December 16, 2011
9:46
WSPC/S0218-1274
03050
International Journal of Bifurcation and Chaos, Vol. 21, No. 11 (2011) 3259–3272 c World Scientific Publishing Company DOI: 10.1142/S0218127411030507
BIFURCATION STRUCTURE FOR MODULATED VIBRATION OF STRINGS SUBJECTED TO HARMONIC BOUNDARY EXCITATIONS JIAZHU HU Black & Veatch Corporation, Overland Park, KS 66201, USA
[email protected] P. FRANK PAI∗ Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA
[email protected] Received September 16, 2010; Revised April 4, 2011 In the studies of nonlinear dynamics, phase plan plot is a most commonly used tool for solution characteristics interpretation. Phase plan plot provides an adequate representation of the dynamic characteristics of single solution, but it does not provide information on the interrelation between neighboring solutions; therefore, the evolution of solutions is studied by examining fragmented information of many individual responses. When the solution space becomes complicated, accurate information of the interrelation between responses is essential for an overall comprehension of the characteristics of the solutions. Noticing the characterizing amplitude variation of the limit cycles for typical modulated responses, bifurcation structure was proposed and developed to examine the overall characteristics of the modulated solutions of nonlinear string vibration. The bifurcation structure depicts all solutions in one plot by recording the upper and lower limits of the modulated vibration limit cycles. All kinds of bifurcations and interrelations between typical solutions can be qualitatively and quantitatively identified using the bifurcation structure. In this study, bifurcations and solution interrelations identified and revealed by bifurcation structures include forward and reverse Hopf bifurcations through period-doubling, appearance of isolated solution branch, solution branch transitions between the Hopf branch and isolated branch, appearance of chaotic attractors, chaotic attractor transitions between the Rossler and Lorenz types, and attractor disappearance by boundary crisis. Keywords: String vibration; modulated vibration; chaotic vibration; attractor transition; bifurcation structure.
1. Introduction Research on string vibration has a long history. As researchers learned more about string dynamics, the topics evolved from planar linear vibration to planar nonlinear vibration, then to nonplanar nonlinear vibration; from periodic to quasi-periodic vibrations, then to chaotic vibrations; and from ∗
stable vibrations to modulated and unstable vibrations. The changes in research topics demonstrate that earlier inaccurate motion equations, which were based on inappropriate assumptions (e.g. longitudinal vibration is negligible), were questioned and reexamined in subsequent studies, which were based on more reasonable assumptions
Author for correspondence 3259
December 16, 2011
3260
9:46
WSPC/S0218-1274
03050
J. Hu & P. F. Pai
[Carrier, 1945, 1949; Harrison, 1948; Lee, 1957; Oplinger, 1960; Murthy & Ramakrishna, 1965; Miles, 1965; Akulenko & Nesterov, 1996]. Considering coupling between two transverse vibrations and those between transverse and longitudinal vibrations, Narashima [1968], Anand [1969a, 1969b] established accurate nonlinear nonplanar string vibration models that were widely adopted by researchers in subsequent decades. Nayfeh et al. [1995] studied the nonlinear response of a taut string subjected to excitation that had components both parallel and perpendicular to its axis. Nayfeh et al. [1995] obtained the nonlinear response curves and studied the basic bifurcations of the response curve. There are no detailed studies on the bifurcation and oscillation characteristics of various modulated solutions of the Hopf branch. Miles [1984] constructed the widely used averaged equations in a four-dimensioned phase space, in which the coordinates are the slowly varying amplitudes of a sinusoidal motion of the dominant mode corresponding to the driving frequency. Using a set of four autonomous first-order nonlinear ordinary differential equations, Miles performed an exhaustive study on the fixed-point solutions and their stabilities. No detailed studies on bifurcation and chaotic solutions were conducted. Using similar autonomous first-order nonlinear ordinary differential equations that were obtained by attacking governing partial differential equations, Johnson and Bajaj [1989] and Bajaj and Johnson [1992] performed extensive bifurcation analyses of nonlinear string vibration and predicted the existence of pitchfork, saddle node, period-doubling and quasiperiodic torus-doubling bifurcations of the Hopf branch, and the appearance of chaotic attractors as well as their disappearances due to the boundary crisis. The bifurcation of the Hopf branch of nonlinear string vibration is abundant and complicated. These previous studies [Johnson & Bajaj, 1989; Bajaj & Johnson, 1992] were primarily focused on the characteristics of each single type of modulated solution. Although the characteristics of all typical solutions have been studied in detail, the interrelations between the solutions have not been well recognized and presented, and therefore, certain aspects of the solution evolutions have been misunderstood. To achieve a more complete understanding of the interrelations between solutions, the authors performed a detailed examination of all typical modulated solutions. Similar detailed results, which can be found in [Johnson & Bajaj, 1989;
Bajaj & Johnson, 1992; Hu, 2006], are not presented in the paper. Based on observations obtained from the detailed study, the bifurcation structure for modulated string vibration was first introduced. The bifurcation structure depicts all representative solutions in one plot by recording the upper and lower limits of the modulated vibration limit cycles. The bifurcation structure presents a comprehensive examination and systematic summarization for all solutions of the branch at specific damping level in one plot. With the bifurcation structure, the differences and interrelations between different modulated solutions at that damping level can be easily identified. When comparing bifurcation structures for different damping levels, the evolution of solutions is revealed in a straightforward and systemic manner. Certain dynamic properties of the system, which could be overlooked or misunderstood when responses were studied individually, were clearly identified by the bifurcation structure. This study is a continuation of the works discussed above, with a focus on examining the development of and interrelation between abundant bifurcations and modulated solutions using the bifurcation structure.
2. The Modulation Equations in Cartesian and Polar Coordinates A simple diagram with a coordinate system for the forced vibration of string subjected to harmonic boundary excitations is shown in Fig. 1. The string is assumed to be made of material that has a longitudinal wave speed much larger than the transverse ones. Theoretical models and equation derivations for modulated vibration of strings have been discussed and obtained by many studies [Nayfeh & Mook, 1979; Miles, 1984; Johnson & Bajaj, 1989; Bajaj & Johnson, 1992; Nayfeh & Pai, 2004; Hu, 2006]. Modulation equations in polar coordinates obtained by Nayfeh and Pai [2004] using multiplescale method were used in this study. Corresponding equations in Cartesian coordinates can be easily obtained by simple substitution [Hu, 2006]. For readers’ convenience, the equations are presented in the following text. For details of equation derivation, please refer to [Nayfeh & Pai, 2004; Hu, 2006]. The study investigates the modulated response of strings subjected to harmonic boundary excitations in a vertical plane, which has fv (x, t) = 0 and fw (x, t) = fw (L, t) = B cos(Ωt). B and Ω are the excitation amplitude and frequency, respectively. The amplitude and phase modulation equations for
December 16, 2011
9:46
WSPC/S0218-1274
03050
Bifurcation Structure for Modulated Vibration of Strings
3261
f v ( x, t )
f w ( x, t )
SB
SA
w
x1 (u )
u
v x3 ( w )
f w ( L, t ) = B cos ( Ωt)
x2 ( v ) Fig. 1.
Forced vibration of string.
the modulated responses of the mth mode in the polar form are: a1 = −µa1 − ψa22 a1 sin(γ1 ) − F sin(γ2 ),
(1)
a2 = −µa2 + ψa21 a2 sin(γ1 ),
(2)
γ2 = σ − ψ(3a21 + 2a22 + a22 cos(γ1 )) F cos(γ2 ) , − a1
(3)
2F cos(γ2 ) γ1 = 2ψ(a21 − a22 )(cos(γ1 ) − 1) − , a1
(4)
F = (−1)m
γ2 ≡ σT2 − β1 ,
B ∗ c2 iσT2 e , L
ψ≡
c21 m3 π 3 . 32L3 c2
(5)
The notations for the above equations are defined as a1 and a2 the in-plane (the plane defined by the excitation direction and string) and out-of-plane modulation amplitude (radius in polar coordinate); β1 and β2 the phase angles of amplitude modulations; c1 the longitudinal wave speed. c2 , the transverse wave speed (assumed to be the same for two transverse directions in this study); µ the damping coefficients per unit length of the string (assumed to be the same for vibrations in both directions); L the total length of the string; B ∗ = B/ε3 the excitation amplitude with ε a small dimensionless measure of the amplitude used as a bookkeeping device in multiple-scale application; σ the excitation frequency detuning parameter; and T2 the second-order time in multiple-scale application. Frequency responses, represented by steady-state solutions corresponding to a1 = a2 = γ1 = γ2 = 0, can be obtained by solving Eqs. (1)–(4) for fixedpoint solutions. The stability of these fixed-point
ψp2 M, 4
(6)
ψq2 M − F, 4
(7)
ψp1 M, 4
(8)
p1 = −σq1 − µp1 + 3ψq1 E + q1 = σp1 − µq1 − 3ψp1 E +
p2 = −σq2 − µp2 + 3ψq2 E − q2 = σp2 − µq2 − 3ψp2 E −
with γ1 ≡ 2β2 − 2β1 ,
solutions can be determined from the eigenvalues of the Jacobian matrix of the right-hand side of Eqs. (1)–(4). The modulation equations in the Cartesian coordinates are:
ψq1 M, 4
(9)
where p1 = a1 cos γ2 , q1 = a1 sin γ2 , p2 = a2 cos γ3 , q2 = a2 sin γ3 , γ3 = γ2 − (γ1 /2), and E ≡ p21 + q12 + p22 + q22 ,
M ≡ p1 q 2 − p2 q 1 .
(10)
3. Modulated Responses and the Bifurcation Structures Both damping and excitation play an important role in the appearance of various nonlinear phenomena in string dynamics. The effects of damping variation are investigated in this study. The damping value is assumed to be the same for the responses of all participating modes in two vibration directions. The forcing value is fixed at a certain level. The system parameters used in this study, except damping, were adopted from the steel string selected by Nayfeh and Pai [2004]. The string parameters are: L = 2.13 m,
f ∗ = 5.39 × 10−5 m/sec,
µ∗ = 0.0341/sec −1 , c2 = 10.5 m/sec,
c1 = 55.7 m/sec, ε=
πc22 = 0.351. c21
December 16, 2011
9:46
WSPC/S0218-1274
03050
J. Hu & P. F. Pai
3262
The authors chose to study vibration that was primarily composed of the sixth mode, i.e. response to harmonic base excitation at the sixth natural frequency ωm = mπc2 /L with m = 6.
3.1. Frequency response curve The frequency response curve shows the nonlinearity of the response, the number of solutions for certain parameters and their properties. Figure 2 shows the (a) in-plane and (b) out-of-plane (a2 ) frequency response curve for a typical case with damping value µ = 0.98µ∗ and fixed excitation amplitude F = f ∗ . In Fig. 2, ∆1 represents the forward pitchfork bifurcation point; ∆5 represents the reverse pitchfork bifurcation point; ∆2 and ∆6 are the tuning points of the planar solution branches; ∆3 and ∆4 are the tuning points of the nonplanar solution branches (cases not shown in Fig. 2). ∆∗1 and ∆∗2 represent the forward and reverse Hopf bifurcation points, respectively. To present a selfcontained study of string dynamics, following is a summary of the response variation as the damping is reduced. At large enough damping, all responses are planar vibrations. All steady-state responses are periodic vibrations in the plane of excitation. For certain frequency detuning range, there are three planar solutions, two stable and one unstable. Of the two stable solutions, one has amplitude that is larger than that of the unstable one and the
other has smaller amplitude. There is only one stable solution for other frequency detuning ranges. As the damping is reduced, the stable planar solution for a specific frequency range becomes unstable. The vibration loses or gains stability depending on how the frequency detuning changes. Increasing the excitation frequency, a nonplanar response, corresponding to a periodic whirling motion, arises after the pitchfork bifurcation point ∆1 where the excitation frequency is slightly beyond the linear natural frequency. After that, the whirling motion continues and loses stability at the reverse pitchfork bifurcation point ∆5 . As damping is reduced even more, one can observe that the stable nonplanar solution becomes unstable due to Hopf bifurcations. The Hopf bifurcation frequency band (∆∗1 , ∆∗2 ) is within the frequency band defined by the two pitchfork bifurcation points (∆1 , ∆5 ). The responses of other frequency bands are the same as cases with larger damping values. The responses corresponding to the fixed-point solutions of the Hopf branch are amplitude- or phase-modulated limit cycles, i.e. whirling or ballooning motions, according to the Hopf bifurcation theorem. All responses stay periodic, with no further bifurcations. All limit-cycle solutions of the Hopf branch are stable of single modulation period over the entire detuning interval. The modulated solutions can be obtained and illustrated in either polar coordinate [Eqs. (1)–(4)] or Cartesian coordinates [Eqs. (6)–(9)]. For a detailed
-3
3.5
-3
x 10
∆5
3
1.8
∆6
x 10
∆*1
1.6
∆*2
1.4
2.5 2
∆1
a2 (m)
a1 (m)
1.2
∆ *1
1.5
∆2
1
1
∆*2
0.8 0.6 0.4
0.5 0 -0.15
0.2 -0.1
-0.05
0
0.05
∆= σ *ε (a)
2
0.1 (HZ)
0.15
0.2
0.25
0 -0.15
∆5
∆1 -0.1
-0.05
0
0.05
∆= σ *ε
2
0.1
0.15
0.2
0.25
(HZ)
(b)
Fig. 2. (a) The in-plane (a1 ) and (b) out-of-plane (a2 ) frequency response curves of the case with damping value µ = 0.98µ∗ and a fixed excitation amplitude f = f ∗ . (—) planar stable solutions; (- · -) nonplanar stable solutions; (- -) unstable solutions; (. . .) modulated solutions.
December 16, 2011
9:46
WSPC/S0218-1274
03050
Bifurcation Structure for Modulated Vibration of Strings x 10-3
2
1.5
1.5
1
1
0.5
0.5
0
q2
a2
2
-0.5
-1
-1
-1.5
-1.5
-2
-1
0 a1
1
2
x 10-3
0
-0.5
-2 -3
3263
-2 -1.5
3
-1
-0.5
0 p2
x 10-3
(a)
0.5
1
1.5 x 10-3
(b)
Fig. 3. Invariance of the modulation in the plane defined by (a) a1 and a2 , and (b) q2 and p2 , where ∆ = 0.108 and µ = 1.015µ∗ . (•) planar solutions; (∇) unstable planar solutions; (◦) limit-cycle solutions.
Bifurcation ends ∆*
2.8
Maximum of a
2.8 2.6 1
Min(a1) & Max(a )
Min(a1) & Max(a ) 1
2.6
1
Bifurcation ends
2
Maximum of a 1
0.1525 HP1
0.1325 HP4 0.1365 HP8 0.1390 HP4 0.1420 HP2
0.1170 HP4 0.1185 HP2
x 10
0.1075 HP4 0.1100 HP8
-3
3
0.1005 HP2
the plots. Because of the trigonometric relations between the vibration amplitudes in two different coordinate systems, there are trajectories in all four quadrants of the coordinate plane defined by a1 and a2 , while only the first and third quadrants have trajectories in the coordinate plane defined by p2 and q2 .
0.1490 HP1
0.1380 HP2
x 10
0.1090 HP4 0.1140 HP8 0.1170 HP4
-3
3
0.1010 HP2
study with phase plan plot of typical responses for all damping levels, please refer to [Johnson & Bajaj, 1989; Bajaj & Johnson, 1992; Hu, 2006]. Figures 3(a) and 3(b) depict the invariance of the modulation limit cycle solutions in both Cartesian and polar coordinate systems. Fixed point solutions in both coordinate systems are shown in
2.4 Bifurcation begins 2.2 2 1.8 1.6 0.08
d xe Fi
t in Po
lu So
ns tio
Minimum of a
1
2.4 2.2 2 1.8
* 1
0.1
0.12 0.14 ∆ = σ *ε 2
(a)
Bifurcation begins
0.16
0.18
1.6 0.08
d xe Fi
0.1
Po
int
S
tio ol u
ns
The isolated branch Minimum of a
0.12 0.14 ∆ = σ *ε 2
1
0.16
0.18
(b)
Fig. 4. The bifurcation structure of the modulated solution of the case with damping value µ and a fixed excitation amplitude f = f ∗ , (a) high damping level case µ = 1.015µ∗ , (b) medium damping level case µ = 1.01µ∗ , (c) medium low damping level case µ = 1.00µ∗ , (d) low damping level case µ = 0.98µ∗ , (e) lower damping level case µ = 0.96µ∗ . (—) Hopf branch; (- -) isolated branch.
December 16, 2011
WSPC/S0218-1274
03050
Bifurcation ends
x 10
0.1705 HP1
0.1545 HP2 0.1585 HP4 0.1630 HP2
-3
3.2
0.0980 HP2 0.1020 HP4 0.1050 HP2 0.1085 HP1
0.1590 HP1
0.1505 HP2
0.1445 HP4
x 10
0.1365 HP2
-3
3
0.1050 HP4 0.1080 HP8 0.1100 HP4 0.1115 HP2 0.1160 HP1 0.1165 HP2 0.1175 HP1
J. Hu & P. F. Pai 0.9950 HP2
3264
9:46
Attractor transition
3
2.5
1
Min(a1) & Max(a )
1
Min(a1) & Max(a )
2.8
Bifurcation begins
2 ed Fix
in Po
tS
ns tio olu
2.6
int Po d e Fix e yp zt n re Lo
2.4 2.2 2 1.8
0.1
0.12 0.14 ∆ = σ *ε 2
0.16
e yp
Rossler type
1.4 0.08
0.18
R
rt sle os
Attractor transition
1.6
1.5 0.08
s ion lu t o S
0.1
0.12 0.14 ∆ = σ *ε 2
0.18
(d) 0.1700 HP2 0.1735 HP4 0.1770 HP2 0.1825 HP1
(c)
0.16
-3
3.2
x 10
Boundary crisis
0.1185
2.4 2.2 2 1.8
0.1450 HP1
2.6
0.0960 HP2
Min(a1) & Max(a ) 1
2.8
0.1540
3
d xe Fi
Po
int
lu t So
s ion
Attractor transition Boundary crisis
1.6 1.4 0.08
0.1
0.12
0.14 ∆ = σ *ε 2
0.16
0.18
0.2
(e) Fig. 4.
3.2. Modulated vibrations, bifurcations, and bifurcation structures Modulated solutions exist for relatively low damping cases. Responses for low damping cases include various modulated solutions and bifurcation. Johnson and Bajaj [1989], Bajaj and Johnson [1992] did an exhaustive study for most typical modulated solutions. However, the authors were primarily focused on the characteristics of each single
(Continued)
type of modulated solution. Although the characteristics of all types of solutions have been studied in detail, the interrelations between the solutions have not been well recognized and presented. Certain aspects of the solution evolution have been misunderstood. Some neighboring bifurcation solutions, which actually belong to the same continuous isolated branch, were considered as those of disconnected isolated branches by some studies. Noticing the complexity of the solutions for the low
December 16, 2011
9:46
WSPC/S0218-1274
03050
Bifurcation Structure for Modulated Vibration of Strings
damping case and the abrupt difference between solutions for various damping levels, the authors conducted an examination to reveal the interrelations between the various solutions. Bifurcation structure was constructed to study the properties of all types of solutions and bifurcations. It was found that bifurcation structure provides a straightforward interpretation for the solutions. Figure 4 shows bifurcation structure plots for each representative low damping case. The bifurcation structures in Fig. 4 were drawn by plotting the maximum and minimum values of the modulated vibration amplitudes of in-plane responses. Circled point indicates detuning ∆ where bifurcations of the Hopf branch occur. The following analysis has investigated the typical modulated responses with bifurcation structures as shown in Fig. 4. Figures 5–9 show additional important characteristics of the representative solutions of all damping level cases shown in Fig. 4.
3.3. High damping level case (µ = 1.015µ∗)
At damping level µ = 1.015µ∗ , the responses between the two Hopf bifurcation points ∆∗1 and ∆∗2 are no longer of a single period, which is true for higher damping cases. The limit-cycle solution may become unstable and undergo period-doubling bifurcations. Figure 5 depicts the 3D view and 2D view phase plan plots of the Hopf branch. Starting from ∆∗1 , the amplitude of the limit cycle increases as the detuning increases and it reaches its maximum around the midpoint of the interval; it then shrinks to zero as the detuning approaches ∆∗2 . The numerical investigation demonstrates that the closer the detuning is to the two bifurcation points, the longer it takes for the integration to converge. Moreover, from the 2D ∆ − q2 view, one can observe a discontinuous increase or decrease of the amplitude q2 where the period-doubling begins or ends. Noticing the geometrical difference between limit cycles of various modulated solutions, the authors constructed a 2D plot that records the maximum and minimum modulation amplitude verses the detuning variation and called it the bifurcation structure. Discontinuous change of amplitude was observed at the bifurcation point in the bifurcation structure. Figure 4(a) shows the plot for the case with µ = 1.015µ∗ . It illustrates that, as the detuning increases, the modulated vibration undergoes forward period-doubling bifurcation and then
3265
shrinks back to a single-period attractor by reverse period-doubling bifurcation after approximately the midpoint of the interval.
3.4. Medium damping level case (µ = 1.01µ∗)
The responses for µ = 1.01µ∗ is basically the same as for those with µ = 1.015µ∗ , except that the frequency range where the Hopf bifurcation exists becomes larger. As the damping decreases, the detuning band (∆∗1 , ∆∗2 ) expands; the lower bound, ∆∗1 , is nearly unchanged, while the upper bound, ∆∗2 , increases. Another important change of the system response when damping decreases to µ = 1.01µ∗ is that the period-doubling bifurcation of the Hopf branch becomes unstable and another coexisting branch, the isolated branch appears through global saddle-node bifurcation. In other words, both stable and unstable limit-cycle solutions exist simultaneously at this damping level. Together with the stable planar branch and the Hopf branch, there are three possible solutions at this frequency range. This case has generally the same bifurcation as the case with µ = 1.015µ∗ , except that, at the middle part of (∆∗1 , ∆∗2 ), the limit cycle shrinks and the oscillator has a smaller amplitude due to the constraints or stabilizing influences of the isolated branch coexisting in the same frequency interval [Johnson & Bajaj, 1989; Bajaj & Johnson, 1992]. Figure 6 depicts the properties of the isolated branch with µ = 1.01µ∗ . Figure 6(a) is the 3D view of the phase plot. At this damping level, the isolated branch undergoes no period-doubling; therefore, the amplitude for the entire branch is nearly constant as shown in the 2D view of the phase plots [Fig. 6(b)]. Figure 6(c) shows the positions and geometries of the Hopf branch solution and the isolated branch solution on the p2 − q2 plane for detuning∆ = 0.1280. The major difference between two different limit cycles is that the isolated branch solution has a cusp at a location prone to the planar fixed point where (p2 , q2 ) = (0, 0). As damping is reduced further, the cusp of the isolated branch limit cycle is more attracted by the planar fixed point and becomes more obvious and sharper. Figure 4(b) shows the bifurcation diagram for the case with µ = 1.01µ∗ . A period-doubling bifurcation sequence of HP(1-2-4-8-4-2-4-2-1) for the Hopf branch solution is shown in the plot. The isolation, where there is absolutely no connection between the two solution branches, is revealed by the plot.
December 16, 2011
3266
9:46
WSPC/S0218-1274
03050
J. Hu & P. F. Pai
x 10-3 2
q2
1 0 -1 -2 2
0.16 1
0.14 0
x 10-3
0.12 -1
0.1
-2
p2
∆ = σ∗ ε2
(a)
2
x 10-3
1.5 1
q2
0.5 Period doubling bifurcation ends
Period doubling bifurcation begins
0 -0.5 -1 -1.5 -2 0.09
Fixed Point Solutions 0.1
0.11
0.12
0.13
∆ = σ ∗ε
0.14
0.15
0.16
2
(b) Fig. 5. Phase plan plots of the Hopf branch of the high damping level case with µ = 1.015µ∗ and f = f ∗ : (a) 3D view of the plots, and (b) 2D ∆ − q2 view of the plots.
3.5. Medium low damping level case (µ = 1.00µ∗) As the damping is reduced to µ = 1.005µ∗ , the bifurcation of the Hopf branch is not much different from those with larger damping values. The isolated branch, however, undergoes a series of perioddoubling bifurcations that lead to a transition to chaos. A Rossler-type chaotic attractor that encircles only one unstable (modulated) nonplanar fixed point solution is formed at certain detuning points.
Studying the trajectories of the isolated branch (typical trajectory plots can be found at [Nayfeh & Mook, 1979; Johnson & Bajaj, 1989; Bajaj & Johnson, 1992; Hu, 2006], one can observe a forward period-doubling bifurcation followed by a Rosslertype attractor and a reverse period-doubling bifurcation as the detuning increases. Further detuning expands the results in a second round of forward period-doubling bifurcations, and the appearance of a Rossler-type attractor is followed by reverse
December 16, 2011
9:46
WSPC/S0218-1274
03050
Bifurcation Structure for Modulated Vibration of Strings
3267
x 10-3
µ = 1.01∗µ ∗
2
q2
1 0 -1 -2 2 1
0.14 0
x 10-3
p2
0.13 -1
0.12 -2
0.11
∆ = σ∗ ε2
(a)
2
x 10-3
2 1.5
1.5
1
1
0.5
q2
q2
0.5 0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
-2
x 10-3
0.116 0.118 0.12 0.122 0.124 0.126 0.128 0.13 0.132 ∆ = σ *ε 2
(b)
-2 -1
-0.5
0 p2
0.5
1 x
10-3
(c)
Fig. 6. Phase plane plots of isolated branch of the medium damping level case with µ = 1.01µ∗ and f = f ∗ : (a) 3D view of the phase plane plots, (b) 2D ∆ − q2 view, and (c) coexistence of the isolated branch and the Hopf branch at ∆ = 0.1280. (•) stable planar solutions; (∇) unstable planar solutions; (◦) limit-cycle solutions.
period-doubling bifurcation. The second sequential bifurcation was considered as a new isolated branch in [Johnson & Bajaj, 1989; Bajaj & Johnson, 1992] and the unstable part of the newly formed isolated branch was merged with the stable part of the previous one by saddle-node bifurcation, which is the same mechanism by which the first isolated branch was formed and merged with the Hopf branch. The authors doubt the existence of the second isolated branch, because there was no indication of the same isolation (disconnection) between this newly formed solution branch and the first solution branch as that
between the isolated solution and the Hopf solution. Figure 4(c) clearly shows the isolation (disconnection) between the Hopf branch and the isolated branch, as well as the continuation between first and second rounds of Hopf bifurcation in the isolated branch. Thus, it is more reasonable to consider the new round of forward and reverse bifurcations as a continuation of the previous round of bifurcation in the same isolated solution branch rather than a new one. Correspondingly, the second (third, etc.) isolated branch in the following analyses actually means the second (third, etc.) round of bifurcations
December 16, 2011
3268
9:46
WSPC/S0218-1274
03050
J. Hu & P. F. Pai
of the isolated branch rather than a newly formed isolated branch. For identification ease, the authors kept the concept of the second isolated branch. When the damping is further reduced, the unstable part of this second isolated branch merges with the stable part of the first isolated branch in the same manner that the first isolated branch merges with the Hopf branch. For the Hopf branch of this case, the bifurcation is the same as those of previous cases, except that a steady-state HP1 solution was obtained at the middle part of the frequency range where only HP2 solutions were observed before. It is considered as a result of stronger influence on the bifurcation of the Hopf branch by the constraint (stabilizing effect) of the multiple isolated branches together. The detuning interval of the stable HP1 solution might be quite small and easy to miss. Similarly, the second isolated branch has the same stabilizing effect or constraint on the first isolated branch, forcing the Rossler-type attractor to undergo reverse bifurcations, which lead to reversed period-doubling bifurcation as the excitation frequency increases further. Figure 4(c) shows all these bifurcations by plotting the maximum and minimum in-plane amplitudes for both the Hopf branch and the isolated branch for the case with µ = 1.00µ∗ . It clearly shows that the stable part of the isolated branch undergoes a cascade of period-doubling bifurcations which lead to the Rossler-type attractor, i.e. trajectory that encircles only one unstable nonplanar fixed point. The Hopf branch undergoes period-doubling bifurcations following the sequence HP(1-2-4-8-4-2-1-2-1-2-4-2-1) for the entire modulated branch. The bifurcation is not symmetric. The period-doubling bifurcation in the second half interval does not bifurcate into HP8 as in the first half. The nonconstant slope curve of the isolated branch shown in Fig. 4(c) indicates that the response is not of a single modulation period as before and period-doubling bifurcation exists.
solutions may not exist for some frequencies. This finding is consistent with the suppositions of Johnson and Bajaj [1989], Bajaj and Johnson [1992] that the stable Hopf branch breaks and merges with the unstable solutions of the isolated branch, leading to a saddle-type bifurcation. Again, this is due to the stabilizing effect of the simultaneously existing isolated branches, which prevents the Hopf branch from period-doubling to infinity in a straightforward manner. As more isolated branches, or more rounds of bifurcation of the isolated branch, appear when the damping decreases, the effects become more influential and finally disconnect the Hopf branch. The bifurcations of isolated branches are influenced in a similar manner by following newly formed isolated branches. At the middle part of the detuning interval, two symmetric trajectories of the newly formed isolated branch (located at the first and third quadrants of the polar coordinate) become connected, creating a new trajectory that undergoes period-doubling bifurcation and resulting in a Lorenz-type chaotic attractor. Studying the forward and reverse bifurcations, the authors found that starting from the Hopf bifurcation point ∆∗1 , as the excitation frequency increases, the limit cycle undergoes period-doubling bifurcation, becomes a Rossler-type attractor, changes to a Lorenz-type limit cycle and attractor, and then shrinks back to a Rossler-type attractor. Somewhere within the first half of the interval, the isolated trajectories become more and more attracted by the lower planar solution until, finally, the two symmetric trajectories are connected at the planar fixed point, creating
2
x 10-3
1.5 1
3.6. Low damping level case (µ = 0.98µ∗)
The frequency response curves with µ = 0.98µ∗ are shown in Fig. 2. There is a small frequency gap on the curve between the stable nonplanar branch and the modulation branch. The gap always exists, although a very small frequency increase step was attempted in the analyses. For a smaller damping value, the entire modulation branch may become even more discontinuous and stable limit-cycle
q2
0.5 0 -0.5 -1 -1.5 -2 -1.5
-1
-0.5
0 p2
0.5
1
1.5 x 10-3
Fig. 7. Transition of limit cycle from Rossler-type to Lorenztype: µ = 0.98µ∗ , f = f ∗ , and ∆ = 0.1140.
December 16, 2011
9:46
WSPC/S0218-1274
03050
Bifurcation Structure for Modulated Vibration of Strings
3269
This phenomenon is shown more clearly by the zoom-in window plots and corresponding frequency spectrum in Figs. 8(a) and 8(b). As the detuning increases further, the two connected symmetric Rossler-type trajectories become a Lorenz-type trajectory that encloses two symmetric nonplanar fixed points. With even further detuning increase, the trajectory departs from the planar fixed point and undergoes period-doubling bifurcation, leading to the Lorenz-type chaotic attractor. As expected, there is a reverse bifurcation along an increased
the homoclinic orbit. Figure 7 shows the transition of limit cycle from Rossler-type to Lorenz-type for the case with damping µ = 0.98µ∗ , force F = f ∗ , and detuning ∆ = 0.1140. When the two symmetric attractors become connected, the dynamics become chaotic, although the trajectories themselves may appear quite simple, due to the dynamic balance caused by the attractions from the two symmetric modulated attractors. This balance makes the oscillator move to the first quadrant some of the time and to the third quadrant at other times.
x 10-4
8
x 10-4
7
1
6
|A(f)|
q2
5 0
4 3 2 1
-1 -8
-6
-4
-2
0
2
4
6
p2
0 0
8 x 10-5
0.002 0.004 0.006 0.008 0.01 Frequency
0.012 0.014 0.016
(a)
x 10-4
x 10-5 4 3 2
2 |A(f)|
q2
1 0 -1 1
-2 -3 -4 -5 -2
-1
0
1 p2
0 0
2 x 10-5
0.002 0.004 0.006 0.008 0.01 Frequency
0.012 0.014 0.016
(b) Fig. 8. (a) Zoom-in views and the corresponding frequency spectrum of Fig. 6, (b) similar plot for the case with µ = 0.98µ∗ , f = f ∗ , and ∆ = 0.1475.
December 16, 2011
3270
9:46
WSPC/S0218-1274
03050
J. Hu & P. F. Pai
(a)
2
x 10-3
1.5 1
q2
0.5 0 -0.5 -1 -1.5 -2 0.1
0.11
0.12
0.13 ∆ = σ *ε 2
0.14
0.15
0.16
(b) Fig. 9. Phase plane plots of the isolated branch for the case with µ = 0.980µ∗ and f = f ∗ : (a) 3D view of the plots, and (b) 2D ∆ − q2 view of the plots.
detuning path. Figure 9 presents (a) 3D and (b) 2D ∆ − q2 views of the phase plane plots of the isolated branch for the case with µ = 0.980µ∗ and F = f ∗ . It can be observed that, for smaller damping, the Hopf branch becomes more discontinuous. Indeed, there is no stable Hopf type limit cycle in the disconnected frequency range at all. Integration solutions for all detuning values in this interval lead to stable
limit cycles of the isolated branch. Two types of attractors exist for the frequency interval where the Hopf branch is discontinuous. One is the limit cycle of the isolated branch and the other is the lower planar one. The bifurcation structure is presented in Fig. 4(d) for the case with µ = 0.98µ∗ and F = f ∗ . All bifurcations and transitions stated above are clearly recorded and indicated in the diagram.
December 16, 2011
9:46
WSPC/S0218-1274
03050
Bifurcation Structure for Modulated Vibration of Strings
3.7. Lower damping level cases (µ =
0.96µ∗)
0.96µ∗ ,
As damping is reduced to µ = the Hopf branch becomes more discontinuous regardless of how small the detuning increment is used for the scanning. Studying the typical phase plots of trajectories (not presented), the authors observed phenomena such as period-doublings both of the Hopf branch and the isolated branch, the coexistence of the solutions of two branches, and the transition of solutions from one branch to the other. A cascade of isolated branch attractors appear via saddlenode bifurcation. The unstable portion merges with the stable portion of the previous isolated branches in the same manner that the first isolated branch was formed and the unstable isolated branch was merged with the stable Hopf branch. This sequence ends with the formation of a homoclinic orbit, a trajectory that asymptotically approaches a saddletype fixed point as t → ±∞. The fixed point has eigenvalues that satisfy Shilnikov’s inequality. Lorenz-type chaotic attractors can be found in proximity to this homoclinic orbit. The most distinctive response feature at this damping level is that Lorenz-type attractors abruptly disappear over a frequency interval and the only stable solution is the lower planar fixed-point solution. This can be explained by the boundary crisis or the hetero-clinic bifurcation in which the chaotic attractor becomes tangent to and then intersects the stable manifold of the saddle-type unstable planar solution existing between the two turning points on the planar solution branch [Johnson & Bajaj, 1989; Bajaj & Johnson, 1992]. For the decrease sweep of detuning, a similar boundary crisis can be observed for detuning values within the second half of the interval. For the frequency range between two detuning values where the boundary crisis occurs, there is only one stable solution — the lower planar solution. Figure 4(e) shows the bifurcation structure for the case with µ = 0.96µ∗ and F = f ∗ . All important characteristics of the responses are recorded in this plot. No limit-cycle solutions were obtained by numerical integration for detuning between the two boundary-crises. Decreasing the damping further, the system is close to a Hamiltonian type and the modulation branch of the frequency response curve becomes even more discrete. The trajectory of the Hopf branch becomes more twisted and distorted.
3271
4. Conclusions The solutions for modulated responses of strings subjected to harmonic boundary excitations are fairly complicated. Most studies on the nonlinear dynamic characteristics of modulated string vibrations have been conducted by investigating the phase plan plots of individual responses. Phase plan plot provides an adequate representation of the dynamic characteristics of single solution, but it does not provide information on the interrelation between neighboring solutions. This study proposed and developed panoramic bifurcation structures for the modulated solutions of string vibration at various damping levels. The bifurcation structure provides an extremely useful framework to classify, systematize and interrelate all of the many different modulated solutions of nonlinear string vibration. Constructed by connecting response amplitude limits, the bifurcation structure presents a clear overall view of various forward and reverse solution evolutions through bifurcations in one plot. This structure is qualitatively illustrative for showing all typical bifurcations and quantitatively accurate for recording bifurcation points of the modulated solution branches. With bifurcation structure, the understanding of solution characteristics, evolutions of and relationships between solutions, and bifurcations of the modulated solutions of the Hopf branch are now more straightforward and accurate. Studying consecutive bifurcation structures, the authors observed that the second round and further bifurcations of the isolated solution were a continuation of previous bifurcations of the same solution branch, rather than a newly isolated solution branch, which had been concluded in previous studies that were based on the examination of individual phase plan plots.
References Akulenko, L. D. & Nesterov, S. V. [1996] “Analyses of the spatial nonlinear vibrations of a string,” J. Appl. Math. Mech. 60, 85–95. Anand, G. V. [1969a] “Large-amplitude damped free vibration of a stretched string,” J. Acoust. Soc. America 45, 1089–1096. Anand, G. V. [1969b] “Stability of nonlinear oscillations of stretched strings,” J. Acoust. Soc. America 46, 667–677.
December 16, 2011
3272
9:46
WSPC/S0218-1274
03050
J. Hu & P. F. Pai
Bajaj, A. K. & Johnson, J. M. [1992] “On the amplitude dynamics and crisis in resonant motion of stretched strings,” Philos. Trans. Roy. Soc. London 338, 1–41. Carrier, G. F. [1945] “On the nonlinear vibration problem of the elastic string,” Quart. Appl. Math. 3, 157–165. Carrier, G. F. [1949] “A note on the vibrating string,” Quart. Appl. Math. VII, 97–101. Harrison, H. [1948] “Plane and circular motion of a string,” J. Acoust. Soc. America 20, 874–875. Hu, J. [2006] Nonlinear Mechanics and Testing of Highly Flexible One-Dimensional Structures Using a Camera-Based Motion Analyses System, PhD thesis, Department of Mechanical and Aerospace Engineering, University of Missouri — Columbia. Johnson, J. M. & Bajaj, A. K. [1989] “Amplitude modulated and chaotic dynamics in resonant motion of strings,” J. Sound Vibr. 128, 87–107. Lee, E. W. [1957] “Nonlinear forced vibration of a stretched string,” British J. Appl. Phys. 8, 411–413.
Miles, J. W. [1965] “Stability of forced oscillations of a vibrating string,” J. Acoust. Soc. America 38, 855–861. Miles, J. W. [1984] “Resonant nonplanar motion of a stretched string,” J. Acoust. Soc. America 75, 1505–1510. Murthy, G. S. S. & Ramakrishna, B. S. [1965] “Nonlinear characters of resonance in stretched strings,” J. Acoust. Soc. America 38, 461–471. Narashima, R. [1968] “Nonlinear vibration of an elastic string,” J. Sound Vibr. 8, 134–146. Nayfeh, A. H. & Mook, D. T. [1979] Nonlinear Oscillations (John-Wiley, Hoboken, NJ). Nayfeh, S. A., Nayfeh, A. H. & Mook, D. T. [1995] “Nonlinear response of a taut string to longitudinal and transverse end excitation,” J. Vibr. Cont. 1, 307–334. Nayfeh, A. H. & Pai, P. F. [2004] Linear and Nonlinear Structural Mechanics (John-Wiley, Hoboken, NJ). Oplinger, D. [1960] “Frequency response of a nonlinear stretched string,” J. Acoust. Soc. America 32, 1529– 1538.
