The role of nonlinear torsional contributions on the stability of flexural–torsional oscillations of open-cross section beams

Journal of Sound and Vibration 358 (2015) 236–250

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The role of nonlinear torsional contributions on the stability of flexural–torsional oscillations of open-cross section beams Angelo Di Egidio a,n, Alessandro Contento b, Fabrizio Vestroni c a b c

Department of Civil, Construction-Architectural, Environmental Engineering, University of L’Aquila, Italy Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, USA Department of Structural and Geotechnical Engineering, University of Rome ’La Sapienza’, Italy

a r t i c l e in f o

abstract

Article history: Received 16 April 2015 Received in revised form 3 August 2015 Accepted 6 August 2015 Handling Editor: M.P. Cartmell Available online 29 August 2015

An open-cross section thin-walled beam model, already developed by the authors, has been conveniently simplified while maintaining the capacity of accounting for the significant nonlinear warping effects. For a technical range of geometrical and mechanical characteristics of the beam, the response is characterized by the torsional curvature prevailing over the flexural ones. A Galerkin discretization is performed by using a suitable expansion of displacements based on shape functions. The attention is focused on the dynamic response of the beam to a harmonic force, applied at the free end of the cantilever beam. The excitation is directed along the symmetry axis of the beam section. The stability of the one-component oscillations has been investigated using the analytical model, showing the importance of the internal resonances due to the nonlinear warping coupling terms. Comparison with the results provided by a computational finite element model has been performed. The good agreement among the results of the analytical and the computational models confirms the effectiveness of the simplified model of a nonlinear open-cross section thin-walled beam and overall the important role of the warping and of the torsional elongation in the study of the one-component dynamic oscillations and their stability. & 2015 Elsevier Ltd. All rights reserved.

1. Introduction The nonlinear dynamic behavior of beams is a topic widely discussed in mechanics. While the early works addressed only planar motion, in the seventies the interest was directed to the three-dimensional behavior of beams. In one of the pioneering papers [1], a one-dimensional polar model of a compact beam is used to describe the non-planar, nonlinear dynamics of an inextensional beam. In the model presented, the torsional component was statically condensed, and the warping neglected. Subsequent papers proposed enriched models, some of which included linear warping, as [2]; however, although the linear warping contribution was considered, in most of the cases beams with compact cross-section were analyzed. Among these paper, flexural–torsional–extensional couplings in the motion of a cantilever beam have been considered in [3–5]. In [6], the complexity deriving from tackling nonlinear warping led to the choice to completely neglect it, thus limiting the applicability of the study to some open cross-section beams with comparable bending and torsional

n

Corresponding author. E-mail addresses: [email protected] (A. Di Egidio), [email protected] (A. Contento), [email protected] (F. Vestroni).

http://dx.doi.org/10.1016/j.jsv.2015.08.004 0022-460X/& 2015 Elsevier Ltd. All rights reserved.

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237

frequencies. Also the papers based on an extension to the nonlinear Vlasov field theory [7], as [8–10], had to rely on several simplifying assumptions such as considering nonlinear flexural quantities of higher order terms compared to torsional deformational quantities. More recently, in [11], the general formulation for torsional–flexural analysis of beams that make use of two-dimensional St. Venant warping functions and one-dimensional independent warping parameters was solved numerically by the finite element method. Some papers, [12–14], addressed the problem of the general flexural–torsional vibration of homogeneous beams of arbitrarily shaped cross-section under arbitrary loading, numerically solving the boundary value problem with a boundary element method approach (BEM). In [15] a modified Vlasov theory is proposed to study the dynamic behavior of thin-walled and variable open section beams. The formulation makes use of the state variable approach in the frequency domain and accounts shear strains and rotatory inertias. An experimental analysis for the study of the free vibrations of nonsymmetrical thin walled beams that also provide a benchmark for [15] is proposed in [16]. The same benchmark has been used more recently in [17] to validate results obtained with a one-dimensional model and to investigate the effects of a coarse warping descriptor on the linear dynamics of a clamped–clamped thin-walled beam with non-symmetric cross-sections. Models accounting for non-uniform warping including the effects of torsion and shear forces are formulated in [18,19], for thin-walled and general crosssections respectively. However, in [19], starting from the assumptions that the cross-section maintains its shape and including three independent warping parameters, distortion was not included even if may be significant for thin-walled cross-sections; some analytical and numerical result are presented in [20]. A linear beam model, where the effects of nonuniform warping related to shear and torsion are considered, is formulated in [21], under the assumption of rigid motion of the section. In [22] the same authors use the Implicit Corotational Method to derive a geometrically nonlinear model for generic cross-sections, which also accounts for non-uniform warping. A finite element is developed to analyze cantilevers with different thin-walled cross-sections. Subsequently a higher order beam model for thin-walled structures that considers higher order warping modes obtained solving the nonlinear eigenvalue problem of the corresponding beam governing equations is presented in [23] In [24,25] the description of the mechanical behavior of beams with open cross-sections is dealt with using a nonlinear beam model developed rigorously starting from an internally constrained three-dimensional continuum, in which torsional and flexural curvatures of the same order of magnitude are considered. The warping is obtained extending the Vlasov theory [7] to the nonlinear field. The effects of the torsional curvature on the elongation of the longitudinal fibers and the nonlinear warping of the section are considered. Due to the complexity of the model and to the several resonance conditions involved in the mechanical system investigated, it has not yet been possible to appreciate completely the role of the new nonlinear contributions. Within a class of open cross section beams with certain geometrical and mechanical characteristics, it can be shown that torsional curvature is greater than the flexural ones. A notable simplification is then obtained in the kinematical relations with respect to the model developed in [24], as shown in [26]. With the model developed in [26], it has been possible to understand the role of the new nonlinear contributions in the static behavior of a cantilever beam and in the stability of static solutions. Numerical and experimental investigations have confirmed the results provided by this model. Although this paper starts with the same assumptions of [26], here the attention is focused on the dynamic behavior of the beam. A simplified set of equations of motion capable of describing the nonlinear dynamics of the open-cross thinwalled beam, also accounting for the nonlinear contribution of the warping, are written for the first time with respect to the cited previous work of the authors and the existing literature. However, it is shown that the reduced equations of motion are capable of preserving the same dynamic characteristics of the rigorous model [24], and consequently are perfectly capable of capturing the main phenomena of the nonlinear response. Three simplified equations of motion are derived to describe the dynamics of inextensional and shear undeformable nonlinear 3D open-cross section thin-walled beam with a section that presents only one symmetry axis. A Galerkin discretization is performed by using a suitable number of shape functions, chosen with special care. The attention is focused on the dynamic response of the beam to a harmonic force, applied at the free end of the cantilever beam and directed along the symmetry axis of the section, which leads to a one-component oscillation. The primary objective of this paper is the study of the stability of the one-component oscillation, by investigating the role of the internal resonances related to the nonlinear warping coupling terms. The results obtained by the analytical model are compared with those provided by a computational finite element model. The behaviors ascertained with and without considering warping are compared to highlight the effects of the nonlinear contributions due to the torsional elongation and the warping. The investigation here performed permits novel stability phenomena to be highlighted, noting that these are strictly connected to the new nonlinear terms due to nonlinear warping and torsional elongation effects. 2. Mechanical model and equations of motion The analytical model (AM), described in [24,25], is here considered to describe the behavior of a thin-walled beam. The beam is assumed to be slender, initially straight, with an open, monosymmetric cross-section and arbitrary restrained at the ends. In order to obtain a model capable of adequately describing the behavior of the beam and, at the same time, of pointing out the main contributions in the equations of motion, the same hypotheses assumed in [26] have been introduced. These hypotheses allow a valid simplification of the displacement field and an easier identification of the leading terms in

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Fig. 1. Reference systems and beam section in the deformed and undeformed configuration.

the equations of motion without any loss of accuracy. The aforementioned hypotheses are: (i) the beam is shear indeformable; (ii) the section in the deformed configuration undergoes non-rigid displacements in its tangential and normal directions due to warping; (iii) the Vlasov condition is verified with the vanishing of the shear strains on the middle surface of the thin-walled beam and, furthermore, the extensional and shear strains of the cross-section plane also vanish (indeformability of the projection of the section in the plane orthogonal to the centroid axis); (iv) the beam is axially inextensible; and (v) the flexural curvatures μ1 ; μ2 are of higher order of magnitude, compared to the torsional curvature μ3 . 2.1. Kinematics Two reference frame are introduced: an external one ðO x1 x2 x3 Þ, suitably chosen with O placed in the centroid G of an end cross-section and oriented so that x1 and x2 are section principal axes while x3 contains the centroid axis; an internal
T one, defined by a unit base vector b ¼ b1 ; b2 ; b3 solid to the not-warped deformed configuration, with b3 tangent to the centroid-axis (Fig. 1). The position of the base b is defined by the orthogonal rotation matrix R, with respect to n oT b ¼ b 1 ; b 2 ; b 3 , that is the triad solid with the section in the undeformed configuration, oriented like the xi-axis, as b ¼ R b. The displacement of the generic point P  ðx1 ; x2 ; x3 Þ is obtained by the superimposition of the non-rigid displacement due to warping to the rigid displacement defined taking as reference point the shear center C  ðx1C ; x2C Þ; uP ¼ uC þ ðR  IÞðx  xC Þ þ Rϕ

(1)


T where I is the identity matrix, uC ¼ fu1 u2 u3 gT are the displacements of the shear center, x ¼ fx1 x2 0gT and ϕ ¼ ϕ1 ϕ2 ϕ3 are the warping functions. The displacement field (1) is defined by a total of nine functions, six of which, ui ðx3 ; tÞ and ϑi ðx3 ; tÞ with (i¼1,2,3), depend only on the abscissa x3 and on the time t, and other three warping functions ϕi ðx1 ; x2 ; x3 ; tÞ (i¼1,2,3). It is worth noticing that the components of the displacement (1) are referred to the external frame ðO x1 x2 x3 Þ, that has the n oT (see Fig. 1). Six orthogonality conditions are imposed to furtherly specialize axes parallel to the unit base b ¼ b 1 ; b 2 ; b 3 the description of the displacements: R ϕ dA ¼ 0 RA 1 A ϕ3 x1 dA ¼ 0;

R

ϕ2 dA ¼ 0 A ϕ3 x2 dA ¼ 0

RA

R ϕ dA ¼ 0  RA  3 A ϕ2 ðx1 x1C Þ ϕ1 ðx2  x2C Þ dA ¼ 0

(2)

These conditions prevent ϕ from being a translation (Eqs. (21–3)), a rotation for any ϕ3 a 0 (Eqs. (24,5)) or a purely torsional rotation (Eq. (26)), thus ensuring the uniqueness of the kinematical description. Given the hypothesis of shear indeformability, the strains εij of the Green–Lagrange strain matrix E ¼ ½εij , that is assumed as deformation measure, can be expressed as functions of the derivatives of the warping functions, of the curvatures μi (i¼1,2,3) and of the elongation eC of the shear center axis. The curvatures μi (i¼1,2,3) and the elongation eC are the

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generalized strain measures of the one-dimensional polar beam model and the antisymmetric curvature matrix C, referred to the undeformed base b, is found as C ¼ RT R0 where ð U Þ0 ¼ ∂=∂x3 . Cases characterized by large torsional curvature and moderately large flexural curvatures are analyzed. Therefore, a suitable ordering μ3 ¼ OðεÞ; μi ¼ Oðε3=2 Þ ði ¼ 1; 2Þ is introduced, also assuming eC and ϕi ði ¼ 1; 2; 3Þ of order ε; where ε is a small parameter. In this case, since only terms up to ε2  order are retained in the model, also quadratic terms in the flexural curvatures (of order ε3 ) have to be neglected. Hence, only quadratic terms in the torsional curvature should be considered (of order ε2 ). Considering terms up-to the ε2 -order Green–Lagrangian strains are defined as ε11 ¼ ϕ1;1 þ 1=2ϕ23;1 ε22 ¼ ϕ2;2 þ 1=2ϕ23;2 γ 12 ¼ ϕ1;2 þ ϕ2;1 þ ϕ3;1 ϕ3;2 γ 13 ¼  μ3 ðx2  x2C Þ þϕ3;1 þϕ1;3 γ 23 ¼ μ3 ðx1  x1C Þ þ ϕ3;2 þ ϕ2;3 ε33 ¼ eC þμ1 ðx2 x2C Þ  μ2 ðx1  x1C Þ þ ϕ3;3 þ 1=2μ23 ½ðx1  x1C Þ2 þðx2  x2C Þ2 

(3)

02 where γ ij ¼ 2εij and eC ¼ u03 þ 1=2ðu02 1 þu2 Þ. In the warping components ϕi;j , the second subscript after the comma, j(j¼1,2,3), stands for derivation with respect to the correspondent variable xj (j¼1,2,3). Details can be found in [24]. It is possible to note that, as expected, due to hypotheses II and IV, the in-plane strains are a direct consequence of the sole warping, while the section torsional rotation affects the longitudinal strain ε33 ; as shown by the nonlinear elongation term proportional to the squared torsional curvature μ3 .

2.2. Evaluation of the warping functions With the aim of having a one-dimensional model, depending on x3 only, it is necessary to find the dependence of the warping functions ϕi ðx1 ; x2 ; x3 ; tÞ on the transversal coordinates x1 and x2 . It is possible to find an expression of the warping functions that depends only on the local abscissa c (see Fig. 1), running along the middle line of the thin-walled section, only if the warping components ϕk ðk ¼ 1; 2; 3Þ, and particularly ϕ3 , can be considered constant over the thickness (i.e. if the beam is thin-walled). This assumption permits to find ϕ1 and ϕ2 after imposing the vanishing of the in-plane strains ε11 ; ε22 and γ 12 on the whole section, as assumed in hypothesis III, even if the plane strain problem is overdetermined. Named n the inward normal to the middle line at P, the final form of the warping is obtained by evaluating the strains εcc ; εnn ; γ cn and γ 3c , and imposing their vanishing together with the conditions given by Eq. (2), as ϕ1 ¼ β1 μ23 ; ϕ2 ¼ β2 μ23 ; ϕ3 ¼ α1 μ3 þ β3 μ03 μ3

(4)

where only ε -order terms are retained, α1 is the Vlasov sectorial area and the functions βi , that can be defined as nonlinear sectorial areas, is defined in [26]. The remaining strain ε33 can be found replacing Eq. (4) in Eq. (36) 2

ε33 ¼ eG þ μ1 x2 μ2 x1 þ μ03 α1 þ 12 μ23 s2 þ ðμ3 μ03 Þ0 β3

(5)

where eG ¼ eC  μ1 x2C þμ2 x1C is the longitudinal strain of the centroid axis and s2 is the square of the distance between the shear center of the section and the generic point P on the middle line (Fig. 1). The differences between Eq. (5) and the expression obtained with the Vlasov linear theory are in the last two terms, which account for the elongation due to torsion and the nonlinear warping, respectively, and in the nonlinear curvatures of the other terms, which however appear formally the same of the linear theory. 2.3. Internal constraints Taking into account the internal constraints of hypotheses I and IV, the number of independent displacement variables can be reduced. Specifically, as shown in [24], the shear indeformability entails that the flexural rotations ϑ1 and ϑ2 can be written as functions of the spatial derivatives of the displacements ui , tgϑ1 ¼ 

u02 ; 1 þu03

u01 ffi tgϑ2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 2 u02 1 þð1 þ u3 Þ

Defining the inextensibility condition as the vanishing of the mean value of ε33 on the cross-section: Z ε33 dA ¼ 0;

(6)

(7)

A

using Eq. (6), u03 can be expressed as u03 ¼ f ðu1 ; u2 ; ϑ3 ; tÞ. As found in [24], the combined use of Eq. (7) and Eq. (36) leads to eG ¼  1=2μ23 ρ2C , so that the longitudinal strain can be written in compact form as the sum of the flexural εf , torsional elongation εt and warping εϕ terms. Specifically ε33 ¼ εf þ εt þ εϕ , where εf ¼ μ1 x2 μ2 x1 ;

εt ¼ 12 μ23 ðs2  ρ2C Þ;

εϕ ¼ μ03 α1 þðμ3 μ03 Þ0 β3

(8)

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Relations (6) and (7) allow to find simplified expressions of the curvatures as functions of the independent displacements u1 , u2 and ϑ3 0 μ1 ¼  u002 þu001 ϑ3 x1C ðu001 u002 þ u000 1 u2 Þ 0 000 μ2 ¼ u001 þ u002 ϑ3 þx1C ðu002 1 þ u1 u1 Þ 0 μ3 ¼ ϑ03  u01 u002  x1C ð2u01 u001 u002 þ u01 u000 1 u2 Þ:

(9)

Specifically, Eqs. (9) are found expanding in series the nonlinear curvatures for a cross-section having x1 as symmetry axis. Moreover, standing the difference of orders between the flexural and torsional curvatures, in μ1 and μ2 only quadratic terms are retained, while μ3 contains up to cubic terms. 2.4. Equations of motion A Hamiltonian approach is used to obtain the equations of motion of the isotropic thin-walled beam having the axis x1 as a symmetry axis. Z t2 Z l
  δH ¼ δLðu1 ; u2 ; ϑ3 ; tÞ þQ 1 δu1 þ Q 2 δu2 þQ 3 δϑ3 þ δ λðf ðu1 ; u2 ; ϑ3 ; tÞÞ  u03 dx3 dt (10) t1

0

where the Lagrangian parameter λ accounts for the inertial longitudinal forces and Q1, Q2 and Q3 are distributed loads that do work in u1 ; u2 and ϑ3 . The kinetic energy can be evaluated as ! Z 3 1 l X 1 2 Tffi m u_ 2i  m x1C u_ 2 ϑ_ 3 þ ℑC ϑ_ 3 dz (11) 2 0 i¼1 2 where m is the mass per unit length of the beam and ℑC the polar mass-moment with respect to the shear center. Eq. (11) is found under the assumption that flexural rotatory and warping inertia terms are negligible and linearizing the rotation _ where the dot denotes time-differentiation. matrix R and the angular velocity matrix W ¼ R T R, Neglecting the effect of the Poisson ratio, the elastic potential energy per unit length reads: Z 1 ½Gðγ 231 þγ 232 Þ þ Eε233 dA V¼ 2 A i 1 1h (12) ¼ GJ μ23 þ EI 1 μ21 þ EI 2 μ22 þEΓ t μ43 þ EΓ 1 μ03 2 þEΓ 2 ðμ3 μ03 Þ02 þ 2EΓ f t μ2 μ23 2 2 where G and E are elastic moduli. Coefficients EI i ði ¼ 1; 2Þ, EΓ 1 and GJ are the flexural stiffness, the Vlasov warping torsional stiffness and the St. Venant torsional stiffness, respectively. Instead, Γ 2 ; Γ t and Γ f t are new coefficients defined as Z Z Z (13) Γ 2 ¼ t β3 ðcÞdc; Γ t ¼ 1=4t ðsðcÞ2  ρ2C Þdc; Γ f t ¼ 1=2t sðcÞ2 x1 ðcÞdc c

c

c

where t is the small thickness of the section; the coefficient Γ 2 can be interpreted as the nonlinear Vlasov sectorial area, the coefficient Γ t is related to the torsional elongation effect εt only, while Γ f t depends on the kinematical coupling between torsional elongation and warping. It has to be noted that Eq. (12) is obtained assuming that γ 31 and γ 32 contribute only to the St. Venant torsional elastic term and that, given the symmetry of the beam, many terms vanish after the integration on the area. The procedure leads to four equations. Finding λ from the simplest one ðmu€ 3  λ0 ¼ 0Þ, and substituting it in the remaining ones, three equations of motion with nonlinear terms of up to cubic order are obtained: 000 00 002 000 0 0002 0 mu€ 1 þ EI 2 f  ½x1C ðu001 u000 1 þu1 u2 ϑ3 þ x1C u1 u1 þ x1C u1 u1 Þ 00 00 002 00 00 0 000 003 0 00 000 00 000 þ ½u1 þ ϑ3 u2 þ x1C ð3u11 þ 2u1 u2 ϑ3 þ u1 u þ 2x1C u1 þ 2x1C u1 u1 u1 Þ  ½x1C ðu01 u001 þ u01 u002 ϑ3 þ x1C u01 u002 1 þ x1C Þ g 00 00 2 002 00 00 000 0 00 002 000 0 00 00 þ EI 1 f½ u2 ϑ3 þu1 ϑ3 þ x1C ðu2  2u1 u2 ϑ3 u1 u2 ϑ3 þ x1C u1 u2 þ x1C u1 u2 u2 Þ 02 000  ½x1C ðu02 u002  u001 u02 ϑ3 þ x1C u001 u02 u002 þx1C u000 1 u2 Þ g 0 002 00 0 00 00 0 000 0 0 0 0 00 0 00 þ GJf ½u1 u2  u2 ϑ3 x1C ð2u1 u2 ϑ3 þ u1 u2 ϑ3 Þ  ½2x1C u1 u2 ϑ3  þ ½x1C u01 u02 ϑ03 000 g 0 0002 000 00 000 00 00 00 000 00 0000 0 00 0 00 002 00 00 þ EΓ 1 f ½u001 u002 u000 2 þ u1 u2  u2 ϑ3  x1C ð3u1 u2 ϑ3 þ 2u1 u2 ϑ3 þu1 u2 ϑ3 Þ þ½u1 u2  u2 ϑ3 0 00 000 00 00 00 000 0 00 0 000 00 00 00 0 00 0 00 00 000 0 0 00 0000 þ x1C ðu1 u2 u2  4u1 u2 ϑ3  u1 u2 ϑ3  2u1 u2 ϑ3 Þ þ ½x1C ðu1 u2 ϑ3 þ 3u1 u2 ϑ3 Þ  ½x1C u1 u2 ϑ2  g 00 02 00 0 02 000 þEΓ f t f½ϑ02 3 þ 2x1C u1 ϑ3   ½x1C u1 ϑ3  g ¼ Q 1 00 00 000 00 000 00 0002 0 0 mu€ 2 þmx1C ϑ€ 3 þ EI 1 f  ½x1C ðu000 1 u1  u1 u1 ϑ3 þ x1C u1 u1 u2 þ x1C u1 u2 Þ 000 0 002 00 00 000 0 00 þ ½u002 ϑ3 u001 þ x1C ð2u001 u002 u002 1 ϑ3 þ u1 u2 þ x1C u1 u2 þ x1C u1 u1 u2 Þ g 0 000 00 þ EI 2 f½  u001 ϑ3 þ u002 ϑ23 þ x1C ðu002 1 ϑ3  u1 u1 ϑ3 Þ g 0 0 02 00 0 0 0 00 0 00 þ GJf½x1C u01 u000 1 ϑ3  þ ½u1 u2 u1 ϑ3  2x1C u1 u1 ϑ3  g 00 0 0000 00 0 002 00 00 00 0 00 000 002 00 0 000 00 00 þ EΓ 1 f½x1C ðu001 u000 1 ϑ3 þu1 u1 ϑ3 Þ þ½u1 u2  u1 ϑ3 þu1 u1 u2 x1C ð2u1 ϑ3 þ 3u1 u1 ϑ3 Þ

(14)

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241

Fig. 2. Mechanical system. Table 1 Cross-section and material characteristics. h ¼ 0:05 m t ¼ 0:002 m xG ¼ 0:00625 m x1C ¼  0:0156 m

b ¼ 0:025 m l ¼ 1:0 m xC ¼ 0:00938 m A ¼ 0:0002 m2

I1 ¼ 8:33 U 10  8 m4

J ¼ 2:67 U 10  10 m4

Γ ¼ 5:29 U 10  12 m6 Γ 2 ¼ 5:714 U 10  21 m10

I 2 ¼ 1:302 U 10  8 m4 ρ2C ¼ 0:000726 m

Γ f t ¼  3:825 U 10  10 m5

Γ tt ¼ 1:265 U 10  11 m6

E ¼ 2:07 U 1011 N=m2 m ¼ 1:56 kg=m

G ¼ 8:61 U 1010 N=m2 ℑC ¼ 0:00114 kg m

000 0 00 00 000 ½u01 u001 u002  u01 ϑ003 þu02 1 u2  2x1C u1 u1 ϑ3  g 00 þEΓ f t f½ϑ3 ϑ02 3  g ¼ Q2

(15)

00 00 002 00 00 000 0 ℑC ϑ€ 3 mx1C u€ 2 þEI 1 fu002 1 ϑ3 u1 u2  x1C ðu1 u2 þu1 u1 u2 Þg 002 00 0 000 00 þ EI 2 fu001 u002 þ u002 2 ϑ3 þ x1C ðu1 u2 þ u1 u1 u2 Þg 0 0 00 0 00 00 0 000 0 0 þ GJf  ½ϑ3  u1 u2 x1C ðu1 u1 u3 þu1 u1 u2 Þ g 002 00 00 000 0 0 000 00 0 00 000 0 0000 0 00 þ EΓ 1 f½ϑ003 u001 u002  u01 u000 2  x1C ðu1 u2 þ u1 u1 u2 þ 3u1 u1 u2 þ 2u1 u1 u2 þ u1 u1 u2 Þ g 000 0 0002 0 002 0 00 000 00 0 002 02 000 000 þ EΓ 2 f  ½ϑ002 3 ϑ3 þ ϑ3 ϑ3  þ 2½ϑ3 þ ϑ3 ϑ3 ϑ3   ½ϑ3 ϑ3 þϑ3 ϑ3  g 00 0 00 0 002 0 0 000 0 0 þEΓ f t fu002 ϑ03 3  2½u1 ϑ3 þ u2 ϑ3 ϑ3 þ x1C ðu1 ϑ3 þ u1 u1 ϑ1 Þ g 00  EΓ t f6ϑ02 3 ϑ3 g ¼ Q 3

(16)

Eqs. (14)–(16) have to be coupled with the relevant boundary conditions, here omitted for brevity. As shown, the nonlinear effects due to the new terms (underlined ones) are mainly related to the interaction between torsional and flexural elongations and depend on EΓ f t . 3. Discretized equations of motion The Galerkin method is used to obtain the discretized equations of motion from Eqs. (14)–(16) and the relevant boundary conditions. 3.1. Geometrical and mechanical characteristics of the beam The same beam used in [26] and shown in Fig. 2 has been considered in the analyses. The geometrical and mechanical characteristics (Table 1), derived from a preliminary study, ensure that the response is characterized by the torsional curvature μ3 prevailing over the flexural ones μ1 ; μ2 . 3.2. Discretization of the equations of motion Due to the characteristics of the section with the symmetry axis x1 (Fig. 2), some eigenfunctions of the linearized Eqs. (14)–(16) are coupled in the displacements u2 and ϑ3 , whereas others are described by the only u1 displacement, which is

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u1

u2 0

t

t [s]

10

Fig. 3. Perturbation along the direction orthogonal to the excitation.

uncoupled from the other two (u2 and ϑ3 ). According to this partial coupling of the linearized equations, the displacements u2 and ϑ3 depend on two modal amplitudes. To reduce the complexity of the discretized equations of motion and to make easier their numerical integration, uncoupled eigenfunctions of the homogeneous, linearized problem given by Eqs. (14)–(16) when x1C ¼ 0 (double symmetric open cross-section) have been used. A convergence analysis is performed to establish the minimum number of uncoupled eigenfunctions that have to be used in the discretization procedure, to obtain accurate results in the evaluation of the stability of the one-component dynamic solutions. The first two uncoupled eigenfunctions of each displacement (a total of six functions) have been used as discretizing functions, which is necessary to guarantee satisfactory results. In other words, the first two flexural modes in each plane of deflection and the first two torsional modes have been taken into account. The displacements vector fu1 ; u2 ; ϑ3 gT is then expressed as a linear combination of the given x3-functions and unknown tfunctions qij ðtÞ: 8 9 2 ϕ11 ðx3 Þ > < u1 > = u2 ¼ 6 4 0 > :ϑ > ; 0 3

0 ϕ21 ðx3 Þ 0

9 2 38 q ðtÞ ϕ12 ðx3 Þ > < 1 > = 6 0 ðtÞ q 0 7 þ4 5 3 > > ϕ31 ðx3 Þ : q5 ðtÞ ; 0 0

first three eigenfunctions

0 ϕ22 ðx3 Þ 0

9 38 q ðtÞ > < 2 > = 0 7 5 q4 ðtÞ > > ϕ32 ðx3 Þ : q6 ðtÞ ; 0

(17)

second three eigenfunctions

where ϕij ðx3 Þ (j ¼ 1; 2) are the first two uncoupled eigenfunctions for each displacement. To obtain the eigenfunctions of a cantilever beam, the relevant homogeneous boundary conditions are: u1 ð0Þ ¼ 0; u01 ð0Þ ¼ 0; EI 2 u001 ðlÞ ¼ 0; EI 2 u000 1 ðlÞ ¼ 0 u2 ð0Þ ¼ 0; u02 ð0Þ ¼ 0; EI 1 u002 ðlÞ ¼ 0; EI 1 u000 2 ðlÞ ¼ 0 0 ϑ3 ð0Þ ¼ 0; ϑ03 ð0Þ ¼ 0; EΓ 1 ϑ003 ðlÞ ¼ 0; EΓ 1 ϑ000 3 ðlÞ GJϑ3 ðlÞ ¼ 0

(18)

The obtained discretized equations are shown in Appendix A. In these equations the parameter γ selects the new terms, introduced in the equations of motion by the nonlinear contributions due to warping and torsional elongation. They correspond to the underlined terms in Eqs. (14)–(16). In the numerical investigations, when γ ¼ 0 is imposed, the results refer to an open-cross section beam where only the linear warping term (Vlasov model) is taken into account. On the contrary, when γ ¼ 1 is imposed, the effects of all the new nonlinear terms are considered. A modal damping equal to 2 percent (ξi ¼ 0:02, i ¼ 1…6) is considered in the numerical investigation.

4. Analytical and computational investigations The aim of the investigations is the study of the stability of the forced one-component oscillations that involve only the displacement u1 . A harmonic force, acting at the free end of the cantilever beam, directed along the symmetry axis x1 (see Fig. 2) is used as excitation: FðtÞ ¼ A sin ðΩtÞ ¼ A sin ð2πf tÞ

(19)

where A is the amplitude, Ω is the driven circular frequency and f is the correspondent frequency (Ω ¼ 2πf ). This excitation defines a one-component oscillation along the same direction of the force, where only the displacement u1 is involved. When this one-component oscillation becomes unstable, also the other components u2 and ϑ3 , are involved in the motion and the oscillations of the beam become a spatial motion. Both the amplitude A and the frequency f are varied to investigate the stability of the forced one-component oscillations.

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Table 2 First three frequencies of the analytical mode (AM) and finite element model (FEM). AM (Hz)

FEM (Hz)

23.13 40.30 81.29

23.38 42.08 82.62

Fig. 4. Modes of the beam FEM.

4.1. Analytical investigation With “analytical investigation” we refer to the two types of analyses performed by using the discretized equations reported in Appendix A. The first analysis consists in the study of the stability of the periodic one-component oscillation occurring when the force F is applied along the symmetry axis, by varying the amplitude A and the frequency f of the harmonic excitation, given by Eq. (19). Use of the Autos [28] bifurcation software is done, by analyzing directly the discretized equations. Moreover, numerical integration of the discretized equations are performed, with the aim to compare the time-histories of the oscillations with those obtained by a computational finite element (FE) model. The numerical integrations have been performed by using the standard NDSolve command of Mathematicas [29]. To check the stability of the one-component oscillation, a perturbation along a direction orthogonal to that of the excitation is imposed to the beam. Specifically, a harmonic force is applied at the free end of the beam along the symmetry axis x1 , to excite the one-component oscillation. After the transitory time, at the time t (see Fig. 3), an impulsive force is applied in the x2 direction, orthogonal to the first one, to introduce a perturbation to the one-component oscillation. If this last is stable, the u2 displacement, after a certain time that depends on the damping of the system, vanishes as shown in Fig. 3b.

4.2. Computational investigation A finite element model has been built in the ADINAs environment [27] using bi-dimensional shell elements to compare FEM results with those of the analytical model. The length of the beam has been divided into 25 parts while the section in 6 parts, two for each wall of the beam, thus meshing the beam with 150 shell elements. To check the efficiency of the computational model the comparison among the first three frequencies of the beam, provided by the analytical and the computational models, is carried out (Table 2). As it is possible to observe, the computational frequencies match very well the analytical ones, thus assuring the efficiency of the computational model. In Fig. 4 the first three computational modes are shown. As expected, the first mode only displacement u1, while the other two involve both displacement u2 and rotation ϑ3 . A nonlinear time-history analysis is performed to check the stability of the one-component solution with the FE model. As in the analytical case, harmonic force is applied at the free end of the beam along the direction of the symmetry axis x1 , to excite the one-component oscillation. Then, at the time t (see Fig. 3), an impulsive force is applied in a direction orthogonal to the first one, to introduce a perturbation to the one-component oscillation.

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Fig. 5. Bifurcation diagrams of the one-component oscillation at Ω ¼ 79:0 Hz: results with (a) and without (b) nonlinear torsional contributions of the elongation and warping; stable analytical solutions (solid lines), unstable analytical solutions (dashed lines), computational stable solutions (dotted lines with asterisks).

5. Stability of the one-component oscillation The stability of the one-component periodic oscillation is strongly influenced by the conditions of internal resonance. Two different types of possible resonance conditions have been considered, both capable of indirectly exciting a mode of vibration. Under a harmonic force along the symmetry axis x1 , the solution after a transient is a one-component u1 periodic oscillation, with the same frequency of the harmonic force. When a perturbation is introduced into the system, the periodic solution forces the other components of the motion as a harmonic excitation, due to couplings.

5.1. Resonance ΩC2ω2 This resonant case occurs when the frequency of the u1 forced motion is twice the frequency of the 2nd mode, due to the presence of terms of the type u1ðnÞ uðmÞ in the equation of motion (15), where ðnÞ and ðmÞ represent generic degrees of 2

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ðmÞ differentiation. Indeed, in this resonant condition the coupling terms uðnÞ are able to excite directly the second mode of 1 u2 vibration. In Fig. 5 the bifurcation diagrams are shown. The curves in the diagrams represent the maximum and the minimum values of the amplitudes of the periodic or quasi-periodic oscillations. They have been obtained by fixing the frequency of the harmonic force (79 Hz) and varying its amplitude A, taken as bifurcation parameters. In Fig. 5a, it is possible to observe that before point S, only the stable, periodic, one-component u1 solution exists (solid lines). At point S a static bifurcation occurs, and a new branch of stable, spatial, periodic solutions bifurcates from this point. The one-component oscillation becomes unstable after the bifurcation (dashed lines). The new stable branch of periodic solution represents a spatial oscillation of the beam with all the three displacement components u1 ; u2 and ϑ3 involved in the motion. At point H, along the new branch of stable, spatial, periodic solutions a Neimark bifurcation takes place. It is worth remarking that, in the amplitude modulation equations, this bifurcation appears as a classical Hopf bifurcation. A branch of stable quasi-periodic solutions bifurcates from point H, and the periodic spatial oscillations become unstable. The curves delimiting the gray regions in Fig. 5a represent the range of the maximum amplitudes of the quasi-periodic oscillations. In Fig. 5b the same analysis is performed without considering the nonlinear torsional elongation effects and the nonlinear contribution of the warping, by letting γ ¼ 0 in the discretized Eqs. (A.1)–(A.6), that correspond to vanish the underlined terms in Eqs. (14)–(16). Comparing the bifurcation diagrams of Fig. 5a and b, it is possible to observe the importance of these nonlinear contributions, which are able to change drastically the behavior of the system. Finally, it is interesting to notice that the curves of one-component oscillation are almost the same in the cases with and without the nonlinear contributions of the torsional elongation and the warping, which exhibit their role on the critical conditions and on the bifurcated response. In Fig. 5a, in the graph of the u1 component, also the results obtained from the computational model are shown (dotted line with asterisks). As it is possible to observe in the zoom of the range around the point S, the FE model admits a bifurcation point (Sc) for an amplitude value A that is slightly greater than the one of the analytical model, from which it follows that along the bifurcated branch the computational amplitude u1 is slightly smaller than the analytical one. This happens since the frequencies of the FE model are on a small scale, different from the analytic ones (see Table 2), hence the external force with fixed frequency produces slightly different effects on the two models.

5.2. Resonance 2ΩCω3  ω2 The stability of the periodic one-component oscillation forced with a frequency 2ΩCω3 ω2 is governed by the resonance which occurs since in the equations of motion (15)–(16) terms u1ðnÞ2 uðmÞ and u1ðnÞ2 ϑðmÞ appear, respectively. These 2 3 terms are capable of exciting the second or third modes of vibration. In Fig. 6 the bifurcation diagrams are shown. Also in this case the curves in the diagrams represent periodic or quasiperiodic oscillations. They have been obtained fixing the frequency of the harmonic force along the symmetry axis (20.5 Hz) and varying its amplitude A, which is taken as the bifurcation parameter. In Fig. 6a, and specifically in the zoom of the circled area (Fig. 6b), it is possible to observe that before point S, only the stable, periodic, one-component u1 solution exists. At point S a static bifurcation occurs, the one-component oscillation becomes unstable, and a new branch of stable spatial periodic solutions bifurcates from this point. At point H, along the new branch of stable spatial solutions a Neimark bifurcation takes place. A branch of stable quasi-periodic solutions bifurcates from point H, and the periodic spatial oscillation becomes unstable. Also in this case, the curves that delimit the gray regions in Fig. 6 represent the range of the maximum amplitudes of the quasi-periodic solutions oscillates. Finally, at point K a homoclinic bifurcation occurs and after this point the quasi-periodic solutions disappear. Observing a Poicarrè map, the periodic solution would appear as a point while the quasi-periodic solution as a closed curve. At a homoclinic bifurcation point, a collision between the periodic solution and the quasi-periodic solution takes place. It is worth remarking that, in the range of the analyses (A A ½0; 45), the system without the nonlinear contribution of the torsional elongation and of the warping does not present any bifurcation point and the one-component oscillations remains always stable due to the absence of the underlined coupling terms in the equations of motion (14)–(16). In Fig. 7 the time-histories of the u1 displacement component, obtained from the analytical and computational FE models, are shown. The analyses have been performed as described in Sections 4.1 and 4.2, by applying harmonic forces along the symmetry axis and, after a transient, a perturbation along the x2 direction. Specifically, Fig. 7a and b refer to the u1 displacement of the analytical and the computational models, respectively, for an external force with Ω ¼ 20:5 Hz and amplitude A ¼ 40:1 kN. These values are in the range between the bifurcation point S and Neimark bifurcation point H (label A1 in the u1 -graph of the Fig. 6b). After a very similar transient, both the analytical and the computational timehistories become stationary with amplitudes slightly smaller than the stable one-component oscillation. When the amplitudes of the external force are above the Neimark bifurcation H (A ¼ 41:1 kN, label A2 in the u1 -graph of Fig. 6b) the time-histories of the u1 component change. In Fig. 7c and d, the u1 displacements of the analytical and the computational models are shown, respectively. Also in this case, after the transient, a periodic modulation of the maximum and the minimum of the amplitudes appear. The maximum and the minimum values of the amplitude u1 are slightly different between the two models. As previously said, this happens since the frequencies of the FE model are slightly different from the analytic ones.

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Fig. 6. Bifurcation diagrams of the one-component oscillation at Ω ¼ 20:5 Hz: (a) results with nonlinear contributions of the elongation and warping; (b) zoom of the previous diagrams; stable analytical solutions (solid lines), unstable analytical solutions (dashed lines) (A1 ¼ 40:1 kN, A2 ¼ 41:1 kN).

Finally, a small asymmetry of the computational time-histories (Fig. 7b, d) with respect the time-axis can be observed, since the absolute values of the maximum and the minimum amplitudes are not the same. This is due to the fact that, when the amplitude of the displacement becomes sufficiently high, the channel section of the beam can lose its shape (Fig. 8), in particular the amplitude is greater in the direction where the free ends of the horizontal walls of the channel section are compressed (see Fig. 2). This kind of behavior cannot be described by the analytical model, where the in-plane shape of the section is fixed.

6. Conclusions In this paper three simplified equations of motion have been deduced by a more complete model of the same authors, describing the dynamics of an inextensional and shear undeformable nonlinear 3D open-cross section thin-walled beam with a section that presents one symmetry axis. The chosen geometrical and mechanical characteristics allow to obtain a beam model where the torsional curvature is greater than the flexural ones. A Galerkin procedure has been performed using a suitable number of shape functions to obtained discretized equations of motion. The dynamic response of the beam, when a harmonic force is applied at the free end of the cantilever beam, has been investigated. The excitation is directed along the symmetry axis of the section of the beam, hence only a one-component oscillation with the same direction of the force occurs. The stability of this one-component oscillation, which occurs in the same direction of the excitation, has been deeply studied. The results obtained by the analytical model have been compared with those provided by a computational finite element model. The good agreement among the results of the analytical and the computational models confirms the effectiveness of

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Fig. 7. Time-histories of the u1 displacement component.

Fig. 8. Loss of shape of the section.

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the simplified nonlinear open-cross section, thin-walled beam model, which maintains the same performances of the complete model. The effects of the nonlinear contributions of the torsional elongation and of the warping have been investigated and the importance to consider these nonlinear effects in the study of the stability of one-component dynamic solutions has been pointed out. The bifurcation diagrams completely change when the nonlinear contributions of the torsional elongation and of the warping are not taken into account. In particular in the scenarios of two different resonance conditions, static bifurcations characterize the evolution of the one-component oscillations in a periodic spatial motion and/or in a quasi-periodic spatial motion. Appendix A 24:9936 F 1 ðt Þ þ 21121:7 q1 ðt Þ 1286:13 q1 ðt Þ2 þ63:7486 q1 ðt Þ3 þ4347:45 q1 ðt Þ q2 ðt Þ þ 20515:6 q3 ðt Þ2 q2 ðt Þ  32617:8 q2 ðt Þ2  64902:5 q1 ðt Þ q2 ðt Þ2  126107:q2 ðt Þ3 þ203:995 q1 ðt Þ q3 ðt Þ2 þ126:74γ 2 q1 ðt Þ q3 ðt Þ2  66709:6 q2 ðt Þ q3 ðt Þ2  10685:8 q1 ðt Þ2 þ 1:02672  106 q3 ðt Þ q4 ðt Þ  1352:44 q1 ðt Þ q3 ðt Þ q4 ðt Þ 295:847γ 2 q2 ðt Þ q3 ðt Þ2 þ 1076:77γ 2 q1 ðt Þ q3 ðt Þ q4 ðt Þ  456583:q2 ðt Þ q3 ðt Þ q4 ðt ÞÞ þ 5:94757γq5 ðt Þ2 þ 2712:66γ 2 q2 ðt Þ q3 ðt Þ q4 ðt Þ þ3:05608  106 q4 ðt Þ2 þ 4991:67 q1 ðt Þ q4 ðt Þ2 þ14613:7γ 2 q1 ðt Þq4 ðt Þ2  811451:q2 ðt Þ q4 ðt Þ2 þ8456:66γ 2 q2 ðt Þq4 ðt Þ2  16713:8 q3 ðt Þ q5 ðt Þ þ3073:3 q1 ðt Þ q3 ðt Þ q5 ðt Þ  6:00852γq1 ðt Þ q3 ðt Þ q5 ðt Þ  93:9447γq3 ðt Þ q5 ðt Þ þ 7702:22q2 ðt Þ q3 ðt Þ q5 ðt Þ  23:8631γq2 ðt Þ q3 ðt Þ q5 ðt Þ 118917:q4 ðt Þ q5 ðt Þ þ150378:q1 ðt Þ q4 ðt Þ q5 ðt Þ þ4:01784γq1 ðt Þ q4 ðt Þ q5 ðt Þ  614:311γq4 ðt Þq5 ðt Þ þ525387:q2 ðt Þ q4 ðt Þ q5 ðt Þ þ 70:898γq2 ðt Þ q4 ðt Þ q5 ðt Þ þ 6658:73 q1 ðt Þ q5 ðt Þ2 þ 76582:4 q2 ðt Þ q5 ðt Þ2  506:65γq2 ðt Þ q5 ðt Þ2  179:287γq1 ðt Þ q5 ðt Þ2  15944:3γq2 ðt Þ q6 ðt Þ2 þ 34796:4q3 ðt Þ q6 ðt Þ þ 130:559γq3 ðt Þ q6 ðt Þ þ 3698:18 q1 ðt Þ q3 ðt Þ q6 ðt Þ þ 412:975γq1 ðt Þ q3 ðt Þ q6 ðt Þ þ 6567:29 q2 ðt Þ q3 ðt Þ q6 ðt Þ þ1414:83γq2 ðt Þ q3 ðt Þ q6 ðt Þ þ 147589:q4 ðt Þ q6 ðt Þ 1648:88γq4 ðt Þ q6 ðt Þ þ 130353:q1 ðt Þ q4 ðt Þ q6 ðt Þ þ1575:74γq1 ðt Þ q4 ðt Þ q6 ðt Þ þ 464163:q2 ðt Þ q4 ðt Þ q6 ðt Þ þ 5591:94γq2 ðt Þ q4 ðt Þ q6 ðt Þ þ 3682:79γq5 ðt Þ q6 ðt Þ  21614:9 q1 ðt Þ q5 ðt Þ q6 ðt Þ  1816:65γq1 ðt Þ q5 ðt Þ q6 ðt Þ þ21666:8 q1 ðt Þ q6 ðt Þ2 169434:q2 ðt Þ q5 ðt Þ q6 ðt Þ 6742:13γq2 ðt Þ q5 ðt Þ q6 ðt Þ þ 7740:08γq6 ðt Þ2 þ 130272:q2 ðt Þ q6 ðt Þ2 þ5:81332 q01 ðt Þ  4428:09gγq1 ðt Þ q6 ðt Þ2 þ 1:q001 ðt Þ ¼ 0

(A.1)

24:9936F 1 ðt Þ þ2173:73 q1 ðt Þ2 þ 187:727q1 ðt Þ3 þ 829533:q2 ðt Þ  65235:6q1 ðt Þq2 ðt Þ þ9538:87 q1 ðt Þ2 q2 ðt Þ  97297:4 q2 ðt Þ2 þ 3076:43 q1 ðt Þ q2 ðt Þ2 þ 98328:5 q2 ðt Þ3 þ 4715:53 q3 ðt Þ2 þ 1877:67 q1 ðt Þ q3 ðt Þ2  295:847γ 2 q1 ðt Þ q3 ðt Þ2  141804:q2 ðt Þ q3 ðt Þ q6 ðt Þ þ 1814:13γ 2 q2 ðt Þ q3 ðt Þ2 þ 902085:q3 ðt Þ q4 ðt Þ þ 19841:3 q1 ðt Þ q3 ðt Þ q4 ðt Þ þ 2712:66γ 2 q1 ðt Þ q3 ðt Þ q4 ðt Þ þ 8734:81 q2 ðt Þ q3 ðt Þ q4 ðt Þ 1190:33γ 2 q2 ðt Þ q3 ðt Þ q4 ðt Þ þ 2:96508  106 q4 ðt Þ2 þ 15890:9 q1 ðt Þ q4 ðt Þ2 þ 314651:q2 ðt Þ q4 ðt Þ2  48:1705γq1 ðt Þ q3 ðt Þ q5 ðt Þ þ106263:γ 2 q2 ðt Þ q4 ðt Þ2 118917:q3 ðt Þ q5 ðt Þ þ 164:517γq3 ðt Þ q5 ðt Þ  1285:79 q1 ðt Þ q3 ðt Þ q5 ðt Þ þ8456:66γ 2 q1 ðt Þ q4 ðt Þ2  45900:9 q2 ðt Þ q3 ðt Þ q5 ðt Þ 382:579γq2 ðt Þ q3 ðt Þ q5 ðt Þ þ 172393:q1 ðt Þ q4 ðt Þ q5 ðt Þ  1:80232  106 q4 ðt Þ q5 ðt Þ  745:137γq4 ðt Þq5 ðt Þ þ814058:q2 ðt Þ q4 ðt Þ q5 ðt Þ 450:85γq2 ðt Þ q4 ðt Þ q5 ðt Þ 13:5248γq1 ðt Þ q4 ðt Þ q5 ðt Þ þ 76582:4 q1 ðt Þ q5 ðt Þ2  84:6021γq1 ðt Þ q5 ðt Þ2 þ 1:26057  106 q2 ðt Þ q5 ðt Þ2  874:038γq5 ðt Þ2 þ 147589:q3 ðt Þ q6 ðt Þ  907:524γq3 ðt Þ q6 ðt Þ  2420:71 q1 ðt Þ q3 ðt Þ q6 ðt Þ 468:996γq2 ðt Þ q5 ðt Þ2 þ 28204:1 q2 ðt Þ q3 ðt Þ2  291:766γq1 ðt Þ q3 ðt Þ q6 ðt Þ  994:137γq2 ðt Þ q3 ðt Þ q6 ðt Þ þ 1:83181  106 q4 ðt Þ q6 ðt Þ 3058:31γq4 ðt Þ q6 ðt Þ  335:294γq1 ðt Þ q4 ðt Þ q6 ðt Þ þ 111169:q1 ðt Þ q4 ðt Þ q6 ðt Þ 154925:q2 ðt Þ q4 ðt Þ q6 ðt Þ 1945:88γq2 ðt Þ q4 ðt Þ q6 ðt Þ þ 5818:03γq5 ðt Þ q6 ðt Þ 169434:q1 ðt Þ q5 ðt Þ q6 ðt Þ  2457:72γq1 ðt Þ q5 ðt Þ q6 ðt Þ þ 22926:8γq6 ðt Þ2  2:07865  106 q2 ðt Þ q5 ðt Þ q6 ðt Þ  5612:06γq2 ðt Þ q5 ðt Þ q6 ðt Þ þ 130272:q1 ðt Þ q6 ðt Þ2 5148:49γq1 ðt Þ q6 ðt Þ2 19619:2γq2 ðt Þ q6 ðt Þ2 þ 1:40318  106 q2 ðt Þ q6 ðt Þ2 þ 36:4315 q02 ðt Þ þ1:q002 ðt Þ ¼ 0 135179: q3 ðt Þ  28746:8 q1 ðt Þ q3 ðt Þ þ 203:995 q1 ðt Þ2 q3 ðt Þ þ 126:74γ 2 q1 ðt Þ2 q3 ðt Þ þ 349:302q1 ðt Þq2 ðt Þq3 ðt Þ  591:694γ 2 q1 ðt Þq2 ðt Þq3 ðt Þ 950573:q2 ðt Þ q3 ðt Þ þ 20355:q4 ðt Þ q6 ðt Þ2 þ28204:1 q2 ðt Þ2 q3 ðt Þ þ 1814:13γ 2 q2 ðt Þ2 q3 ðt Þ þ 4:27354  10  9 q4 ðt Þ  48326:9q1 ðt Þ q4 ðt Þ

(A.2)

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249

þ 8011:71q1 ðt Þq2 ðt Þq4 ðt Þ  676:218q1 ðt Þ2 q4 ðt Þ  16851:5γ 2 q1 ðt Þ2 q4 ðt Þ 3:26483  106 q2 ðt Þq4 ðt Þ þ 4367:4q2 ðt Þ2 q4 ðt Þ 210363:γ 2 q2 ðt Þ2 q4 ðt Þ  118082γ 2 q1 ðt Þq2 ðt Þq4 ðt Þ 2558:43γq5 ðt Þq6 ðt Þ2 þ 551:388γq1 ðt Þq5 ðt Þ  3914:75q1 ðt Þ2 q5 ðt Þ þ 1:35988γq1 ðt Þ2 q5 ðt Þ  16713:8q1 ðt Þq5 ðt Þ þ 2405:84γq2 ðt Þq5 ðt Þ  160274:q1 ðt Þq2 ðt Þq5 ðt Þ þ 6:45133γq1 ðt Þq2 ðt Þq5 ðt Þ 118917:q2 ðt Þq5 ðt Þ þ 1040:43q3 ðt Þq5 ðt Þ2  524988:q2 ðt Þ2 q5 ðt Þ  96:435γq2 ðt Þ2 q5 ðt Þ þ11966:q4 ðt Þq5 ðt Þ2 þ 45:8301γq5 ðt Þ3 þ 34796:4q1 ðt Þq6 ðt Þ þ 8350:08γq1 ðt Þq6 ðt Þ  3602:31q1 ðt Þ2 q6 ðt Þ þ147589:q2 ðt Þq6 ðt Þ  161409:q1 ðt Þq2 ðt Þq6 ðt Þ  20:1868gq1 ðt Þ2 q6 ðt Þ  159:703γq1 ðt Þq2 ðt Þq6 ðt Þ þ 27639:9γq2 ðt Þq6 ðt Þ  572940:q2 ðt Þ2 q6 ðt Þ  267:734γq2 ðt Þ2 q6 ðt Þ  3377:32q3 ðt Þq5 ðt Þq6 ðt Þ þ 3385:44q3 ðt Þq6 ðt Þ2  26474:1q4 ðt Þq5 ðt Þq6 ðt Þ þ1037:96γq5 ðt Þ2 q6 ðt Þ  16694:5γq6 ðt Þ3 þ14:7067q03 ðt Þ þ 1:q003 ðt Þ þ0:0182387q005 ðt Þ þ 0:00110549q006 ðt Þ ¼ 0

(A.3)

9196:32q1 ðt Þq3 ðt Þ 676:218q1 ðt Þ2 q3 ðt Þ þ981:17γ 2 q1 ðt Þ2 q3 ðt Þ 1:00567  106 q2 ðt Þq3 ðt Þ þ8011:71q1 ðt Þq2 ðt Þq3 ðt Þ þ5788:35γ 2 q1 ðt Þq2 ðt Þq3 ðt Þ þ 4367:4q2 ðt Þ2 q3 ðt Þ þ4745:97γ 2 q2 ðt Þ2 q3 ðt Þ þ 5:30901  106 q4 ðt Þ  313926:q1 ðt Þq4 ðt Þ þ4991:67q1 ðt Þ2 q4 ðt Þ þ 14613:7γ 2 q1 ðt Þ2 q4 ðt Þ 3:58779  106 q2 ðt Þq4 ðt Þ  9303:92q1 ðt Þq2 ðt Þq4 ðt Þ þ 16913:3γ 2 q1 ðt Þq2 ðt Þq4 ðt Þ þ314651:q2 ðt Þ2 q4 ðt Þ þ106263:γ 2 q2 ðt Þ2 q4 ðt Þ þ 76:8511γq1 ðt Þq5 ðt Þ  118917:q1 ðt Þq5 ðt Þ þ188:882q1 ðt Þ2 q5 ðt Þ þ17:1661γq1 ðt Þ2 q5 ðt Þ þ 1655:36γq2 ðt Þq5 ðt Þ  1:80232  106 q2 ðt Þq5 ðt Þ 125694:q1 ðt Þq2 ðt Þq5 ðt Þ þ 176:184γq1 ðt Þq2 ðt Þq5 ðt Þ 336561:q2 ðt Þ2 q5 ðt Þ þ104:016γq2 ðt Þ2 q5 ðt Þ þ 11966:q3 ðt Þq5 ðt Þ2 þ196965:q4 ðt Þq5 ðt Þ2  200:002γq5 ðt Þ3 þ 147589:q1 ðt Þq6 ðt Þ þ7124:98γq1 ðt Þq6 ðt Þ 9823:91q1 ðt Þ2 q6 ðt Þ þ0:601992γq1 ðt Þ2 q6 ðt Þ þ1:83181  106 q2 ðt Þq6 ðt Þ 186918:q1 ðt Þq2 ðt Þq6 ðt ÞÞ 26474:1q3 ðt Þq5 ðt Þq6 ðt Þ þ 123:375γq1 ðt Þq2 ðt Þq6 ðt Þ 821052:q2 ðt Þ2 q6 ðt Þ  176:43γq2 ðt Þ2 q6 ðt Þ þ 27414:4γq2 ðt Þq6 ðtÞ þ 7054:8γq5 ðt Þ2 q6 ðt Þ þ20355:q3 ðt Þq6 ðt Þ2 þ 219247:q4 ðt Þq6 ðt Þ2 0

þ 22528:γq5 ðt Þq6 ðt Þ2  324788:q4 ðt Þq5 ðt Þq6 ðt Þ þ 6420:28γq6 ðt Þ3 þ92:1651q4 ðt Þ þ 1:q004 ðt Þ þ 0:0162889 q006 ðt Þ  0:00245653 q005 ðt Þ ¼ 0

(A.4)

2667:99γq1 ðt Þ2 q3 ðt Þ 1:66657  107 q1 ðt Þq3 ðt Þ  187894:γq1 ðt Þq3 ðt Þ  279679:q1 ðt Þ2 q3 ðt Þ þ 1:0238  106 γq2 ðt Þ2 q3 ðt Þ 1:18575  108 q2 ðt Þq3 ðt Þ  163191:γq2 ðt Þq3 ðt Þ þ333454:γq1 ðt Þq2 ðt Þq3 ðt Þ  2:91775  107 q2 ðt Þ2 q3 ðt Þ 4:90591  106 q1 ðt Þq2 ðt Þq3 ðt Þ þ 3:81216  106 q1 ðt Þ2 q4 ðt Þ  1:18575  108 q1 ðt Þq4 ðt Þ  4:31291  106 γq1 ðt Þq4 ðt Þ þ 21673:6γq1 ðt Þ2 q4 ðt Þ 1:79713  109 q2 ðt Þq4 ðt Þ  1:35948  107 γq2 ðt Þq4 ðt Þ þ2:95753  107 q1 ðt Þq2 ðt Þq4 ðt Þ þ 1:31146  106 γq1 ðt Þq2 ðt Þq4 ðt Þ þ1:58708  108 q2 ðt Þ2 q4 ðt Þ þ 3:99348  106 γq2 ðt Þ2 q4 ðt Þ  1:46332  106 γq6 ðt Þ3 þ6:63958  106 q1 ðt Þ2 q5 ðt Þ  8608:13γq1 ðt Þ2 q5 ðt Þ 573984:γq1 ðt Þq5 ðt Þ þ 1:52725  108 q1 ðt Þq2 ðt Þq5 ðt Þ 3:77777  106 γq2 ðt Þq5 ðt Þ þ 171608:γq1 ðt Þq2 ðt Þq5 ðt Þ þ 1:25695  109 q2 ðt Þ2 q5 ðt Þ þ 123350:γq2 ðt Þ2 q5 ðt Þ þ 1:03743  106 q3 ðt Þ2 q5 ðt Þ 5:3979  106 γq2 ðt Þq6 ðt Þ 5:2405  106 γq4 ðt Þq5 ðt Þq6 ðt Þ þ 8796:05γq5 ðt Þ3 276236:gγq3 ðt Þq5 ðt Þ2  3:18492  106 γq4 ðt Þq5 ðt Þ2  117685:q6 ðt Þ þ447675:γq1 ðt Þq6 ðt Þ 1:07764  107 q1 ðt Þ2 q6 ðt Þ  42009:5γq1 ðt Þ2 q6 ðt Þ þ1:96399  108 q4 ðt Þ2 q5 ðt Þ  1:68947  108 q1 ðt Þq2 ðt Þq6 ðt Þ  723244:γq1 ðt Þq2 ðt Þq6 ðt Þ þ201789:γq2 ðt Þ2 q6 ðt Þ  1:03633  109 q2 ðt Þ2 q6 ðt Þ 1:68381  106 q3 ðt Þ2 q6 ðt Þ þ 158645:γq3 ðt Þq5 ðt Þq6 ðt Þ  2:6398  107 q3 ðt Þq4 ðt Þq6 ðt Þ  1:61927  108 q4 ðt Þ2 q6 ðt Þ þ 2:38632  107 q3 ðt Þq4 ðt Þq5 ðt Þ  139875:γq5 ðt Þ2 q6 ðt Þ  549317:γq3 ðt Þq6 ðt Þ2 þ 73861:4q5 ðt Þ  5:45623  106 γq4 ðt Þq6 ðt Þ2  643295:γq5 ðt Þq6 ðt Þ2 0

00

00

00

00

þ 11:5395q5 ðt Þ þ 18:1863q3 ðt Þ  2:44946q4 ðt Þ þ 1:q5 ðt Þ  0:068977q6 ðt Þ ¼ 0 4:5043  107 q1 ðt Þq3 ðt Þ þ 46689:6γq1 ðt Þq3 ðt Þ þ 41366:4q1 ðt Þ2 q3 ðt Þ þ 1:9105  108 q2 ðt Þq3 ðt Þ  1:59958  106 γq2 ðt Þq3 ðt Þ 33066:7γq1 ðt Þ2 q3 ðt Þ þ 728742:γq2 ðt Þ2 q4 ðt Þ 151473:γq1 ðt Þq2 ðt Þq3 ðt Þ  9:99506  107 q2 ðt Þ2 q3 ðt Þ

(A.5)

250

A. Di Egidio et al. / Journal of Sound and Vibration 358 (2015) 236–250

þ1:9105  108 q1 ðt Þq4 ðt Þ  6:93826  106 γq1 ðt Þq4 ðt Þ  70993:7γq2 ðt Þ2 q3 ðt Þ þ2:37122  109 q2 ðt Þq4 ðt Þ  8:01232  106 q1 ðt Þ2 q4 ðt Þ 23308:6γq1 ðt Þ2 q4 ðt Þ þ 351626:γq1 ðt Þq2 ðt Þq4 ðt Þ  2:06432  107 γq2 ðt Þq4 ðt Þ  4:08579  107 q1 ðt Þq2 ðt Þq4 ðt Þ þ 581174:γq1 ðt Þq5 ðt Þ  4:21125  108 q2 ðt Þ2 q4 ðt Þ 12304:8q5 ðt Þ  7:83801  106 q1 ðt Þq2 ðt Þq3 ðt Þ þ 1:81638  109 q2 ðt Þ2 q6 ðt Þ 54536:9γq1 ðt Þ2 q5 ðt Þ  7:00758  106 γq2 ðt Þq5 ðt Þ þ 4:38236  106 q3 ðt Þ2 q6 ðt Þ 2:19328  108 q1 ðt Þq2 ðt Þq5 ðt Þ  938918:γq1 ðt Þq2 ðt Þq5 ðt Þ þ 261964:γq2 ðt Þ2 q5 ðt Þ  2:18593  106 q3 ðt Þ2 q5 ðt Þ  3:427  107 q3 ðt Þq4 ðt Þq5 ðt Þ þ 102977:γq3 ðt Þq5 ðt Þ2 3:40162  106 γq4 ðt Þq5 ðt Þ2 2:10215  108 q4 ðt Þ2 q5 ðt Þ þ 2:80471  107 q1 ðt Þ2 q6 ðt Þ  541:365γq5 ðt Þ3 þ957388:q6 ðt Þ  3:00194  106 γq1 ðt Þq6 ðt Þ þ3:37267  108 q1 ðt Þq2 ðt Þq6 ðt Þ  81319:3γq1 ðt Þ2 q6 ðt Þ 2:06667  107 γq2 ðt Þq6 ðt Þ þ2:83809  108 q4 ðt Þ2 q6 ðt Þ  2:02771  106 γq1 ðt Þq2 ðt Þq6 ðt Þ  1:39899  107 q1 ðt Þ2 q5 ðt Þ þ 5:26979  107 q3 ðt Þq4 ðt Þq6 ðt Þ  5:77083  106 γq2 ðt Þ2 q6 ðt ÞÞ 1:34537  109 q2 ðt Þ2 q5 ðt Þ þ 155395:γq5 ðt Þ2 q6 ðt Þ  1:42625  106 γq3 ðt Þq5 ðt Þq6 ðt Þ  247173:γq5 ðt Þq6 ðt Þ2 00

þ 41:5138q6 ðt Þ  1:41666  107 γq4 ðt Þq5 ðt Þq6 ðt Þ þ 1:3599  106 γq3 ðt Þq6 ðt Þ2 407431:γq6 ðt Þ3 00

00

00

00

þ 5:65076  106 γq4 ðt Þq6 ðt Þ2 þ 1:43102q3 ðt Þ þ 21:0855q4 ðt Þ  0:0895462q5 ðt Þ þ1:q6 ðt Þ ¼ 0

(A.6)

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