sivo, Analytical study of'similar normal modes and their bifurcations in a class of strongly nonlinear systems, hlternat. J. Non-Linear Mech., 25, 5.(1990), 521--533. [i0] T. L. Jonson and R. Rand, On the existence and bifurcation of minimal modes, haternat. J. Non-Lhzear Mech., 14, 1 (1979), 1~12. [11] Chen Yushu and Xu Jian, Universal classification of bifurcating solutions to a primary parametric resonance in Van del Pol-Duffing-Mathieu's systems, Science hi Chhta, Series A, 39, 4 (1996), 405~417.