Limit cycle prediction for ballooning strings

Limit cycle prediction for ballooning strings

International Journal of Non-Linear Mechanics 35 (2000) 373}383 Limit cycle prediction for ballooning strings Fang Zhu, Christopher D. Rahn* Departme...

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International Journal of Non-Linear Mechanics 35 (2000) 373}383

Limit cycle prediction for ballooning strings Fang Zhu, Christopher D. Rahn* Department of Mechanical Engineering, Clemson University, Box 340921, Clemson, SC 29634-0921, USA Received 5 February 1998; accepted 12 January 1999

Abstract This paper uses the Hopf bifurcation and center manifold theorems to predict the limit cycles of ballooning strings observed in many textile manufacturing processes. The steady state and linearized solutions of the non-linear governing equations are reviewed. The non-linear dynamic equations are discretized and projected to a "nite-dimensional linear normal mode space, resulting in a set of non-linearly coupled but linearly uncoupled ordinary di!erential equations. The "rst two equations, corresponding to the lowest normal mode, are analyzed using the Hopf bifurcation theorem. The "rst Lyapunov coe$cient is calculated to prove the existence of stable limit cycles for double-loop balloons with small string length. Analysis of the "rst two modes using the center manifold method veri"es the e!ectiveness of the two-dimensional approximation. The bifurcation theorem, however, fails to apply to the large string length Hopf bifurcation point because the "rst Lyapunov coe$cient is indeterminable. Numerical simulation of the non-linear two-dimensional equations agrees with experimental results quantitatively for small string length and qualitatively for large string length. ( 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction Rotating strings form balloons similar to those observed in textile manufacturing process such as spinning, unwinding, and twisting. Recent studies on rotating strings have shown a variety of interesting dynamic behaviors. The numerical studies by Batra et al. [1] and Fraser [2] and theoretical and experimental studies by Zhu et al. [3] show balloons may have multiple shapes for "xed parameters due to the highly non-linear governing equations. The number of loops in the balloon shape increases with increasing string length or decreasing tension. Linear dynamic investigations by Zhu et al. [3] and Stump et al. [4] show that

* Corresponding author.

single-loop balloons are stable, one and a half-loop balloons are divergent unstable, and double-loop balloons may be #utter unstable for su$ciently low air drag. String extensibility has limited e!ect on balloon stability for most of the textile yarns [5]. The linear vibration analysis in [3] captures the steady state and the experimentally-observed jump phenomena and #utter instabilities. The experimental observation of stable limit cycles in the #utter regions, however, is not predicted. In this paper, a non-linear vibration analysis is used to predict limit cycles in double-loop balloons. Generally, three approaches are used to study non-linear dynamical systems. First, the Hopf bifurcation theorem and center manifold theorem provide qualitative results [6,7]. Holmes [8] applies the center manifold theory to the study of the bifurcations in a continuous gyroscopic system. Second, perturbation methods [9,10] allows

0020-7462/99/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 9 9 ) 0 0 0 0 7 - 4

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quantitative study of the local behaviors of nonlinear dynamic systems. Applications of perturbation methods to the limit cycles in strings and cables include the papers by Perkins [11] and Mockensturm et al. [12]. Finally, direct simulation of the dynamic equations provides global results. Recently, Clark et al. [13] used this method to treat the dynamics of rotating strings. In this paper, the non-linear dynamics of ballooning strings is investigated. First, the equations of motion, steady-state solutions, and linear stability are reviewed. Second, the Hopf bifurcation theorem on a single-mode (two-dimensional) model is used to predict the existence of stable limit cycles for double-loop balloons with small string length. Third, a four-dimensional (two-mode) model and the center manifold theorem verify the two-dimensional prediction. Finally, simulations of the two-dimensional model are compared with experimental results.

2. Equations of motion

Fig. 1. Schematic diagram of a rotating string system.

Fig. 1 shows a schematic diagram of the ballooning string system. The string is modeled as a perfectly #exible one-dimensional continuum pinned at the upper boundary and attached to an eyelet rotating with constant angular velocity ) at the bottom boundary. The unstressed, s0, steady state, s*, and "nal, s&, con"gurations of the string are shown. The coordinate system axes e , e , and 1 2 e rotate about e with speed ). In this coordinate 3 3 system, the steady state s* is stationary. The vector R*(S , ¹)"X(S )e #>(S )e # 0 0 1 0 2 Z(S )e locates s* where S is the arc length coordi0 3 0 nate measured along s0 and ¹ is time. The "nal con"guration s& is located by R&(S , ¹). The relative 0 displacement of the string between s* and s& is U(S , ¹)"R&!R*"; e #; e #; e . (1) 0 1 1 2 2 3 3 Under the assumption of small strains and negligible gravitational and tangential drag forces [3], the non-dimensional equation of motion for the ballooning string is

AA

L Ls

BA

Lr Lu c Lu Lu p#c ) # ) Ls Ls 2 Ls Ls

BB

Lr Lu # Ls Ls

"e ](e ]r)#e ](e ]u)#e 3 3 3 3 3 Lu L2u d ] # # n Dv& Dv& , Lt Lt2 16 n n

(2)

where R* S r" "xe #ye #ze , s" 0, 1 2 3 a a U u" , a

t")¹,

16aD n, d" n oA 0

(3)

P E p" , c" , oA a2)2 oa2)2 0

L(r#u) ] v& " n Ls

AA

B

(4)

B

Lu L(r#u) #k](u#r) ] , (5) Lt Ls

where a is the length of the rotating eyelet, D is the n air drag coe$cient, o is the string mass density, E is the string Young's modulus, A is the unstressed 0 string cross-sectional area, and P is the steady-state

F. Zhu, C.D. Rahn / International Journal of Non-Linear Mechanics 35 (2000) 373}383

string tension. The corresponding boundary conditions are r(0)"0, r(l)"e #he , 1 3

(6)

u(0, t)"u(l, t)"0,

(7)

where l is the non-dimensional unstressed string length, h"H/a, and H is the vertical distance between the two eyelets.

3. Steady state and linear dynamics In steady state, the displacement u"0, so the equilibrium equation of the string is

A B

d dr d p "k](k]r)# n Dv* Dv* , ds ds 16 n n

(8)

375

where

A

B

dr dr v* " ] (k]r)] . n ds ds

(9)

Due to the neglect of tangential drag, the string tension p may be related to the eyelet tension p as % follows: p"p !1 (x2#y2). (10) % 2 Eqs. (8) and (10) are solved using a shooting technique detailed in Ref. [3]. Fig. 2 shows the solution curves of eyelet tension versus the string length parameter *"1!h/l for three d . Each point on n the curve represents a steady-state balloon. Representative balloon shapes for d "1 and *"0.153 n are shown in the inset plots in which r"(x2#y2)1@2. Balloons between A and B have a single-loop. From B to F, the balloon shapes change from one and a half-loops to triple-loops.

Fig. 2. Non-dimensional eyelet tension versus string length: h"10, d "1 (solid), d "2 (dash-dotted), d "4 (dashed). Capital letters n n n indicate turning points. Small letters indicate the locations of balloons shown in the inset plots.

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F. Zhu, C.D. Rahn / International Journal of Non-Linear Mechanics 35 (2000) 373}383

The linearized equations for small motions around the steady-state solution are obtained by the cancellation of the steady-state terms and the non-linear u terms in Eq. (2), resulting in

CA B

L Lr Lu Lr Lu c ) #p Ls Ls Ls Ls Ls

D

Lu L2u d "k](k]u)#2k] # # n Jv , Lt Lt2 16 n

(11)

where J is the Jacobian Matrix [3] and (12) v "v& !v* . n n n The displacement "elds are approximated by the series expansions: N u (s, t)" + g (t)h (s), k"1, 2, 3, k jk j j/1 where the comparison functions

(13)

S

h (s)" j

2 sin( jps/l) l

satisfy the boundary conditions (7). Substitution of Eq. (13) into Eq. (11) and application of Galerkin's method provides the discretized equation of motion gK #Cg5#Kg"0, or in the "rst-order form xR "Ax,

4. Non-linear dynamics (14)

where dots indicate time di!erentiation, g"[g , 11 g , g , g ,2, g ]T, x"[g, g5]T, and 12 13 21 3N 0 I . (15) A" !K !C

C

one-and-a-half and two-and-a-half-loop balloons remain unstable. A typical stability result is plotted in Fig. 3 for d "2. The insert plot of Fig. 3 shows n the "rst eigenvalue root locus associated with the steady-state solutions from A to D. In this range, all higher eigenvalues have negative real parts and thus have no e!ect on the system stability. From A to B, the eigenvalue stays in the left half plane. From B to C, the eigenvalue is on the real axis in the right half plane. At C, d becomes less than zero. 1 At h , the eigenvalue crosses the imaginary axis, 1 signifying a Hopf bifurcation. The eigenvalue crosses the imaginary axis again towards the left half plane at h . Thus, a Hopf bifurcation occurs if the 2 string length decreases from D to h . Obviously, 2 balloons between h and h are #utter unstable. 1 2 Increasing the air drag decreases the #utter region. Clark et al. [13] get the same results via a perturbation technique. Detail discussions of the linear stability can be found in [3,5]. The linear analysis also shows that Lu/Ls is nearly perpendicular to Lr/Ls for low-order modes. In most applications, c is very large so Lr/Ls ) Lu/Ls+0 for c(Lr/Ls ) Lu/Ls) to be comparable to other terms in Eq. (11). The balloon is signi"cantly sti!er in the tangential direction de"ned by Lr/Ls. Thus, vibration in the low-order modes is expected to be perpendicular to Lr/Ls (i.e. Lr/Ls ) Lu/Ls+0).

D

The eigenvalues of A (j "d #iu ) are obtained j j j numerically. The numerical result shows that, for d (4 all n single-loop balloons are stable (d(0), one-anda-half and two-and-a-half-loop balloons are divergent unstable (d '0, u "0), double-loop balloons 1 1 may be stable or #utter unstable (d '0, u O0), 1 1 and triple-loop balloons may be stable or higher mode #utter unstable. For d *4, all single-, n double-, and triple-loop balloons are stable but

Cancellation of the steady-state terms and linearization of the air drag term in Eq. (2) yields

C

A

L Lu Lr Lu c Lu Lu p # c ) # ) Ls Ls Ls Ls 2 Ls Ls

BA

BD

Lr Lu # Ls Ls

Lu L2u d "k](k]u)#2k] # # n Jv , Lt Lt2 16 n

(16)

which contains quadratic and cubic non-linearities. The neglect of non-linear air drag eliminates all velocity-dependent non-linearities from Eq. (16), simplifying the analysis. This is reasonable for small d because J has the same order as v& . Thus, the n n linear air drag is third-order. Substitution of expansion (13) into Eq. (16) and application of Galerkin's approach gives the non-linear discretized equation

F. Zhu, C.D. Rahn / International Journal of Non-Linear Mechanics 35 (2000) 373}383

377

Fig. 3. Root locus for d "2 and c"2000. n

in state-space form xR "Ax#F(x, t),

(17)

where F is a vector containing velocity independent quadratic and cubic non-linearities. Eq. (17) can be cast into Jordan canonical form [15] yR "By#f(y, t)

(18)

The linear model is least predictive in the #utter region h !h . Experiments [3] show that the 1 2 string either limits cycles around the double-balloon shape or jumps to a single-loop balloon. Limit cycling causes tension #uctuations that can reduce the quality of the "nished textile product. Thus, it is important to study the non-linear response of the system in the #utter region. Equation (18) can also be written as

through the transformation u5 "Au#q (u, v), 1

x"Py, where the columns of P are the real and imaginary parts of the eigenvectors of A, f"P~1F, and

v5 "Bv#q (u, v) 2 with

B"P~1AP

C

d

1 u 1 " F

!u d 1 F

1 2 2 F

0

0

2

0

0

0

0

0

0

F

F

D

d !u 3N 3N u d 2 3N 3N

.

C

A"

(19)

(20)

D

GH

d !u y 1 1 , u" 1 u d y 1 1 2

and

G

H

f (u, v) . q " 1 1 f (u, v) 2

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F. Zhu, C.D. Rahn / International Journal of Non-Linear Mechanics 35 (2000) 373}383

From the linear analysis, u is associated with a center (at h and h ) or unstable (between h and 1 2 1 h ) manifold and v with a stable manifold. Note 2 that u and v are linearly decoupled.

terms, except for z2z6 which is the resonant term and plays a key role in the Hopf bifurcation analysis, are transformed into higher-order non-linear terms by introducing the second transformation

4.1. Two-dimensional approximation

g8 g8 g8 03 w6 3, z"w# 30 w2# 12ww6 2# 12j 4jM 6(3jM !j)

First, it is assumed that y to y decay to zero 3 6N and thus have little e!ect on the system response. This assumption is checked in the next section by including y and y in the non-linear analysis. In 3 4 this section, the reduced Eq. (18) is studied:

GH C

DG H G

H

y5 d !u y f (u, 0) 1 " 1 1 1 # 1 . (21) f (u, 0) y5 u d y 2 2 1 1 2 Note that f , f , d and u depend on the string 1 2 1 1 length. At h , d "0 and u "u '0. For con1 1 1 0 venience, a string length parameter a is introduced which is zero at h , negative to the left of h , and 1 1 positive to the right of h . Eq. (21) is rewritten in 1 complex form z5 "j(a)z#f

(22)

w5 "jw#c w2w6 #O(DwD4), (26) 1 in which c is explicitly given by 1 g g (2j#jM ) Dg D2 Dg D2 g 02 # 21. c " 20 11 # 11 # 1 2DjD2 j 2(2j!jM ) 2 (27) Eq. (26) is the PoincareH normal form for the Hopf bifurcation. Note that c is a function of the string 1 length parameter a and Eq. (26) is valid for all su$ciently small DaD. Based on the Hopf bifurcation theorem [7], the system described by Eq. (26) (and thus Eq. (21)) is topologically equivalent near the origin to

GH C

DG H

k5 b !1 1 " 1 b k5 2

with z"y #iy , j"d #iu 1 2 1 1 and f"f (u, 0)#if (u, 0) 1 2 or in power series form

yielding

(23)

1 g zjz6 k, (24) z5 "jz# + j!k! jk 2xj`kx3 where z6 is the complex conjugate of z, and the power series coe$cients g "g (a) are calculated jk jk numerically. For small a, the transformation g g g 02 z6 2 z"z# 20z2# 11zz6 # 2j 2(2jM !j) jM eliminates the quadratic terms in Eq. (24), yielding 1 g8 zjz6 k#O(DzD4), (25) z5 "jz# + j!k! jk j`k/3 where jM is the conjugate of j and the g8 are transjk formed coe$cients. In addition, all cubic non-linear

k 1 k 2

GH

k #sign(l (0))(k2#k2) 1 , (28) 1 1 2 k 2 if the eigenvalues cross the imaginary axis with non-zero speed d (a"0)"0, 1

dd 1 da

K

O0 a/0

(29)

and

K

1 Re(c ) 1 " Re(ig g #u g )O0, l (0)" 20 11 0 21 1 2u2 u 0 0 a/0 (30) where l (a) is the "rst Lyapunov coe$cient. 1 The equilibrium point (0, 0) of Eq. (28) is stable when b(0. The Hopf bifurcation occurs at b"0. When b is positive and near zero, the equilibrium point is unstable, but the system has stable limit cycles around the equilibrium point if l (0)(0. If 1 l (0)'0, no stable limit cycle exists. Equivalently, 1

F. Zhu, C.D. Rahn / International Journal of Non-Linear Mechanics 35 (2000) 373}383

an equilibrium point (or double-loop balloon) of system (21) is stable when a(0, Hopf bifurcates at a"0, and stably limit cycles when a'0 if l (0) is 1 negative. Thus, l (0) determines the existence of 1 stable limit cycles in the #uttering double-balloon response. Note that only g (0), g (0), and g (0) 20 11 21 are necessary for the calculation of l (0). 1 For the left bifurcation point (h : *"0.0525) 1 when d "2 and c"2000, d "0 and d@ O0. Nu1 n 1 merical calculation yields u "0.337, g "0.095#0.136i, 1 20 g "!0.024!0.020i, g "!57.01!12.6i, 11 21 (31) leading to l (0)"!84.58(0. 1 Thus, this analysis predicts the existence of stable limit cycles near the bifurcation point h . Also, Eq. 1 (31) implies the dominance of the z2z6 cubic term (g ) in the non-linear response. 21 Unfortunately, the Hopf bifurcation theorem does not apply to the second bifurcation point at h because l (0)"0. The existence of stable limit 2 1 cycles near h can be determined analytically by 2 including fourth or higher-order resonant terms in Eq. (26) or numerically via simulation of Eq. (21). 4.2. Four-dimensional approximation Four equations are investigated to check the prediction obtained from the two-dimensional analysis. This corresponds to the inclusion of the "rst two modes in the non-linear system (16). Eq. (20) applies with u"[y , y ]T3R2 and v"[y , 1 2 3 y ]T3R2. Again, y ,2, y are assumed zero. At 4 5 6N the bifurcation point h , u and v are the center and 1 stable manifolds, respectively, with q (0, 0)"0, Dq (0, 0)"0, 1 1 q (0, 0)"0, Dq (0, 0)"0, 2 2 where D"L/Lu#L/Lv. The center manifold theory [14] is applied to reduce the four-dimensional system to a two-dimensional center manifold.

379

A time-invariant manifold is called a center manifold for system (20) if it can be locally represented as follows: =#(0)"M(u, v)3R2]R2D v"<(u), <(0)"0, D<(0)"0N for su$ciently small DuD. The conditions <(0)"0 and D<(0)"0 imply that =#(0) is tangent to the eigenspace formed by the eigenvectors of matrix A. For su$ciently small DuD, the dynamics of Eq. (20), restricted to the center manifold, is u5 "Au#q (u, <(u)), u3R2, 1

(32)

or, in complex form with z"y #iy , 1 2 z5 "iu z#G(z, z6 , v), 0 v5 "Cv#H(z, z6 , v).

(33)

A quadratic approximation of the center manifold =#(0) is v"<(z, z6 )"1w z2#w zz6 #1w z6 2#O(DzD3) 2 20 2 02 11 (34) with unknowns w , w 3R, and w "w6 . Using ij 11 20 02 the Taylor expansion in Eq. (33) gives z5 "iu z#1G z2#G zz6 #1G z6 2#1G z2z6 2 20 2 02 2 21 0 11 #SG , vTz#SG , vTz6 #2, 10 01 v5 "Cv#1H z2#H zz6 #1H z6 2#2, 2 20 2 02 11

(35)

where H is real, H "HM , and SG, vT" 11 20 02 +2 GM i. In terms of the functions G and H from i/1 iv Eq. (33) Li`j G " G(z, z6 , 0) ij LziLz6 j

K

, i#j*2, z/0

K K

L2 GM " G(z, z6 , v) , i"1, 2, 10,i Lv Lz i z/0,v/0 L2 G(z, z6 , v) , i"1, 2, GM " 01,i Lv Lz6 z/0,v/0 i Li`j H(z, z6 , 0) , i#j*2. H " ij LziLz6 j z/0

K

(36)

380

F. Zhu, C.D. Rahn / International Journal of Non-Linear Mechanics 35 (2000) 373}383

Substitution of Eq. (34) into Eq. (35) and equation of quadratic terms yields w "(2iu I!C)~1H , 20 0 20 w "!C~1H , 11 11 w "(!2iu I!C)~1H , 02 0 02 where I is the 2]2 identity matrix and the matrices (2iu I!C), C, (!2iu I!C) are invertible, be0 0 cause 0 and $2iu are not eigenvalues of C. Re0 striction of Eq. (20) to the center manifold, up to cubic terms, is z5 "iu z#1G z2#G zz6 #1G z6 2 2 20 0 2 02 11 #1(G !2SG ,C~1H T 2 21 10 11 #SG ,(2iu I!C)~1H T)z2z6 #2, (37) 01 0 20 where only the cubic term needed for the Hopf bifurcation analysis is displayed. Eq. (37) can be rewritten in the form of Eq. (24): 1 z5 "iu z# + g zjz6 k#2, (38) 0 j!k! jk 2xj`kx3 and the "rst Lyapunov coe$cient is given by 1 l (0)" Re(ig g #u g ). (39) 1 20 11 0 21 2u2 0 Again, if l (0)(0, there exists a stable limit cycle at 1 the bifurcation point. For the case of d "2 and c"2000, at h , nun 1 merical calculation yields u "0.337, g "0.0946#0.136i, 1 20 g "!0.0235!0.0196i, g "!56.5!12.2i, 11 21 (40) leading to l (0)"!84.8, 1 proving the existence of stable limit cycles close to the bifurcation point h . Note the close agreement 1 with the two-dimensional approach. Again, however, the "rst Lyapunov coe$cient is zero at the right bifurcation point h . 2

4.3. Simulation and experimental results First, a limit cycle close to h is simulated and 1 compared with experimental results. Fig. 4 shows the response of Eq. (21) with c"2000, d "2, *"0.060 n and p "4.72. % A fourth-order Runge}Kutta integrator solves the equation and the phase portrait is shown in Fig. 4a. The orbit either starts inside or outside the limit cycle at (0.12, 0) or (0.22, 0), respectively. Both trajectories end on a limit cycle with an approximate radius of 0.15. Thus, the simulation result agrees with the analytically predicted limit cycle behavior. The time response of y shown in Fig. 4b has 1 a frequency (0.40) very close to the corresponding linear frequency (0.39). The response of dynamic eyelet tension p versus $ time t is plotted in Fig. 4c. The dynamic eyelet tension is calculated as follows: Lr Lu c Lu Lu p "p #c ) # ) , at s"0. $ % Ls Ls 2 Ls Ls

(41)

The three terms in Eq. (41) are constant, linear, and quadratic in the displacement. The displacement response is nearly sinusoidal from Fig. 4b. Thus, the dynamic tension response should include non-dimensional frequency components at zero, the displacement limit cycle frequency (u "0.40), LC and twice the displacement frequency (u " 2LC 0.80). The magnitudes of these components depend on the steady-state and dynamic displacements. In the simulation, the "rst linear mode approximates the spatial distribution of string displacement. For this mode, Lr/Ls is almost perpendicular to Lu/Ls, greatly reducing the u frequency component in LC the simulated response. Accordingly, Fig. 4c shows the u frequency dominating with a small, low2LC frequency ripple at u . LC The experimental setup is detailed in [5] and shown in Fig. 5. The string starts from the bottom eyelet on a four-bar linkage and terminates at an inductive tension sensor above the upper eyelet.

F. Zhu, C.D. Rahn / International Journal of Non-Linear Mechanics 35 (2000) 373}383

381

Fig. 4. A limit cycle for d "2, *"0.132, and p "4.50. (a) Phase portrait of (y y ), (b) y versus time t, (c) simulated dynamic eyelet n % 1 2 1 tension p versus time t, (d) experimental eyelet tension p versus time t. $ $

The four-bar linkage is driven through a PC controlled permanent magnet DC motor so that the eyelet in the rotating link undergoes constant speed circular motion. In the experiment, a low extensibility (EA "58.3 N) single strand of con0 tinuous "lament polyester yarn with linear density oA "0.033 g/m is used. The test stand has a" 0 2.54 cm, H"25.4 cm, ¸"27.1 cm, and )" 370 rad/s, yielding h"10, *"0.060, d "2, and n c"2000. Note that the air drag parameter is chosen to match the steady-state response. Thus, no free parameters are available to tune the simulation result to match the experimental results. All model parameters are derived from independent measurements. Using a 2000 Hz sampling rate, the non-dimensional eyelet tension signal is "ltered by a "fth-order Butterworth lowpass "lter with a nondimensional cut-o! frequency of 0.88. The resulting signal is shown in Fig. 4d. The "lter reduces highfrequency noise in the tension signal but retains the dominant frequency components observed in the un"ltered tension FFT at 0.35 and 0.82. These frequencies closely match u and u , verifying LC 2LC

the linear and, to some extent, non-linear analyses. The experimental tension amplitude matches the simulation to within 3.5%. The shape of the experimental and simulated responses di!er signi"cantly, however, because the "rst- and second-frequency components in the experimental response are approximately equal whereas the second-frequency component dominates the simulated response. Fig. 6 shows the simulated and experimental responses for the same string near h (*"0.132). 2 Although the Lyapunov exponent at h is indeter2 minate, the simulation and experiment clearly show a limit cycle response. The limit cycle amplitude is considerably larger than at h . The tension frequen1 cies are close to those at h with the u " 1 2LC 0.92 component dominating the tension response. The u frequency component, however, dominLC ates the experimental tension response and the amplitude is considerably larger than the simulation. The experimental results qualitatively verify the theory. First, the theoretically predicted limit cycles are experimentally veri"ed. Second, the experimental tension data contains the two dominant

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F. Zhu, C.D. Rahn / International Journal of Non-Linear Mechanics 35 (2000) 373}383

Constraint of the spatial distribution of limit cycle motion to the #utter mode causes the poor agreement between simulation and experiment for large amplitude limit cycles. Clark et al. [13] used the method of characteristics to simulate the same model used in this analysis and obtain very good agreement with the experiment. The single-mode simulation used here constrains the spatial motion in the limit cycle to the linear #utter mode, reducing the limit cycle amplitude and distorting the relative magnitude of the two frequencies in the tension response. At low amplitudes, the linear and non-linear modes agree. The experimental results show that the non-linear mode shape changes with amplitude because the relative magnitudes of the two frequency components change. This e!ect cannot be captured with a single-mode simulation because the mode shape is independent of amplitude. Unfortunately, adding one or two modes to the simulation does not appear promising because all of the low-order linear modes have Lr/Ls nearly perpendicular to Lu/Ls.

5. Conclusions

Fig. 5. Experimental setup.

frequency components predicted by the theory. Third, the limit cycle amplitude matches the theory for the small amplitude case in Fig. 4. Finally, as predicted by theory, the limit cycle amplitude grows with increasing *. Unfortunately, it is not possible to collect additional experimental data for smaller amplitude limit cycles than that shown in Fig. 4 due to disturbances and sensor noise in the current test stand. The fairly good agreement in Fig. 4 indicates the approach is useful for predicting tension limit cycle amplitudes less than 10% of the average tension. In Fig. 6, the experimental limit cycle amplitude increases to 45% of the average value. For these large amplitude limit cycles, this approach provides poor quantitative prediction of the response.

Application of the Hopf bifurcation theorem to a reduced two-dimensional model of a double-loop ballooning string predicts the existence of stable limit cycles for small air drag and string length. Use of the center manifold reduction theorem to project a four-dimensional model onto a two-dimensional center manifold produces a similar result. The Hopf bifurcation theorem, however, cannot be applied to the large string length bifurcation point because the "rst Lyapunov coe$cient is indeterminate. Numerical simulation of the two-dimensional model, however, demonstrates the existence of stable limit cycles for large string length double-loop balloons. The frequencies and amplitudes of the tension limit cycle predicted by the two-dimensional model closely match experimental results for small amplitudes. For large amplitudes, however, the theory underestimates the limit cycle amplitude. The error arises from constraint of the limit cycle motion to the single, linear #utter mode used in the two-dimensional analysis.

F. Zhu, C.D. Rahn / International Journal of Non-Linear Mechanics 35 (2000) 373}383

383

Fig. 6. A limit cycle for d "2, *"0.060, and p "4.72. (a) Phase portrait of (y y ), (b) y versus time t, (c) simulated dynamic eyelet n % 1 2 1 tension p versus time t, (d) experimental eyelet tension p versus time t. $ $

Acknowledgements This research was supported by the National Textile Center.

References [1] S. Batra, T. Ghosh, M. Zeidman, An integrated approach to dynamic analysis of the ring spinning process - III, Text. Prax. Int. 47 (1992) 793}800. [2] W. Fraser, On the theory of ring spinning, Phil. Trans. Roy. Soc. London A 342 (1993) 439}468. [3] F. Zhu, K. Hall, C.D. Rahn, Steady-state response and stability of ballooning strings in air, Int. J. Non-linear Mech. 33 (1) (1998) 33}46. [4] D. Stump, W. Fraser, Transient solutions of the ring-spinning balloon equations, ASME J. Appl. Mech. 63 (1996) 523}528. [5] F. Zhu, R. Sharma, C.D. Rahn, Vibrations of ballooning elastic strings. ASME J. App. Mech. 64 (3) (1997) 676}683.

[6] J.E. Marsden, M. McCracken, The Hopf Bifurcation and Its Applications, Springer, New York, 1976. [7] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, 1995. [8] P.J. Holmes, Bifurcations to divergence and #utter in #owing-induced oscillations: a "nite-dimensional analysis, J. Sound Vib. 55 (4) (1977) 471}503. [9] P. Hagedorn, Non-Linear Oscillations, Oxford Science Publications, New York, 1988. [10] A.H. Nayfeh, D.T. Mook, Non-linear Oscillations, Wiley, New York, 1979. [11] N.C. Perkins, Modal interactions in the non-linear response of elastic cables under parametric/external excitation, Int. J. Non-Linear Mech. 27 (2) (1992) 233}250. [12] E.M. Mockensturm, N.C. Perkins, A.G. Ulsoy, Stability and limit cycles of parametrically excited, axially moving strings, ASME J. Appl. Mech. 118 (1996) 346}351. [13] J.D. Clark, W.B. Fraser, R. Sharma, C.D. Rahn, The dynamic response of a ballooning yarn: Theory and Experiment 454 (1978) (1998) 2767}2789. [14] S. Wiggins, Introduction to Applied Non-linear Dynamical Systems and Chaos, Springer, New York, 1990. [15] L. Perko, Di!erential Equations and Dynamical Systems, Springer, New York, 1996.