ANALYTICAL PREDICTIONS OF CHAOS IN A NON-LINEAR ROD

Journal of Sound and <ibration (1999) 227(3), 523}544Article No. jsvi.1999.2386, available online at http://www.idealibrary.com on

002

ANALYTICAL PREDICTIONS OF CHAOSIN A NON-LINEAR ROD

ALBERT C. J. LUO

Department of Mechanical Engineering & Industrial Engineering,Southern Illinois ;niversity, Edwardsville, Edwardsville, I¸62026-1805, ;.S.A.

AND

RAY P. S. HAN

Department of Mechanical Engineering, ;niversity of Iowa, Iowa City,IA52242-1527, ;.S.A.

(Received 14 October 1998, and in ,nal form 28 April 1999)

The resonant and chaotic conditions for non-dampened, non-linear, planar rodsare developed through the Chirikov criterion, and the subharmonic bifurcationconditions for weakly dampened, non-linear, planar rods are also presentedthrough the Melnikov method. The analytical conditions are based on a simplysupported, geometrically non-linear, planar rod model with a speci"c single-moderesponse, but these conditions are applicable to geometrically non-linear planarrods with di!erent supports. Chaos and transient motion from chaos to periodicmotion in the non-linear rod are simulated through the approximate rod model,and they are illustrated through the Poincare mapping section.

( 1999 Academic Press

1. INTRODUCTION

The accurate prediction of dynamic responses of an elastic rod experiencing largedeformation is very signi"cant in practice, and it relies on a more accurate beamtheory describing such large deformation. The earlier beam theory developed byKirchho! was presented by Love [1]. In 1958, Erisksen and Truesdell [2] used theCosserat approach [3] to develop an exact theory of stress and strain in rods andshells through the oriented bodies. In 1965, Green and Laws [4] extended thisconcept and developed a general theory of rods through two directors at each pointin rods. Such theory is very di$cult to apply in practice. In 1972, Reissner [5, 6]developed a one-dimensional "nite-strain, static beam theory but how to treat themoment was not given. In 1973, Wempner [7] presented mechanics of curved rodsbut the strain is the Almansi}Hamel strain. The strain energy of non-linear rodswas presented in reference [8]. In 1983, Maewal [9] gave strain}displacementrelations in non-linear rods and shells. In 1987, Danielson and Hodges [10]discussed non-linear beam kinematics through the deposition of the rotationtensor, and a mixed variational formulation for dynamics of moving beams was

2-460X/99/430523#22 $30.00/0 ( 1999 Academic Press

524 ALBERT C. J. LUO AND RAY P. S. HAN

presented in reference [11]. Simo and Vu-Quoc [12] used the exact strain todevelop a theory for geometrically nonlinear, planar rods, and several higher-orderapproximate theories were also given. Recently, this approach was used fordevelopment of the 3-D composite beam theory and numerical approaches weredeveloped for prediction of dynamic responses in references [13}18]. The "niteelement modelling of geometrically non-linear beams in helicopters was alsoreported in references [19, 20]. The other derivation of equations of motion forgeometrically non-linear rods can be found in references [21}25].

In 1972 the vibration of non-linear, planar rods based on an accurate beamtheory was investigated through a perturbation approach [26]. The free, non-lineartransverse vibration of beams was investigated in reference [27] when the beamproperties varied along with length. Ho et al. [28, 29] discussed the non-linearvibration of rods through a single-mode model and a perturbation approach. Theforced vibration of non-linear, torsional, inextensional beams was investigated inreference [30]. The planar, forced oscillations of shear indeformable beams wasinvestigated through a speci"c, single-mode response and perturbation method[31] and the planar motion of an elastic rod under a compressive force wasanalyzed [32]. In 1981, Holmes and Marsden [33] used the Melnikov method toinvestigate the chaotic oscillation of a forced beam. Maewal [34] investigatedchaotic motion in a harmonically excited elastic beam through the perturbationapproach and Lyapunov exponent method. The dynamical potential for thenon-linear vibration of cantilevered beams was discussed in reference [35], and thenumerical simulations of chaotic motions in non-dampened non-linear rods werealso presented. In 1984, Reichl and Zheng [36] used the Chirikov criterionpresented in reference [37] to present an analytical condition for a stochastic layerin perturbed double-well systems. From the above literatures, the analyticalprediction of chaos in non-linear rods still needs to be developed.

In this paper, an approximate model governing the large deformation of planarrods is presented. Based on this approximate theory, the conditions for chaos innon-dampened rods will be developed through the Chirikov overlap criterion, andthe conditions for subharmonic bifurcation in dampened rods will be derivedthrough the Melnikov method. Chaos in vicinity of resonant scparatrix for thenon-dampened rod and transient motion to steady state periodic oscillation in thedampened rod are simulated numerically.

2. EQUATIONS OF MOTION

Consider the planar, non-linear vibration of a simply suported, initially straight,slender rod experiencing large deformation, and this rod is subjected to an axiallycompressive force P at the two ends, and a transverse, distributed, periodic forceq(x, t), as shows in Figure 1. The other planar rod models are found in references[25, 31, 32]. The geometrical relation and the exact axial strain [31] are

sin h"w,x

J(1#u,x

)2#w2,x

, cos h"1#u

,xJ(1#u

,x)2#w2

,x

(1)

ex"e

0#J(1#u

,x)2#w2

,x!1, (2)

Figure 1. A non-linear, planar rod subjected to the compressive force P and transverselydistributed loading q (x, t).

PREDICTIONS OF CHAOS 525

where u and w are the longitudinal and transverse displacements of the middlesurface, e

0"P/EA is the initial strain and the (d)

,xdenotes partial derivative with

respect to x.Similar to references [5, 26], the equations of motion for the large deformation of

the planar rod are

oAuK#muK!(N cos h!Q sin h),x"0, oAwK#mwK!(N sin h!Q cos h)

,x"q(t),

(3, 4)

!M,x#Q[(1#u

,x) cos h#w

,xsin h]!N[w

,xcos h!(1#u

,x) sin h]+0, (5)

where N, Q and M are the normal force, shear force and bending moment; o andm are the density and damping coe$cient of the rod, and the dot represents thederivative with respect to time. The slender rod is linear elastic and the moment andnormal force [26] are

M"!

EI[w,xx

(1#u,x

)!u,xx

w,x]

[J(1#u,x

)2#w2,x

]3(6)

N(x, t)"!P#EAex"!P#EA[J(1#u

,x)2#w2

,x!1], (7)

where E, I and A are the Young's modulus, moment of inertia and cross-sectionalarea, respectively. Substitution of equation (1) into equation (5) gives

Q"

M,x

J(1#u,x

)2#w2,x

(8)

Substitution of equations (6)}(8) into equations (3) and (4) yields accurateequilibrium equations governing the large deformation of rod, i.e.,

oAuK#muR#AC(P#EA)

J(1#u,x

)2#w2,x

!EAD (1#u,x

)!w

,x(1#u

,x)2#w2

,x

]GEI[w

,xx(1#u

,x)!u

,xxw

,x]

[J(1#u,x

)2#w2,x

]3 H,xB,x

"0, (9)


oAwK#mwR #AC(P#EA)

J(1#u,x

)2#w2,x

!EADw,x!

1#u,x

(1#u,x

)2#w2,x

]GEI[w

,xx(1#u

,x)!u

,xxw

,x]

[J(1#u,x

)2#w2,x

]3 H,xB,x

"q(t). (10)

The boundary conditions for the simply supported rod in Figure 1 is

u"w"u,xx

"w,xx

"0 at x"0, l. (11)

For development of an approximate theory for the rod, the followingassumptions are adopted:

o (u)@o(w), o (u,x

)@o(w,x

), o(u,xx

)@o(w,xx

). (12)

Use of the Taylor series expansion in equations (6) and (7), and retention of thelowest-order non-linear terms give

M+!EI[w,xx

(1!2u,x!3

2w2

,x)!u

,xxw,x

], (13)

N(x, t)+!P#EA(u,x#1

2w2,x

). (14)

From equation (12), the inertia and damping forces in equation (3) (or equation (9))can be neglected, and integration of equation (3) gives

N cos h!Q sin h+e(t). (15)

Because the rod is very slender, Q sin h@N cos h can be proved through thenon-dimensionalization analysis with equations (6), (8) and (12). When Q sin h isneglected in equation (15), equations (14) and (15) give

u,x#

12

w2,x+

c (t)#PEA

"

e (t)EA

. (16)

The integration constant c (t)"e (t)!P is determined through the axial force atthe boundary in Figure 1. With equation (11), integration of equation (16) yields

u+!

12 P

x

0

w2,x

dx#x2l P

1

0

w2,x

dx. (17)

Therefore, equation (17) becomes

u,x+

12l P

1

0

w2,x

dx!12

w2,x

. (18)

Note that the reduction of equation (3) can be completed as in references [26, 32].Substitution of equation (18) into equation (10), use of the Taylor series expansionand retention of cubic terms gives an approximate equation of motion for theplanar, geometrically non-linear rod, i.e.,

oAwK#mwR #Pw,xx A1!

12l P

l

0

w2,x

dxB!EAw,xx

12l P

l

0

w2,x

dx

#EICw,xxxxA1!32l P

l

0

w2,x

dxB#2w,xxx

w,xx

w,x#w3

,xxD+q (x, t) (19)


For the other boundary conditions of rods, the approximate equations of motioncan be derived in a similar manner.

3. GALERKIN PROCEDURE

With equation (11), a solution to equation (19) is expressed as

w(x, t)"=+

m/1

Fm(t) sin

mnxl

. (20)

For a speci"c m, application of the Galerkin method to equation (19) yields

f G#d fQ#a1f#a

2f 3+Q(t), (21)

where

f (t)"mnF

m(t)

l, a

1"

Pmcr

!Pa

, a2"

EA!4Pmcr

#P4a

,

d"m

mna, a"

oAlmn

, Pmcr

"

(mn)2EIl2

, Q(t)"12a P

l

0

q (x, t) sinmnx

ldx. (22)

Consider Q(t)"Q0cos Xt as an example for illustration of chaotic motion. The

coe$cients a1

and a2

vary with the mode number m, the compressive force P andthe material and geometrical properties. Therefore, there are four possible cases, i.e.,Case I: a

1'0 a

2'0; Case II: a

1'0 and a

2(0; Case III: a

1"0 and a

2'0; Case

IV: a1(0 and a

2'0.

4. CHAOTIC CONDITIONS FOR A NON-DAMPENED ROD

4.1. CHAOTIC MOTION NEAR RESONANT SEPARATRIX

Consider chaotic motion in Case I (a1'0 and a

2'0) of the non-linear rod. The

"rst-order equations of equation (21) at d"0 are

f Q"y, yR "!a1

f!a2

f 3#Q0cosXt, (23)

and its Hamiltonian,

H"12y2#1

2a1

f 2#14a2

f 4!f Q0cos Xt, (24)

is expressed by H"H0#H

1, where

H0"1

2y2#1

2a1

f 2#14a2

f 4 and H1"!f Q

0cos Xt, (25)

For the conservative system in equation (23), its orbits in the phase plane for a givenenergy are qualitatively sketched in Figure 2. The solution of the conservativesystem for given H

0"E

0in equation (25) is [36}38]

f"h cnC2Khn

, kD and y"$Sa22

h2

ksn C

2Khn

, kDdnC2Khn

, kD , (26)

Figure 2. Conservative trajectories in the phase plane for the given energy of Case I.


where &&cn'', &&sn'', and &&dn'' are the Jacobian-elliptic functions, K"K (k) is thecomplete elliptic integral of the "rst kind and k is its modulus. The amplitude h, thenatural frequency u and the phase h of the solutions are

h"S2k2a

1(1!2k2)a

2

, u"

12S

a22

hnkK

and h"ut. (27)

The energy E0

and the action variable J for a speci"c orbit are expressed by

E0"

k2(1!k2 )a21

(1!2k2)2a2

and J"J2a

2h3

3nk3[(1!k2)K#(2k2!1)E], (28)

where E"E(k) is the complete elliptic integral of the second kind. The period iscomputed by ¹"2n/u. Substitution of equation (26) into equation (24) and theFourier expansion of the time-dependent term give

H"H0(J )!Q

0

=+n/1

Q2n~1

Mcos[(2n!1)u!X]t cosXt

#cos[(2n!1)u#X]t cosXtN, (29)

where

Q2n~1

"

nh2kK

sech CnAn!12B

K@KD , K@"K (k@) and k@"J1!k2. (30)

Except for the (2n!1)th primary resonance, all other terms in H will average tozero over one period ¹ in references [38, 39]. Hence, with equation (27), theresonant condition is

u"

X2n!1

"

n2K (k)S

a1

1!2k2. (31)


From the foregoing the speci"c k for the (2n!1)th primary resonance and thegiven X can be determined, and this value is represented by k

2s~1. Further, the

amplitude h and the action J corresponding to k2n~1

are represented by h2n~1

andJ2n~1

. Chaotic motion in the neighborhood of the (2n!1)th resonant separatrix isof great interest. Only the contribution of the (2n#1)th resonance to the chaoticmotion relative to the (2n!1)th resonant separatrix is considered, and theHamiltonian H

0in the neighborhood of the (2n!1)th resonant separatirx is

expanded by a Taylor series. Therefore, equation (29) becomes

H+H0(J

2n~1)#A

dH0

dJ BJ2n~1

(J!J2n~1

)#12 A

d2H0

dJ2 BJ2n~1

(J!J2n~1

)2#2

!Q0[Q

2n~1(J

2n~1) cos[(2n!1)u!X]t

#Q2n`1

(J2n`1

) cos[(2n#1)u!X]tN. (32)

For introduction of a new canonical co-ordinate system (pN , /M ) requiring pN "0when J"J

2n~1, a generating function is de"ned as

G(J, /M )"!( J!J2n~1

) A/M #Xt2n!1B . (33)

Therefore,

pN "!

LGL/M

"

J!J2n~1

2n!1and h"u"!

LGLJ

"

/M #Xt2n!1

. (34)

Introduce a new Hamiltonian

HM "H#

LGLt

+H0(J

2n~1)!

12

B0(2n!1)2pN 2

!Q0GQ2n~1

(J2n~1

) cos/M #Q2n`1

( J2n`1

) cosC2n#12n!1

/M #X1tDH , (35)

where

B0"A

d2H0( J )

dJ2 BJ2n~1

"

n2

4(h2n~1

)2(K2n~1

)3 CK2n~1!

1!2(k2n~1

)21!(k

2n~1)2

E2n~1D

X1"

2X(2n!1)

,LGLt

"AdH

0( J )

dJ BJ2n~1

(J!J2n~1

). (36)

The variables in equation (35),

/"/M , p"(2n!1)2(2n#1)B

0X

pN , H"

(2n!1)2(2n#1)2B0

X2[HM !H

0(J

2n~1)],

(37)

are re-scaled. Thus the re-scaled Hamiltonian is

H"

12

p2!;0cos/!<

0cos A

2n!12n#1

/#XltB , (38)


where

;0"

(2n!1)2(2n#1)2B0

X2Q

0Q

2n~1and <

0"

(2n!1)2(2n#1)2B0

X2Q

0Q

2n`1,

(39)

Q2n~1

"

J2X

Ja2(2n!1) cosh [n (n!1

2)K@

2n~1/K

2n~1]

and

Q2n`1

"

J2X

Ja2(2n#1) cosh [n(n#1

2)K@

2n`1/K

2n`1]. (40)

The Chirikov resonance overlap condition in references [36}38] is written as

2J;0#2J<

0"1. (41)

From the foregoing, the chaotic condition for the onset of global stochasticity nearthe (2n!1)th resonant separatrix with the in#uence of the (2n#1)th resonance is

Q0"

X2

4(2n!1)2(2n#1)2B0A

1

JQ2n~1

#JQ2n`1

B2. (42)

For a given excitation frequency X, equation (31) gives the elliptic modulus k2n~1

(e.g., the (2n!1)th resonance). From equation (28) the constant E0

for a speci"cresonance can be computed, and also the complete elliptic integral of the "rst andsecond kinds (K

2n~1and E

2n~1) and h

2n~1in equation (27) can be evaluated as

well. From equations (30) and (36), the intermediate parameters Q2n`1

, Q2n~1

, B0

can be determined. Finally, the excitation amplitude Q0

is determined throughequation (42). For a given X, the critical value of Q

0is also constant for the speci"c

resonance. For illustration of such a relationship, the resonant condition inequation (31) is illustrated in Figure 3(a) through excitation frequency X andconservative energy E

0at a

1"a

2"1)0. The chaotic condition in equation (42) is

shown in Figure 3(b) through excitation amplitude Q0

and excitation frequency X.The derivations for Cases II, III, IV(a) and (b) are similar to Case I; hence, the

results are placed in Appendix A. Figure 4 shows the conservative orbits in thephase plane for the aforementioned four cases. The Case IV(c) will be discussed inthe next section because it possesses a chaotic motion di!erent from the aboveCases.

The resonant conditions for Cases II}IV(b) are illustrated in the left plots ofFigures 5}8 through the excitation frequency X and the conservative energy E

0at

Da1D"Da

2D"1)0 The chaotic conditions for Case: II}IV(b) are shown in the right

plots of Figures 5}8 through the excitation amplitude Q0

and the excitationfrequency X.

4.2. CHAOTIC MOTION NEAR HOMOCLINIC ORBIT

The chaotic motion in the neighborhood of the homoclinic orbit possessingH

0"0 is di!erent from the Cases I}IV(b). The conservative orbits in the

neighborhood of the homoclinic orbit are sketched in Figure 9, and the solution of

Figure 3. (a) Resonant condition and (b) chaotic condition for Case I of a non-dampened rod ata1"a

2"1.

Figure 4. Conservative trajectories in phase plane: (a) Case II (heteroclinic orbit), (b) Case III,(c) Case IV(a) (homoclinic orbit), and (d) Case IV(b) (homoclinic orbit).


the homoclinic orbit is

f"$S2 Da

1D

a2

sech (Ja2t) and y"$S

2a2

Da1D sech (Ja

2t) tanh (Ja

2t).

(43)

Figure 5. (a) Resonant condition and (b) chaotic condition for Case II of a non-dampened rod ata1"a

2"1.

Figure 6. (a) Resonant condition and (b) chaotic condition for Case III of a non-dampened rod ata1"a

2"1.


Because kP1 as E0P0, the solutions of the conservative systems in Cases IV(a)

and IV(b) approach to the homoclinic orbit. From Appendix A, the amplitude h,the conservative energy E

0and the period ¹ for Case IV(a) are approximated from

h+S2 Da

1D

a2

, E0+

a21

a2

(1!k2 ) and ¹+

4J2kK

Ja2h

. (44)

As in references [38, 40], the complete elliptic integral of the "rst kind as kP1 isapproximated by

K+

12

ln A16

1!k2B . (45)

Figure 7. (a) Resonant condition and (b) chaotic condition for Case IV(a) of a non-dampened rodat a

1"a

2"1.

Figure 8. (a) Resonant condition and (b) chaotic condition for Case IV(b) of a non-dampened rodat a

1"a

2"1.


Substitution of equation (45) into equation (44) results in

¹+

2

JDa1Dln A

16a21

a2E

0B . (46)

The energy increment for one period ¹ in reference [38] is

DH" limT?=

PT`t0

t0ALH

0Ly

LH1

L f!

LH0

L fLH

1Ly B dt

+!Q0nXS

2a2

sech AnX

2J Da1DB sin(Xt

0), (47)

Figure 9. Homoclinic orbit and conservative trajectories in its neighborhood for Case IV(c).


where t0is the initial time and DH"E

i`1!E

i. Let E

0"E

i`1in equation (46); the

phase change of the trajectory for one period is

D/"/i`1

!/i"X¹+

2X

JDa1Dln A

16a21

a2E

i`1B . (48)

The energy and phase relationships between the ith and (i#1)th iterations are

Ei`1

+Ei!Q

0nXS

2a1

sech AnX

JDa1DB sin (/

i) and

/i`1

"/i#

2X

JDa1Dln A

16a21

a2E

i`1B , (49)

which is a separatrix map. The period-1 "xed point of equation (49) requires

2X

JDa1Dln A

16a21

a2E1B"2n(2n!1) and /

1"0, n, (50)

where E1

is the energy at the period-1 "xed point. Let Ei"E

1#DE

i, linearization

of the separatrix map in equation (49) at E1

gives a standard map, i.e.,

wi`1

"wi#D sin(/

i) and /

i`1"/

i#w

i`1, (51)

where

wi"!

2X

JDa1D

DEi

Ei

and D"

2Q0nX

E1JDa

1DS

2a2

sech AnX

2JDa1DB . (52)

The parameter D is termed the strength of the stochasticity. For the standardmapping, the transition to chaos occurs at D*+0)97162 in reference [38]. TheKAM torus will disappear and thus, the chaotic condition relative to the (2n!1)th

Figure 10. (a) Resonant condition and (b) chaotic condition for a non-dampened rod for Case IV(c)Da

1D"a

2"1 (**) Case IV(a); **, Case IV(b)).


resonance outside the homoclinic orbit is

Q0+

4D*n A

Da1D

X B SDa

1D

2a2

e~*(2n~1)nJDa1D/X+ cosh A

nX

2JDa1DB . (53)

In a similar manner, the chaotic condition relative to the mth resonance inside thehomoclinic orbit is

Q0+

8D*n A

Da1D

X B SDa

1D

2a2

e~(2mnJDa1D /X) cosh A

nX

2JDa1DB . (54)

As in Case I, the resonant conditions in neighborhood of the homoclinic orbit andthe chaotic conditions in equation (53) and (54) are given in Figure 10 forDa

1D"Da

2D"1)0.

5. BIFURCATION CONDITION FOR A WEAKLY DAMPENED ROD

5.1. RESONANCE-BASED BIFURCATION

From equation (21), the equivalent system of "rst order equations is

f Q"y, yR "!a1

f!a2

f 3#e (Q0cosXt!d f Q ). (55)

As in reference [41], the Melnikov function for the foregoing at e"1 is

M(2n~1)(t0, Q

0, d, X)"P

t0`T

t0

y[Q0cos (Xt!dy)] dt"!dI

1#Q

0I2sinXt

0,

(56)


where ¹"2n/u is the period of the non-damped system and

I1"

83a

2AS

a1

1!2k2B3[(1!k2)K#(2k2!1)E],

I2"

2nJ2X

Ja2

sechCAn!12B n

K@KD . (57)

Figure 11. Subharmonic bifurcation conditions for the dampened rod (d"0)5) at Da1D"Da

2D"1:

(a) Case I, (b) Case II, (c) Case III, (d) Case IV(a, b), (e) Case IV(b) and, (f ) Case IV(c).


If Q0/d'I

1/I

2, there is a (2n!1)th subharmonic periodic motion with

¹"2(2n!1)n/X. Thus, the subharmonic bifurcation condition in reference [41]is

Q0"

I1

I2

d. (58)

Similarly, the conditions for the subharmonic bifurcations of Cases II}IV(b) isderived and the results are summarized in Appendix B.

5.2. HOMOCLINIC-ORBIT-BASED BIFURCATION

The Melnikov function in reference [41] is

M (t0, Q

0, d, X)"P

=

~=

y[Q0cos MX (t#t

0)!d] dt"!dI

1#Q

0I2sinXt

0,

(59)

where

I1"

4(JDa1D)3

3a2

and I2"

2nJ2X

Ja2

sech AnX

2JDa1DB . (60)

The bifurcation condition for Case IV(c) can be expressed in equation (58). Thesebifurcation conditions for Cases I}IV(c) are also plotted in Figure 11 at d"0)5.

6. NUMERICAL SIMULATIONS

An adaptive Runge}Kuta integration scheme is used for illustration of chaoticmotion in the non-linear rod through the Poincare mapping section. Forobservation of chaos in vicinity of a speci"c resonant separatrix relative to the givenconserative energy E

0, the initial condition, excitation frequency and excitation

amplitude should be computed analytically. The (2n!1)th resonance of Case I isused for explanation of the computation of input parameters for numericalsimulations. For the given E

0, the (2n!1)th resonant separatrix in Case I can be

determined through the conservative system of equation (38). As in reference[36}38], the hyperbolic points are at /"0, 2m

1n (m

1"1, 2,2 ). The initial phase

h for speci"ed m1

can be computed by equation (34), and the initial condition( f

0, y

0) can be computed from equation (26), and ( f

0, y

0) also satisfy H

0"E

0in

equation (25). The elliptic modulus can be obtained from equation (28). Theexcitation frequency X can be computed through equation (31) and the criticalexcitation amplitude Q

0for chaotic motion can be computed through equation

(42). For the weakly damped rod, the excitation amplitudes Q0

in thenon-dampened rod are used, and the dampened coe$cients are chosen forsatisfaction of bifurcation condition. The computed input data is tabulated in Table1. Because the chaotic motion of Cases III and IV(a) are similar to Case I, theirnumerical simulations will not be presented.

TABLE 1

Input data for numerical simulations (a1"a

2"1)

Figures Types Order of System input parametersresonance

d ( f0, y

0) X Q

0

Non-dampened rod

12(a) Case I 3rd / (!2)75723145, 0)00000000) 7.64562 13)092*12(b) Case II 3rd / (0)32752924, 0)45959614) 2)51708 0)305s

12(c) Case I(b) 2nd / (1)27721125, 0)00000000) (R) 2)57721 0)105t

(!0)88066134, 0)41721017) (L)12(d) Case IV(c) 3rd (inner) / (0)00000000, 0)00000000) 3)00 0)06A

5th (outer)

Dampened rod13(a) Case I 3rd 0)001B (2)75723145, 0)00000000) 7)64562 13)09213(b) Case II 3rd 0)001B (!0)32752924, 0)45959614) 2)51708 0)30513(c) Case IV(b) 2nd 0)001B (0)88066134, 0)41721017) (R) 2)57721 0)105

(!1)27721125, 0)00000000) (L)13(d) Case IV(c) 3rd (inner) 0)001B (0)00000000, 0)00000000) 3)00 0)06

*Predicted by Chirikov overlap crierionsLess than Q

0"0)2598 obtained through the Chirikov overlap criterion,

tLess than Q0"0)1257 obtained by the analytical prediction,

AGreater than the critical values Q0+0)0294 for the 5th resonant separatrix outside of homoclinic

orbit and Q0+0)202 for jthe 3rd resonant-separatrix inside of homoclinic orbit,

BLess than critical damping value for periodic motion relative to the resonance.


The Poincare mapping sections of chaotic motions for Cases I, II, IV(b) and (c) ofthe non-dampened, non-linear rod are shown in Figure 12. From selection of theinput data in Table 1, the chaotic motion in the neighborhood of the third resonantseparatrix should be observed for Case I. In Figure 12(a) such chaotic motionrelative to the third resonant separatrix is clearly shown. The chaotic motion in theneighborhood of the third resonant-separatrix of Case II is also simulated, asshown in Figure 12(b). For Case IV(b), the chaotic motions based on the secondresonant sepatratix in the two potential wells are illustrated in Figure 12(c). Theasymmetry of the chaotic motion in the two wells is observed even though theconservative trajectories in the two wells are symmetric. For Case IV(c), the chaoticmotion in the neighborhood of the homoclinic orbit pertains to Cases IV(a) and (b).As analytically predicted, the 5th resonant separatrix outside the homoclinic orbitand the 3rd resonant separatrix in the two potential wells are observed in Figure12(d), and the asymmetry of chaotic motions in the two wells is also observed.

The Poincare mappings of the transient motion from the chaotic motion to thesteady state, periodic motion relative to the subharmonic resonance for Cases I, II,IV(b) and (c) of the weakly dampened rod are illustrated in Figure 13. Forobservation of periodic motion pertaining to subharmonic resonance, the dampingfor numerical simulations is less than the critical damping. For Case I, the transient

Figure 12. Poincare mapping sections of chaos in a non-dampened rod at Da1D"Da

2D"1: (a) Case

I, (b) Case II, (c) Case IV(b), and, (d) Case IV(c).


motion from the chaotic motion to a steady state periodic motion relative to the3rd resonance is illustrated in Figure 13(a). The period-3, subharmonic motion willbe obtained as tPR. If d*0)195, such periodic motion under the chosenexcitation will not exist. Similarly, for Case II, a transient motion from the chaoticmotion to a steady state, period-3 motion is shown in Figure 13(b). For Case IV(b),a transient motion from the chaotic motion to the steady-state, period-2 motions inthe two potential wells are given in Figure 13(c). The locations of the period-2motion in the two wells are very distinguishing. For Case IV(c), a transient motionfrom the chaotic motion to period-3 motion in the right potential wells is shown inFigure 13(d).

7. CONCLUSION

The resonant and chaotic conditions for non-dampened, non-linear, planar rodsare derived, and the conditions for subharmonic bifurcation in weakly dampened,

Figure 13. Poincare mapping sections of transient motion from chaotic to steady-state periodicmotions in a dampened rod at Da

1D"Da

2D"1: (a) Case I, (b) Case II, (c) Case IV(b), and, (d) Case IV(c).


non-linear, planar rods are also presented. The derivation of the analyticalconditions is based on a simply supported, planar rod model, but these conditionsare applicable to the planar rod models with di!erent supports.

ACKNOWLEDGMENTS

The authors would like to thank the anonymous reviewers for their usefulsuggestions.

REFERENCES

1. A. E. H. LOVE 1944 A ¹reatise on the Mathematical ¹heory of Elasticity. New York:Dover Publications, fourth edition.

2. E. COSSERAT and F. COSSERAT 1909 ¹heorie des Corps Deformables, 953}1173. Paris:Appendix.


3. J. L. ERICKENSEN and C. TRUESDELL 1958 Archieves for Rational Mechanics and Analysis1, 295}323. Exact theory of stress and strain in rods and shells.

4. A. E. GREENE, F. R. S. and N. LAWS 1966 Royal Society of ¸ondon Proceeding A 293,145}155. A general theory of rods.

5. E. REISSNER 1972 Journal of Applied Mathematics and Physics (ZAMP) 23, 759}804. Onone-dimensional "nite-strain beam theory: the plane problem.

6. E. REISSNER 1973 Studies in Applied Mathematics LII, 87}95. On one-dimensionallarge-displacement "nite-strain beam theory: the plane problem.

7. A WAEWAL 1983 Journal of Structural Mechanics 10, 393}401. A set of strain-displacement relations in nonlinear rod and shells.

8. V. L. BERDICHEVSKY 1982 PMM 45, 518}529. On the energy of an elastic rod.9. G. WEMPNER 1973 Mechanics of Solids with Application to ¹hin Body. New York:

McGraw-Hill.10. D. A. DANIELSON and D. H. HODGE 1987 ASME Journal of Applied Mechanics 54,

258}262. Nonlinear beam kinematics by decomposition of the rotation tensor.11. D. H. HODGE 1990 International Journal of Solids and Structures 26, 1253}1273. A mixed

variational formulation based on exact intrinsic equations for dynamics of movingbeams.

12. J. C. SIMO and L. VU-QUOC 1987 Journal of Sound and <ibration 119, 487}508. The roleof nonlinear theories in transient dynamics analysis of #exible structures.

13. J. C. SIMO nd L. VU-QUOC 1991 International Journal of Solids and Structures 27,371}393. A geometrically-exact rod model incorporating shear and torsion-warpingdeformation.

14. L. VU-QUOC and I. K. EBCIOGLU 1995 ASME Journal of Applied Mechanics 62, 756}763.Dynamics formulation for geometrically-exact sandwich beams and 1-D plates.

15. L. VU-QUOC and H. DENG 1995 ASME Journal of Applied Mechanics 62,479}488. Galerkin projection for geometrically-exact sandwich beams allows for playdrop-o!.

16. L. VU-QUOC, H. DENG and I. K. EBCIOGLU 1996 Journal of Nonlinear Science 6, 239}270.Sandwich beams: a geometrically-exact formulation.

17. L. VU-QUOC and I. K. EBCIOGLU 1996 Zeitschrift fur Angewande Mathematik undMechanik (ZAMM) 76, 756}763. General multilayer geometrically-exact beams/1-Dplate with piecewise linear section deformation.

18. L. VU-QUOC and H. DENG 1997 Computer Methods in Applied Mechanical andEngineering 146, 135}172. Dynamics of geometrically-exact sandwich beams:computational aspects.

19. M. BORRI and P. MANTEGAZZA 1985 l1Aerotecnica Missili e Spazio 64, 143}159. Somecontributions on structural dynamic modeling of helicopter rotor blades.

20. O. A. BAUCHAU and N. K. KANG 1993 Journal of the American Helicopter Society 38,3}14. A multibody formulation for helicopter structural dynamic analysis.

21. M. R. M. CRESPO DA SILVA and C. C. GLYNN 1978 Journal of Structural Mechanics 6,437}448. Nonlinear #exural}#exural}torsional dynamics of inextensional beams*I:equations of motion.

22. M. R. M. CRESPO DA SILVA 1991 Applied Mechanics Review 44, 51}59. Equations fornonlinear analysis of 3D motions of beams.

23. P. F. PAI and A. H. NAYFEH 1990 Nonlinear Dynamics 1, 477}502. Three-dimensionalnonlinear vibrations of composite beams*I: equation of motion.

24. P. F. PAI and A. H. NAYFEH 1992 Nonlinear Dynamics 3, 273}303. A nonlinearcomposite beam theory.

25. P. F. PAI and A. H. NAYFEH 1994 International Journal of Solids and Structures 31,1309}1340. A fully nonlinear theory of curved and twisted composite rotor bladesaccounting for warping and three-dimensional stress e!ects.

26. G. R. VERMA 1972 Studies in Applied Mathematics LII, 805}814. Nonlinear vibrations ofbeams and membranes.


27. A. H. NAYFEH 1973 Journal of the Accoustical Society of America 53, 766}770. Nonlineartransverse vibration of beams with properties that vary along the length.

28. C. H. HO, R. A. SCOTT and J. G. EISLEY 1975 International Journal of NonlinearMechanics 10, 113}127. Non-planar, nonlinear oscillations of beams*I: forcedmotions.

29. C. H. HO, R. A. SCOTT and J. G. EISLEY 1976 International Journal of Sound and<ibration 47, 333}339. Non-planar, nonlinear oscillations of beams*II: free motions.

30. M. R. M. CRESPO DA SILVA and C. C. GLYNN 1978 Journal of Structural Mechanics 6,449}641. Nonlinear #exural}#exural}torsional dynamics of inextensional beams*II:forced motion.

31. A. LUONGO, G. REGA and F. VESTRONI 1986 ASME Journal of Applied Mechanics 53,619}624. On nonlinear dynamics of planar shear indeformable beams.

32. T. M. ATANACKOVIC and L. J. CVETICANIN 1996 ASME Journal of Applied Mechanics 63,392}398. Dynamics of plane motion of an elastic rod.

33. P. J. HOLMES and J. MARSDEN 1981 Archives for Rational Mechanics and Analysis 76,135}166. A partial di!erential equation with in"nitely many periodic orbits: chaoticoscillations of a forced beam.

34. A. MAEWAL 1986 ASME Journal of Applied Mechanics 53, 625}631. Chaos ina harmonically excited elastic beam.

35. V. L. BERDICHEVSKY, W. W. KIM and A. OZBEK 1995 Journal of Sound and<ibration 179,151}164. Dynamics potential for nonlinear vibrations of cantilevered beams.

36. L. E. REICHI and W. M. ZHENG 1984 Physical Review A 30, 1068}1077. Perturbeddouble-well system: the pendulum approximation and low-frequency e!ects.

37. B. V. CHIRIKOV 1979 Physics Reports 52, 263}379. A universal instability of manydimensional oscillator systems.

38. A. J. LICHTENBERG and M. A. LIEBERMAN 1992 Regular and Chaotic Dynamics. NewYork: Springer-Verlag.

39. A. C. J. LUO 1995 Analytical modeling of bifurcations, chaos, and multifractals in nonlineardynamics. Ph.D. Dissertation, University of Manitoba, Winnipeg, Manitoba, Canada.

40. A. CAYLEY 1895 Elliptic Functions. London: Bells and Sons.41. J. GUCKENHEIMER and P. HOLMES 1983 Nonlinear Oscillations, Dynamical Systems and

Bifurcation of <ector Fields. Springer-Verlag: New York.

APPENDIX A: RESULTS FOR A NON-DAMPENED ROD

The results for Cases II, III, IV(a) and (b) are listed as follows.Case II (a

1'0 and a

2(0):

E0"

k2a21

(k#1)2 Da2D, h"S

2k2a1

(k#1)2 Da2D, u"

JDa2Dhn

2J2kK (k), (A.1)

B0"

n2

4h2(k@2n~1

)2 (K2n~1

)3 CK2n~1!

1#(k2n~1

)21!(k

2n~1)2

E2n~1D , (A.2)

Q2n~1

"

J2X

JDa2D (2n!1) sinhCnAn!

12B

K@2n~1

K2n~1

D,

Q2n`1

"

J2X

JDa2D (2n#1) sinhCnAn#

12B

K@2n`1

K2n`1

D. (A.3)


The chaotic condition for this case is identical to that of Case I.Case III: a

1"0 and a

2'0:

k"1

J2, h"A

4E0

a2B1@4

, u"

Ja2hn

2K(k), B

0"

n2

2h2(K2n~1

)2, (A.4)

Q2n~1

"

J2X

JDa2D (2n!1) cosh [n (n!1

2)]

,

Q2n`1

"

J2X

JDa2D (2n#1) cosh [n (n#1

2)]

. (A.5)

The chaotic condition for this case is expressed by equation (42).Case I<(a): a

1(0 and a

2'0: Consider the conservative energy of Case IV: (a)

E0'0, (b) E

0(0, (c) E

0"0. Only the results for Cases IV(a) and (b) are given

herein because the results for Case IV(c) have been presented in the paper.For Case IV(a), we have

E0"

(1!k2 )k2a21

(2k2!1)2a2

, h"S2k2Da

1D

(2k2!1)a2

, u"

Ja2hn

2J2kK, (A.6)

B0"

n2

4h2(k2n~1

)3 CK2n~1!

1!2(k2n~1

)21!(k

2n~1)2

E2n~1D , (A.7)

Q2n~1

"

J2X

Ja2(2n!1) cosh [n (n!1

2)K@

2n~1/K

2n~1],

Q2n`1

"

J2X

Ja2(2n#1) cosh [n (n#1

2)K@

2n`1/K

2n`1]. (A.8)

The chaotic condition for this case is expressed by equation (42).For Case IV(b), we have

E0"

(k2!1)a21

(2!k2)2a2

, h"S2 Da

1D

(2!k2)a2

, u"

Ja2hn

2J2K, (A.9)

B0"

n2

h2(kn)4(k

n)3 CKn

!

2!(kn)2

1!(kn)2

EnD , (A.10)


Qn"

J2X

Ja2n cosh[nnK@

n/K

n], Q

n`1"

J2X

Ja2(n#1) cosh[n(n#1)K@

n/K

n],

(A.11)

Q0"

X2

4n2(n#1)2B0A

1

JQn#JQ

n`1B2. (A.12)

For this case, the chaotic condition presented in equation (A.13) is di!erent fromequation (41).

APPENDIX B: RESULTS FOR A WEAKLY DAMPENED ROD

The results for Cases II, III, IV(a) and (b) of a weakly dampened rod are listedherein.

Case II: a1'0 and a

2(0:

I1"

83 Da

2D AS

a11

k2B2[(1#k2)E!(1!k2 )K],

I2"

2nJ2X

JDa2D

csch CAn!12Bn

K@K D sin(Xt

0). (B.1)

Case III: a1"0 and a

2'0:

I1"

16[K(k)]4X3

3[(2n!1)n]3a2

, I2"

2nX

Ja2

sech CAn!12BnD sin(Xt

0). (B.2)

Case I<(a): a1(0, a

2'0 and E

0'0:

I1"

83a

2AS

Da1D

2k2!1B3[(2k2!1)E#(1!k2)K],

I2"

2nJ2X

Ja2

sech CAn!12Bn

K@K D sin(Xt

0). (B.3)

Case I<(a): a1(0, a

2'0 and E

0(0:

I1"

43a

2AS

Da1D

2!k2B3[(2!k2)E!(1!k2 )K],

I2"

nJ2X

Ja2

sech CnnK@K D sin(Xt

0). (B.4)

The bifurcation condition for all the cases is identical to equation (58).

ANALYTICAL PREDICTIONS OF CHAOS IN A NON-LINEAR ROD

Documents

Transcript of ANALYTICAL PREDICTIONS OF CHAOS IN A NON-LINEAR ROD