Non-linear non-planar vibrations of geometrically imperfect inextensional beams. Part II—Bifurcation analysis under base excitations

International Journal of Non-Linear Mechanics 44 (2009) 161 -- 179

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Non-linear non-planar vibrations of geometrically imperfect inextensional beams. Part II—Bifurcation analysis under base excitations 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

I N F O

Article history: Received 1 January 2008 Received in revised form 28 October 2008 Accepted 28 October 2008 Keywords: Geometric imperfection Inextensional beam Fixed-point Bifurcation

A B S T R A C T

The non-linear non-planar steady-state responses of a near-square cantilevered beam (a special case of inextensional beams) with general imperfection under harmonic base excitation is investigated. By applying the combination of the multiple scales method and the Galerkin procedure to two non-linear integro-differential equations derived in part I, two modulation non-linear coupled first-order differential equations are obtained for the case of a primary resonance with a one-to-one internal resonance. 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 are 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 is also observed. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction The non-linear response of a long, slender beam is of great importance and the subject of many works since engineering structures like helicopter rotor blades, spacecraft antennae, flexible satellites, airplane wings, gun barrels, robot arms, high-rise buildings, longspan bridges, and subsystems of more complex structures can be modeled as a beam-like slender member. When the amplitude of excitation is small, linear modeling of beams may yield results coinciding with the experimental observations but when the amplitude of excitation is large, it does not predict non-linear phenomena such as jumps, saturation, modal interactions, sub- and super-harmonic resonances, chaos and other non-linear phenomena. Furthermore, it may yield misleading results such as planar response under planar excitation all of which are the consequence of ignoring several nonlinearities such as inertia, curvature, mid-plane stretching, natural geometric imperfection and various beam effects like shear deformation, warping and rotary inertia. In the following survey, a summary of the most relevant works are presented. Haight and King [1] experimentally investigated the response of near-circular and near-square cross-section beams by only including the inertia non-linearities. They found that planar motions can lose stability and change to non-planar steady-state whirling motions. Crespo da Silva and Glynn [2] investigated the

∗ Corresponding author. Tel.: +98 311 3915242; fax: +98 311 3912628. E-mail address: [email protected] (H. Nahvi). 0020-7462/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2008.10.008

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 different and response curves for higher modes are approximately independent of non-linear curvature terms. Whirling responses of near-square and near-circular cross-section beams to lateral base excitation, using the equations derived in [2] and ignoring the effect of damping, is investigated by Hyer [3]. He found that whirling motions exist near resonance excitation but could not locate unstable whirling motions. Crespo da Silva [4] 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 [5] 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 [6] 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. Crespo da Silva and Zaretzky [7] studied the non-linear responses of compact cantilever and

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clamped-pinned/sliding beams in the presence of a one-to-three internal resonance. Zaretzky and Crespo da Silva [8] experimentally investigated the non-linear modal coupling in the response of compact cantilever beams and obtained excellent agreement with the theoretical predictions of Ref. [7]. Crespo da Silva and Zaretzky [9] examined the flexural–torsional coupling in inextensional beams for a case of one-to-one internal resonance between an in-plane bending mode and a torsional mode and excited the in-plane mode. They found that within certain ranges of the excitation amplitude, the inplane bending component of the coupled response saturates so that, any further energy pumped into the system is transferred to the torsional motion via the internal resonance. Huang et al. [10] studied the dynamic stability of a cantilever beam attached to a translational/rotational base. They identified instability regions of the system for various combinations of the excitation frequencies and amplitudes of the oscillations. Luczko [11] used a geometrically accurate model to investigate the bifurcations and internal resonances in extensional space-curved rods. The model includes non-linear expressions of strains as functions of generalized coordinates, nonlinear material equations, as well as all non-linear terms in the equations of motion derived from the balance of momentum and angular momentum. He found a vast variety of internal resonance phenomena, which are caused by the coupling due to the inclusion of geometrical non-linearities. Avramov [12] 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 almostperiodic oscillations. Dwivedy and Kar [13] investigated the nonlinear dynamics of a base-excited slender beam carrying a lumped mass subjected to simultaneous combination parametric resonance of sum and difference type 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. Lacarbonar et al. [14] 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.` Luongo and Egidio [15] 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. [16] 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. In this paper, first by applying the combination of multiple scales method and Galerkin procedure [6,12,17–20] the modulation equations for a primary-resonant base excitation with a one-to-one internal resonance are obtained from the two coupled integro-differential non-linear equations of geometrically imperfect cantilever beams in [21]. Then, by using the numerical algorithms and the pseudo-arclength method [22], the variation of the steadystate solutions and their stability are shown in bifurcation diagrams for both the imperfect and perfect cases. The bifurcation diagrams are then compared and the influence of the small imperfection is discussed. 2. Problem formulation and equations The same notations as in [21] are used to analyze the non-linear vibrations of an imperfect, cantilevered, base-excited beam with

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

near-square cross-section, as shown in Fig. 1. In this figure, XYZ and  denote the inertial and the cross-section principle coordinate systems, respectively, D , D and D are the principle stiffnesses,  ≡ D /D and  ≡ D /D , t is time, u(x, t), v(x, t) and w(x, t) are the components of cross-section centroid displacement after deformation. The length of the beam L and the characteristic time  L2 m/D are used to non-dimensionalize the variables, where m is

mass per unit arc length of the imperfect beam. The general imperfection is assumed to be the sum of three functions, namely, v0 (x), w0 (x) and 0 (x). The first two are small initial deflection functions of the beam neutral axis and the third one is a small initial rotation angle function. The acceleration due to gravity is denoted by g. To simplify the problem we use the following assumptions: (i) the viscous damping coefficients along Y- and Z-axes of the inertial coordinate system are equal; (ii) the cantilevered end of the imperfect beam is excited harmonically along the Z-direction of the inertial coordinate system by F cos t, as shown in Fig. 1, where F and  are constant amplitude and frequency of excitation, respectively. Replacing w(x, t) with w∗ (x, t) + F cos t in Eqs. (23a) and (24a) in [21], where w∗ (x, t) denotes the displacement with respect to the base along the Z-direction and dropping the superscript ∗, the nondimensional equations of non-planar vibrations of inextensional imperfect beams can be written as v¨ + cv˙ +  viv = − [v (v v + w w ) ]  +

1  (v (−v0 2 + w0 2 )) −  (v0 (w0 w ) ) 2 

− (1 −  )(0 v − 0 w ) 2



  1 2 2         − v0 v + v v + 2v (v v ) + (v w w ) 0 0 2 0 

− [v (w0 w ) )] −

(1 −  )2



  w

  +(1 −  ) w

x

0 x 1

  1

 v w d d

v w d − w

 0

x

 w v d

O. Aghababaei et al. / International Journal of Non-Linear Mechanics 44 (2009) 161 -- 179

+

(1 −  )



  w

x

  1

0

163

  x  −(1 −  ) v (v0 w + w0 v ) d

 (−v0 w +  w0 v ) d d

0

   x −(1 −  ) w (v0 w + w0 v ) d

   x + v (−v0 w +  w0 v ) d

   x + w (v0 w −  w0 v ) d

−

1

0

(1 −  )



1

 (1 −  )    w0 +

x 

  + w0





−

0

x 0

1

w v d

v w d d

  − v0



  +(1− ) w0

x 1

+

v w d



−

1

−

1  v 2

x 1

 

j jt 2

0

 2

0

1

w v d 

x

 

0

1

 v w d d

  −(1 −  ) v0

x 1

 v w d

 (−v0 w +  w0 v ) d d

1

0

2

 

0

   x + w0 (v0 w −  w0 v ) d 



x

   x + v0 (−v0 w +  w0 v ) d

  x  + w0 (v0 w + w0 v ) d



x

v0



   x − v0 (v0 w + w0 v ) d

     x   w0 (−v0 w +  w0 v ) d d  0 1



1



v0



0







1 w 2 

2

(v + w ) d d



1 − w0 2

x



1



x

j2 jt2



1

 

j2 jt2

0



  0

2



2

(v + w ) d d  (2v0 v

+ 2w0 w ) d

 d

−

       1  x j2     (2v v + 2w w ) d d v 0 0 2 jt 2 0 1

−

      x j2    1   (2v v + 2w w ) d d w 0 0 2 jt2 0 1

−

       1  x j2     (2v v + 2w w ) d d v0 0 0 2 jt 2 0 1

−

      x j2   2 1 2 (v + w ) d d w0 2 jt2 0 1

       1  x j2 2 2 − v0 (v + w ) d d 2 jt 2 0 1

1 ¨ 0 2 + w0 2 ) + F 2 cos t + cF  sin t − w(v 2

  1 L3 1 2 2 ¨ 0 2 +w0 2 )− − v(v mg 1+ (v0 +w0 ) 2 D 2

(1a)

¨ + cw ˙ + wiv = −[w (w w + v v ) ] w  1 2 2 2 + −w0 w0 w + w (v0 + w0 ) + v0 w 2 2

2

 1 2 + −w0 w w − w w − w (w0 w ) 2 0 

− [w (v v0 ) ] −

(1 −  )2





v



x

1

(1 −  )



  1

0

  −(1 −  ) v +

x

  v 0

v w d d

v w d − v

x

  1





x 0

v = w = v = w = 0 at x = 0

(2a)

v = w = v = w = 0

(2b)

at x = 1

To analyze Eqs. (1a) and (1b), the method of multiple scales is applied [6,12,17–20]. For the following perturbation analyses, a small parameter is introduced. Also, let c = 2 , F = 3 f ,  = 1 + 2  (i.e. ∗ near-square beams), 0 (x) = 0 (x) (i.e. initial twist angle function of the undeformed beam [21]) v0 (x) = v∗0 (x), and w0 (x) = w∗0 (x). In what follows, the superscript ∗ is dropped for simplicity. In Eq. (1a), the constant gravitational term (i.e. the last term) produces a constant static deflection, v = vd (x, t) + vs (x) where vd and vs denote the dynamic and static responses, respectively. Again, the subscript d is omitted for the sake of simplicity. As the work of Pai and Nayfeh [6], we let (L3 /D )mg = 2 g0 , therefore, vs (x) = O( 2 ). Let us also assume an approximation to v(x, t) and w(x, t) as



− 2w0 w w −  w0 (v v )

And the boundary conditions up to the first order are

3. Analyses of base-excited resonant motions

+v0 (v0 w ) − w0 w −  w0 (v0 v ) +(1 −  )(0 w + 0 v )

(1b)



w v d

(−v0 w +  w0 v ) d d



v(x, t; ) = v1 (x, T0 , T2 ) + 3 v3 (x, T0 , T2 ) + · · ·

(3)

w(x, t; ) = w1 (x, T0 , T2 ) + 3 w3 (x, T0 , T2 ) + · · ·

(4)

In Eqs. (3) and (4), Tk = k t, k = 0, 2 are fast and slow time scales, respectively. Substituting Eqs. (3) and (4) in Eqs. (1a) and (1b) and equating coefficients of like powers of yields

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O. Aghababaei et al. / International Journal of Non-Linear Mechanics 44 (2009) 161 -- 179

Order :

+

1



  v0

0

x

  1

 (−v0 w1 + w0 v1 ) d d

(5a)

   x − v0 (v0 w1 + w0 v1 ) d

D20 w1 + wiv 1 =0

(5b)

   x + v0 (−v0 w1 + w0 v1 ) d

v1 = w1 = v1 = w1 = 0 at x = 0

(6a)

   v 1 = w1 = v1 = w1 = 0

(6b)

D20 v1

+ viv 1

=0

0

1

at x = 1

    x   1 2 2 D20 w1 − (v1 + w1 ) d d 2 1 0   x     1 w0 D20 − (2v0 v1 + 2w0 w1 ) d d 2 1 0     x   1 − (2v0 v1 + 2w0 w1 ) d d D20 w1 2 1 0     x   1 2 2 D20 w0 − (v1 + w1 ) d d 2 1 0

Order 3 : iv        D20 v3 + viv 3 = − 2D0 D2 v1 − D0 v1 − v1 − [v1 (v1 v1 + w1 w1 ) ]

 +

1  2 2 (v (−v0 + w0 )) − (v0 (w0 w1 ) ) 2 1



  1 2 2         − v0 v1 + v 0 v1 + 2v1 (v0 v1 ) + (v0 w1 w1 ) 2

−

v3 = w3 = v3 = w3 = 0 at x = 0

    x   − v1 (w0 w1 ) + w1 (v0 w1 − w0 v1 ) d

−

1





w0



x 0

  1

2

2

(−v0 w1 + w0 v1 ) d d

0

0

 = −(w1 v1 + v0 w1 + w0 v1 )x=1

1

      1  x 2  2 2 D0 v1 (v1 + w1 ) d d 2 1 0

−

      1  x 2  D0 v1 (2v0 v1 + 2w0 w1 ) d d 2 1 0

−

      1  x 2  v0 D0 (2v0 v1 + 2w0 w1 ) d d 2 1 0

−

      1  x 2  2 2 (v1 + w1 ) d d D0 v0 2 1 0

−

1 2 2 (v + w0 )D20 v1 2 0

(7a)

       D20 w3 + wiv 3 = − 2D0 D2 w1 − D0 w1 − [w1 (w1 w1 + v1 v1 ) ]  1 2 2 2 + −w0 w0 w1 + w (v + w0 ) + v0 w1 2 1 0  2     + v0 (v0 w1 ) − w0 w − w (v v ) 1 0 0 1

 1 2 + −w0 w1 w1 − w w − w1 (w0 w1 ) 2 0 1      − 2w0 w1 w − w (v v ) 1 0 1 1    x 

− w1 (v1 v0 ) + v1 (−v0 w1 + w0 v1 ) d

0

x



w1 v1 d

1

(8b)

At x = 1  1  1  = − (w1 v1 +v0 w1 +w0 v1 ) d−

   x + w0 (v0 w1 − w0 v1 ) d −

2

w3 = v0 − 12 w0 w1 + v0 v0 w1



   x + w0 (v0 w1 + w0 v1 ) d

  − v0

2

       1   w 3 = (v0 ) + v0 w1 + v0 v0 w1 − 2 w0 w1 ,

w1 v1 d

0

(8a)

v3 = −w0  − w0 v1 w1 − v0 w0 w1 − 12 v0 v1



x

(7b)

         v 3 = −(w0 ) − w0 v1 w1 − (v0 w0 ) w1 ,

1

  + w0

1 2 2 (v + w0 )D20 w1 + f 2n cos T0 2 0



0

1

  1

(v0 w1 −w0 v1 ) d d (8c)

where Dk () ≡ j()/ jTk . The boundary conditions (8a) and (8b) are derived from the general boundary conditions stated in [21]. The order of coupled terms with vs (x) and 0 (x) are higher than 3 and do not appear in Eqs. (7a) and (7b). The general solution of Eqs. (5a) and (5b) consists of two infinite sequences of natural modes corresponding to two infinite sequences of natural frequencies. Any mode that is not directly or indirectly excited will decay to zero with time due to the presence of damping [17]. Considering Eqs. (5a) and (5b), for near-square beams, the natural modes and the corresponding natural frequencies of vibrations along Y- and Z-directions of the inertial coordinate system are the same. In this paper, the primary-resonant excitation of the n th mode in the Z-direction ( nz ),  = nz (1 + 2 ), which in turn excites the n th mode in the Y-direction ( ny ) through a one-to-one internal resonance ny = nz with zero external detuning is analyzed. There exist such an internal resonance for near-square beams, i.e.  = 1 + 2 , where  is the out-of-squareness detuning parameter. Since the former and latter 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 solution of Eqs. (5a) and (5b). It is assumed that none of the other modes in the Y- and Z-directions is involved in any internal resonance with the n th mode in the Y- or Z-directions. Thus, the solution of (5a) and (5b) can be expressed as v1 = Xn (x)Avn (T2 ) exp(i n T0 ) + cc

(9a)

w1 = Xn (x)Awn (T2 ) exp(i n T0 ) + cc

(9b)

In Eqs. (9a) and (9b), cc stands for complex conjugate of the preceding terms, Xn (x) and n are the n th natural normal mode shape and the n th natural frequency of linear vibrations along Y- and Z-directions, respectively, which are shown in Appendix A.

O. Aghababaei et al. / International Journal of Non-Linear Mechanics 44 (2009) 161 -- 179

The Galerkin procedure is used to reduce Eqs. (7a) and (7b) into ordinary differential equations. Toward this end, let us define

vn = Avn (T2 ) exp(i n T0 ) + cc

(10a)

wn = Awn (T2 ) exp(i n T0 ) + cc

(10b)

vn =

 0

1

and wn =

Xn (x)v3 (x, T0 , T2 ) dx



1 0

Xn (x)w3 (x, T0 , T2 ) dx

− 2 n awn + [C1 − C2 a2vn sin 2(vn − wn )]awn − C7 avn sin(vn − wn ) + 21 f 2n sin(wn ) = 0

(14c)

[2 n wn + C8 + C5 a2wn + C6 a2vn + C2 a2vn cos 2(vn − wn )]awn + C7 avn cos(vn − wn ) + 21 f 2n cos(wn ) = 0

(14d)

where the coefficients Ci are defined in Appendix C and

vn = n T2 − vn and wn = n T2 − wn

Multiplying Eqs. (7a) and (7b) by Xn (x) and integrating both sides over the domain of x (i.e. 0–1), one obtains

Alternatively, by expressing Avn and Awn in Cartesian form

D20 vn + 2n vn = − 2D0 D2 vn − D0 vn − rn4 vn − 1 3vn

Avn = 12 (pvc − ipvs ) exp(i n T2 ) and

−1 vn 2wn +2 vn −3 wn −4 2vn +5 2wn

Awn = 12 (pwc − ipws ) exp(i n T2 )

+6 vn wn − 12 7 vn D20 2vn − 12 7 vn D20 2wn

The modulation equations are written in Cartesian form which is more suitable for stability analysis of fixed points

− 12 8 vn D20 vn − 12 9 vn D20 wn − 12 10 D20 vn

− C5 [(p2vc + p2vs )pvs − 2p2vc pvs ]

(11a)

+ (C6 − C2 )[pvs (p2wc + p2ws )] + 2C2 pvc pwc pws

D20 wn + 2n wn = − 2D0 D2 wn − D0 wn − 1 3wn − 1 wn 2vn

− C5 [(p2vc + p2vs )pvc + 2pvc p2vs ]

+17 vn wn − 12 7 wn D20 2vn − 12 7 wn D20 2wn

− 2C2 pvs pwc pws − (C2 + C6 )[(p2wc + p2ws )pvc ]

− 12 8 wn D20 vn − 12 9 wn D20 wn − 12 18 D20 vn −2 n pwc

− 12 19 D20 wn − 12 20 D20 2vn − 12 20 D20 2wn

Awn = 12 awn (T2 ) exp(iwn (T2 )) (13)

Substituting Eq. (13) into Eqs. (12a) and (12b) and separating real and imaginary parts yields − 2 n avn + [C1 + C2 a2wn sin 2(vn − wn )]avn (14a)

[2 n vn + C4 + C5 a2vn + C6 a2wn + C2 a2wn cos 2(vn − wn )]avn + C3 awn cos(vn − wn ) = 0

−2 n pws = − (C8 + C9 )pwc − C1 pws − C7 pvc − C5 [(p2wc + p2ws )pwc + 2pwc p2ws ] − 2C2 pws pvc pvs − (C2 + C6 )[(p2vc + p2vs )pwc ] − 21 f 2n

(16d)

where pic = ain cos in

and

pis = ain sin in , i = v, w

4. Bifurcation analyses of steady-state solutions

To obtain the phase-amplitude modulation equations, Avn and Awn are expressed in polar form as

+ C3 awn sin(vn − wn ) = 0

(16c)

(12b)

− 1 [A2vn A¯ wn + 2Avn A¯ vn Awn ] + 27 2n A¯ wn A2vn

and

+ 2C2 pwc pvc pvs + (C6 − C2 )[(p2vc + p2vs )pws ]

It is notable that the system of Eqs. (16a)–(16d) is not invariant under any transformation of (pvc , pvs , pwc , pws ). This can be compared by the number of invariant transformations for the perfect beam system of equations which are three, thus for any asymmetric solution of those system of equations found, three other solutions can be obtained [6]. In the next section, the bifurcations of fixed points of the phaseamplitude modulated equations are studied. Also, the influence of various parameters on the fixed points is investigated.

  dAwn 1 − i n Awn + 13 + 19 2n Awn dT2 2   1 + 14 + 18 2n Avn + [−31 + 27 2n ]A2wn A¯ wn 2

Avn = 12 avn (T2 ) exp(ivn (T2 ))

= (C8 + C9 )pws − C1 pwc + C7 pvs

(12a)

− 2i n

1 21 f 2n exp(i n T2 ) = 0 2

(16b)

− C5 [(p2wc + p2ws )pws − 2p2wc pws ]

(11b)

where the coefficients i are defined in Appendix B. Substituting Eqs. (10a) and (10b) into (11a) and (11b), extracting the secular terms and equating them to zero yields the modulation equations which govern Avn (T2 ) and Awn (T2 ) as   dAvn 1 − 2i n − i n Avn + −rn4 + 2 + 10 2n Avn dT2 2   1 + −3 + 11 2n Awn + [−31 + 27 2n ]A2vn A¯ vn 2 − 1 (A¯ vn A2wn + 2Avn A¯ wn Awn ) + 27 2n A¯ vn A2wn = 0

(16a)

−2 n pvs = − (C4 + C9 )pvc − C1 pvs − C3 pwc

+13 wn +14 vn +15 2wn +16 2vn

+ 21 f 2n cos T0

(15)

−2 n pvc = (C4 + C9 )pvs − C1 pvc + C3 pws

− 12 11 D20 wn − 12 12 D20 2vn − 12 12 D20 2wn

+

165

(14b)

Steady-state solutions of the system correspond to the fixed points (i.e. constant phase-amplitude solutions) of Eqs. (14a)–(14d), namely, the solutions in which avn = awn = vn = wn = 0

(17a)

Or equivalently the solutions in which pvc = pvs = pwc = pws = 0

(17b)

Thus, by substituting Eq. (17b) into Eqs. (16a)–(16d), they reduce to a set of algebraic equations that can be solved numerically. In this paper, a pseudo-arclength scheme [22] is used to trace the fixedpoint curves versus various parameters. Since Eqs. (16a)–(16d) are first-order autonomous differential equations, the eigenvalues of the

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O. Aghababaei et al. / International Journal of Non-Linear Mechanics 44 (2009) 161 -- 179

Fig. 2. Frequency–response amplitude curves of the first flexural modes (n = 1) for a near-square beam when  = 0.04,  = 0, c1 = c1 = 0.05, c2 = c2 = 0; av , aw =response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive, SN, saddle-node; PF, pitchfork, (a) 21 f = 0.001, (b) 21 f = 0.002.

Fig. 3. Frequency–response amplitude curves of the first flexural modes (n = 1) for a near-square beam when  = 0.04,  = 0, c1 = c1 = c2 = c2 = 0; av , aw =response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive, SN, saddle-node; PF, pitchfork, (a) 21 f = 0.001, (b) 21 f = 0.002.

Jacobian matrix of their right-hand sides at a particular fixed point can be used to determine its stability [22]. Unlike the case of perfect beam analyzed in [6], the steady-state responses of the imperfect beam may always be non-planar (i.e. whirling motion) which means that imperfection may activate the indirectly excited mode in the whole range of the detuning parameter . Investigating Eqs. (16a) and (16b), one finds that if the coefficient C3 vanishes, both planar and non-planar steady-state responses are possible (planar responses correspond to avn = 0, awn
0). On the other hand, if C3 does not vanish, planar steady-state response is impossible and just non-planar responses exist. Considering C3 as given in Appendix C, two cases of vanishing C3 , i.e. (v0 (x) = 0, w0 (x)
0) and (v0 (x)
0, w0 (x) = 0), are possible. Also, if v0 (x) = w0 (x) = 0, Eqs. (16a)–(16d) reduce to the equations reported in [6]. In what follows, the beam is assumed to have a near-square uniform cross-section with homogeneous and isotropic material and obey Hooke's law. Assuming Poisson's ratio  = 0.3 and using Ref. [21],  = 0.6489 for a near-square cross-section metallic beam.

Furthermore, it is assumed that the imperfection has the general form of v0 (x) = c1 X1 (x) + c2 X2 (x) and w0 (x) = c1 X1 (x) + c2 X2 (x) where c1 , c2 , c1 and c2 are arbitrary constants and X1 (x) and X2 (x) are first and second undamped shape functions of the linear flexural beam vibrations. In Sections 4.1 and 4.2, the primary-resonant excitation of the first and second flexural modes in the Z-direction with the first and second flexural modes in the Y-direction being autoparametrically excited through a one-to-one internal resonance are investigated, respectively. Variations of the response amplitudes with different parameters are presented and compared. 4.1. Case  = 1 (1 + 2 ) Parts (a) and (b) in Fig. 2 show the variation of steady-state response amplitudes with the detuning parameter for the case of a pure square cross-section beam (i.e.  = 0) with  = 0.04 and two different base excitation amplitudes 0.001 and 0.002, respectively. The shape of imperfection is assumed to be v0 (x) = w0 (x) = 0.05X1 .

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Fig. 4. Frequency–response amplitude curves of the first flexural modes (n = 1) for a near-square beam when  = 0.04,  = −0.01, 21 f = 0.002; av , aw =response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive; SN, saddle-node; PF, pitchfork: (a) c1 = c2 = c2 = 0, c1 = 0.05, (b) c1 = c2 = 0, c1 = c2 = 0.05, (c) c1 = c2 = 0, c1 = c2 = 0.05, (d) c1 = c1 = c2 = 0, c2 = 0.05.

Fig. 5. Frequency–response amplitude curves of the first flexural modes (n = 1) for a near-square beam when  = 0.04,  = −0.01, 21 f = 0.002; av , aw =response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive; SN, saddle-node; PF, pitchfork: (a) c1 = c1 = 0.05, c2 = c2 = 0, (b) c1 = c2 = c1 = c2 = 0.05, (c) c1 = c2 = c1 = 0.05, c2 = 0 (d) c1 = c2 = c2 = 0.05, c1 = 0.

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Fig. 6. Frequency–response amplitude curves of the first flexural modes (n = 1) for a near-square beam when  = 0.04,  = −0.01, 21 f = 0.002; av , aw =response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive; SN, saddle-node; PF, pitchfork: (a) c2 = c1 = 0.05, c1 = c2 = 0, (b) c1 = c1 = c2 = c2 = 0.

Fig. 7. Frequency–response amplitude curves of the first flexural modes (n = 1) for a near-square beam when  = 0.02,  = −0.01, 21 f = 0.002, c1 = c1 = 0.05, c2 = c2 = 0; av , aw =response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive; SN, saddle-node, PF, pitchfork.

Parts (a) and (b) in Fig. 3 are the counterparts of the corresponding parts in Fig. 2 for the case of a perfect beam (i.e. v0 (x)=w0 (x)=0) which are the same as the results reported in [6]. When the amplitude of base excitation is small, the response amplitudes are small, single valued and stable. As f increases, different bifurcations appear in Fig. 2(b) as in Fig. 3(b) and response curves of aw are moderately bent to right, implying that the non-linear geometric terms are dominant. Unlike the predictions of Pai and Nayfeh [6] for a perfect beam, the imperfect beam response is non-planar for the whole interval of detuning parameter , i.e. the indirectly excited mode is always activated (i.e. av
0), see Figs. 2 and 3. In contrast to Fig. 3(b), in Fig. 2(b), the saturation phenomenon is predicted over a larger interval of detuning parameter and the response curves do not lose stability through any pitchfork bifurcation. Also, a frequency shift to left is observed in Figs. 2(a) and (b) with respect to Figs. 3(a) and (b), respectively. Another interesting phenomenon observed in Fig. 2(b) is the frequency island generation [14] (i.e. the creation of closed isolated response curves). Figs. 4(a)–(d), 6(a) and (b) show the variation of steady-state response amplitudes with the detuning parameter for the case of a

near-square cross-section beam with  = −0.01 and various imperfection shapes. Damping and base excitation amplitude is constant for all of the above-mentioned figures. In parts (a)–(d) of Fig. 4, either v0 (x) = 0 or w0 (x) = 0, in parts (a)–(d) of Fig. 5 and in Fig. 6(a), neither v0 (x) = 0 nor w0 (x) = 0 and in Fig. 6(b), v0 (x) = w0 (x) = 0. By decreasing the detuning  from 0.0 to −0.01 (see Figs. 3(b) and 6(b)), the indirectly excited mode av is activated over a larger interval of detuning parameter because when  is negative, the beam is more flexible in the Y-direction than in the Z-direction. Therefore, the vibration in the Y-direction can more easily be auto-parametrically excited [6]. Figs. 4(a)–(d), 5(a)–(d) and 6(a) (i.e. the imperfect cases) should be compared with Fig. 6(b) (i.e. the perfect case). A frequency shift to left is observed in Figs. 4(a)–(d), 5(a)–(d) and 6(a). In parts (b)–(d) of Fig. 4 and (b)–(d) of Fig. 5, the saturation phenomenon is not observed. The response curves in parts (a)–(c) of Fig. 4 undergo sub- and super-critical pitchfork and saddle-node bifurcations as in Fig. 6(b) while the response curves in parts (a)–(d) of Fig. 5 undergo saddle-node bifurcations only. The response curves in Figs. 4(d) and 6(a) do not lose stability. In part (a) of Fig. 4, (a) of Fig. 5 and (b) of Fig. 6, the response amplitude

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Fig. 8. Frequency–response amplitude curves of the first flexural modes (n = 1) for a near-square beam when  = 0.04,  = −0.01, 21 f = 0.003, c2 = c2 = 0.05, c1 = c1 = 0; av , aw =response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive, (. . .) unstable with the real part of a complex conjugate pair of eigenvalues being positive, SN, saddle-node, PF, pitchfork, HF, Hopf bifurcation.

curves in the Y-direction saturate approximately at the same values. The corresponding detuning interval length of saturation is approximately the same in part (a) of Fig. 4 and (b) of Fig. 6 but larger to some extent in Fig. 5(a). In Fig. 4(c), the non-planar range of vibrations is narrowed down. In Fig. 4(d), the indirectly excited mode is not activated and the response is planar and stable for the whole range of detuning parameter. The pick of response amplitude in Fig. 4(d) is much less than its counterpart in Fig. 6(b) because the special shape of imperfection (i.e. v0 = 0.05X2 and w0 = 0) makes the beam stiffer in both Y- and Z-directions. In parts (a) and (d) of Fig. 5, two solution branches for each of directly and indirectly excited modes are predicted. Depending on the initial conditions, by sweeping the frequency quasi-statically, the response amplitudes can vary through any of the isolated curves. For example, suppose the response amplitude in the Y-direction is located on the stable portion of the lower curve for av in Fig. 5(a) and consequently the response amplitude in the Z-direction is located on the stable portion of the upper curve for aw . By increasing the frequency (detuning parameter),

Fig. 9. Response amplitude curves of the first flexural modes (n = 1) versus the base excitation amplitude for a near-square beam when  = 0.04,  = −0.01, = −0.015, c1 = c1 =0.05, c2 =c2 =0 ; av , aw =response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive; SN, saddle-node, PF, pitchfork.

av dose not change considerably since it is saturated while aw increases and/or decreases slightly. By further increasing the frequency and reaching the saddle-nodes SN3 and SN3 , respectively (see Fig. 5(a)), two scenarios are possible. Depending on the domain of attraction of equilibrium points on the right side of SN2 and SN2 , and the ones on the left side of SN4 and SN4 , av and aw may jump to the stable equilibrium points on the right of SN2 and SN2 or to the stable equilibrium points on the left of SN4 and SN4 , respectively. If the first scenario occurs, by further increasing the frequency, av and aw decreases monotonously without undergoing any other bifurcation while upon the occurrence of the second scenario, av and aw jump from SN4 and SN4 to their lower branches, respectively, and decrease monotonously.

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In part (b) of Fig. 5, the special shape of imperfection makes the beam stiffer in the Z-direction because the pick of aw response curves are less than their counterparts in Fig. 6(b) while in Fig. 5(c), the situation is reverse. Also, in parts (b) and (c) of Fig. 5, in a small range of detuning parameter, the aw response amplitude is less than that of the av . It means that although the base excitation is along the Z-direction, the beam vibrations amplitude along the Y-direction is larger. In Fig. 6(a), the indirectly excited mode is activated over the whole range of detuning parameter, namely, the response is nonplanar always. It should be noted that the major consequence of imperfection is the non-planarity of steady-state responses. Parts (a) and (b) of Fig. 7 show the response amplitude curves of av and aw versus the detuning parameter for the parameters  =0.02,  =−0.01, 21 f =0.002 and v0 =w0 =0.05X1 . When the damping is decreased from 0.04 to 0.02, the response curves undergo more saddlenode bifurcations comparing with the curves in Fig. 5(a). Also, the pick of response amplitudes increases considerably. As in Fig. 5(a), two solution branches for each of the directly and indirectly excited modes are predicted in parts (a) and (b) of Fig. 7. Two saddle-node bifurcations on the right side of parts (a) and (b) of Fig. 7 are not shown. Parts (a) and (b) of Fig. 8 show that by increasing the amplitude of base excitation from 0.002 to 0.003, the closed isolated response amplitude curves of av and aw undergo one Hopf bifurcation resulting in the creation of limit cycles for amplitudes and phases. Also, in contrast to Fig. 5(a), the response curve of av does not saturate. Parts (a) and (b) of Fig. 9 show the response amplitude curves versus the amplitude of base excitation for the parameters  = 0.04,

= −0.015,  = −0.01 and v0 = w0 = 0.05X1 . Fig. 9(a) shows that by increasing the base excitation amplitude from zero, the indirectly excited mode amplitude av grows through the open isolated curve while aw remains constant after increasing to a certain threshold (i.e. saturated). By further increasing the base excitation amplitude, a downward jump occurs through the saddle-node SN4 after which, av decreases and aw increases to smaller and larger values, respectively. Then, by further increasing the base excitation amplitude, av decreases and aw increases monotonously. Depending on the initial conditions, the response amplitudes av and aw can vary through each of the closed isolated curves in parts (a) and (b) of Fig. 9, respectively; furthermore, the response amplitude av may be larger than aw . Fig. 10 shows the variation of the response amplitude curves with the amplitude of base excitation for the same parameters as in Fig. 9 except that the imperfection is zero (i.e. v0 =w0 =0). In contrast to Fig. 9, the solution curves in Fig. 10 are single-branch and undergo no bifurcation; furthermore, the indirectly excited mode is not activated and the amplitude of directly excited mode aw increases monotonously from zero. Parts (a)–(d) of Fig. 11 show the variation of response amplitudes with the amplitude of imperfection c1 for the parameters  = 0.04,  = −0.01, 21 f = 0.002 and four different detuning parameters. The shape of the imperfection is assumed to be similar to the first flexural mode shape function in both Y- and Z-directions v0 = w0 = c1 X1 . In Fig. 11, the response amplitudes at c1 = 0 correspond to the case of perfect beam response amplitudes shown in Fig. 6(b). By increasing c1 from zero, av increases from zero monotonously until it saturates at approximately av = 0.08 in parts (a) and (b) of Fig. 11. Saturation occurs over different detuning intervals. The response amplitude aw also starts from a non-zero value but not monotonously until the saddle-node SN4 is reached. By further increasing c1 , a downward jump occurs to the lower open isolated response amplitude curves of av and aw . The response amplitudes decrease monotonously after jump because by further increasing the imperfection amplitude c1 , the beam behaves stiffer. Parts (a) and (b) of Fig. 11 show that beam flexibility

Fig. 10. Response amplitude curves of the first flexural modes (n=1) versus the base excitation amplitude for a near-square beam when  = 0.04,  = −0.01, = −0.015, c1 = c2 = c1 = c2 = 0; av , aw =response amplitudes along Y- and Z-directions; (——) stable.

in the Y-direction (i.e. the indirectly excited direction) over a limited range of imperfection amplitude c1 increases because the vibration in the Y-direction is activated considerably. In Fig. 11(c), by increasing c1 from zero, av remains almost constant (saturated at 0.083) while aw increases monotonously but slowly. Whether initiating on any of the related isolated curves depending on the initial conditions, av and aw reach the saddle-nodes SN2 /SN3 and SN2 /SN3 , respectively. Then, by further increasing c1 , a downward jump occurs to the lower isolated response amplitude curves passing through (SN3 , SN1 ) and (SN3 , SN1 ), respectively, and the response amplitudes decrease monotonously because the beam behaves stiffer. In Fig. 11(d), the indirectly excited mode is not activated (i.e. av ≈ 0) and aw decreases monotonously from a smaller value than those in parts (a)–(c) of Fig. 11 undergoing no bifurcation. Parts (a)–(d) of Fig. 12 show the variation of response amplitudes with the detuning parameter  for the parameters  = 0.04, 21 f = 0.002, v0 = w0 = 0.05X1 and four different values of detuning . In Fig. 12, the response amplitudes aw are saturated except for a limited range of . In Fig. 12(a), the open solution branches undergo no bifurcation, av increases upon decreasing aw and vice versa. The closed isolated curves undergo two saddle-node and two Hopf bifurcations resulting in the creation of limit cycles. In parts (b) and (c) of Fig. 12, the open and closed isolated curves undergo two saddle-node bifurcations which results in the jump phenomena. In Fig. 12(d), the single-branch response amplitude curves undergo no bifurcation. Parts (a)–(c) of Fig. 12 show that when  < 0, activation of the indirectly excited mode is easier and the non-planar vibrations is more serious; this is because when  < 0, the beam is more flexible in the Y-direction (i.e. the indirectly excited direction), whereas when  > 0, the beam is more flexible in the Z-direction. In either case, the influence of  on the stability and boosting the non-planar vibrations is noteworthy. Fig. 13 shows the variation of response amplitudes with the detuning parameter  for the same parameters as in part (a) of Fig. 12 except that v0 = w0 = 0. In contrast to part (a) of Fig. 12, the solution curves in Fig. 13 are single-branch and undergo no bifurcation; furthermore, the indirectly excited mode is not activated and the

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Fig. 11. Response amplitude curves of the first flexural modes versus the imperfection (v0 = w0 = c1 X1 (x), n = 1) for a near-square beam when  = 0.04,  = −0.01, 21 f = 0.002; av , aw =response amplitudes along Y- and Z- directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive, SN, saddle-node, PF, pitchfork: (a)

= −0.008, (b) = −0.005, (c) = 0.005, (d) = 0.015.

Fig. 12. Response amplitude curves of the first flexural modes (n = 1) versus the detuning  for a near-square beam when  = 0.04, 21 f = 0.002, c1 = c1 = 0.05, c2 = c2 = 0; av , aw =response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive, (. . .) unstable with the real part of a complex conjugate pair of eigenvalues being positive, SN, saddle-node, PF, pitchfork, HF, Hopf bifurcation: (a) = −0.02, (b) = −0.015, (c) = −0.008, (d) = −0.005.

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Fig. 13. Response amplitude curves of the first flexural modes (n = 1) versus the detuning  for a near-square beam when  = 0.04, 21 f = 0.002, = −0.02, c1 =c2 =c1 =c2 =0; av , aw =response amplitudes along Y- and Z-directions; (——) stable.

Fig. 14. Response amplitude curves of the first flexural modes (n=1) versus damping for a near-square beam when  = −0.01, 21 f = 0.002, = −0.008, c1 = c1 = 0.05, c2 = c2 = 0; av , aw =response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive, SN, saddle-node.

Fig. 15. Response amplitude curves of the first flexural modes (n=1) versus damping for a near-square beam when  = −0.01, 21 f = 0.002, = −0.008, c1 = c2 = c1 = c2 = 0; av , aw =response amplitudes along Y- and Z-directions; (——) stable.

Fig. 16. Frequency–response amplitude curves of the second flexural modes (n = 2) for a near-square beam when  =0.04,  =0.002, 21 f =6×10−5 , c1 =c1 =c2 =c2 =0; av , aw =response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive, (. . .) unstable with the real part of a complex conjugate pair of eigenvalues being positive, SN, saddle-node, PF, pitchfork, HF, Hopf bifurcation.

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Fig. 17. Frequency–response amplitude curves of the second flexural modes (n = 2) for a near-square beam when  = 0.04,  = 0.002, 21 f = 6 × 10−5 ; av , aw =response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive, (. . .) unstable with the real part of a complex conjugate pair of eigenvalues being positive, SN, saddle-node, PF, pitchfork, HF, Hopf bifurcation: (a) and (b) c1 = 0.05, c1 = c2 = c2 = 0, (c) and (d) c1 = 0.05, c2 = c1 = c2 = 0.

amplitude of directly excited mode aw is saturated which can be justified by investigating Eqs. (16a)–(16d). When the indirectly excited mode is not activated (i.e. pvc = pvs = 0), since the only -dependent coefficient is C4 (see Appendix C) then the amplitude of the directly excited mode aw is independent of  and constant. Physically, while the vibration along the Y-axis is not activated, the vibration along the Z-axis is independent of the detuning . Fig. 14 shows the variation of response amplitude curves with damping for the parameters = −0.008,  = −0.01, 21 f = 0.002 and v0 = w0 = 0.05X1 . Each part in Fig. 14 consists of five branches which undergo saddle-node bifurcations and one bifurcation-free branch. When  = 0, there are five possible stable steady-state responses depending on the initial conditions. The branches going through the SN4 and SN5 are very close to each other; furthermore, the stable portions of these branches in parts (a) and (b) are almost constant and saturated. If  is less than 0.0047884, the response amplitude of indirectly excited mode through the internal resonance can be 3–4 times larger than that of the resonantly excited mode. The bifurcation-free branch in part (a) is saturated for  < 0.045 and has a gradual decrease for values more than  = 0.045 while it has a gradual decrease in part (b). Fig. 15 shows the variation of response amplitude curves with damping for the same parameter values as in Fig. 14 except that v0 = w0 = 0 (i.e. perfect beam). In contrast to Fig. 14, the response amplitude curves in Fig. 15 are single-branch and bifurcation-free; furthermore, the indirectly excited mode is not activated. 4.2. Case  = 2 (1 + 2 ) The second mode resonance excitation is investigated as a representative of the high-frequency modes. Fig. 16 shows the variation of response amplitudes with the detuning parameter for the

Fig. 18. Frequency–response amplitude curves of the second flexural modes (n = 2) for a near-square beam when  = 0.04,  = 0.002, 21 f = 6 × 10−5 , c1 = c2 = 0.05, c1 = c2 = 0; av , aw =response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive, SN, saddle-node.

parameters  = 0.04,  = 0.002, 21 f = 6 × 10−5 and v0 = w0 = 0 (i.e. perfect beam) which coincide with the results of Pai and Nayfeh [6]. Fig. 16 shows that the non-planar vibration may be activated by the one-to-one internal resonance through a subcritical or supercritical pitchfork bifurcation when increasing the frequency detuning from values less than (PF 1 ) or decreasing it from values more than (PF 2 ), respectively. In Figs. 17 and 18, either v0 = 0 or w0 = 0

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Fig. 19. Frequency–response amplitude curves of the second flexural modes (n = 2) for a near-square beam when  = 0.04,  = 0.002, 21 f = 6 × 10−5 , c1 = c1 = 0.05, c2 = c2 = 0; av , aw =response amplitudes along Y- and Z-directions; (——) stable,

Fig. 20. Frequency–response amplitude curves of the second flexural modes (n = 2) for a near-square beam when  = 0.04,  = 0.002, 21 f = 6 × 10−5 ; av , aw =response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at

(- - - - -) unstable with at least one eigenvalue being positive, (. . .) unstable with the real part of a complex conjugate pair of eigenvalues being positive, SN, saddle-node, PF, pitchfork, HF, Hopf bifurcation.

least one eigenvalue being positive, (. . .) unstable with the real part of a complex conjugate pair of eigenvalues being positive, SN, saddle-node, PF, pitchfork, HF, Hopf bifurcation: (I) and (II) (a) c1 = c2 = c1 = 0.05, c2 = 0, (b) c1 = c2 = 0, c2 = c1 = 0.05.

while in Figs. 19–21, neither v0 = 0 nor w0 = 0. Fig. 17 shows the variation of response amplitudes with the detuning parameter for  = 0.04,  = 0.002, 21 f = 6 × 10−5 and the imperfection cases (v0 = 0, w0 = 0.05X1 ) and (v0 = 0.05X1 , w0 = 0), respectively. In contrast to Fig. 16, a frequency detuning shift is observed in Figs. 17–21. In Fig. 17, the response amplitude curves undergo the same bifurcation points as in Fig. 16 (i.e. the perfect beam case) except that the first pitchfork bifurcation in parts (c) and (d) of Fig. 17 is of supercritical type while it is of subcritical type in Fig. 16. Namely, the transition from planar to non-planar vibrations through the above-mentioned pitchfork bifurcations occurs gradually and suddenly (jump), respectively. Furthermore, there is a small interval of the detuning parameter in Fig. 17 as in Fig. 16 in which no stable fixed point exists. Hence, the response of the beam in this interval is expected to be aperiodic including chaotic. Fig. 18 shows that the indirectly excited mode is not activated by the one-to-one internal resonance for the imperfection case v0 = 0.05X1 + 0.05X2 , w0 = 0. In contrast to Fig. 16(b), the aw response amplitude curve in Fig. 18 has a simple form of frequency–amplitude curve corresponding to resonance

excitation of one-degree-of-freedom systems with cubic nonlinearity since it has two saddle-node bifurcations and is moderately bent to left indicating that the effective non-linearity is of the softening type. Similar figures to Fig. 18 are obtained for the cases (v0 = 0.05X2 , w0 =0), (v0 =0, w0 =0.05X2 ) and (v0 =0, w0 =0.05X1 +0.05X2 ) with a frequency shift with respect to Fig. 18. Fig. 19 shows the variation of response amplitude curves with the detuning parameter for the imperfection case v0 =w0 =0.05X1 . Since the C3 coefficient in Eqs. (16a) and (16b) does not vanish for this case of imperfection (see Appendix C), the indirectly excited mode is activated by the oneto-one internal resonance over the whole range of detuning and the corresponding response amplitude curves undergo no pitchfork bifurcation as shown in part (a) of Fig. 19. Open isolated curves in Fig. 19 undergo four saddle-nodes SN1 , SN3 , SN4 and SN6 and two Hopf bifurcations HF1 and HF2 which lead to limit cycles while the closed isolated curves undergo two saddle-nodes SN2 and SN5 . Unlike Fig. 16 for the perfect beam and Fig. 17 for the two imperfect beam cases, there is no frequency detuning range without

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Fig. 22. Response amplitude curves of the second flexural modes (n = 2) versus the base excitation amplitude for a near-square beam when  = 0.04,  = −0.018,

= −0.0219, c1 = c1 = 0.05, c2 = c2 = 0; av , aw = response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive, (. . .) unstable with the real part of a complex conjugate pair of eigenvalues being positive, SN, saddle-node, PF, pitchfork, HF, Hopf bifurcation.

Fig. 21. Frequency–response amplitude curves of the second flexural modes (n = 2) for a near-square beam when  = 0.04,  = −0.018, 21 f = 6 × 10−5 , c1 = c1 = 0.05, c2 = c2 = 0 ; av , aw =response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive, (. . .) unstable with the real part of a complex conjugate pair of eigenvalues being positive, SN, saddle-node, PF, pitchfork, HF, Hopf bifurcation.

steady-state responses in Fig. 19. Parts (I) and (II) of Fig. 20 show the variation of response amplitude curves with the detuning parameter

for the cases (v0 = 0.05X1 + 0.05X2 , w0 = 0.05X1 ), and (v0 = 0.05X2 , w0 = 0.05X1 ), respectively. The av and aw curves in parts (I) and (II) of Fig. 20 have a frequency shift with respect to each other, respectively. The aw response curves, as in Fig. 18, have a simple form of a frequency–amplitude curve corresponding to resonance excitation of one-degree-of-freedom systems with cubic non-linearity. The av response curves in part (I) undergo two saddle nodes followed by a pick and a small saturation range. Fig. 21 shows the variation of response amplitudes with the detuning parameter for  = 0.04,  = −0.018, 21 f = 6 × 10−5 and v0 = w0 = 0.05X1 . The response amplitudes include two open and closed curves. The open curve undergoes four saddle-node and one Hopf bifurcations (SN2 , SN3 , SN4 , SN6 and HF1 ) while the closed curve undergoes two saddle-node and one Hopf bifurcations (SN1 , SN5 and HF2 ). Comparing Figs. 21 and 19, one finds that by decreasing the detuning  from 0.002 to −0.018, a region of frequency

detuning appears in which no stable fixed point exists. Furthermore, the number of Hopf bifurcations that the open curves undergo reduces to one in parts (a) and (b) of Fig. 21. Fig. 22 shows the variation of response amplitudes with the base excitation amplitude for  = 0.04,  = −0.018, = −0.0219 and v0 = w0 = 0.05X1 . By increasing the base excitation amplitude from zero, aw increases rapidly while av remains almost constant until a jump occurs from SN3 and SN3 , respectively. There exists no stable fixed point for the base excitation amplitude range corresponding to SN3 and HF1 and the beam response is predicted to be aperiodic, including chaotic. When the base excitation amplitude is increased to values greater than the value corresponding to HF1 , the beam response may be steady state with the predicted amplitudes. In this case, by further increasing the base excitation amplitude, aw and av increase monotonously. Fig. 23 shows the variation of response amplitudes with the base excitation amplitude for the same parameters as in Fig. 22 except that v0 = w0 = 0. By increasing the amplitude of base excitation from zero, aw increases monotonously while av remains inactivated. Since no stable fixed-point is predicted for larger values of base excitation corresponding to SN3 , by further increasing the base excitation amplitude, a jump occurs through SN3 to non-steadystate solutions. It should be noted that in Fig. 22, stable steady-state responses are predicted for base excitation amplitudes larger than the value corresponding to HF2 , while in Fig. 23, no stable steadystate response is predicted for base excitation amplitudes larger than the value corresponding to SN3 . The variation of response amplitudes with the detuning  are presented in Fig. 24 for  = 0.04, 21 f = 6 × 10−5 , = −0.0219 and v0 = w0 = 0.05X1 . The open curves undergo two saddle-node bifurcations SN1 and SN3 and four Hopf bifurcations HF2 , HF3 , HF6 and HF7 while the closed curves undergo two saddle-node bifurcations SN2 and SN4 and three Hopf bifurcations HF1 , HF4 and HF5 . The response amplitudes are saturated for enough small and large values of . There are two small regions of detuning  in which no stable fixed point exists and hence, the beam response is predicted to be aperiodic, including chaotic. Fig. 24 shows that when  < 0, activation of the indirectly excited mode is

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Fig. 24. Response amplitude curves of the second flexural modes (n = 2) versus the base excitation amplitude for a near-square beam when  = 0.04, 21 f = 6 × 10−5 ,

= −0.0219, c1 = c1 = 0.05, c2 = c2 = 0; av , aw = response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive, (. . .) unstable with the real part of a complex conjugate pair of eigenvalues being positive, SN, saddle-node, PF, pitchfork, HF, Hopf bifurcation. Fig. 23. Response amplitude curves of the second flexural modes (n = 2) versus the base excitation amplitude for a near-square beam when  = 0.04,  = −0.018,

=−0.0219, c1 =c2 =c1 =c2 =0; av , aw =response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive, (. . .) unstable with the real part of a complex conjugate pair of eigenvalues being positive, SN, saddle-node, PF, pitchfork, HF, Hopf bifurcation.

easier and the non-planar vibrations is serious. Fig. 25 shows the variation of response amplitudes with the detuning  for the same parameters as in Fig. 24 except that v0 = w0 = 0. In contrast to Fig. 24, the response amplitude curves in Fig. 25 are single-branch and undergo no bifurcation; furthermore, the indirectly excited mode is not activated and the amplitude of directly excited mode aw is saturated and independent of  as in Fig. 13. The variation of response amplitudes with damping are presented in parts (a) and (b) of Fig. 26 for  = −0.018, 21 f = 6 × 10−5 ,

= −0.0219 and v0 = w0 = 0.05X1 . Each part consists of two curves, one of which undergoes two saddle-node bifurcations SN1 and SN2 and one Hopf bifurcation HF1 that leads to limit cycles and the other one undergoes one saddle-node bifurcation SN3 . There exists no stable fixed point for damping values less than a certain threshold

Fig. 25. Response amplitude curves of the second flexural modes (n = 2) versus the base excitation amplitude for a near-square beam when  = 0.04, 21 f = 6 × 10−5 ,

= −0.0219, c1 = c2 = c1 = c2 = 0; av , aw =response amplitudes along Y- and Zdirections; (——) stable.

O. Aghababaei et al. / International Journal of Non-Linear Mechanics 44 (2009) 161 -- 179

Fig. 26. Response amplitude curves of the second flexural modes (n = 2) versus damping for a near-square beam when  = −0.018, 21 f = 6 × 10−5 , = −0.0219, c1 = c1 = 0.05, c2 = c2 = 0; av , aw =response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive, (. . .) unstable with the real part of a complex conjugate pair of eigenvalues being positive, SN, saddle-node, PF, pitchfork, HF, Hopf bifurcation.

corresponding to SN1 and the beam response is predicted to be aperiodic, including chaotic. Since the detuning is not zero, the response amplitudes for  = 0 are bounded. The variation of response amplitudes with damping for the same parameters as in Fig. 26 and v0 = w0 = 0 are presented in Fig. 27. In contrast to Fig. 26 for the imperfect beam that the stable steady-state responses are predicted for damping values greater than a certain threshold, stable steady-state responses are predicted for any value of damping in Fig. 27 for the perfect beam; furthermore, the indirectly excited mode in Fig. 27 cannot be activated for damping values greater than the value corresponding to SN1 . Unlike the stable branch in Fig. 26, the variation of aw response amplitude along the bifurcation-free path in Fig. 27 is gradual. Fig. 28(a) and (b) show the trajectories of the beam tip center of cross-section (i.e. v = av cos(t − v ) versus w = aw cos(t − w ) for = 1) for the resonant excitations of first and second flexural modes, respectively. The shape of imperfection is assumed to be v0 = w0 = 0.05X1 . In Fig. 28(a) and (b), the phase differences of inplane and out-of-plane vibrations are 0.2780 and 1.4877 radians,

177

Fig. 27. Response amplitude curves of the second flexural modes (n = 2) versus damping for a near-square beam when  = −0.018, 21 f = 6 × 10−5 , = −0.0219, c1 =c2 =c1 =c2 =0; av , aw =response amplitudes along Y- and Z-directions; (——) stable, (- - - - -) unstable with at least one eigenvalue being positive, SN, saddle-node.

respectively. Since the phase differences in Fig. 28(a) and (b) are close to 0 and /2, the shape of trajectories are inclined narrow and approximately right ellipses, respectively. Different effects of geometric imperfection may be summarized as: elimination of pitchfork bifurcations, Figs. 2(b), 4(d), 5(a)–(d), 7(a) and (b), 8(a) and (b), (18) and 19(a) and (b), the phenomenon of frequency island generation, i.e. creation of closed isolated curves, Figs. 2(b), 5(a) and (d), 7(a) and (b), 8(a) and (b), 19(a) and (b) and 21(a) and (b), shifting effect to left (all imperfect cases), narrowing or widening the saturation ranges predicted by perfect beam model, Figs. 4(b)–(d), 5(b)–(d) and 2(b), respectively, predicting larger amplitudes of vibration for the indirectly excited mode than the directly excited one, Figs. 5(b) and 8(a) and (b), and changing the subcritical pitchfork bifurcation to a supercritical one, Fig. 17(c). 5. Conclusions A complete static bifurcation analysis of steady-state responses of laterally base-excited near-square cantilever beams with and without natural geometric imperfection is carried out. The primary resonances of the first and second flexural modes with a one-to-one internal resonance are considered. It is shown that ignoring the

178

O. Aghababaei et al. / International Journal of Non-Linear Mechanics 44 (2009) 161 -- 179

Fig. 28. Trajectories (out-of-plane versus in-plane vibrations) of beam tip center of cross for c1 = c1 = 0.05, c2 = c2 = 0 and = 1, (a) n = 1,  = 0.04,  = −0.01, = −0.025, 21 f = 0.002 (b) n = 2,  = 0.04,  = 0.002, = −0.015, 21 f = 6 × 10−5 .

geometric imperfection in such beams and use of perfect beam model may result in the prediction of different dynamic behaviors, even for small imperfection. The stable or unstable steady-state non-planar responses of a cantilever beam with small imperfection may be mistaken with stable or unstable planar responses of a perfect beam as a consequence of perfect beam modeling since the indirectly excited mode in the case of perfect beam is activated in limited ranges of parameter values. Multi-branch (frequency island generation [14]) non-planar response amplitude curves without pitchfork bifurcation is the main feature of the generally imperfect beams steady-state response curves (i.e. beams with v0
0 and w0
0) under base excitation while, the response amplitude curves of the perfect beam undergo at least one pitchfork bifurcation. In the stable steady-state responses of the imperfect beam, larger amplitude values for the indirectly excited mode through the internal resonance is possible than the directly excited mode through the primary resonance while, the perfect beam modeling results in the prediction of planar motions. Also, the saturation of the stable steady-state responses is predicted differently by imperfect and perfect beam models. Investigation of the dynamic solutions, i.e. periodic, quasi-periodic and chaotic responses, of the imperfect cantilever beams under primary-resonant base excitation with a one-to-one internal resonance and comparison with that of the perfect beams is suggested for further work.

cos rn + cosh rn Xn (x) = cosh rn x − cos rn x + (sin rn x − sinh rn x) sin rn + sinh rn  1  1 n = rn2 , Xn2 (x) dx=1, Xn (x)Xm (x) dx=0, n
m = 1, 2, 3, . . . 0

0

1 = 2 =

0

1 2



1





0

 −

Xn (v0 (w0 Xn ) ) dx

  1  x  1  − Xn w0 v0 Xn d d dx



 −

0

1

2



0

 Xn w0

 1 Xn w0

0

x 0 x 1

1



 w0 Xn d





dx

w0 Xn d

x

v0 Xn d

0

0



 dx −

6 = −

 

+

7 =

 0

1

0

 Xn w0

x 1

  v0 Xn d dx



1



 Xn w0

 Xn Xn d

0

x

 

1

0

 1 Xn Xn



1

0





0

x 

1

0

 Xn v0

x

1

 Xn v0

x

 Xn v0

 

1 0 1

0

  w0 Xn d dx

dx

1 0 x 

 dx



w0 Xn d d

 

0

x

1

v0 Xn d d

  0

1

 Xn Xn

1

 2 Xn d d dx,

 

1

0 1

0

0 x 1





x

 Xn Xn

1

0

11 = 2

0

1

Xn (Xn (w0 Xn ) ) dx −

 Xn Xn



10 = 2

1

0

1

8 = 2

0

dx,



v0 Xn d d

 dx +

 w0 Xn d d dx

1

0

2

2

(v0 + w0 )Xn2 dx

 2 Xn d d dx

1 2



Xn (w0 w0 Xn ) dx +

0

− −

0

1 0

0

Xn (v0 Xn ) dx +





1 0



Xn ((v0 + w0 )Xn ) dx 2

2

1

Xn (v0 (v0 Xn ) ) dx

  1  1  x  1  2  − Xn (w0 Xn ) dx − Xn v0 v0 Xn d d dx +

 dx

1

0

 Xn w0

1





0



 1  2 2     Xn v0 Xn + v dx, 0 Xn + 2Xn (v0 Xn ) 2 0

  1  1  x    5 = − Xn (v0 Xn Xn ) dx + Xn Xn v0 Xn d dx

4 =

13 = − 2

1 0



Xn (Xn (Xn Xn ) ) dx

Xn ((−v0 + w0 )Xn ) dx 0

  1  x  1  − Xn w0 w0 Xn d d dx

+

1

0

12 =

Appendix B. 1



9 = 2

Appendix A.



3 =

2

 Xn v0

 1 Xn v0

0



x 0 x 1



v0 Xn d 

0

dx



v0 Xn d

dx

0

1

O. Aghababaei et al. / International Journal of Non-Linear Mechanics 44 (2009) 161 -- 179



1



Xn (w0 (v0 Xn ) ) dx 0

  1  x  1  + Xn v0 w0 Xn d d dx

14 = −



 −

15 = −



0



0



0



0



0

−

16 = − 17 = − −

18 = 2 19 = 2 20 =



0

1

1

1

1

1 0





1

1 0 1

0 1

0



Xn v0

0



x 0

1

  w0 Xn d dx +

0



1 2



Xn (w0 Xn Xn ) dx −

1 0



Xn (Xn (w0 Xn ) ) dx − 2 

Xn (w0 (Xn Xn ) ) dx +  Xn (Xn (v0 Xn ) ) dx −



Xn v0



x 0

 Xn w0

Xn w0

x

 

x

 

 Xn w0

1

1 x

0

0 2









 Xn Xn d

1



1



 Xn v0

x

1



w0 Xn d

 dx

  Xn (w 0 Xn ) dx 2

1



Xn (w0 Xn Xn ) dx  x

   Xn Xn w0 Xn d dx

0 1

0 1

0

  Xn Xn

1 x

1

  v0 Xn d dx

dx

 v0 Xn d d dx, w0 Xn d d

Xn d d

 dx,



 dx +

21 =



1

0 1

0

2

2

(v0 + w0 )Xn2 dx

Xn dx

Appendix C. C1 = − n ,

C2 = − 14 1 + 12 7 2n , C3 = −3 + 12 11 2n 4 C4 = −rn + 2 + 12 10 2n C5 = 14 (−31 + 27 2n ), C6 = − 12 1 , C7 = 14 + 12 18 2n C8 = 13 + 12 19 2n , C9 = 2 2n References [1] E.C. Haight, W.W. King, Stability of non-linear oscillations of an elastic rod, J. Acoust. Soc. Am. 52 (1972) 899.

179

[2] 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. [3] M.W. Hyer, Whirling of a base-excited cantilever beam, J. Acoust. Soc. Am. 65 (1979) 931. [4] M.R.M. Crespo da Silva, On the whirling of a base-excited cantilever beam, J. Acoust. Soc. Am. 67 (1980) 704. [5] 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. [6] 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. [7] M.R.M. Crespo da Silva, C.L. Zaretzky, Non-linear modal coupling in planar and non-planar responses of inextensional beams, Int. J. Non-Linear Mech. 25 (1990) 227. [8] C.L. Zaretzky, M.R.M. Crespo da Silva, Experimental investigation of non-linear modal coupling in the response of cantilever beams, J. Sound Vib. 174 (1994) 145. [9] M.R.M. Crespo da Silva, C.L. Zaretzky, Nonlinear flexural–flexural–torsional interactions in beams including the effect of torsional dynamics, I: primary resonance, Nonlinear Dyn. 5 (1994) 3. [10] J.-S. Huang, R.-F. Fung, C.-R. Tseng, Dynamic stability of a cantilever beam attached to a translational/rotational base, J. Sound Vib. 224 (2) (1999) 221. [11] J. Luczko, Bifurcations and internal resonances in space-curved rods, Comput. Methods Appl. Mech. Eng. 191 (2002) 3271. [12] K.V. Avramov, Non-linear beam oscillations excited by lateral force at combination resonance, J. Sound Vib. 257 (2) (2002) 337. [13] 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. [14] W. Lacarbonar, H.N. Arafat, A.H. Nayfeh, Non-linear interactions in imperfect beams at veering, Int. J. Non-Linear Mech. 40 (2005) 987. [15] A. Luongo, A. Di Egidio, Divergence, Hopf and double-zero bifurcations of a non-linear planar beam, Comput. Struct. 84 (2006) 1596. [16] 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. [17] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley, New York, 1979. [18] W. Zhang, Chaotic motion and its control for nonlinear nonplanar oscillations of a parametrically excited cantilever beam, Chaos, Solutions & Fractals 26 (3) (2005) 731. [19] L.D. Zavodney, A.H. Nayfeh, The non-linear response of a slender beam carrying a lumped mass to a principal parametric excitation: theory and experiment, Int. J. Non-Linear Mech. 24 (2) (1989) 105. [20] A.H. Nayfeh, P.F. Pai, Non-linear non-planar parametric responses of an inextensional beam, Int. J. Non-Linear Mech. 24 (2) (1989) 139. [21] O. Aghababaei, H. Nahvi, S. Ziaei-Rad, Non-linear non-planar vibrations of geometrically imperfect inextensional beams, part I: equations of motion, Int. J. Non-linear Mech. (2008), in press, doi:10.1016/j.ijnonlinmec.2008.10.006. [22] A.H. Nayfeh, B. Balachandran, Applied Nonlinear Dynamics, Wiley, New York, 1995.