Dynamic bifurcation and sensitivity analysis of non-linear non-planar vibrations of geometrically imperfect cantilevered beams

ARTICLE IN PRESS International Journal of Non-Linear Mechanics 45 (2010) 121–139

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Dynamic bifurcation and sensitivity analysis of non-linear non-planar vibrations of geometrically imperfect cantilevered beams O. Aghababaei, H. Nahvi , S. Ziaei-Rad Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran

a r t i c l e in fo

abstract

Article history: Received 14 February 2009 Received in revised form 6 October 2009 Accepted 6 October 2009

The non-linear non-planar dynamic responses of a near-square cantilevered (a special case of inextensional beams) geometrically imperfect (i.e., slightly curved) and perfect beam under harmonic primary resonant base excitation with a one-to-one internal resonance is investigated. The sensitivity of limit-cycles predicted by the perfect beam model to small geometric imperfections is analyzed and the importance of taking into account the small geometric imperfections is investigated. This was carried out by assuming two different geometric imperfection shapes, fixing the corresponding frequency detuning parameters and continuation of sample limit-cycles versus the imperfection parameter. The branches of periodic responses for perfect and imperfect (i.e. small geometric imperfection) beams are determined and compared. It is shown that branches of periodic solutions associated with similar limitcycles of the imperfect and perfect beams have a frequency shift with respect to each other and may undergo different bifurcations which results in different dynamic responses. Furthermore, the imperfect beam model predicts more dynamic attractors than the perfect one. Also, it is shown that depending on the magnitude of geometric imperfection, some of the attractors predicted by the perfect beam model may collapse. Ignoring the small geometric imperfections and applying the perfect beam model is shown to contribute to erroneous results. & 2009 Published by Elsevier Ltd.

Keywords: Bifurcation Limit-cycle Chaos Sensitivity Geometric imperfection

1. Introduction The non-linear dynamic response of a long, slender beam has been the subject of many theoretical and experimental efforts due to the fact that engineering structures like helicopter rotor blades, spacecraft antennae, flexible satellites, airplane wings, gun barrels, robot arms, high-rise buildings, long-span bridges, and subsystems of more complex structures can be modeled as a beam-like slender member. Linear perfect (i.e. ignoring geometric imperfections) modeling of small-amplitude vibrating beams may contribute to results that match the experimental observations but when the amplitude of excitation is large, it does not predict the dynamic responses correctly. This is the consequence of ignoring several non-linearities such as inertia, curvature, midplane stretching, natural geometric imperfection and various beam effects like shear deformation, warping and rotary inertia. In the following survey, a brief summary of the most relevant works is presented. Crespo da Silva and Glynn [1] investigated the flexural– flexural–torsional dynamics of beams to primary resonances accounting for both geometric and inertia non-linearities. They found that the first and the second mode response curves are

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E-mail address: [email protected] (H. Nahvi). 0020-7462/$ - see front matter & 2009 Published by Elsevier Ltd. doi:10.1016/j.ijnonlinmec.2009.10.002

different and response curves for higher modes are approximately independent of non-linear curvature terms. Crespo da Silva [2] used the same equations including damping and investigated the whirling motions of base-excited cantilever beams. He found that some whirling motions are unstable; furthermore, neither planar nor non-planar stable steady state motions existed in some ranges of frequency detuning. Crespo da Silva [3] investigated the planar response of an extensional beam to a periodic excitation. He found that the effect of the non-linearity due to midplane stretching is dominant and that neglecting the non-linearities due to curvature and inertia does not introduce significant error in the results. Also, unlike the response of an inextensional beam, the single-mode response of an extensional beam is always hardening. Pai and Nayfeh [4] investigated the non-planar oscillations of compact (i.e. near-square) beams under lateral base excitations. They located Hopf bifurcations and found that the system can exhibit quasi-periodic or chaotic motions; furthermore, the low-frequency modes are dominated by geometric non-linearities while the high-frequency modes are dominated by inertia non-linearities. Shyu et al. [5] used the equation in [1] to investigate the stationary whirling responses of a cantilever beam with static deflection to a subharmonic resonance of order one-half and a superharmonic resonance of order two. Restuccio et al. [6] investigated the planar and non-planar motions of the clamped– clamped/sliding beam to a principle parametric resonance when the cross-section is nearly square using the equations in [1].

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Nomenclature m mass per unit length of the beam Dii, i = Z,z flexural stiffness constants v(x,t),w(x,t) beam neutral axis deflections along Y and Z axes v0(x), w0(x) beam neutral axis initial deflections along Y, Z axes di, i= v,w damping coefficients F base excitation amplitude

Depending on the excitation amplitude, the damping, and the ratio of the principal flexural rigidities, they uncovered various frequency-response diagrams and found periodically and chaotically modulated motions. They found that the transition from periodic motions to chaotically modulated motions is via the process of torus doubling and subsequent destruction of the torus. Arafat et al. [7] used the equations in [1] to investigate the planar and non-planar dynamics of a cantilever beam to a principal parametric resonance of one of the modes in the presence of a one-to-one internal resonance between two in-plane and out-ofplane modes. They investigated the possible bifurcations in the planar and non-planar responses when the excitation frequency is varied. They found that over some frequency interval, the nonplanar periodic motions can bifurcate to amplitude- and phasemodulated motions with the modulation being either periodic or chaotic. Avramov [8] investigated the non-linear oscillations of a simply supported beam subjected to a periodic force at a combination resonance. He used the center manifold method and discovered the Naimark–Sacker bifurcations leading to almost-periodic oscillations. Dwivedy and Kar [9] investigated the non-linear dynamics of a base-excited slender beam carrying a lumped mass subjected to simultaneous combination parametric resonance of sum and difference types along with 1:3:5 internal resonances. They observed interesting phenomena like blue sky catastrophe, jump down phenomena and simultaneous occurrence of periodic and chaotic orbits. Lacarbonara et al. [10] investigated non-linear interactions in a hinged–hinged uniform moderately curved beam with a torsional spring at one end. The beam mixed-mode response is shown to undergo several bifurcations, including Hopf and homoclinic bifurcations, along with the phenomenon of frequency island generation and mode localization. Zhang [11] conducted an analysis of the chaotic motion and its control for the non-linear non-planar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. Based on the averaged equations that he obtained, numerical simulation was utilized to discover the periodic and chaotic motions for the non-linear nonplanar oscillations of the cantilever beam. The numerical results indicate that the transverse excitation in the Z direction at the free end can control the chaotic motion to a period n motion or a static state for the non-linear non-planar oscillations of the cantilever beam. Luongo and Egidio [12] applied the multiple scales method to a one-dimensional continuous model of planar, inextensible and shear-undeformable straight beam to derive the equations governing the system asymptotic dynamic around a bifurcation point. They studied the post-critical behavior around the bifurcations. Paolone et al. [13] analyzed the stability of a cantilever elastic beam with rectangular cross-section under the action of a follower tangential force and a bending conservative couple at the free end. The linear stability of the trivial equilibrium is studied, revealing the existence of buckling, flutter and double-zero critical points. Zhang [14] investigated the multi-pulse global bifurcations and chaotic dynamics for the non-linear non-planar oscillations of a

O

excitation frequency

oni, i= z,ybeam nth natural frequency along Z and Y axes s excitation frequency detuning parameter d

Zzx XYZ

beam cross section out-of-squareness detuning parameter current principle axes coordinate system of beam cross-section inertial reference coordinate system

cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end by using an extended Melnikov method in the resonant case. The results of numerical simulation show that the Shilnikov-type multi-pulse chaotic motions can occur for the non-linear non-planar oscillations of the cantilever beam, which verifies the analytical prediction. Aghababaei et al. [15] derived the non-linear equations and boundary conditions of non-planar (two bending and one torsional) vibrations of inextensional isotropic geometrically imperfect beams (i.e. slightly curved and twisted beams) using the extended Hamilton’s principle. The order of magnitude of the natural geometric imperfection was assumed to be the same as the first order of vibrations amplitude. Although the natural imperfection is small, their study shows that in contrast to the case of straight beams (i.e. geometrically perfect beams), the vibration equations are linearly coupled and have linear and quadratic terms in addition to cubic terms. Also, in the case of near-square or near-circular beams, coupling terms between lateral and torsional vibrations exist. Furthermore, a problem of parametric excitation in the case of perfect beams changes to a problem of mixed parametric and external excitation in the case of imperfect beams. They have also investigated the validity of the proposed model using the existing experimental data. Aghababaei et al. [16] investigated the non-linear non-planar steady-state responses of a near-square cantilevered beam (a special case of inextensional beams) with geometric imperfection under harmonic base excitation using the equations in [15]. By applying the combination of the multiple scales method and the Galerkin procedure to two non-linear integro-differential equations derived in [15], two modulation non-linear coupled first-order differential equations were obtained for the case of a primary resonance with a one-to-one internal resonance. They showed that the modulation equations contain linear imperfection-induced terms in addition to cubic geometric and inertial terms. Variations of the steady-state response amplitude curves with different parameters were presented. Bifurcation analyses of fixed points show that the influence of geometric imperfection on the steady-state responses can be significant to a great extent although the imperfection is small. The phenomenon of frequency island generation was also observed. In this paper, first the limit-cycles of the perfect cantilever beam are determined. Then, by assuming two different geometric imperfection shapes, two sample determined limit-cycles are continued versus the imperfection parameter (this is done by assuming that the corresponding frequency detuning parameter is fixed) and the sensitivity of those limit-cycles to small geometric imperfections is investigated and the importance of taking into account the small geometric imperfections is discussed for the first time. Then, by invoking a special geometric imperfection shape, the limit-cycles of the imperfect beam are computed by incorporating the imperfect beam model. By continuation of the computed limit-cycles for the imperfect beam and those for the perfect beam, the branches of dynamic solutions for both beams are determined and compared. The effect of small geometric imperfections on the dynamic bifurcations is extensively studied.

ARTICLE IN PRESS O. Aghababaei et al. / International Journal of Non-Linear Mechanics 45 (2010) 121–139 0 2on pvs ¼  ðC4 þ C9 Þpvc  C1 pvs  C3 pwc

2. Problem description and equations The same notations as in [16] are used to analyze the dynamic solutions in non-linear vibrations of an imperfect and perfect, cantilevered, base-excited beam with near-square cross-section, as shown in Fig. 1. It should be noted that for a perfect beam, the geometric imperfection (small initial non-dimensional deflection functions of the beam neutral axis, v0 ðxÞ and w0 ðxÞ), as shown in Fig. 1, are zero (i.e. v0 ðxÞ ¼ w0 ðxÞ ¼ 0). In this figure, XYZ and Zzx denote the inertial and the cross-section principle coordinate systems, respectively. In this paper, the primary-resonant base excitation of the nth mode in the Z-direction (i.e. O ¼ onz ð1 þ e2 sÞ which in turn excites the nth mode in the Y-direction ðony Þ through a one-to-one internal resonance ony ¼ onz  on with zero external detuning) is analyzed (s is the frequency detuning parameter). There exists such an internal resonance for near-square beams [16]. Since the former and later modes are excited directly by the primary resonance and indirectly through the one-to-one internal resonance, respectively, they are the only non-decaying contributing modes in the response of the beam [16]. It is assumed that none of the other modes in the Y- and Z-directions are involved in any internal resonance with the 2nd modes in the Y- or Z-directions. Expanding the in-plane and out-of-plane non-dimensional deflection variables (i.e. v(x,t) and w(x,t), respectively) in power series involving a bookkeeping perturbation parameter e as wðx; t; eÞ ¼ ew1 ðx; T0 ; T2 Þ þ e3 w3 ðx; T0 ; T2 Þ þ    where w ¼ v; w and w1 ðx; T0 ; T2 Þ ¼ Xn ðxÞawn ðT2 Þcos½on T0 þ rwn ðT2 Þ, Xn ðxÞ is the nth normalized flexural mode shape (i.e. single mode approximation of the response, it is assumed that the 2nd in-plane and out-of-plane flexural modes are interacting under the primary resonance O ¼ on ð1 þ e2 sÞ) and Ti ¼ ei t ði ¼ 0; 2Þ, replacing the power series in the governing equations of motion in [16], performing a perturbation analysis and equating the secular terms to zero, the following modulation equations are obtained (for the sake of briefness, the modulation equations are directly adopted, for a detailed perturbation analysis, the reader is referred to Ref. [16]): 0 ¼ ðC4 þC9 Þpvs  C1 pvc þ C3 pws 2on pvc



C5 ½ðp2vc þ p2vs Þpvs



2p2vc pvs  þ ðC6

 C2 Þ½pvs ðp2wc þ p2ws Þþ 2C2 pvc pwc pws

123

ð1aÞ

Fig. 1. Geometrically imperfect cantilever beam under base excitation, XYZ: inertial coordinate system, Zzx: principle axes of the beam cross-section. Deflections are shown exaggeratedly.

 C5 ½ðp2vc þ p2vs Þpvc þ 2pvc p2vs   2C2 pvs pwc pws  ðC2 þC6 Þ½ðp2wc þ p2ws Þpvc 

ð1bÞ

0 2on pwc ¼ ðC8 þC9 Þpws  C1 pwc þ C7 pvs

 C5 ½ðp2wc þp2ws Þpws  2p2wc pws  þ 2C2 pwc pvc pvs þ ðC6  C2 Þ½ðp2vc þ p2vs Þpws 

ð1cÞ

0 2on pws ¼  ðC8 þ C9 Þpwc  C1 pws  C7 pvc

 C5 ½ðp2wc þp2ws Þpwc þ2pwc p2ws   2C2 pws pvc pvs  ðC2 þC6 Þ½ðp2vc þ p2vs Þpwc   a21 f o2n

ð1dÞ

where pic ¼ ain cos gin , pis ¼ ain sin gin and gin ¼ on sT2  rin , i ¼ v; w. In the above equations, ð0 Þ denotes derivative with respect to the time scale T2 , f denotes the scaled non-dimensional amplitude R1 of base excitation (i.e. e3 f ¼ F, see Fig. 1), a21 ¼ 0 Xn dx and C1 to C9 are parametrically defined in the Appendix B of [16]. The numerical values of Ci ; i ¼ 1; 2; . . . ; 9 are given in the Appendix A. It should be noted that the ratio Dzz =DZZ is ordered to 1 þ e2 d for near square beams where d denotes the out-of-squareness of beam cross-section and appears in the constant C4 . The damping factors in both directions are assumed to be equal dw ¼ dv ¼ d and ordered to e2 m which appears in the constant C1 . The effect of geometric imperfection appears in the constants C3 ; C4 ; C7 ; and C8 . In Section 3, the Eqs. (1a)–(1d) are applied to determine the limit-cycles in non-linear non-planar (bending–bending) vibrations of an imperfect base-excited cantilever beam with the primary resonant excitation of the second flexural mode in the Z-direction (i.e. o2z ) which in turn excites the second flexural mode in the Y-direction (i.e. o2y ) through a one-to-one internal resonance o2y ¼ o2z  o2 . For the sake of simplicity, aw , rw , and o are used instead of aw2 , rw2 , and o2 , w ¼ v; w, respectively. 3. Perfect beam limit-cycles Among the methods of constructing limit-cycles such as bruteforce approach, harmonic balance method and time-domain methods [17], it was decided to use the brute-force method. In this method, one chooses an initial condition, integrates the system of Eqs. (1a)–(1d) and ultimately converges to an attractor that may or may not be a limit-cycle. Since the frequency- and time-domain methods have their own restrictions to converge [17], the brute-force method was used to program for its convenience. The algorithm of finding limit-cycles is presented in the Appendix B. Applying the Eqs. (1a)–(1d), invoking v0 ðxÞ ¼ w0 ðxÞ ¼ 0 (i.e. the perfect beam), and using the brute-force method, it was possible to determine 12 distinct limit-cycles for the perfect beam as shown in Fig. 2. The first, second and third columns in Fig. 2 show the phase portraits of the determined limit-cycles in pvc pvs -, pwc pws -, and av aw - planes, respectively. The forth column shows the logarithmic-scale FFT of the first state variable (i.e. pvc ) versus the non-dimensional frequency. The FFT of the 2nd, 4th, 5th and 6th limit-cycles shown in Fig. 2 contains only odd harmonics indicating that the corresponding pvc pvs phase portraits are symmetric, in other words, the corresponding limit-cycles are symmetric. The other FFTs in Fig. 2 contain odd and even harmonics indicating asymmetric limit-cycles. Fig. 3 shows the time traces of the determined limit-cycles (the state variable pvc ) in Fig. 2 within a three-period non-dimensional time interval. However, for the 7th and 8th limit-cycles, the

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Fig. 2. The phase portraits and FFTs of the perfect beam model ðv0 ðxÞ ¼ w0 ðxÞ ¼ 0Þ predicted limit-cycles when m ¼ 0:04; d ¼ 0:002; a21 f ¼ 6  105 .

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Fig. 3. The time traces of the limit-cycles in Fig. 2.

results are presented over a one-period time interval to keep the visibility. Care should be taken that the picks in the 7th and 8th time traces shown in Fig. 3 have a very small difference with respect to each other which is not clear enough. It should be noted that one of the Floquet multipliers associated with the 7th and 8th limit-cycles is unity which ascertains the minimal periods shown in Fig. 3.

3.1. Sensitivity to small geometric imperfections In this section, the effect of small geometric imperfections on the dynamic response of the perfect beam is analyzed by assuming two different sample imperfection shapes v0 ðxÞ ¼ cðX1 ðxÞ þ X2 ðxÞÞ; w0 ðxÞ ¼ 0

ð2aÞ

v0 ðxÞ ¼ 0; w0 ðxÞ ¼ cðX1 ðxÞ þ X2 ðxÞÞ

ð2bÞ

Note that in Eqs. (2a) and (2b), X1 ðxÞ and X2 ðxÞ are the first and second mode shapes of natural undamped vibrations of the perfect beam, respectively. The imperfection shapes in Eqs. (2a) and (2b) resembles a slightly arched cantilever beam in XY and XZ planes, respectively. By a similar approach in Section 3., it was possible to determine different limit-cycles for the fixed frequency detunings indicated in Fig. 2 and within the imperfection amplitude (i.e. c) interval of (0–0.001) for the two imperfection shapes in Eqs. (2a) and (2b) that are not shown here. The branches of periodic solutions for the two imperfection cases in Eqs. (2a) and (2b) are computed by continuation of the 1st and 2nd determined

limit-cycles in Fig. 2 and the above mentioned limit-cycles using the AUTO [18] continuation software. Figs. 4 and 6 show the variation of limit-cycle non-dimensional period with respect to the imperfection amplitude c and the chaotic bands for the imperfect beams. In Figs. 4 and 6, the frequency detuning parameter s is assumed to be fixed and equals to the values corresponding to the limit-cycles shown in Fig. 2.

3.1.1. Sensitivity analysis of the 1st limit-cycle As Fig. 4 shows, the perfect beam model predicts a limit-cycle at s ¼  0:011249 and c ¼ 0. By introducing an imperfection of the shape defined in Eq. (2a) into the perfect beam, applying the imperfect beam model modulation Eqs. (1a)–(1d) and continuing the predicted limit-cycle versus the imperfection amplitude c from zero, one arrives at a stable branch of periodic solutions that is terminated by a supercritical Hopf-bifurcation at cHF ¼ 1:64678  104 (see Fig. 4a). The post-bifurcation state is attracted by stable steady state non-planar vibrations. At the imperfection amplitude c ¼ 7:0034  104 , the system state jumps to steady state planar vibrations (XZ) since the system poses a saddle-node bifurcation at the specified imperfection amplitude. Fig. 4b shows the branches of periodic solutions and chaotic bands for the imperfection shape defined in Eq. (2b). In this case, by continuing the perfect beam model predicted limit-cycle at s ¼  0:011249, a stable branch of periodic solutions is obtained which undergoes a period-doubling bifurcation. Continuation of the period-doubled stable branch (not shown) reveals the existence of another period-doubling bifurcation. The sequence

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Fig. 4. Branches of periodic solutions and the chaotic bands when s ¼  0:011249, m ¼ 0:04; d ¼ 0:002; a21 f ¼ 6  105 (see the 1st limit-cycle in Fig. 2) for two geometric imperfection shapes, (——) stable, (———) unstable, HF, Hopf-bifurcation; PD, Period-Doubling; PD S., Period-Doubling Sequence; CF, Cyclic-Fold; HB, Homoclinic Bifurcation (a) v0 ðxÞ ¼ cðX1 ðxÞ þ X2 ðxÞÞ; w0 ðxÞ ¼ 0, (b) w0 ðxÞ ¼ cðX1 ðxÞþ X2 ðxÞÞ; v0 ðxÞ ¼ 0.

of period-doubling bifurcations persists which culminates in chaos. Another branch of periodic solutions in Fig. 4b is obtained by forward and backward continuation of the limit-cycle shown in Fig. 5 for c ¼ 0:00061. When c ¼ 6:1935  104  cHB , the limitcycles become homoclinic to the saddle-focus ðpvc ; pvs ; pwc ; pws Þ ¼ ð0; 0; 0:0158123; 0:0117225Þ. This is confirmed by the fact that the period of limit-cycle approaches infinity as c reaches the cHB . As discussed in [16], the coefficient C3 in Eqs. (1a) and (1b) vanishes for the perfect beam and also for both the imperfection cases defined in Eqs. (2a) and (2b). Thus, care should be taken that the modulation Eqs. (1a)–(1d) poses the symmetry ðpvc ; pvs ; pwc ; pws Þ3ðpvc ; pvs ; pwc ; pws Þ. This indicates that for any asymmetric limit-cycle of the perfect and imperfect beams discussed in this paper, another limit-cycle may be obtained by reversing the signs of pvc and pvs state variables. Here, by saying the limit-cycles becoming homoclinic to saddle-focus it means that the two asymmetric limit-cycles (one of them obtained by the above-mentioned method) come close to each other as c approaches cHB from the left until they meet at the saddle-focus, become merged, and form a single symmetric limit-cycle after homoclinic bifurcation. The asymmetric branch of periodic solutions before homoclinic bifurcation changes to a symmetric branch after the homoclinic bifurcation. The symmetric branch undergoes two cyclic-fold bifurcations and then loses symmetry through a symmetrybraking bifurcation. A stable asymmetric branch emanates from the symmetry-braking bifurcation. Finally, a sequence of perioddoubling bifurcations occurs culminating in chaos. The chaotic motion transits to steady state planar (XZ plane) motion through a boundary crisis at about c ¼ 6:7  104 . It is interesting to note that for imperfection amplitudes greater than c ¼ 6:7  104 , the out-of-plane vibrations is suppressed. In other words, modal interaction between the primary resonantly excited mode (i.e. the second flexural mode) in XZ plane (see Fig. 1) and the second out-of-plane flexural mode in XY plane through an internal resonance is canceled. Fig. 5 shows dynamic responses for six sample imperfection amplitudes c and s ¼  0:011249. Any of the dynamic responses in Fig. 5 is possible to occur instead of the 1st limit-cycle shown in Fig. 2 when s ¼  0:011249 at the presence of geometric

imperfection of the shape in Eq. (2b). The presence of subharmonic of order 1=2 in the second FFT ðc ¼ 0:00039Þ in Fig. 5 is a characteristic of period-two limit-cycle, i.e. the period of the limitcycle is doubled with respect to the period before period-doubling bifurcation. The fifth FFT ðc ¼ 0:00062Þ in Fig. 5 contains only odd harmonics which certifies the symmetry of its corresponding limit-cycle. The broadband nature in the third and sixth FFTs is the characteristic of chaotic motions.

3.1.2. Sensitivity analysis of the 2nd limit-cycle Fig. 6 shows the branches of periodic solutions and chaotic bands that may appear instead of the 2nd limit-cycle in Fig. 2 for s ¼  0:01105 with values of imperfection amplitude up to c ¼ 0:001. Fig. 6a reveals the extent of sensitivity of the perfect beam model predicted limit-cycle to small geometric imperfections of the shape defined in Eq. (2a) since the period of limit-cycle tends to infinity as c-1:10147  104 . The imperfect beam model poses a saddle-focus at c ¼ 1:10147  104  cHB and the limit-cycles experience a homoclinic bifurcation when c ¼ cHB . The symmetric limit-cycle changes to an asymmetric one after homoclinic bifurcation. The period decreases for imperfection amplitudes larger than cHB and the branch of periodic solutions undergoes a number of cyclic-fold and period-doubling bifurcations. For values larger than cCF3 , the system response is chaotic. The chaotic band is followed by periodic motions that their period decreases through an infinite number of reverse period-doubling bifurcations until the system state is attracted by the last period-one branch of periodic solutions. The last branch is terminated by a super-critical Hopf-bifurcation at c ¼ 7:57904  104 . The postbifurcation state is attracted by steady state non-planar vibrations. At the imperfection amplitude c ¼ 9:9781  104 , the system state jumps to steady state planar vibrations (XZ) since the system poses a saddle-node bifurcation at the specified imperfection amplitude. As Fig. 6b shows, the period of limit-cycle decreases considerably by increasing the imperfection amplitude from zero. After passing through two cyclic-fold bifurcations, the stable branch of symmetric periodic solutions loses symmetry through a symmetry-braking bifurcation. A stable branch of asymmetric periodic

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Fig. 5. The phase portraits, time traces, and FFTs of the limit-cycles depicted from the branches in Fig. 4 for the imperfect beam model with w0 ðxÞ ¼ cðX1 ðxÞþ X2 ðxÞÞ; v0 ðxÞ ¼ 0 when s ¼  0:011249, m ¼ 0:04; d ¼ 0:002; a21 f ¼ 6  105 .

solutions emanates from the later bifurcation, undergoes four period-doubling bifurcations that results in stable period-doubled branches of periodic solutions, and finally ends at the symmetrybraking bifurcation SB2. The unstable part of a branch of symmetric periodic solutions that passes through the SB2 is not clear in Fig. 6b. The stable part (to the right of SB2) undergoes two cyclic-fold bifurcations and loses its symmetry through SB3. A stable branch of asymmetric periodic solutions emerges from SB3 and goes through PD5, PD6, and CF5. An infinite number of period-doubling bifurcations occur after and before PD5 and PD6, respectively, culminating in chaos.

Care should be taken that the previous branches of periodic solutions are located within c ¼ 0 and c ¼ 2:05089  104 . As Fig. 6b shows, for certain ranges of c, more than one periodic solution exists. For the values of imperfection amplitude between cCF5 and cCF6 the dynamic response is chaotic. After the two remaining branches in Fig. 6b, chaotic motion is predicted. As shown in Fig. 4b, the chaotic motion transits to steady state planar motion through a boundary crisis at about c ¼ 2:60791  104 . Comparing Figs. 4 and 6b, one finds that transition from nonplanar chaotic motions to planar steady state motions occurs at different values of imperfection amplitudes. This indicates that

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Fig. 6. Branches of periodic solutions and the chaotic bands when s ¼  0:01105, m ¼ 0:04; d ¼ 0:002; a21 f ¼ 6  105 (see the 2nd limit-cycle in Fig. 2) for two geometric imperfection shapes, (——) stable, (———) unstable, HF, Hopf-bifurcation; PD, Period-Doubling; PD S., Period-Doubling Sequence; CF, Cyclic-Fold; HB, Homoclinic Bifurcation; BC, Boundary Crisis; SB, Symmetry-Braking (a) v0 ðxÞ ¼ cðX1 ðxÞ þ X2 ðxÞÞ; w0 ðxÞ ¼ 0, (b) w0 ðxÞ ¼ cðX1 ðxÞþ X2 ðxÞÞ; v0 ðxÞ ¼ 0.

the frequency detuning parameter may play an important role on the amount of sensitivity to geometric imperfections. Actually, one finds the importance of taking into account the geometric imperfection by considering the predicted possible dynamic responses of the imperfect beam model in Figs. 4 and 6 (and those obtained by continuation of the ten remaining limitcycles which are not presented in this paper for the sake of brevity) and comparing them with the perfect beam model predicted limit-cycles in Fig. 2. Figs. 4 and 6 show that presence of small geometric imperfections in a near-square cantilevered beam may lead to unexpected and completely different dynamic responses with respect to those predicted by the perfect beam model.

to the right Hopf-bifurcation HF2 in Fig. 12 while a few were found to be located between HF1 and SN3. Also, there is no stable steady-state solution within the sSN3  sHF2 (see Fig. 12) frequency detuning range. On the other hand, although the frequency response diagram for the imperfect beam is similar and very close to that for the perfect beam (see Fig. 12), with respect to the perfect beam, a larger number of limit-cycles for the imperfect beam were found to be located within the sHF1  sSN3 and sSN3  sHF2 frequency detuning ranges. Once again, the number of limit-cycles to the right of SN3 is greater than those located to its left. This may be due to the fact that there is no stable steady state solution within the frequency range sSN3  sHF2 in Fig. 12.

4. Imperfect beam limit-cycles

4.1. Branches of periodic solutions

With the effect of small geometric imperfections proved to be important on the dynamic response of a cantilevered near square beam, the imperfection shape v0 ðxÞ ¼ cðX1 ðxÞ þ X2 ðxÞÞ; w0 ðxÞ ¼ 0 with the imperfection amplitude c ¼ 0:0008 was introduced in the perfect beam and by a similar approach in Section 3 (i.e. the brute force method), 26th limit-cycles (more than two times the number of limit-cycles for the perfect beam in Fig. 2) were found to exist as shown in Figs. 7 and 8. In addition to twelve recognized limit-cycles that are similar in shape to those determined for the perfect beam (i.e. the 1st, 8–13th, 19th and 20th, and 24–26th limit-cycles in Figs. 7 and 8), 14 new limit-cycles are determined (2nd–7th limit-cycles in Fig. 7 and 14–18th, 21st–23rd limit-cycles in Fig. 8). The numbering of limit-cycles in Figs. 7 and 8 is continuous. Among the new limit-cycles, the 17th and 18th one in Fig. 8 are symmetric. This is certified by the fact that only odd harmonics are present in the associated FFTs. Figs. 9–11 show the time traces of the determined limit-cycles (the state variable pvc ) in Figs. 7 and 8 within a three-period time interval. It is interesting to note that most of the perfect beam limitcycles computed in Section 3. were found to be located within the frequency detuning range from the saddle-node bifurcation SN3

In this Section the branches of dynamic solutions for the imperfect (the imperfection shape v3 ðxÞ ¼ cðX1 ðxÞ þ X2 ðxÞÞ; w3 ðxÞ ¼ 0 and c ¼ 0:0008) and perfect beams are computed by continuation of the limit-cycles in Fig. 2 and Figs. 7 and 8 versus the frequency detuning parameter s using the AUTO [18] continuation software. This is done to compare the dynamic responses and dynamic bifurcations along the branch of periodic solutions for the perfect and imperfect beams. Figs. 13–17 and 19–22 represent the branches of periodic solutions and the chaotic bands for the imperfect and perfect beams. Parts (a) and (b) in Fig. 13 show the first branch of similar limitcycles for the imperfect and perfect beams, respectively (the first one in Figs. 7 and 2). Comparing parts (a) and (b) in Fig. 13, one finds that for the frequency detuning range less than sHF in part (a) and more than sHF in part (b), the imperfect and perfect beam models predict completely different dynamic responses, i.e. while the response is predicted to be steady state non-planar or planar (XZ) motions by the imperfect beam model, the non-planar chaotic or periodic motions shown in Fig. 13b and Figs. 14 and 15 are predicted by the perfect beam model to occur. For the frequency detuning range more than sHF in part (a) and less than sBC in part (b), the response is predicted to be periodic and chaotic

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Fig. 7. The phase portraits and FFTs of the imperfect beam model ðv0 ðxÞ ¼ 0:0008ðX1 ðxÞ þ X2 ðxÞÞ; w0 ðxÞ ¼ 0Þ predicted limit-cycles when m ¼ 0:04; d ¼ 0:002; a21 f ¼ 6  105 .

by the imperfect and perfect beam models, respectively. As Figs. 13, 16a and b represent, in the frequency range more than sBC in Fig. 13b and less than sBC in Fig. 16a, the perfect beam model predicts planar steady state motions while the imperfect beam model predicts periodic (branch A in Fig. 13a, new branches obtained by continuation of the 2nd–7th limit-cycles of the imperfect beam that are not shown, and branch B in Fig. 16a) and

chaotic non-planar (i.e. both directly and indirectly excited flexural modes interact) motions. Since the 2nd and 8th limit-cycles of the perfect and imperfect beams, respectively, are similar (see Figs. 2 and 7), the associated branches (branch B in Figs. 14 and 15 and branch B in Fig. 16a) obtained by continuation of them versus s may be comparable. As Figs. 13b, 14, 15, and 16a indicate, the frequency detuning range for

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Fig. 8. The phase portraits and FFTs of the imperfect beam model ðv0 ðxÞ ¼ 0:0008ðX1 ðxÞþ X2 ðxÞÞ; w0 ðxÞ ¼ 0Þ predicted limit-cycles when m ¼ 0:04; d ¼ 0:002; a21 f ¼ 6  105 .

the branch B of the perfect and imperfect beams are ð0:011102; 0:011020Þ and ð0:0108678; 0:01080153Þ, respectively. Furthermore, the associated branch B of the perfect and imperfect beams undergoes different bifurcations. Let us trace the branch B of the perfect beam from the stable extreme left side in Fig. 14a, i.e. a point between CF1 and PD1. By

increasing the frequency detuning, a period-doubling bifurcation occurs. The sequence of period-doublings continues culminating in chaos. By further increasing s, the chaotic motions change to periodic ones and a sequence of reverse period-doublings takes place resulting in settling back to branch B between PD2 and CF3. A jump from CF3 (sCF3 ¼  0:0110802) to the stable period-four

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Fig. 9. The time traces of the limit-cycles in Fig. 7.

Fig. 10. The time traces of the limit-cycles in Figs. 7 and 8.

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sub-branch of branch B (between PD3 and PD4) in Fig. 14b occurs. Then, by further increasing s, a reverse period-doubling followed by a period-doubling at PD5 and two reverse period-doublings at PD6 and PD2 occurs, again resulting in settling back to branch B between PD2 and CF1. By further increasing s, a jump to the stable upper portion of branch B occurs (from CF1). Then, the period of limit-cycles increases and approaches infinity as shown in Fig. 15. This is the evidence of a homoclinic bifurcation at sHB ¼  0:0110557. By tracing the branch B of the imperfect beam in Fig. 16a from its extreme stable left side (between CF and PD1), one finds another scenario. By increasing the frequency detuning s, a period-doubling and a reverse period-doubling at PD1 and PD2, respectively, occurs. By further increasing s, the period of the limit-cycles approaches infinity due to a homoclinic bifurcation at sHB ¼  0:0108168. The post-homoclinic bifurcation state is similar to that of branch B for the perfect beam in Fig. 15. For the sake of brevity, the branches of dynamic solutions obtained by continuation of the 2nd–7th limit-cycles of the imperfect beam versus the frequency detuning parameter (the

Fig. 11. The time traces of the limit-cycles in Fig. 8.

limit-cycles exclusively predicted by the imperfect beam model) are not shown. One may find that these branches are located between the branch A in Fig. 13a and the branch B in Fig. 16a. Branches C obtained by continuation of the 9th (imperfect) and 3rd (perfect) limit-cycles in Figs. 7 and 2 (similar limit-cycles) are presented in parts (b) and (c) of Fig. 16, respectively. Comparing the branches C in Fig. 16b and c over the indicated frequency detuning ranges reveals that the imperfect and perfect beam models express different dynamics at the same frequency detuning, i.e. while the perfect beam model predicts periodic motions within the frequency detuning range sCF4 o s o sCF , the imperfect beam model predicts chaotic motions. Furthermore, the imperfect beam model demonstrates a period-doubled subbranch over sPD1 o s o sPD2 in Fig. 16b while period-four, period-eight, and ultimately; chaotic motions are predicted by the perfect beam model over the same frequency detuning range in Fig. 16c. Another interesting difference is the different bifurcations along the branches. Suppose an experiment is conducted from the extreme stable right side of the branches C in Fig. 16b and c (the left of CF4 and CF). By decreasing the frequency detuning, two different scenarios are predicted by the two beam models. The imperfect beam model predicts: decreasing the limit-cycle period—jump at CF2 to the upper stable portion of the branch—increasing the limit-cycle period—period-doubling at PD2—reverse period-doubling at PD1—increasing limit-cycle period and finally, transition to chaos through a cyclic-fold bifurcation in Fig. 16b. The perfect beam model predicts: increasing the limit-cycle period—period-doubling at PD3— period-doubling at PD2—reverse trans-critical bifurcation at TC2—trans-critical bifurcation at TC1 and finally, a sequence of period-doublings routes to chaos from PD1 in Fig. 16c. Branches D obtained by continuation of the 10th (imperfect) and 4th (perfect) limit-cycles in Figs. 7 and 2 (similar limit-cycles) are presented in parts (a) and (b) of Fig. 17, respectively. Besides the frequency detuning shift of the branches with respect to each other, there exist some inconsistencies in the predictions of the two beam models (imperfect and perfect) that are demonstrated in Fig. 18. Fig. 18 shows the history of an experiment that may be conducted along the branches. The first and fourth columns show the phase portrait of the limit-cycles in the pvc pvs -plane for the

Fig. 12. Frequency-response amplitude curves (steady state solutions of Eqs. (1a)–(1d)) for a near-square beam when m ¼ 0:04; d ¼ 0:002; a21 f ¼ 6  105 , v0 ðxÞ ¼ w0 ðxÞ ¼ 0; av ; aw ¼ response amplitudes along Y and Z directions; (——) stable, (———) unstable, SN =Saddle-Node, PF= Pitchfork, HF =Hopf bifurcation [16].

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Fig. 13. Branches of periodic solutions and the chaotic bands when m ¼ 0:04; d ¼ 0:002; a21 f ¼ 6  105 obtained by continuation of the similar limit-cycles of the imperfect (1 in Fig. 7) and perfect beams (1 in Fig. 2), (——) stable, (———) unstable, HF, Hopf-bifurcation; PD, Period-Doubling; PD S., Period-Doubling Sequence; BC, Boundary Crisis, (a) v0 ðxÞ ¼ 0:0008ðX1 ðxÞ þ X2 ðxÞÞ; w0 ðxÞ ¼ 0, (b) v0 ðxÞ ¼ w0 ðxÞ ¼ 0.

Fig. 14. Branches of periodic solutions and the chaotic bands when m ¼ 0:04; d ¼ 0:002; a21 f ¼ 6  105 obtained by continuation of the 2nd limit-cycle of the perfect beam ðv0 ðxÞ ¼ w0 ðxÞ ¼ 0Þ (2 in Fig. 2), (——) stable, (———) unstable, PD, Period-Doubling; PD S., Period-Doubling Sequence; CF, Cyclic-Fold.

Fig. 15. Branches of periodic solutions and the chaotic bands when m ¼ 0:04; d ¼ 0:002; a21 f ¼ 6  105 obtained by continuation of the 2nd limit-cycle of the perfect beam ðv0 ðxÞ ¼ w0 ðxÞ ¼ 0Þ (2 in Fig. 2), (——) stable, (———) unstable, HB, Homoclinic Bifurcation; SB, Symmetry-Braking; PD, Period-Doubling; PD S., Period-Doubling Sequence.

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Fig. 16. Branches of periodic solutions and the chaotic bands when m ¼ 0:04; d ¼ 0:002; a21 f ¼ 6  105 obtained by continuation of the limit-cycles of the imperfect (8 and 9 in Fig. 7) and perfect (3 in Fig. 2) beams, (——) stable, (———) unstable, HB, Homoclinic Bifurcation; SB, Symmetry-Braking; PD, Period-Doubling; PD S., Period-Doubling Sequence; BC, Boundary Crisis; TC, Trans-critical; (a), (b) v0 ðxÞ ¼ 0:0008ðX1 ðxÞ þ X2 ðxÞÞ; w0 ðxÞ ¼ 0, (c) v0 ðxÞ ¼ w0 ðxÞ ¼ 0, branches C in parts (b) and (c) should be compared.

Fig. 17. Branches of periodic solutions and the chaotic bands when m ¼ 0:04; d ¼ 0:002; a21 f ¼ 6  105 obtained by continuation of the similar limit-cycles of the imperfect (10 in Fig. 7) and perfect beams (4 in Fig. 2), (——) stable, (———) unstable, PD, Period-Doubling; PD S., Period-Doubling Sequence; CF, Cyclic-Fold; SB, Symmetry-Braking; (a) v0 ðxÞ ¼ 0:0008ðX1 ðxÞþ X2 ðxÞÞ; w0 ðxÞ ¼ 0, (b) v0 ðxÞ ¼ w0 ðxÞ ¼ 0.

imperfect and perfect beams. The corresponding FFTs of the pvc state variable are presented in the second and third columns. Suppose the system state is attracted by a limit-cycle located to the right of CF1 in Fig. 17 (see the first row in Fig. 18). By increasing the frequency detuning, the imperfect and perfect beam models predict a super-critical and sub-critical symmetry-

braking bifurcation to occur, respectively. The post-bifurcation state is predicted to be stable asymmetric limit-cycles by both the beam models. The asymmetry is certified by the FFTs in the second row in Fig. 18. By further increasing s, a period-doubling and a reverse period-doubling is predicted by the perfect beam model while a large number of period-doublings (no chaotic

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Fig. 18. The phase portraits and FFTs of the limit-cycles depicted from the branches D in Fig. 17 when m ¼ 0:04; d ¼ 0:002; a21 f ¼ 6  105 ; Left two columns, the imperfect beam v0 ðxÞ ¼ 0:0008ðX1 ðxÞþ X2 ðxÞÞ; w0 ðxÞ ¼ 0, Right two columns, the perfect beam v0 ðxÞ ¼ w0 ðxÞ ¼ 0, Pn, Period-n limit-cycle.

motion was determined) and a large number of reverse perioddoublings over a limited frequency detuning range is predicted by the imperfect beam model. The appearance of sub-harmonic of order 1=2 in the FFTs of the third row is the evidence of periodtwo limit-cycles. Generally, the appearance of sub-harmonics of order 1=k; k ¼ 2; 4; 8; . . . in a FFT is the characteristic of period-k periodic motion [17]. The fourth FFT in Fig. 18 (second column) corresponds to a limit-cycle of period 3922.4. Care should be taken not to interpret this FFT as the power spectrum of a chaotic motion since it contains picks at equally spaced frequencies. As Fig. 17 indicates, by further increasing the frequency detuning, the imperfect and perfect beam models predict a jump from CF3 to a period-two limit-cycle and a period-doubling bifurcation at PD3, respectively (see the seventh row of the first two columns and the fifth row of the second two columns in Fig. 18). The remaining scenario is predicted to be the same by both the beam models, i.e. a period-doubling route to chaos, then

a sequence of reverse period-doublings and finally, a reverse symmetry-braking bifurcation. The broadband character in the eighth and seventh FFTs in the second and third columns certifies the chaotic response. As a significant characteristic of chaotic motions FFT, the preceding FFTs do not contain picks at certain frequencies [17]. The presence of only odd harmonics in the last row of Fig. 18 confirms the symmetry of the corresponding limitcycles. The minimum inconsistency between the imperfect and perfect beam branches of dynamic solutions has been found to exist in the branches obtained by the continuation of the eleventh and 5th limit-cycles in Figs. 7 and 2 (i.e. similar limit-cycles) which are not presented here (branches E). Part (a) in Fig. 19 shows the continued 12th and 13th limitcycles of the imperfect beam in Fig. 7 named as branches F and G, respectively. The branch obtained by continuation of the 6th limit-cycle of the perfect beam in Fig. 2 (similar to the 12th

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Fig. 19. Branches of periodic solutions and the chaotic bands when m ¼ 0:04; d ¼ 0:002; a21 f ¼ 6  105 obtained by continuation of the similar limit-cycles of the imperfect (12 and 13 (Branch G) in Fig. 7) and perfect beams (6 in Fig. 2), (——) stable, (———) unstable, PD, Period-Doubling; PD S., Period-Doubling Sequence; CF, CyclicFold; SB, Symmetry-Braking; (a) v0 ðxÞ ¼ 0:0008ðX1 ðxÞ þ X2 ðxÞÞ; w0 ðxÞ ¼ 0, (b) v0 ðxÞ ¼ w0 ðxÞ ¼ 0.

Fig. 20. Branches of periodic solutions and the chaotic bands when m ¼ 0:04; d ¼ 0:002; a21 f ¼ 6  105 obtained by continuation of (a), (b), (c) the 7th and (d) the 8th limit-cycles (7 and 8 in Fig. 2) of the perfect beam (v0 ðxÞ ¼ w0 ðxÞ ¼ 0), (——) stable, (———) unstable, PD, Period-Doubling; PD S., Period-Doubling Sequence; CF, Cyclic-Fold; SB, Symmetry-braking; TC, Trans-critical.

limit-cycle of the imperfect beam in Fig. 7) is shown in Part (b) of Fig. 19. The reader finds the branch obtained by continuation of the 7th limit-cycle of the perfect beam (similar to the 13th limitcycle of the imperfect beam) in parts (a), (b), and (c) of Fig. 20 called as branch G. The branches F in parts (a) and (b) of Fig. 19 undergo the same bifurcations except that the stable sub-branch emanating from SB3 terminates at SB4 in part (b) while it undergoes two period-doublings and terminates at SB4 in part (a). It is interesting to note that the imperfect beam predicts two different branches of periodic solutions over a certain frequency detuning range. In other words, as the imperfect beam model

predicts, the system state may be attracted by two different limitcycles at the same frequency depending on the initial conditions. Let us suppose that the system state is attracted by a symmetric stable limit-cycle on branch G (between SB1 and SB2). By decreasing the frequency detuning, the symmetric periodic response loses symmetry through a super-critical symmetry-braking bifurcation (it should be reminded that the stable symmetric branch continues as an unstable symmetric branch while an asymmetric stable branch emerges at a supercritical symmetry-braking bifurcation [17]). Then by further decreasing the frequency detuning, a sequence of period-doubling

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Fig. 21. Branches of periodic solutions and the chaotic bands when m ¼ 0:04; d ¼ 0:002; a21 f ¼ 6  105 obtained by continuation of (a) the 20th, (b) the 21st, (c) the 22nd, and (d) the 23rd limit-cycles of the imperfect beam ðv0 ðxÞ ¼ 0:0008ðX1 ðxÞ þ X2 ðxÞÞ; w0 ðxÞ ¼ 0Þ in Fig. 8 and (e) the 9th limit-cycle of the perfect beam v0 ðxÞ ¼ w0 ðxÞ ¼ 0 in Fig. 2, (——) stable, (———) unstable, PD, Period-Doubling; PD S., Period-Doubling Sequence; CF, Cyclic-Fold.

Fig. 22. Branches of periodic solutions and the chaotic bands when m ¼ 0:04; d ¼ 0:002; a21 f ¼ 6  105 obtained by continuation of (a) the 24th (branch J), the 25th (branch K) and the 26th (branch L) limit-cycles of the imperfect beam ðv0 ðxÞ ¼ 0:0008ðX1 ðxÞþ X2 ðxÞÞ; w0 ðxÞ ¼ 0Þ in Fig. 8 and (b) the 10th (branch J), the 11th (branch K) and the 12th (branch L) limit-cycles of the perfect beam v0 ðxÞ ¼ w0 ðxÞ ¼ 0 in Fig. 2, (——) stable, (———) unstable, PD, Period-Doubling; PD S., Period-Doubling Sequence; CF, Cyclic-Fold.

route to chaos occurs. The chaotic motion transits to a symmetric periodic limit-cycle through boundary crisis. The same scenario happens by increasing the frequency detuning when the system state is attracted by a symmetric stable limit-cycle on branch G. The branch G of the perfect beam shown in Figs. 20a, 20b, and 20c should be compared with the branch G of the imperfect beam in Fig. 19a. One finds the frequency shift of the two branches with respect to each other as the first discrepancy. Furthermore, the branch G in Fig. 20 goes through two close cyclic-folds CF3 and CF4 between SB1 and SB2. The stable asymmetric branch that emerges from SB2 undergoes different bifurcations with respect

to its counterpart in Fig. 19a, i.e. period-doubling PD2, perioddoubling PD3, trans-critical TC1, reverse trans-critical TC2, transcritical TC3, and finally, cyclic-fold CF5. The chaotic response at the right of Fig. 19a is followed by six branches of dynamic solutions obtained by continuation of the 14th–19th limit-cycles of the imperfect beam. These branches are not presented to keep concision. The branch H in Fig. 20 is obtained by continuation of the eighth (see Fig. 2) limit-cycle predicted by the perfect beam model. The asymmetric branch in Fig. 20d undergoes supercritical period-doubling bifurcation. The similar branch of periodic

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solutions obtained by continuation of the 19th limit-cycle of the imperfect beam undergoes a sub-critical period-doubling bifurcation (not shown). The post sub-critical period-doubling bifurcation state is attracted by chaos while there is a period-doubling route to chaos in the case of super-critical period-doubling bifurcation. The continuation of the 20th limit-cycle of the imperfect beam (see Fig. 8) and its similar limit-cycle predicted by the perfect beam (the 9th one in Fig. 2) yields the branches I in Figs. 21a and 21e which undergo the same bifurcations. As Fig. 21b–21d presents, the branches of periodic solutions I1, I2, and I3 resulted from the continuation of the 21st–23rd limit-cycles of the imperfect beam (exclusively predicted by the imperfect beam model, see Fig. 8) appear over a frequency range in which chaotic motion is predicted by the perfect beam model. The branches of periodic solutions calculated by continuation of the last three similar limit-cycles in Figs. 2 and 8 are presented in Fig. 22. Branches J, K, and L in parts (a) and (b) of Fig. 22 experience the same bifurcations but at slightly different frequency detunings. The branch L in Fig. 22 terminates at a Hopf-bifurcation. The post-bifurcation state may be attracted by stable steady state non-planar motions or periodic motions associated with the branches J and K. It was found that by increasing the imperfection amplitude c, the branches A and B of the imperfect beam move to right while the branches C, D, E, F, G, H, I, J, K, and L move to left and the frequency shift of them with respect to the perfect beam branches of periodic solutions become larger. As an example, the effect of increasing the imperfection amplitude on branch K was studied. Fig. 23 shows the loci of CF3 and CF2 of branch K in Fig. 22a. It is understood from Fig. 23 that by increasing the imperfection amplitude, as the cyclic-folds of branch K in Fig. 22a move to left, their distance diminishes until the stable branch K collapses at the imperfection amplitude c ¼ 0:00278. Fig. 24 shows the constant-period (T ¼ 1051:9) branch of periodic solutions obtained by continuation of the 2nd limit-cycle of the perfect beam (See Fig. 2). Fig. 24 indicates that as the beam becomes more imperfect, the branch B in Fig. 16a contracts more. This is understood by considering the frequency detuning

Fig. 24. The loci (location in c  s plane) of the 2nd limit-cycle of the perfect beam (see Fig. 2) for the imperfection shape v0 ðxÞ ¼ cðX1 ðxÞþ X2 ðxÞÞ; w0 ðxÞ ¼ 0, SB, symmetry-braking (——) stable, (———) unstable.

difference of the symmetry-braking bifurcation and the limitcycle of period 1051.9 on branch B in Fig. 16a and the fact that the constant-period stable branch of periodic solutions in Fig. 24 loses stability through a symmetry-braking bifurcation. In other words, by increasing the imperfection amplitude from zero, the homoclinic and symmetry-braking bifurcations in Fig. 16a become closer and at the same time they move to right. The exact study of branches of periodic solutions for different geometric imperfection shapes and amplitudes is cumbersome but what is important is that ignoring geometric imperfection and applying the perfect beam model may result in erroneous predictions for dynamic responses.

5. Conclusions

Fig. 23. The loci (location in c  s plane) of cyclic-folds CF3 and CF2 on branch K in Fig. 22b for the imperfection shape v0 ðxÞ ¼ cðX1 ðxÞ þ X2 ðxÞÞ; w0 ðxÞ ¼ 0.

Investigation of the effect of small geometric imperfections on the dynamic response of cantilevered near-square beams should be conducted based on two points of view. First is the extent to which the geometric imperfection may affect a predicted periodic response by the perfect beam model and the second, is the comparison of branches of periodic solutions obtained by continuation of similar limit-cycles predicted by the perfect and imperfect beam models. Depending on the frequency detuning parameter, the sensitivity of the predicted limit-cycles by the perfect beam model to small geometric imperfections may be to a great extent, thus, ignoring the small geometric imperfections and applying the perfect beam model may result in completely different and unexpected results. The comparison of the branches of dynamic solutions associated with the similar limit-cycles predicted by the perfect and imperfect beam models reveals that there exists a frequency shift of them with respect to each other, furthermore, in some cases; they undergo different bifurcations resulting in different dynamic responses. By increasing the imperfection amplitude, some of the branches of the imperfect beam move to right while some other move to left and the frequency shift of them with respect to the perfect beam branches of similar periodic solutions become larger. It was found that depending

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on the imperfection amplitude, some more branches of periodic solutions may emerge in addition to the branches associated with the perfect beam or some of the branches may disappear.

Appendix A

m =0.04, d =0.002 v0(x)=w0(x)=0 C1 =  0.8814, C2 = 31779.7261, C3 = 0, C4 =  0.9710 C5 =  25070.3746, C6 =  31779.7261, C7 = 0, C8 = 0, C9 =971.0450s v0(x)=0.0008(X1(x) +X2(x)), w0(x)=0 C1 =  0.8814, C2 = 31779.7261, C3 = 0, C4 =  0.9360 C5 =  25070.3746, C6 =  31779.7261, C7 = 0, C8 = 0.0589, C9 =971.0450s

Appendix B Generally, no specific method exists to choose the initial conditions for the brute-force approach to converge but we have run the brute-force method as the following: 1. Choosing a frequency detuning s very close and to the right of the frequency detuning associated with the HF1 (Hopf bifurcation) in Fig. 12. 2. Choosing the state vector ðpvc ; pvs ; pwc ; pws Þ that corresponds to HF1. This vector may be determined by incorporating the algorithm of finding Hopf bifurcations noted in Ref. [17]. 3. Integrating the system of modulation Eqs. (1a)–(1d) by the above chosen s and state vector. Performing the steps 1–3 results in the convergence of the modulation equations to the 1st limit-cycle of the perfect beam. To obtain the 2nd limit-cycle

139

[7] H.N. Arafat, A.H. Nayfeh, C.-M. Chin, Nonlinear nonplanar dynamics of parametrically excited cantilever beams, Nonlinear Dyn. 281 (15) (1998) 31. [8] K.V. Avramov, Non-linear beam oscillations excited by lateral force at combination resonance, J. Sound Vib. 257 (2) (2002) 337. [9] S.K. Dwivedy, R.C. Kar, Simultaneous combination and 1:3:5 internal resonances in a parametrically excited beam-mass system, Int. J. Non-Linear Mech. 38 (2003) 585. [10] W. Lacarbonar, H.N. Arafat, A.H. Nayfeh, Non-linear interactions in imperfect beams at veering, Int. J. Non-Linear Mech. 40 (2005) 987. [11] A. Luongo, A. Di Egidio, F. Divergence, Hopf and double-zero bifurcations of a non-linear planar beam, Comput. Struct. 84 (2006) 1596. [12] A. Paolone, M. Vasta, A. Luongo, Flexural-torsional bifurcations of a cantilever beam under potential and circulatory forces I: non-linear model and stability analysis, Int. J. Non-Linear Mech. 41 (2006) 586. [13] W. Zhang, Chaotic motion and its control for nonlinear nonplanar oscillations of a parametrically excited cantilever beam, Chaos Solutions Fractals 26 (3) (2005) 731. [14] W. Zhang, M.H. Yao, J.H. Zhanga, Using the extended Melnikov method to study the multi-pulse global bifurcations and chaos of a cantilever beam, J. Sound Vib. 319 (2009) 541. [15] O. Aghababaei, H. Nahvi, S. Ziaei-Rad, Non-linear non-planar vibrations of geometrically imperfect inextensional beams, part I: equations of motion and experimental validation, Int. J. Non-Linear Mech. 44 (2009) 146. [16] O. Aghababaei, H. Nahvi, S. Ziaei-Rad, Non-linear non-planar vibrations of geometrically imperfect inextensional beams. Part II—bifurcation analysis under base excitations, Int. J. Non-Linear Mech. 44 (2009) 160. [17] A.H. Nayfeh, B. Balachandran, Applied Nonlinear Dynamics, Wiley, New York, 1995. [18] E.J. Doedel et al., AUTO 2000: continuation and bifurcation analysis software for ordinary differential equations, California Institute of Technology, Pasadena, California, USA, 2001.

Manuel F.M. Barros received his Licenciatura degree in Electrical Engineering from Faculty of Sciences and Technology, University of Coimbra (UC), Portugal in 1987 and concluded his Ph.D. from Instituto Superior Te´cnico, Lisbon, in 2009, in Computer and Electrical Engineering. Since 2004, he is a professor at the Electrical Engineering Department from Instituto Polite´cnico de Tomar. Since 2003, he is, also, a researcher ~ at the Instituto de Telecomunicac- oes. From 1987 to 2004 he joined the Laboratorio de Gesta~ o de Energia at UC, working as a researcher in a technological R&D project involving electrical metering devices, remote communications and handheld devices. His research interests are in analog IC design automation and emerging evolutionary computation techniques including multi-objective optimization algorithms and learning strategies.

1. The previously chosen s is incremented by a small amount. 2. A state vector of the previously computed limit-cycle is chosen. 3. The modulation equations are integrated numerically. This may converge to the 2nd limit-cycle or not. If no convergence is obtained, then 4. A new state vector is chosen by randomly perturbing the previously chosen state vector and the integration is repeated. If the integration fails to converge after some trials (repeating step 4), the steps 1–3 above are repeated.

Jorge Guilherme received his M.Sc. and Ph.D. degrees from the Instituto Superior Tecnico, Lisbon, in 1994 and 2003, in the area of microelectronics. He joined the Electrical and Computer Engineering Department at the Instituto Politecnico Tomar in 1996. Since 2003, he is, also, a researcher at the Instituto de Telecomunicac~ oes. His research interests are in the areas of microelectronics, data conversion and power management. He has worked in the microelectronics area since 1990 and has published more than 50 papers. He has worked for 10 years in Arakit in the area of electronics and was the head of the technical department. He joined Chipidea in 1999 as an analogue designer working in the field of data converters and power management.

References [1] M.R.M. Crespo da Silva, C.C. Glynn, Nonlinear flexural–flexural–torsional dynamics of inextensional beams, II: forced motions, J. Struct. Mech. 6 (1978) 449. [2] M.R.M. Crespo da Silva, On the whirling of a base-excited cantilever beam, J. Acoust. Soc. Am. 67 (1980) 704. [3] M.R.M. Crespo da Silva, Non-linear flexural–flexural–torsional–extensional dynamics of beams, II: response analysis, Int. J. Solids Struct. 24 (1988) 1235. [4] P.F. Pai, A.H. Nayfeh, Non-linear non-planar oscillations of a cantilever beam under lateral base excitations, Int. J. Non-Linear Mech. 25 (1990) 455. [5] I.-M.K. Shyu, R.H. Plaut, D.T. Mook, Whirling of a forced cantilevered beam with static deflection. II: Superharmonic and subharmonic resonances, Nonlinear Dyn. 194 (4) (1993) 337. [6] J.M. Restuccio, C.M. Krousgrill, A.K. Bajaj, Nonlinear nonplanar dynamics of a parametrically excited inextensional elastic beam, Nonlinear Dyn. 196 (2) (1991) 263.

Nuno Horta received his Licenciado, M.Sc. and Ph.D. degrees in Electrical Engineering from Instituto Superior Te´cnico (IST), Technical University of Lisbon, Portugal, in 1989, 1992 and 1997, respectively. In March 1998, he joined the IST Electrical and Computer Engineering Department. Since 1998, he is, also, a ~ researcher at the Instituto de Telecomunicac- oes. His research interests are mainly in analog and mixedsignal IC design, analog IC design automation, symbolic analysis and soft computing. Dr. Horta has authored or co-authored more than 50 papers in international journals and conferences. He has also participated as researcher or coordinator in several National and European R&D projects.