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Global bifurcations and chaotic motions of a flexible multi-beam structure
Accepted Manuscript Global bifurcations and chaotic motions of a flexible multi-beam structure Tian-Jun Yu, Wei Zhang, Xiao-Dong Yang
PII: DOI: Reference:
S0020-7462(17)30023-9 http://dx.doi.org/10.1016/j.ijnonlinmec.2017.06.015 NLM 2870
To appear in:
International Journal of Non-Linear Mechanics
Received date : 15 January 2017 Revised date : 11 June 2017 Accepted date : 20 June 2017 Please cite this article as: T. Yu, W. Zhang, X. Yang, Global bifurcations and chaotic motions of a flexible multi-beam structure, International Journal of Non-Linear Mechanics (2017), http://dx.doi.org/10.1016/j.ijnonlinmec.2017.06.015 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Global bifurcations and chaotic motions of a flexible multi-beam structure Tian-Jun Yu, Wei Zhang, Xiao-Dong Yang Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, P. R. China
Abstract Global bifurcations and multi-pulse chaotic motions of flexible multi-beam structures derived from an L-shaped beam resting on a vibrating base are investigated considering one to two internal resonance and principal resonance. Base on the exact modal functions and the orthogonality conditions of global modes, the PDEs of the structure including both nonlinear coupling and nonlinear inertia are discretized into a set of coupled autoparametric ODEs by using Galerkin's technique. The method of multiple scales is applied to yield a set of autonomous equations of the first order approximations to the response of the dynamical system. A generalized Melnikov method is used to study global dynamics for the "resonance case". The present analysis indicates multi-pulse chaotic motions result from the existence of Šilnikov's type of homoclinic orbits and the critical parameter surface under which the system may exhibit chaos in the sense of Smale horseshoes are obtained. The global results are finally interpreted in terms of the physical motion of such flexible multi-beam structure and the dynamical mechanism on chaotic pattern conversion between the localized mode and the coupled mode are revealed.
Keywords: Flexible multi-beam structures; Autoparametric system; Global dynamics; Melnikov method; Localized mode; Coupled mode
Corresponding Author, Email: [email protected] 1
1. Introduction Flexible multi-beam structures are typical components of large space station, aeronautical engineering, civil engineering and other mechanical systems. A profound understanding of the dynamical characteristics of these structural systems is essential for their design and control [1]. With the development of the more flexible materials, the nonlinearities in such structural systems cannot be neglected in dynamics research. The nonlinear dynamic analysis is the key in modeling, analysis and control of these large space structures after deployment and during service [2, 3]. Weak quadratic nonlinearities arise in many multi degree-of-freedom systems due to inertial effects of large motions, the geometry of motion, the centrifugal and Coriolis forces. Such systems, as known, include ships, arches, liquid in containers, beams and plates under static loadings [4]. In fact, flexible multi-beam structures are typical quadratic nonlinearity systems due to the nonlinear coupling and nonlinear inertia. In the earlier study, Haddow et al. [5], Bux and Roberts [6], Ashworth and Barr [7], Balachandran and Nayfeh et al. [8-11] have made important contributions to the nonlinear dynamic characteristics of such structures by using the method of multiple scales [12] with assistance of experiments. Crespo da Silva [13] constituted a hybrid formulation that combines the advantages of both analytical and numerical methods to analyze a class of flexible multi-beam structures which each beam in the structure behaves as inextensional. In the last decade, Apiwattanalunggarn et al. [14] developed a nonlinear component mode synthesis method proposed by Hurty [15] and modified by Craig and Bampton [16], and studied the dynamic properties of complex structures composed of several simple substructures. Wang and Bajaj [17] used the method of multiple scales as well as a numerical shooting technique to investigate the bifurcations of a multi-beam elastic structure. Warminski et al. [18] investigated an autoparametric system composed of two beams with rectangular cross sections. Their results showed that certain modes in the stiff and flexible directions of both beams may interact, and intuitively unexpected out-of-plane motion may appear. Recently, Bochicchio [19] found the existence of a global regular attractor of solutions in an elastically coupled double-beam system. 2
Awrejcewicz et al. [20] applied numerical methods to discover novel chaotic synchronizations of a beam-plate interacting structure. The presence of chaotic oscillations in many structural engineering is undesirable as they can impair the optimal performance and even destabilize the structure resulting in fatigue failure. Therefore, it’s especially important to explore corresponding methods to predict the chaotic motions. Some investigators have studied the global bifurcations and chaotic motions in engineering systems by using numerical methods [21-25]. On the other hand, many researchers have developed some global perturbation methods for detecting chaotic dynamics. Originally this method was attributed to the former Soviet scholars Melnikov [26]. By applying this classical method, Chen and Liu [27] dealt with the chaotic attitude motion of a magnetic rigid spacecraft in a circular orbit and Yao and Li [28] investigated the global bifurcation of a composite laminated plate with geometric nonlinearity, respectively. The details of the framework of higher-dimensional Melnikov method can be found in Wiggins [29]. Kovačič and Wiggins [30] developed a new global perturbation technique which may be utilized to detect the Šilnikov-type single-pulse homoclinic and heteroclinic orbits in four-dimensional autonomous ordinary differential equations. Malhotra and Namachchivaya [31] then used the method to investigate the global dynamics of a shallow arch structure. Furthermore, Kaper and Kovačič [32] established the existence of several classes of multi-bump orbits homoclinic to resonance bands for completely-integrable Hamiltonian systems subject to small-amplitude Hamiltonian or dissipative perturbations. Camassa et al. [33] presented an extension of the Melnikov method which can be used for ascertaining the existence of homoclinic and heteroclinic orbits with many pulses in the class of near-integrable systems. Then, Aceves et al. [34] used the theory to find two families of Šilnikov's type of homoclinic orbits in a second-harmonic generating optical cavity. Awrejcewicz and Holicke [35] made outstanding contributions to the development and application of Melnikov-type methods applied to high dimensional smooth and nonsmooth dynamical systems. Recently, Li et al. [36], Yu and Chen [37], Yu et al. [38] played active roles for applying higher-dimensional Melnikov method in nonlinear mechanical systems. Moreover, Yu et al. [39, 40] and Zhou et al. [41] investigated the multi-pulse chaotic motions in rotor systems and gyroscopic 3
continua by using Energy-Phase method [42]. In addition, for the past few years, there have been increasing interests in "hidden attractors" reported by Leonov et al. [43], Sprott et al. [44] and Wei et al. [45]. In this paper, we extend the work in [38] for the "resonance case" to study the multi-pulse chaotic motions of a flexible multi-beam structure resting on a vibrating base considering one to two internal resonance and principal resonance and try to explore the physical mechanism which may lead to complex dynamical behaviors. The structure selected in the present paper due to the widely applications in a large class of space and robotic systems [46]. The organization of this paper is as follows: In section 2, the PDEs of the structure considering nonlinear coupling and nonlinear inertia are introduced and then, the exact modal functions obtained in [38] are adopted to truncate the PDEs into a set of coupled nonlinear ODEs via the Galerkin method and the orthogonality conditions of global modes. In the next sections, we investigate the global dynamics of the structure using the method of multiple scales in combined with a global perturbation method. Specifically, in section 3.1, the method of multiple scales is applied to yield a set of autonomous equations of the first order approximations to the response of the dynamical systems. By using canonical transformations, these perturbation equations are reduced to a standard form which is appropriate for applying the generalized Melnikov theory. In section 3.2, the dynamics of unperturbed systems is examined and time parameterized expressions for heteroclinic orbits are solved. In section 3.3, invariant manifolds and associated dynamics are presented and then the effects of small perturbations are discussed. In section 3.4, N-pulse Melnikov function is calculated analytically and the parameter conditions for the occurrence of chaotic motions in the dynamical systems are obtained. In section 4, the global results are finally interpreted in terms of the physical motion. In section 5, we end the paper with concluding remarks. 2. Governing equations As shown in Fig. 1, the flexible multi-beam structures derived from an L-shaped beam consists of two flexible beams connected with right-angle. The horizontal beam is fixed to the base which is subjected to a harmonic excitation along the Y1 direction. 4
Y1
2
X2 s2
Iy
F F cos t
O
Ix
c
1
O2
s1
Y2
O1
X1
Fig. 1 Flexible multi-beam, coordinate systems and displacements.
The basic assumptions made for the flexible multi-beam are as follows (1). The diameter of multi-beam is small compared to its length, so that each beam behaves like an Euler-Bernoulli beam; (2). The motion of structure is planar; (3). Rotary inertia, shear deformation and warping are neglected; (4). The centerline of multi-beam in each section is inextensible; (5). The deflections of multi-beam in each section are small but finite amplitudes; (6). The influence of gravitational force can be neglected for the flexible structure serving in space environment.
The dimensionless PDEs by retaining nonlinear terms up to 2 and corresponding linear boundary conditions can be cast into the following form [38] Equation of motion of the horizontal beam is
1 2 1 0 (1a) 1 f 2 d 2 1 1 0
where 2 2 2 c , and we assume that the material of the structure is subjected to internal dissipation of the Kelvin-Voigt type [47], which denoted by a stress-strain relationship of the form E , where represents the viscoelastic material damping factor. Here, we arrange the base excitation term with the inertia term together in order to expressing concisely. The second term accounts for the inertia of flexible body 5
impacting on the horizontal beam as a parametric excitation and the last term is the stiffness term considering structural damping. Equation of motion of the vertical beam is
..
1 1 1 2 c f 2 c f d 2 2 1 d1 3 1 2 0 (1b) 2 2 0
2
where the first term is the inertia term, the second term is the component of the inertia at the junction point c, the third term accounts for the inertia of point c impacting on the vertical beam as a parametric excitation, the forth term accounts for the nonlinear inertial (shortening effect) which was neglected in the study of structural dynamics sometimes. The last term is the stiffness term considering structural damping. Boundary conditions of the horizontal beam are 1 0, 0 , 1 0, 0 ,
c
1 1 1 11, 0 , 4 20, 3 1 11, 0
(2a)
Boundary conditions of the vertical beam are 2 0, 0 , 2 0, c 0 , 1 21, 0 , 1 2 1, 0 (2b) Then the natural frequencies and mode shapes can be determined from the solutions of the differential equations with boundary value problems. The first two mode shapes are shown in Fig. 2.
1.5
1.5
1
1 Y
2
Y
2
0.5
0.5
0
0 0
0.5
1
1.5
0
X
0.5
1
1.5
X
Fig. 2 (a) The first mode shape of vibration.
Fig. 2 (b) The second mode shape of vibration.
Moreover, we present the orthogonality conditions for this flexible multi-beam structure ignoring specific process of proofs as follows 6
1
1
3 S 2 m S 2 n d 2 S1m S1n d 1 S1m 1S1n 1 M m mn (3a) 0
0
1
0
S 2m S 2n d 2
1 S1m S1n d 1 K m mn (3b) 0
where mn is Kronecker's delta. It is clearly that the orthogonality conditions for such flexible multi-beam structure are aimed at the global modes based on the physical space. The analytical mode shapes and orthogonality conditions are critical prerequisites when the Galerkin method is utilized to discretize the nonlinear PDEs. 3. Global bifurcations and multi-pulse chaotic motions
In this section we use a generalized version of Melnikov method to study the global dynamics for such flexible multi-beam structure. The generalized Melnikov method is a global perturbation technique that allows one to analytically obtain the parameter range for which multi-pulse chaotic motions exist and reveal the dynamical mechanism of energy flow between modes. According to the result of linear modal analysis, the higher modes of vibration are far separated from the first two modes of vibration, so we choose the first two modes of vibration of the structure as comparison function and trial function. On the basis of orthogonality of the global modes, we consider the nonlinear dynamics of the structure as a two-degree-of-freedom system as follows u1 21u1 12 u1 f2 cos p11u1 p12 u 2 g11u12 g 12 u1u 2 g 22 u 22 s11u1u1 s12 u1u 2 s 21u2 u1 s 22 u2 u 2 e11 f2 cos
(4a)
u2 2 2 u 2 22 u 2 f 2 cos p 21u1 p22 u 2 g11u12 g 12 u1u 2 g 22 u 22 s11u1u1 s12 u1u 2 s 21u2 u1 s 22 u2 u 2 e12 f 2 cos
(4b)
We see that u2 is essentially a parametric excitation for u1 and vice versa. Such internal resonances are sometimes referred to as autoparametric resonances. 3.1 Perturbation analysis and standard form We seek a first-order uniform expansion by using the method of multiple scales in the form u1 u11 T0 , T1 2 u12 T0 , T1
(5a)
u 2 u 21 (T0 , T1 ) 2 u 22 (T0 , T1 )
(5b)
7
where Tn n . In this paper, the principal resonance of the second mode in the presence of one to two internal resonance are considered and we use the detuning parameters 1 and 2 to describe the nearness of 2 to 21 and to 2 , respectively. Thus the frequency relations can be expressed by 2 21 1 , 2 2 2
(6)
which means that the forcing frequency is tuned to the second eigenfrequency of the system with higher precision. Letting k1 s12 12 s 21 22 g12 1 2 , k 2 g1112 s1112 , the solvability conditions can be obtained
2i1 D1 A 2i11 A k1 A Be i T 0
(7a)
1 1
2i2 D1 B 2i 2 2 B k 2 A 2 e i T 1 1
e12 f2 i T e 0 2 2 1
(7b)
where D1 T1 , A and B are complex-valued functions and the over bar of A and B indicates the complex conjugate. For the solvability conditions, we introduce the polar forms A
2a1 k1k 2
e i , B 1
2a 2 k1
e i
2
(8)
where ai , i (i 1, 2) are real functions of time T1 . Substituting Eqs. (8) into Eqs. (7) and separating the real and imaginary parts of the resulting equations yields 2 a1 2 a 2 sin 2 21 (9a) 2 1 a1 1 1 a1 a1 2a 2 cos 2 21 (9b) 2 e k f 1 a 2 2 2 a 2 a1 2a 2 sin 2 21 12 1 2a 2 sin 2 (9c) 4 e k f 1 a 2 2 2 a 2 a1 2a 2 cos 2 21 12 1 2a 2 cos 2 (9d) 2 8 2 where 1 1 1 T1 , 2 2 2T1 . 2 In Eqs. (9) the overdot indicates differentiation with respect to T1, the slow time scale a1 21 a1
variable. To avoid confusion with the symbol of one dimensional torus T 1 , T1 will be replaced by t in this section. In order to reduce the averaged system to the form appropriate for applying the generalized Melnikov theory, we introduce the following canonical transformations to Eqs. (9)
8
1 1 p1 a1 , q1 1 2 , p 2 a 2 a1 , q2 2 (10) 2 2 In addition, the forcing and dissipation terms are rescaled one order smaller
compared to the nonlinear terms by 1 1 , 2 2 , f f . We have 2 p 2 p1
p 1 2 1 p1
q1
2 p1 sin 2 q1 (11a)
2 p 2 p1 1 p1 e12 k1 f cos 2q1 cos 2q1 cos q 2 (11b) 2 2 2 p 2 p1 8 2 p 2 p1 p 2 2 2 p 2 p1 1 p1
q 2
p1 2 p 2 p1
e12 k1 f 2 p 2 p1 sin q 2 (11c) 4
cos 2q1 2
e12 k1 f 4 2 p 2 p1
cos q 2
(11d)
The objective in this section is to study the global dynamics associated with system (11). Next, we examine the structure of geometry in the unperturbed system when 0 . 3.2 Dynamics of the unperturbed system We obtain the equations that describe the dynamical system without damping and forcing. They are 2 p 2 p1
p 1
q1
2 p1 sin 2 q1 (12a)
2 p 2 p1 1 p1 cos 2q1 cos 2q1 2 2 2 p 2 p1
(12b)
p 2 0 (12c) q 2
p1 2 p 2 p1
It is clear that the dynamics in the
p1 , q1
cos 2q1
(12d)
plane is completely independent of q 2 .
Since p 2 0 , we have set p 2 p 20 I , and thus it is sufficient to study the phase flow in
p1 , q1
plane. Also note that p 2 is a combination of the actions a1 and a 2 of the
averaged equations, which is a measure of the mechanical energy of the system and thus
0 p1 2 I . Further, q1 is periodic and can be restricted to 0 q1 . The Hamiltonian corresponding to the two-dimensional vector field is given as 9
H p1 , q1 , I
1 p p1 1 2
It is obvious that the orbits in the
p1 , q1
2 I p1 cos 2q1
(13)
phase space depend only on the two
parameters I , 1 . In subsequent section, we describe the geometric structure in the unperturbed
p1 , q1
phase space. The equilibrium points of the reduced vector field can
be obtained for two different cases depending on the quantity 1 .
2 12 (a) I Case 8 When 1 0 , the equilibrium points are as follows 1)
1 1 * p1 0 , q1 cos 1 q1 2 2 2I
2)
p1 0 , q1 q1* 1
3)
p1 p
4)
p1 p1
5)
1
p1 p
24 I 2 12 4 14 24 2 12 I 18 24 I 2 12 4 14 24 2 12 I 18 24 I 2 12 4 14 24 2 12 I 18
, q1 0 , q1
(14a) (14b)
(14c)
(14d) 2
, q1
(14e)
When 1 0 , the equilibrium points are as follows 1)
p1 0 , q1 q1*
2)
p1 0 , q1 q1* (15b)
3)
p1 p1 , q1 0
4)
p1 p1 , q1
5)
p1 p1 , q1
(b) I
(15c)
2
(15d) (15e)
2 12 Case 8
When 1 0 , the equilibrium points are as follows 10
(15a)
1)
p1 p1 , q1 0
(16a)
2)
p1 p1 , q1
2
(16b)
3)
p1 p1 , q1
(16c)
When 1 0 , the equilibrium points are as follows 1)
p1 p1 , q1 0
2)
p1 p1 , q1
3)
p1 p1 , q1
(17a)
2
(17b) (17c)
0.2
0.15
0.15
p
1
p
1
0.2
0.1
0.1
0.05 *
(0,q1 ) 0
0
0.5
0.05
A1 1
A2 1.5 q
2
2.5
*
A1
*
(0,π-q1 )
0
3
0
(0,q1 ) 0.5
1
1
1.5 q
*
(0,π-q1 ) 2
2.5
3
1
(a) Fig. 3 Unperturbed phase flow in
A2
(b)
p1 , q1 phase space. (a)
1 0 , (b) 1 0 .
The stability analysis indicates that the phase flow has the qualitatively different behavior for the case (a) comparing with the case (b). One can easily calculate the eigenvalues of the Jacobian matrix of the unperturbed system in the
p1 , q1 vector field
and find out that the equilibrium points (0, q1* ) and (0, q1* ) exist as saddle-type fixed
points for the case (a) and the equilibrium points ( p1 ,0) , ( p1 , 2) and ( p1 , ) exist
as center-type fixed points for the two cases. One can also deduce that there are no homoclinic/heteroclinic connections for the case (b). The phase portraits for the case (a) 11
are shown in Fig. 3 (a) and Fig. 3 (b). These phase portraits indicate the existence of a heteroclinic cycle ( A1 A2 ) connecting the two saddle points (0, q1* ) and (0, q1* ) . In the subsequent section we assume that the aforementioned condition is satisfied. The orbits in the
p1 , q1 phase
space are the level curves of the unperturbed
Hamiltonian H restricted to it. The equations for heteroclinic orbits are obtained as follows Orbit A 1 : p1 2 I
2 12 4 cos 2 2q1
(18a)
Orbit A2: p1 0
(18b)
For orbit A1, in order to get a explicit form for q1 as a function of time t, substituting (18a) into (12b) and integrating with respect to t, we obtain
q1 (t )
8I 2 2 8I 2 2 1 1 1 arctan tanh 2 1
t (19a) 2
where 1 0 and the initial condition is q1 0 . 8I 2 2 8I 2 2 1 1 1 q (t ) arctan tanh 1 2 1
t
(19b)
where 1 0 with the initial condition is q1 0 0 . Hence, the explicit expressions of p1 t along the orbit A1 is given by 2 2 8I 2 12 2 8 I 1 sec h p1 (t ) t (20) 4 Substituting the expression q1 t and p1 t into the Eq. (12d) and integrating with
respect to t, we obtain
q 2 t q 20 2q1 (t )
(21)
where q 20 is a constant obtained from initial conditions. Therefore, the phase difference after each heteroclinic excursion is equal to
8I 2 2 1 q 2 q 2 q 2 2 arctan (22) 1 In fact, this quantity gives the difference on phase as a trajectory leaves and returns to the invariant manifold 0 as we will discuss in the next section. 12
3.3 Invariant mannifolds and associated dynamics A As has beenn noted prev viously, thee equilibrium m points 0, q1* and 0, q1* , which
are coonnected by heterocliniic orbits, exxist as hyperrbolic fixed points in thhe
p1 , q1
phase
space.. However,, in the fo our-dimensioonal phase space these fixed p oints repreesent a two-dimensional invariant manifold m thaat is normallly hyperbolic 2 12 011 p1 , q1 , p 2 , q 2 p1 0, q1 q1* , p 2 I ,q2 T1 8
(23a)
and 2 122 02 p1 , q1 , p 2 , q 2 p1 0, q1 q1* , p 2 I , q 2 T 1 (23b) 8
D Define a tw wo-dimensional maniffold 0 (Fig. ( 4 (a))), invariantt under thee flow generaated by the unperturb bed vector field. Thee invariant manifold 0 is no ormally hyperbbolic, whicch means that underr linearized d dynamics, rates off expansio on and contraaction transsverse to 0 dominaate those taangent to 0 . Moreovver, the no ormally hyperbbolic invariiant manifolld persists uunder small perturbations.
Fig. 4 (a) The unperturrbed system aand manifolds 0 in
p1 , q1 , p 2 phhase space.
T The manifoold 0 has h a threee-dimension nal stable manifold W s 0 and a three-dimensionaal unstable manifold W u 0 . The T existence of the hheteroclinic orbits conneecting 01 and 02 implies tthe nontran nsversal inttersection oof W s 0 and dimensionall heteroclinic manifo old, denoteed by . The W u 0 along a three-d 13
trajecttories in approach a trajectoryy in 0 assymptoticallly as t . T The dynam mics restrictted to the invariant manifold 0 is deetermined by b the equatiions p 2 0 (24a) q 2 0
(24b)
W We note thatt 0 is in n fact a planne of fixed points. p This is called thhe “resonancce case” (Wigggins [29]) duue to the vaanishing of q 2 .
Fig. 4 (bb) The perturb bed system annd manifolds in
p1 , q1 , p 2 phaase space.
H Having exaamined the dynamical motions for f the fourr-dimensionnal system in the unpertturbed casee, we now study the eeffects of sm mall perturbation whicch arise fro om the base eexcitation ass well as thee structural damping. The T normal hyperboliciity property y of the maniffold 0 im mplies that it persists aalong with its stable and unstablee manifoldss under sufficiiently smalll perturbations. But, in general,, the dynam mics withinn these man nifolds changge dramaticaally. Under small pertuurbation ( 0 ), the manifold 00 persists as a (Fig. 4 (b)) due too its normall hyperboliccity property y and which h is defined by 2 12 1 p1 , q1 , p 2 , q 2 p1 0 (), q1 q1* ( ), p 22 p 2 p 12 , q 2 T 1 (25a) 8
and
2 p1 , q1 , p 2 , q 2 p1 0 ( ), q1 q1* ( ), 14
p 22 p 2 p12
2 12 , q2 T 1 8
(25b)
The flow restricted to is governed by the following equations p 2 2 2 p 2
e12 k1 f 2 p 2 sin q 2 (26a) 4
q 2 2
e12 k1 f 4 2 p2
(26b)
cos q 2
where is locally invariant manifold with boundaries. There is a single equilibrium in this plane at the point e 2 k 22 f 2 p p 2 , q 2 12 12 , tan 1 2 2 2 32 2 2
(27)
which is a spiral sink if 2 0 and 2 0 . Dynamics in this plane evolve on a slow, 1 time scale.
3.4 N - pulse Melnikov function For dynamical system (11), a homoclinic orbit with N pulses consists of two pieces. The first piece of such a homoclinic orbit is close to a string of N consecutive heteroclinic
p1 , q1 , p 2 , q2 t , I , q 20
orbits
jq 2 ,
with j 0,1, , N 1 .
These
heteroclinic orbits are called pulses. The string begins with the unstable manifold of the equilibrium point
p
1
, q1 , p 2 , q 2 . And the equations of p2 and q 2 fix the parameters
I and q 20 in all N pulses of the string. The last heteroclinic orbit in the string runs
back in plane at the point
p 2 , q 2 Nq 2 . The second piece of a homoclinic orbit
with N pulses is close to the trajectory of system (26) that connects the landing point
p 2 , q 2 Nq 2 of the last pulse back to the equilibrium p 2 , q 2 , as shown in Fig. 5.
15
p2 q2
Fig. 5 A multi-pulse jumping orbit homoclinic to p .
Moreover, the homoclinic orbit with N pulses only exists if two further conditions are satisfied. First, its second piece, or rather the corresponding trajectory of system (11) that connects the point
p 2 , q 2 Nq 2 to
the equilibrium p 2 , q 2 , must be above
the line p 2 2 12 8 . Otherwise, the existence of such homoclinic orbits cease to be valid. Second, we must choose the parameters 1 , 2 and f so that the N - pulse Melnikov function M N I , q 2 , to be described next, vanishes along the string of heteroclinic orbits
p1 , q1 , p 2 , q 2 t , I , q 20 , …, p1 , q1 , p 2 , q 2 t , I , q 20
jq 2 .
The N - pulse Melnikov function is the sum of the ordinary Melnikov functions calculated along the heteroclinic pulses, that is M N I , q 2
N 1
M t , I , q 2 jq 2
j 0
N 1
DH j 0
N 2 1 8 I 2 12 2
0
, g dt
2I N 1 N 1 q 2 sin q2 e12 k 1 f sin q 2 2 2 2
(28)
H 0 H 0 H 0 H 0 2 p 2 p1 cos 2q1 , DH 0 , , , p1 q1 p 2 q 2
,
where H 0 p1 , q1 , p 2 , q 2
1 p p1 1 2
g p 21 p1 , g q 1
1
e12 k1 f 8 2 p 2 p1
cos q 2 , g q 2
g p 2 2 p 2 p1 1 p1 2
16
e12 k1 f 4 2 p 2 p1
cos q 2 ,
e12 k1 f 2 p 2 p1 sin q 2 4
(29)
Simple zeros of the N - pulse Melnikov function, namely, values of I and q 2 for which M N I , q 2 0 and D q M N I , q 2 0 , pick out the survivors under perturbation 2
form among these unperturbed homoclinic orbits. Conditions for the existence of simple zeros of the N - pulse Melnikov function are obtained as N 2 1 8 I 2 12 2
N 1 N 1 2I e12 k 1 f sin q 2 sin 0 2 2 2
N 1 N 1 2I e12 k 1 f cos q 2 sin 0 2 2 2
(30a)
(30b)
From Eq. (30a), it is easy to see that the conditions for the manifolds to intersect in terms of parameters 1 , 2 , f is given by d
2 2 I e12 k1 2 2 f N 8 I 1 1
(31)
The threshold value surface obtained from the analytical expressions is plotted in Fig. 6 and chaotic dynamics may be generated below the surface.
d
1
N
Fig. 6 The threshold value surface of d with respect to N and 1 .
4. Physical interpretation
In this section, we try to interpret the global solutions in terms of physical motions of such flexible multi-beam structure based on the analytical mode shapes shown in Fig. 2. The global dynamics in the physical space will be described for the observer who is standing at the moving reference frame X1O1I1. The first two modes of the structure which 17
are related to the action-angle variables can be expressed as 2
u1
k1 k 2
u2
1 2 p1 cos q1 q 2 2 2 2
2 2 p 2 p1 sin q 2 2 k1
(32a)
(32b)
The heteroclinic structure in the unperturbed system was shown existing in Fig. 3 whenever the system energy is above the critical value. In the absence of any perturbation, The dynamics can occur either on or off the invariant manifold. If the trajectory lies entirely on the manifold 0 , the motion of the structure only occurs in the second mode which means a localized nonlinear normal mode exists. The snapshots of corresponding modal motion for different time are presented in Fig. 7. In the resonant case, a plane of fixed points in the
p2 , q2
phase space leads to a periodic motion in this localized mode.
If the trajectories are outside of 0 , actually, the trajectories can be on the center-type fixed point, any periodic manifold, or the heteroclinic manifold. The center-type fixed point located at q1 0 and q1 2 in the
p1 , q1
phase space are two nonlinear
normal modes characterized by a coupled periodic motion. If the trajectory lies on any periodic manifold, then the actual motion of the structure would consist of amplitude and phase modulated oscillation which manifests as a coupled quasi-periodic motion of two modes , see Fig. 8. However, for a trajectory lying on the heteroclinic manifold, the physical motion would start as an amplitude modulated coupled modal motion of two modes, as time tends to infinity, shift towards the localized modal motion, see Fig. 9. 2
1.5
Y
1
0.5
0 0
0.5
1
1.5
X
Fig. 7 The snapshots of nonlinear modal motion for the flexible structure. 18
2
1.5
1.5
1
1 Y
Y
2
0.5
0.5
0
0
-0.5
0
0.5
1
-0.5
1.5
X
0
0.5
(a)
X
1
1.5
(b)
Fig. 8 The snapshots of nonlinear modal motion for the flexible structure. (a) q1 0 . (b) q1 2 .
2 1.5
Y
1 0.5 0 -0.5
0
0.5
1
1.5
X
Fig. 9 The snapshots of physical motion when a trajectory lying on the heteroclinic manifold.
Under small perturbations due to damping dissipation and base excitation, single or multi-pulse orbits might arise relying on system parameters. We try to interpret the dynamical behaviors associated with the trajectories coming out of hyperbolic sink p . In this case, 0 persists as an invariant subspace for the perturbed system. The initial conditions on lead the phase flow towards the attracting sink p . As time goes on, the amplitude starts decreasing slowly and eventually the motion becomes a constant amplitude oscillation governed by the second modal motion, see Fig. 7. In the 19
four-dimensional phase space, if the initial conditions are set close to W u p , and the transversal condition is satisfied, the dynamical response starts as a coupled modal motion whose amplitude and phase are modulated, but soon shifts towards to a localized modal motion, which means the motion of the flexible multi-beam structure is localized in the second modal motion for a long time. As the trajectory approaches W u p , it takes off again and repeats the similar motion when a single-pulse orbit exists. When a multi-pulse orbit exist, as shown in Fig. 5, one can replace the single-pulse process as multi-pulse orbits in the vicinity of N-chains of homoclinic orbits in the resonance plane. Physically, such multi-pulse jumping orbits mean the first modal motion with the subharmonic frequency are involved suddenly. While doing this, energy flow between the localized mode and the coupled mode in a chaotic fashion will be observed. 5. Conclusions
In this paper, we have employed a generalized Melnikov method to study the global behaviors of a class of flexible multi-beam structures resting on a vibrating base for the principal resonance of the second mode with one to two internal resonance. First, it is shown that the dynamical system has heteroclinic connections for certain values of the system parameters in the absence of any perturbations. Then, under small perturbations arising from damping dissipation and base excitation, complex and ordered dynamics may occur in the case of "resonance", and the system may exhibit chaotic dynamics in the sense of Smale horseshoes due to the formation of Šilnikov-type homoclinic orbits. The conditions on system parameters under which such autoparametric system exhibits multi-pulse chaotic motions are also explored. Finally, The physical interpretation of global solutions provides an intuitive understanding on the dynamical mechanism of energy flow between the localized mode and the coupled mode in a chaotic fashion. Acknowledgements
This research is supported by National Natural Science Foundation of China (NNSFC) through Grant Nos.11290150, 11290152, 11290154, 11322214 and 11672007, the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB).
20
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24
Highlights
Global dynamics of a class of flexible multi-beam structures under autoparametric resonance is explored.
A generalized Melnikov method demonstrates its powerful utility for gaining a comprehensive insight into the global bifurcations and multi-pulse chaotic dynamics of the nonlinear dynamical system.
How energy may flow between different modes is investigated by checking the existence of multi-pulse homoclinic orbits.
Localized and coupled nonlinear normal modes are observed in terms of the physical motion.
Chaotic pattern conversion between localized mode and coupled mode are revealed in such autoparametric system.
PII: DOI: Reference:
S0020-7462(17)30023-9 http://dx.doi.org/10.1016/j.ijnonlinmec.2017.06.015 NLM 2870
To appear in:
International Journal of Non-Linear Mechanics
Received date : 15 January 2017 Revised date : 11 June 2017 Accepted date : 20 June 2017 Please cite this article as: T. Yu, W. Zhang, X. Yang, Global bifurcations and chaotic motions of a flexible multi-beam structure, International Journal of Non-Linear Mechanics (2017), http://dx.doi.org/10.1016/j.ijnonlinmec.2017.06.015 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Global bifurcations and chaotic motions of a flexible multi-beam structure Tian-Jun Yu, Wei Zhang, Xiao-Dong Yang Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, P. R. China
Abstract Global bifurcations and multi-pulse chaotic motions of flexible multi-beam structures derived from an L-shaped beam resting on a vibrating base are investigated considering one to two internal resonance and principal resonance. Base on the exact modal functions and the orthogonality conditions of global modes, the PDEs of the structure including both nonlinear coupling and nonlinear inertia are discretized into a set of coupled autoparametric ODEs by using Galerkin's technique. The method of multiple scales is applied to yield a set of autonomous equations of the first order approximations to the response of the dynamical system. A generalized Melnikov method is used to study global dynamics for the "resonance case". The present analysis indicates multi-pulse chaotic motions result from the existence of Šilnikov's type of homoclinic orbits and the critical parameter surface under which the system may exhibit chaos in the sense of Smale horseshoes are obtained. The global results are finally interpreted in terms of the physical motion of such flexible multi-beam structure and the dynamical mechanism on chaotic pattern conversion between the localized mode and the coupled mode are revealed.
Keywords: Flexible multi-beam structures; Autoparametric system; Global dynamics; Melnikov method; Localized mode; Coupled mode
Corresponding Author, Email: [email protected] 1
1. Introduction Flexible multi-beam structures are typical components of large space station, aeronautical engineering, civil engineering and other mechanical systems. A profound understanding of the dynamical characteristics of these structural systems is essential for their design and control [1]. With the development of the more flexible materials, the nonlinearities in such structural systems cannot be neglected in dynamics research. The nonlinear dynamic analysis is the key in modeling, analysis and control of these large space structures after deployment and during service [2, 3]. Weak quadratic nonlinearities arise in many multi degree-of-freedom systems due to inertial effects of large motions, the geometry of motion, the centrifugal and Coriolis forces. Such systems, as known, include ships, arches, liquid in containers, beams and plates under static loadings [4]. In fact, flexible multi-beam structures are typical quadratic nonlinearity systems due to the nonlinear coupling and nonlinear inertia. In the earlier study, Haddow et al. [5], Bux and Roberts [6], Ashworth and Barr [7], Balachandran and Nayfeh et al. [8-11] have made important contributions to the nonlinear dynamic characteristics of such structures by using the method of multiple scales [12] with assistance of experiments. Crespo da Silva [13] constituted a hybrid formulation that combines the advantages of both analytical and numerical methods to analyze a class of flexible multi-beam structures which each beam in the structure behaves as inextensional. In the last decade, Apiwattanalunggarn et al. [14] developed a nonlinear component mode synthesis method proposed by Hurty [15] and modified by Craig and Bampton [16], and studied the dynamic properties of complex structures composed of several simple substructures. Wang and Bajaj [17] used the method of multiple scales as well as a numerical shooting technique to investigate the bifurcations of a multi-beam elastic structure. Warminski et al. [18] investigated an autoparametric system composed of two beams with rectangular cross sections. Their results showed that certain modes in the stiff and flexible directions of both beams may interact, and intuitively unexpected out-of-plane motion may appear. Recently, Bochicchio [19] found the existence of a global regular attractor of solutions in an elastically coupled double-beam system. 2
Awrejcewicz et al. [20] applied numerical methods to discover novel chaotic synchronizations of a beam-plate interacting structure. The presence of chaotic oscillations in many structural engineering is undesirable as they can impair the optimal performance and even destabilize the structure resulting in fatigue failure. Therefore, it’s especially important to explore corresponding methods to predict the chaotic motions. Some investigators have studied the global bifurcations and chaotic motions in engineering systems by using numerical methods [21-25]. On the other hand, many researchers have developed some global perturbation methods for detecting chaotic dynamics. Originally this method was attributed to the former Soviet scholars Melnikov [26]. By applying this classical method, Chen and Liu [27] dealt with the chaotic attitude motion of a magnetic rigid spacecraft in a circular orbit and Yao and Li [28] investigated the global bifurcation of a composite laminated plate with geometric nonlinearity, respectively. The details of the framework of higher-dimensional Melnikov method can be found in Wiggins [29]. Kovačič and Wiggins [30] developed a new global perturbation technique which may be utilized to detect the Šilnikov-type single-pulse homoclinic and heteroclinic orbits in four-dimensional autonomous ordinary differential equations. Malhotra and Namachchivaya [31] then used the method to investigate the global dynamics of a shallow arch structure. Furthermore, Kaper and Kovačič [32] established the existence of several classes of multi-bump orbits homoclinic to resonance bands for completely-integrable Hamiltonian systems subject to small-amplitude Hamiltonian or dissipative perturbations. Camassa et al. [33] presented an extension of the Melnikov method which can be used for ascertaining the existence of homoclinic and heteroclinic orbits with many pulses in the class of near-integrable systems. Then, Aceves et al. [34] used the theory to find two families of Šilnikov's type of homoclinic orbits in a second-harmonic generating optical cavity. Awrejcewicz and Holicke [35] made outstanding contributions to the development and application of Melnikov-type methods applied to high dimensional smooth and nonsmooth dynamical systems. Recently, Li et al. [36], Yu and Chen [37], Yu et al. [38] played active roles for applying higher-dimensional Melnikov method in nonlinear mechanical systems. Moreover, Yu et al. [39, 40] and Zhou et al. [41] investigated the multi-pulse chaotic motions in rotor systems and gyroscopic 3
continua by using Energy-Phase method [42]. In addition, for the past few years, there have been increasing interests in "hidden attractors" reported by Leonov et al. [43], Sprott et al. [44] and Wei et al. [45]. In this paper, we extend the work in [38] for the "resonance case" to study the multi-pulse chaotic motions of a flexible multi-beam structure resting on a vibrating base considering one to two internal resonance and principal resonance and try to explore the physical mechanism which may lead to complex dynamical behaviors. The structure selected in the present paper due to the widely applications in a large class of space and robotic systems [46]. The organization of this paper is as follows: In section 2, the PDEs of the structure considering nonlinear coupling and nonlinear inertia are introduced and then, the exact modal functions obtained in [38] are adopted to truncate the PDEs into a set of coupled nonlinear ODEs via the Galerkin method and the orthogonality conditions of global modes. In the next sections, we investigate the global dynamics of the structure using the method of multiple scales in combined with a global perturbation method. Specifically, in section 3.1, the method of multiple scales is applied to yield a set of autonomous equations of the first order approximations to the response of the dynamical systems. By using canonical transformations, these perturbation equations are reduced to a standard form which is appropriate for applying the generalized Melnikov theory. In section 3.2, the dynamics of unperturbed systems is examined and time parameterized expressions for heteroclinic orbits are solved. In section 3.3, invariant manifolds and associated dynamics are presented and then the effects of small perturbations are discussed. In section 3.4, N-pulse Melnikov function is calculated analytically and the parameter conditions for the occurrence of chaotic motions in the dynamical systems are obtained. In section 4, the global results are finally interpreted in terms of the physical motion. In section 5, we end the paper with concluding remarks. 2. Governing equations As shown in Fig. 1, the flexible multi-beam structures derived from an L-shaped beam consists of two flexible beams connected with right-angle. The horizontal beam is fixed to the base which is subjected to a harmonic excitation along the Y1 direction. 4
Y1
2
X2 s2
Iy
F F cos t
O
Ix
c
1
O2
s1
Y2
O1
X1
Fig. 1 Flexible multi-beam, coordinate systems and displacements.
The basic assumptions made for the flexible multi-beam are as follows (1). The diameter of multi-beam is small compared to its length, so that each beam behaves like an Euler-Bernoulli beam; (2). The motion of structure is planar; (3). Rotary inertia, shear deformation and warping are neglected; (4). The centerline of multi-beam in each section is inextensible; (5). The deflections of multi-beam in each section are small but finite amplitudes; (6). The influence of gravitational force can be neglected for the flexible structure serving in space environment.
The dimensionless PDEs by retaining nonlinear terms up to 2 and corresponding linear boundary conditions can be cast into the following form [38] Equation of motion of the horizontal beam is
1 2 1 0 (1a) 1 f 2 d 2 1 1 0
where 2 2 2 c , and we assume that the material of the structure is subjected to internal dissipation of the Kelvin-Voigt type [47], which denoted by a stress-strain relationship of the form E , where represents the viscoelastic material damping factor. Here, we arrange the base excitation term with the inertia term together in order to expressing concisely. The second term accounts for the inertia of flexible body 5
impacting on the horizontal beam as a parametric excitation and the last term is the stiffness term considering structural damping. Equation of motion of the vertical beam is
..
1 1 1 2 c f 2 c f d 2 2 1 d1 3 1 2 0 (1b) 2 2 0
2
where the first term is the inertia term, the second term is the component of the inertia at the junction point c, the third term accounts for the inertia of point c impacting on the vertical beam as a parametric excitation, the forth term accounts for the nonlinear inertial (shortening effect) which was neglected in the study of structural dynamics sometimes. The last term is the stiffness term considering structural damping. Boundary conditions of the horizontal beam are 1 0, 0 , 1 0, 0 ,
c
1 1 1 11, 0 , 4 20, 3 1 11, 0
(2a)
Boundary conditions of the vertical beam are 2 0, 0 , 2 0, c 0 , 1 21, 0 , 1 2 1, 0 (2b) Then the natural frequencies and mode shapes can be determined from the solutions of the differential equations with boundary value problems. The first two mode shapes are shown in Fig. 2.
1.5
1.5
1
1 Y
2
Y
2
0.5
0.5
0
0 0
0.5
1
1.5
0
X
0.5
1
1.5
X
Fig. 2 (a) The first mode shape of vibration.
Fig. 2 (b) The second mode shape of vibration.
Moreover, we present the orthogonality conditions for this flexible multi-beam structure ignoring specific process of proofs as follows 6
1
1
3 S 2 m S 2 n d 2 S1m S1n d 1 S1m 1S1n 1 M m mn (3a) 0
0
1
0
S 2m S 2n d 2
1 S1m S1n d 1 K m mn (3b) 0
where mn is Kronecker's delta. It is clearly that the orthogonality conditions for such flexible multi-beam structure are aimed at the global modes based on the physical space. The analytical mode shapes and orthogonality conditions are critical prerequisites when the Galerkin method is utilized to discretize the nonlinear PDEs. 3. Global bifurcations and multi-pulse chaotic motions
In this section we use a generalized version of Melnikov method to study the global dynamics for such flexible multi-beam structure. The generalized Melnikov method is a global perturbation technique that allows one to analytically obtain the parameter range for which multi-pulse chaotic motions exist and reveal the dynamical mechanism of energy flow between modes. According to the result of linear modal analysis, the higher modes of vibration are far separated from the first two modes of vibration, so we choose the first two modes of vibration of the structure as comparison function and trial function. On the basis of orthogonality of the global modes, we consider the nonlinear dynamics of the structure as a two-degree-of-freedom system as follows u1 21u1 12 u1 f2 cos p11u1 p12 u 2 g11u12 g 12 u1u 2 g 22 u 22 s11u1u1 s12 u1u 2 s 21u2 u1 s 22 u2 u 2 e11 f2 cos
(4a)
u2 2 2 u 2 22 u 2 f 2 cos p 21u1 p22 u 2 g11u12 g 12 u1u 2 g 22 u 22 s11u1u1 s12 u1u 2 s 21u2 u1 s 22 u2 u 2 e12 f 2 cos
(4b)
We see that u2 is essentially a parametric excitation for u1 and vice versa. Such internal resonances are sometimes referred to as autoparametric resonances. 3.1 Perturbation analysis and standard form We seek a first-order uniform expansion by using the method of multiple scales in the form u1 u11 T0 , T1 2 u12 T0 , T1
(5a)
u 2 u 21 (T0 , T1 ) 2 u 22 (T0 , T1 )
(5b)
7
where Tn n . In this paper, the principal resonance of the second mode in the presence of one to two internal resonance are considered and we use the detuning parameters 1 and 2 to describe the nearness of 2 to 21 and to 2 , respectively. Thus the frequency relations can be expressed by 2 21 1 , 2 2 2
(6)
which means that the forcing frequency is tuned to the second eigenfrequency of the system with higher precision. Letting k1 s12 12 s 21 22 g12 1 2 , k 2 g1112 s1112 , the solvability conditions can be obtained
2i1 D1 A 2i11 A k1 A Be i T 0
(7a)
1 1
2i2 D1 B 2i 2 2 B k 2 A 2 e i T 1 1
e12 f2 i T e 0 2 2 1
(7b)
where D1 T1 , A and B are complex-valued functions and the over bar of A and B indicates the complex conjugate. For the solvability conditions, we introduce the polar forms A
2a1 k1k 2
e i , B 1
2a 2 k1
e i
2
(8)
where ai , i (i 1, 2) are real functions of time T1 . Substituting Eqs. (8) into Eqs. (7) and separating the real and imaginary parts of the resulting equations yields 2 a1 2 a 2 sin 2 21 (9a) 2 1 a1 1 1 a1 a1 2a 2 cos 2 21 (9b) 2 e k f 1 a 2 2 2 a 2 a1 2a 2 sin 2 21 12 1 2a 2 sin 2 (9c) 4 e k f 1 a 2 2 2 a 2 a1 2a 2 cos 2 21 12 1 2a 2 cos 2 (9d) 2 8 2 where 1 1 1 T1 , 2 2 2T1 . 2 In Eqs. (9) the overdot indicates differentiation with respect to T1, the slow time scale a1 21 a1
variable. To avoid confusion with the symbol of one dimensional torus T 1 , T1 will be replaced by t in this section. In order to reduce the averaged system to the form appropriate for applying the generalized Melnikov theory, we introduce the following canonical transformations to Eqs. (9)
8
1 1 p1 a1 , q1 1 2 , p 2 a 2 a1 , q2 2 (10) 2 2 In addition, the forcing and dissipation terms are rescaled one order smaller
compared to the nonlinear terms by 1 1 , 2 2 , f f . We have 2 p 2 p1
p 1 2 1 p1
q1
2 p1 sin 2 q1 (11a)
2 p 2 p1 1 p1 e12 k1 f cos 2q1 cos 2q1 cos q 2 (11b) 2 2 2 p 2 p1 8 2 p 2 p1 p 2 2 2 p 2 p1 1 p1
q 2
p1 2 p 2 p1
e12 k1 f 2 p 2 p1 sin q 2 (11c) 4
cos 2q1 2
e12 k1 f 4 2 p 2 p1
cos q 2
(11d)
The objective in this section is to study the global dynamics associated with system (11). Next, we examine the structure of geometry in the unperturbed system when 0 . 3.2 Dynamics of the unperturbed system We obtain the equations that describe the dynamical system without damping and forcing. They are 2 p 2 p1
p 1
q1
2 p1 sin 2 q1 (12a)
2 p 2 p1 1 p1 cos 2q1 cos 2q1 2 2 2 p 2 p1
(12b)
p 2 0 (12c) q 2
p1 2 p 2 p1
It is clear that the dynamics in the
p1 , q1
cos 2q1
(12d)
plane is completely independent of q 2 .
Since p 2 0 , we have set p 2 p 20 I , and thus it is sufficient to study the phase flow in
p1 , q1
plane. Also note that p 2 is a combination of the actions a1 and a 2 of the
averaged equations, which is a measure of the mechanical energy of the system and thus
0 p1 2 I . Further, q1 is periodic and can be restricted to 0 q1 . The Hamiltonian corresponding to the two-dimensional vector field is given as 9
H p1 , q1 , I
1 p p1 1 2
It is obvious that the orbits in the
p1 , q1
2 I p1 cos 2q1
(13)
phase space depend only on the two
parameters I , 1 . In subsequent section, we describe the geometric structure in the unperturbed
p1 , q1
phase space. The equilibrium points of the reduced vector field can
be obtained for two different cases depending on the quantity 1 .
2 12 (a) I Case 8 When 1 0 , the equilibrium points are as follows 1)
1 1 * p1 0 , q1 cos 1 q1 2 2 2I
2)
p1 0 , q1 q1* 1
3)
p1 p
4)
p1 p1
5)
1
p1 p
24 I 2 12 4 14 24 2 12 I 18 24 I 2 12 4 14 24 2 12 I 18 24 I 2 12 4 14 24 2 12 I 18
, q1 0 , q1
(14a) (14b)
(14c)
(14d) 2
, q1
(14e)
When 1 0 , the equilibrium points are as follows 1)
p1 0 , q1 q1*
2)
p1 0 , q1 q1* (15b)
3)
p1 p1 , q1 0
4)
p1 p1 , q1
5)
p1 p1 , q1
(b) I
(15c)
2
(15d) (15e)
2 12 Case 8
When 1 0 , the equilibrium points are as follows 10
(15a)
1)
p1 p1 , q1 0
(16a)
2)
p1 p1 , q1
2
(16b)
3)
p1 p1 , q1
(16c)
When 1 0 , the equilibrium points are as follows 1)
p1 p1 , q1 0
2)
p1 p1 , q1
3)
p1 p1 , q1
(17a)
2
(17b) (17c)
0.2
0.15
0.15
p
1
p
1
0.2
0.1
0.1
0.05 *
(0,q1 ) 0
0
0.5
0.05
A1 1
A2 1.5 q
2
2.5
*
A1
*
(0,π-q1 )
0
3
0
(0,q1 ) 0.5
1
1
1.5 q
*
(0,π-q1 ) 2
2.5
3
1
(a) Fig. 3 Unperturbed phase flow in
A2
(b)
p1 , q1 phase space. (a)
1 0 , (b) 1 0 .
The stability analysis indicates that the phase flow has the qualitatively different behavior for the case (a) comparing with the case (b). One can easily calculate the eigenvalues of the Jacobian matrix of the unperturbed system in the
p1 , q1 vector field
and find out that the equilibrium points (0, q1* ) and (0, q1* ) exist as saddle-type fixed
points for the case (a) and the equilibrium points ( p1 ,0) , ( p1 , 2) and ( p1 , ) exist
as center-type fixed points for the two cases. One can also deduce that there are no homoclinic/heteroclinic connections for the case (b). The phase portraits for the case (a) 11
are shown in Fig. 3 (a) and Fig. 3 (b). These phase portraits indicate the existence of a heteroclinic cycle ( A1 A2 ) connecting the two saddle points (0, q1* ) and (0, q1* ) . In the subsequent section we assume that the aforementioned condition is satisfied. The orbits in the
p1 , q1 phase
space are the level curves of the unperturbed
Hamiltonian H restricted to it. The equations for heteroclinic orbits are obtained as follows Orbit A 1 : p1 2 I
2 12 4 cos 2 2q1
(18a)
Orbit A2: p1 0
(18b)
For orbit A1, in order to get a explicit form for q1 as a function of time t, substituting (18a) into (12b) and integrating with respect to t, we obtain
q1 (t )
8I 2 2 8I 2 2 1 1 1 arctan tanh 2 1
t (19a) 2
where 1 0 and the initial condition is q1 0 . 8I 2 2 8I 2 2 1 1 1 q (t ) arctan tanh 1 2 1
t
(19b)
where 1 0 with the initial condition is q1 0 0 . Hence, the explicit expressions of p1 t along the orbit A1 is given by 2 2 8I 2 12 2 8 I 1 sec h p1 (t ) t (20) 4 Substituting the expression q1 t and p1 t into the Eq. (12d) and integrating with
respect to t, we obtain
q 2 t q 20 2q1 (t )
(21)
where q 20 is a constant obtained from initial conditions. Therefore, the phase difference after each heteroclinic excursion is equal to
8I 2 2 1 q 2 q 2 q 2 2 arctan (22) 1 In fact, this quantity gives the difference on phase as a trajectory leaves and returns to the invariant manifold 0 as we will discuss in the next section. 12
3.3 Invariant mannifolds and associated dynamics A As has beenn noted prev viously, thee equilibrium m points 0, q1* and 0, q1* , which
are coonnected by heterocliniic orbits, exxist as hyperrbolic fixed points in thhe
p1 , q1
phase
space.. However,, in the fo our-dimensioonal phase space these fixed p oints repreesent a two-dimensional invariant manifold m thaat is normallly hyperbolic 2 12 011 p1 , q1 , p 2 , q 2 p1 0, q1 q1* , p 2 I ,q2 T1 8
(23a)
and 2 122 02 p1 , q1 , p 2 , q 2 p1 0, q1 q1* , p 2 I , q 2 T 1 (23b) 8
D Define a tw wo-dimensional maniffold 0 (Fig. ( 4 (a))), invariantt under thee flow generaated by the unperturb bed vector field. Thee invariant manifold 0 is no ormally hyperbbolic, whicch means that underr linearized d dynamics, rates off expansio on and contraaction transsverse to 0 dominaate those taangent to 0 . Moreovver, the no ormally hyperbbolic invariiant manifolld persists uunder small perturbations.
Fig. 4 (a) The unperturrbed system aand manifolds 0 in
p1 , q1 , p 2 phhase space.
T The manifoold 0 has h a threee-dimension nal stable manifold W s 0 and a three-dimensionaal unstable manifold W u 0 . The T existence of the hheteroclinic orbits conneecting 01 and 02 implies tthe nontran nsversal inttersection oof W s 0 and dimensionall heteroclinic manifo old, denoteed by . The W u 0 along a three-d 13
trajecttories in approach a trajectoryy in 0 assymptoticallly as t . T The dynam mics restrictted to the invariant manifold 0 is deetermined by b the equatiions p 2 0 (24a) q 2 0
(24b)
W We note thatt 0 is in n fact a planne of fixed points. p This is called thhe “resonancce case” (Wigggins [29]) duue to the vaanishing of q 2 .
Fig. 4 (bb) The perturb bed system annd manifolds in
p1 , q1 , p 2 phaase space.
H Having exaamined the dynamical motions for f the fourr-dimensionnal system in the unpertturbed casee, we now study the eeffects of sm mall perturbation whicch arise fro om the base eexcitation ass well as thee structural damping. The T normal hyperboliciity property y of the maniffold 0 im mplies that it persists aalong with its stable and unstablee manifoldss under sufficiiently smalll perturbations. But, in general,, the dynam mics withinn these man nifolds changge dramaticaally. Under small pertuurbation ( 0 ), the manifold 00 persists as a (Fig. 4 (b)) due too its normall hyperboliccity property y and which h is defined by 2 12 1 p1 , q1 , p 2 , q 2 p1 0 (), q1 q1* ( ), p 22 p 2 p 12 , q 2 T 1 (25a) 8
and
2 p1 , q1 , p 2 , q 2 p1 0 ( ), q1 q1* ( ), 14
p 22 p 2 p12
2 12 , q2 T 1 8
(25b)
The flow restricted to is governed by the following equations p 2 2 2 p 2
e12 k1 f 2 p 2 sin q 2 (26a) 4
q 2 2
e12 k1 f 4 2 p2
(26b)
cos q 2
where is locally invariant manifold with boundaries. There is a single equilibrium in this plane at the point e 2 k 22 f 2 p p 2 , q 2 12 12 , tan 1 2 2 2 32 2 2
(27)
which is a spiral sink if 2 0 and 2 0 . Dynamics in this plane evolve on a slow, 1 time scale.
3.4 N - pulse Melnikov function For dynamical system (11), a homoclinic orbit with N pulses consists of two pieces. The first piece of such a homoclinic orbit is close to a string of N consecutive heteroclinic
p1 , q1 , p 2 , q2 t , I , q 20
orbits
jq 2 ,
with j 0,1, , N 1 .
These
heteroclinic orbits are called pulses. The string begins with the unstable manifold of the equilibrium point
p
1
, q1 , p 2 , q 2 . And the equations of p2 and q 2 fix the parameters
I and q 20 in all N pulses of the string. The last heteroclinic orbit in the string runs
back in plane at the point
p 2 , q 2 Nq 2 . The second piece of a homoclinic orbit
with N pulses is close to the trajectory of system (26) that connects the landing point
p 2 , q 2 Nq 2 of the last pulse back to the equilibrium p 2 , q 2 , as shown in Fig. 5.
15
p2 q2
Fig. 5 A multi-pulse jumping orbit homoclinic to p .
Moreover, the homoclinic orbit with N pulses only exists if two further conditions are satisfied. First, its second piece, or rather the corresponding trajectory of system (11) that connects the point
p 2 , q 2 Nq 2 to
the equilibrium p 2 , q 2 , must be above
the line p 2 2 12 8 . Otherwise, the existence of such homoclinic orbits cease to be valid. Second, we must choose the parameters 1 , 2 and f so that the N - pulse Melnikov function M N I , q 2 , to be described next, vanishes along the string of heteroclinic orbits
p1 , q1 , p 2 , q 2 t , I , q 20 , …, p1 , q1 , p 2 , q 2 t , I , q 20
jq 2 .
The N - pulse Melnikov function is the sum of the ordinary Melnikov functions calculated along the heteroclinic pulses, that is M N I , q 2
N 1
M t , I , q 2 jq 2
j 0
N 1
DH j 0
N 2 1 8 I 2 12 2
0
, g dt
2I N 1 N 1 q 2 sin q2 e12 k 1 f sin q 2 2 2 2
(28)
H 0 H 0 H 0 H 0 2 p 2 p1 cos 2q1 , DH 0 , , , p1 q1 p 2 q 2
,
where H 0 p1 , q1 , p 2 , q 2
1 p p1 1 2
g p 21 p1 , g q 1
1
e12 k1 f 8 2 p 2 p1
cos q 2 , g q 2
g p 2 2 p 2 p1 1 p1 2
16
e12 k1 f 4 2 p 2 p1
cos q 2 ,
e12 k1 f 2 p 2 p1 sin q 2 4
(29)
Simple zeros of the N - pulse Melnikov function, namely, values of I and q 2 for which M N I , q 2 0 and D q M N I , q 2 0 , pick out the survivors under perturbation 2
form among these unperturbed homoclinic orbits. Conditions for the existence of simple zeros of the N - pulse Melnikov function are obtained as N 2 1 8 I 2 12 2
N 1 N 1 2I e12 k 1 f sin q 2 sin 0 2 2 2
N 1 N 1 2I e12 k 1 f cos q 2 sin 0 2 2 2
(30a)
(30b)
From Eq. (30a), it is easy to see that the conditions for the manifolds to intersect in terms of parameters 1 , 2 , f is given by d
2 2 I e12 k1 2 2 f N 8 I 1 1
(31)
The threshold value surface obtained from the analytical expressions is plotted in Fig. 6 and chaotic dynamics may be generated below the surface.
d
1
N
Fig. 6 The threshold value surface of d with respect to N and 1 .
4. Physical interpretation
In this section, we try to interpret the global solutions in terms of physical motions of such flexible multi-beam structure based on the analytical mode shapes shown in Fig. 2. The global dynamics in the physical space will be described for the observer who is standing at the moving reference frame X1O1I1. The first two modes of the structure which 17
are related to the action-angle variables can be expressed as 2
u1
k1 k 2
u2
1 2 p1 cos q1 q 2 2 2 2
2 2 p 2 p1 sin q 2 2 k1
(32a)
(32b)
The heteroclinic structure in the unperturbed system was shown existing in Fig. 3 whenever the system energy is above the critical value. In the absence of any perturbation, The dynamics can occur either on or off the invariant manifold. If the trajectory lies entirely on the manifold 0 , the motion of the structure only occurs in the second mode which means a localized nonlinear normal mode exists. The snapshots of corresponding modal motion for different time are presented in Fig. 7. In the resonant case, a plane of fixed points in the
p2 , q2
phase space leads to a periodic motion in this localized mode.
If the trajectories are outside of 0 , actually, the trajectories can be on the center-type fixed point, any periodic manifold, or the heteroclinic manifold. The center-type fixed point located at q1 0 and q1 2 in the
p1 , q1
phase space are two nonlinear
normal modes characterized by a coupled periodic motion. If the trajectory lies on any periodic manifold, then the actual motion of the structure would consist of amplitude and phase modulated oscillation which manifests as a coupled quasi-periodic motion of two modes , see Fig. 8. However, for a trajectory lying on the heteroclinic manifold, the physical motion would start as an amplitude modulated coupled modal motion of two modes, as time tends to infinity, shift towards the localized modal motion, see Fig. 9. 2
1.5
Y
1
0.5
0 0
0.5
1
1.5
X
Fig. 7 The snapshots of nonlinear modal motion for the flexible structure. 18
2
1.5
1.5
1
1 Y
Y
2
0.5
0.5
0
0
-0.5
0
0.5
1
-0.5
1.5
X
0
0.5
(a)
X
1
1.5
(b)
Fig. 8 The snapshots of nonlinear modal motion for the flexible structure. (a) q1 0 . (b) q1 2 .
2 1.5
Y
1 0.5 0 -0.5
0
0.5
1
1.5
X
Fig. 9 The snapshots of physical motion when a trajectory lying on the heteroclinic manifold.
Under small perturbations due to damping dissipation and base excitation, single or multi-pulse orbits might arise relying on system parameters. We try to interpret the dynamical behaviors associated with the trajectories coming out of hyperbolic sink p . In this case, 0 persists as an invariant subspace for the perturbed system. The initial conditions on lead the phase flow towards the attracting sink p . As time goes on, the amplitude starts decreasing slowly and eventually the motion becomes a constant amplitude oscillation governed by the second modal motion, see Fig. 7. In the 19
four-dimensional phase space, if the initial conditions are set close to W u p , and the transversal condition is satisfied, the dynamical response starts as a coupled modal motion whose amplitude and phase are modulated, but soon shifts towards to a localized modal motion, which means the motion of the flexible multi-beam structure is localized in the second modal motion for a long time. As the trajectory approaches W u p , it takes off again and repeats the similar motion when a single-pulse orbit exists. When a multi-pulse orbit exist, as shown in Fig. 5, one can replace the single-pulse process as multi-pulse orbits in the vicinity of N-chains of homoclinic orbits in the resonance plane. Physically, such multi-pulse jumping orbits mean the first modal motion with the subharmonic frequency are involved suddenly. While doing this, energy flow between the localized mode and the coupled mode in a chaotic fashion will be observed. 5. Conclusions
In this paper, we have employed a generalized Melnikov method to study the global behaviors of a class of flexible multi-beam structures resting on a vibrating base for the principal resonance of the second mode with one to two internal resonance. First, it is shown that the dynamical system has heteroclinic connections for certain values of the system parameters in the absence of any perturbations. Then, under small perturbations arising from damping dissipation and base excitation, complex and ordered dynamics may occur in the case of "resonance", and the system may exhibit chaotic dynamics in the sense of Smale horseshoes due to the formation of Šilnikov-type homoclinic orbits. The conditions on system parameters under which such autoparametric system exhibits multi-pulse chaotic motions are also explored. Finally, The physical interpretation of global solutions provides an intuitive understanding on the dynamical mechanism of energy flow between the localized mode and the coupled mode in a chaotic fashion. Acknowledgements
This research is supported by National Natural Science Foundation of China (NNSFC) through Grant Nos.11290150, 11290152, 11290154, 11322214 and 11672007, the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB).
20
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24
Highlights
Global dynamics of a class of flexible multi-beam structures under autoparametric resonance is explored.
A generalized Melnikov method demonstrates its powerful utility for gaining a comprehensive insight into the global bifurcations and multi-pulse chaotic dynamics of the nonlinear dynamical system.
How energy may flow between different modes is investigated by checking the existence of multi-pulse homoclinic orbits.
Localized and coupled nonlinear normal modes are observed in terms of the physical motion.
Chaotic pattern conversion between localized mode and coupled mode are revealed in such autoparametric system.