Coupled flexural-torsional vibrations of a composite beam attached to ...

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Porto, Portugal, 30 June - 2 July 2014 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: 2311-9020; ISBN: 978-972-752-165-4

Coupled flexural-torsional vibrations of a composite beam attached to a rotating hub 1

Jerzy Warminski1, Jaroslaw Latalski1, Zofia Szmit1 Department of Applied Mechanics, Faculty of Mechanical Engineering, Lublin University of Technology, Nadbystrzycka 36 Str., 20-618 Lublin, Poland email: [email protected]; j.latalski@ pollub.pl ; [email protected]

ABSTRACT: The presented research discusses the coupled flexural-torsional vibrations of a thin walled composite box beam made of a ply composite material attached to a rotating hub. The observed coupling between flexural and torsional modes is resulting from the directional properties of the composite material with assumed a priori fibers’ orientation. Moreover, several non-classical effects like shear deformation, cross-sectional rotatory inertia and driving hub inertia are taken into account. Based on authors previous research a system of partial differential equations for the discussed structure is given. Next, this system is solved and the eigenvalues and eigenmodes are obtained by the Extended Galerkin method. A simple parametric analysis of hub’s inertia impact on systems dynamic characteristics is performed by considering aluminium and steel hub design. Within this analysis resonance curves obtained for excited vibrations have been plotted. Next, graphs representing natural frequencies of the beam-hub system against hub’s mass moment of inertia are shown. Finally, time histories of the transversal displacement, shear deformation angle and twist angle are given for both considered rotor assembly variants. KEY WORDS: Rotating beam; Timoshenko beam model; Non-classical effects; Composite material; Coupled vibrations. 1

INTRODUCTION

Rotating beams are important structural elements widely used in mechanical and aerospace engineering as turbine blades, various cooling fans, windmill blades, helicopter rotor blades, airplane propellers, flexible robotic arms etc. Introduction of composite materials technology has significantly influenced their design and opened new directions for the scientific research. Various elastic couplings, resulting from the directional-dependent properties of composites and plystacking sequences combinations are becoming commonly exploited to enhance structural response. One of the most striking applications of this idea with respect to rotating beams is the XV-15 tilt rotor aircraft design. A proper tension–twist elastic coupling has been implemented in the structural design to ensure two different rotor’s blades twist distributions corresponding to various systems operation modes (i.e. airplane and helicopter flight mode) [3], [10]. Due to the complex static and dynamic behavior of composite blades and possible shaft-hub-blades interactions the proposed topic needs to be extensively studied to provide the accurate prediction of rotor’s assembly characteristics, which are essential for the reliable design of modern mechanical structures and aeronautical crafts. In the past few years, a number of analytical models of anisotropic thin-walled beams have been proposed and validated either numerically or experimentally. Numerous results concerning the structural dynamic behavior of composite thin-walled beams can be found in the professional literature; also several review papers containing an extended assessment of rotating beam modelling methods with special regard to helicopter rotor blades have been published e.g. [9] [20].

A versatile and comprehensive work devoted to thin-walled composite beam analysis has been done by Librescu and Song [18], [17], [11]. Authors elaborated a theory of thin-walled composite beams of an arbitrary, closed or open cross-section. The work-out approach encompassed a number of nonclassical effects such as the material anisotropy and circumferential stiffness non-uniformity, transverse shear deformation, Vlasov effect etc. under the initial assumption of the beam's cross-section to be rigid in its own plane. In the performed analysis of rotating systems [17], [11] the effects of centrifugal and Coriolis forces were taken into account; also effects of pretwist and presetting were studied [13]. A very extensive theory of rotating slender beams has been developed since the nineties by D. Hodges and his coworkers. It was presented in a series of papers and later collected in a book by D. Hodges [5]. The modelling was based on asymptotic procedures that exploited the magnitude of system's parameters such as strain and slenderness. The proposed approach derived from a three-dimensional elasticity formulation the two sets of analyses: one over the cross section, providing elastic constants that might be used in a suitable set of beam equations, and the other set being the beam equations themselves. Authors developed a software code called Variational Asymptotic Beam Sectional Analysis (VABS) which used the originally worked-out analysis method. The VABS software was later verified and validated within the frame of several works e.g. by Yu et al. in [23], by Kovvali et al. in [8]. The worked out theory is especially well suited for helicopter rotor blades modelling since it accounts for initial twist and geometrical nonlinearities. Moreover, it allows for arbitrary cross-sectional geometry and material properties as well. The elaborated theory in its current state seems to miss explicit dependencies for the case when

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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

nonlinear couplings arise from nonconstant angular velocity. This phenomenon is essential in studying the so called nonideal system when energy source e.g. DC motor interacts with the rotating structure. Coupled bending-twist vibrations were also studied by other researchers. By way of example Kaya and Ozgumus in [7] investigated the free vibration response of an axially loaded, closed-section composite Timoshenko beam which featured material coupling between flapwise bending and torsional vibrations due to ply orientation. The governing differential equations of motion were derived and next, the impact of various couplings, as well as the slenderness ratio on the natural frequencies was investigated. Only the cases of constant rotating speed were considered and interactions with a hub-shaft subassembly were omitted. Similarly Sina et al. in [15] and [16] analyzed a rotating thin-walled composite Timoshenko beam in linear regime by following Librescu's approach. In the performed analysis centrifugal and Coriolis forces were taken into account; however no variable rotating speed nor hub-beam interactions were taken into account. A discussion of taper and slenderness ratio impact on natural frequencies and mode shapes was furthermore presented in [15], while in [16] effect of beam pretwist was studied. Apart from analytical based models composite thin walled beams are studied also by means of finite element method. Altenbach et al. [2] based on a generalized Vlasov theory for thin-walled composite beam developed an isoparametric finite element with arbitrary nodal degrees of freedom. The element was tested for multiple cases of open and closed cross-section cantilever designs, although only static cases were considered. The 3D model of a rotating beam with geometric nonlinearities was investigated in [19] by the p-version of finite element method. The two models, Bernoulli-Euler and Timoshenko, were studied and the importance of warping function for different rectangular cross-sections was shown. Moreover, authors concluded that additional shear stresses which appeared while bending and torsion were coupled have an essential influence on the beam's dynamics. A number of researchers investigated the dynamics of a whole rotor assembly by taking into account interaction of a blade bending vibration with shaft torsional modes. As a way of example a comprehensive analysis was done by Huang and Ho in [6]. In the analytical model authors incorporated shaft flexibility and analyzed the dynamic coupling between shaft torsion and blade bending of a rotating shaft-hub-blade unit. The given approach allowed the shaft to vibrate freely around its rotation axis instead of assuming a periodic perturbation of the shaft speed that might accommodate the shaft flexibility only to a limited extent. Numerical examples were given for cases with between two and six symmetrically arranged blades. The results showed not only coupling between the shaft, the disk, and blades, but also coupling between individual blades where the shaft acted only as a rigid support and experienced no torsional vibration. Finally, the effect of shaft speed on the modal frequencies was investigated. Furthermore, plots illustrating the occurrence of critical speeds and flutter instabilities were presented. Also Al-Bedoor [1] analyzed elastic blade attached to a disk driven by a shaft flexible in torsion. The shaft torsional

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flexibility is lumped in the form of flexible coupling that is usually employed in rotor systems. The Lagrangean approach in conjunction with the finite element method was employed in deriving the equations of motion. The dynamic coupling terms between the system reference rotational motion, shaft torsional deformations and blade bending deformations were accounted for. The simulation results showed strong dependence and interaction between the shaft torsional deformations and blade bending deformations. The presented study confirmed the necessity for including both the shaft torsional flexibility and the blade lead-lag deformations when a reliable model for either rotating blades or rotor torsional dynamics is requested. Research on hub inertia effect and payload on the vibration of a flexible slewing link was done by Low in [12]. The author analyzed an actuator at the base represented by a hub with given inertia and an attached flexible beam with uniform linear mass density and an additional payload. The beam was modelled by Euler Bernoulli approach and lead-lag bending was considered. Natural frequencies of a system with respect to hub’s and payload inertia’s were determined. These by virtue of the developed model were next compared with results obtained experimentally. The obtained findings suggested that the exact natural frequencies for the discussed systems are intermediate between the clamped and pinned cases. Warminski and Balthazar in [21] modelled a rotating EulerBernoulli beam made of isotropic material driven by a nonideal energy source. In the performed analysis geometrical nonlinearities and nonconstant rotating speed were taken into account; moreover the hub inertia was considered. The authors showed transitions through the resonances for a reduced order discrete system. More research on blades and hub interactions on complete rotor assembly dynamic characteristics was done by Warminski et al. in [22]. In that paper a model of a nonlinear system composed of a hub with attached two pendula rotating in a horizontal plane was studied. Each single pendulum was treated as a massless, non-deformable rod with a lumped mass at the tip and connected to the hub by a flapping hinge. The system has been excited by an external torque generated by a DC motor which has been considered as an ideal system with torque given by a harmonic function. An influence of the structural parameters like mass of the hub and pendula length on natural end excited vibrations was. The complete synchronization, phase synchronization, bifurcations and transition through resonances were analyzed considering the influence of the mass of the hub. The existence of chaotic oscillations of the system and paths leading to chaos was demonstrated as well. In the current paper the research is extended to the analysis of a system with deformable beam exhibiting coupled lead-lag bending and torsional vibrations. In the analysis several nonclassical effects like shear deformation, cross-sectional rotatory inertia and driving hub inertia are taken into account. 2

PROBLEM FORMULATION

The structural model used in this paper is a straight, prismatic, single-cell, fiber reinforced composite thin-wall box beam clamped at the rigid hub of radius R0 experiencing rotational


motion as shown in Figure 1. The length of the beam is denoted by l, its wall thickness by h and it is assumed to be constant spanwise. The composite material is assumed to be linearly elastic.

• the stress in transverse normal σnn direction and the hoop stress resultant (Nss) are very small and can be neglected, • the beam is attached to the hub with a given ‘a priori’ inertia as defined with respect and no torsional deformations of the hub’s shaft are accounted for. 2.2

Figure 1. A model of a rotating thin-walled beam. The following coordinate systems are defined to describe the motion of the beam: • global and fixed in space Cartesian coordinate system (X0, Y0, Z0) attached at the center of the hub O, • Cartesian coordinate system (X1, Y1, Z1) with origin O set at the center of the hub, rotating with arbitrary angular velocity dψ/dt about axis OZ1=OZ0 (Figure 1a), • beam Cartesian coordinate system (x,y,z) located at the blade root and oriented with respect to plane of rotation X1Y1 at angle θ denoting blade presetting (pitch) angle (Figure 1b). Axis ox is directed along beam span and oz axis is normal to the beam chord. The origin o of the (x,y,z) coordinates is set at the center of the beam crosssection, therefore axes ox and OX1 coincide, • local, curvilinear coordinate system (x,n,s) related to blade cross-section---see Figure 1c. Its origin is set conveniently at the point on a mid-line contour. The circumferential coordinate s is measured along the tangent to the middle surface in a counter-clockwise direction, whereas n points outwards and along the normal to the middle surface. 2.1

Lamination scheme and mode couplings

As reported in the professional literature two fabric configurations that induce special elastic mechanics are commonly encountered in composite structural design. These are achievable by skewing the ply-angles with respect to the longitudinal axis x. Considered first by Rehfield and Atilgan [14], these structural configurations are referred to as circumferentially uniform stiffness (CUS) and circumferentially asymmetric stiffness (CAS) configurations. For a thin-walled beam of rectangular cross-section CUS arrangement implies the ply-angle distribution α1(y) = α1(-y) of the top and bottom walls of the box beam (flanges) and α1(z) = α1(-z) on the lateral walls (webs). On the other hand CAS design corresponds to the formula α1(y) = - α1(-y) and α1(z) = - α1(-z) respectively. In the above α denotes the dominant ply orientation measured from the positive s-axis toward the positive x-axis (see Figure 2). To induce the intended elastic coupling between flapwisebending and twist, the use the circumferentially asymmetric stiffness ply-angle distribution [11] is necessary. This scheme decouples the full set of equations of motion (6 d.o.f.) into two sub-systems: one exhibiting flapwise bending/shear – twisting coupling and the second one where axial and chordwise bending/shear modes are coupled.

Assumptions

For the development of the equations of motion the following kinematic and static assumptions are postulated: • the original shape of the cross-section is maintained in its plane, but is allowed to warp out of the plane, • the concept of the non-uniform torsional model is adopted, so the rate of beam twist ϕ'=dϕ/dx depends in general on the spanwise coordinate x, • in addition to the primary warping effects (related to the cross-section shape) the secondary warping related to the wall thickness is also considered, the transverse beam shear deformations γxy, γxz, are taken into account. These are assumed to be uniform over the beam cross-section, • the ratio of wall thickness to the radius of curvature at any point of the beam wall is negligibly small while compared to unity. In a special case of the prismatic beams made of planar segments this ratio is exactly 0,

Figure 2. Lamination schemes commonly encountered in structural composite design.

2.3

Analytical model of the beam; equations of motion

The equations of motion and boundary conditions of the rotating beam are derived according to the extended Hamilton's principle of the least action:

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|

0 where J is the action, T is the kinetic energy, U is the potential energy and the work of the external forces is given by the Wext term. Full derivation of the equations of motion for a complete model including both transversal/lateral bending directions, shear deformations and primary and secondary warping effect, as well as arbitrary presetting angle and non-constant rotational speed can be found in authors paper [4]. Here, the simplified system of equations is just given, where presetting angle θ is fixed at π/2. To discard out-of-plane bending CAS fabric arrangement is considered as reported in the previous section. Moreover, warping restraint is neglected in further analysis. After accounting for the given above simplifications one arrives at the following equations: •

In the foregoing relations the prime symbol corresponds to spatial derivative and overdot one to time derivative; moreover the following notation is used: •

Inertia coefficients ,

,

,

,

1,

,

,

,

/

2 /

where ρ is material density, •

Stiffness coefficients

Extensional

Hub rotation 2

Flapwise bending

2 •

0,

2

0 Axial direction

Flapwise transverse shear

2 0

Twist with boundary conditions,

•

|

0,

|

0,

Flapwise bendingtwist is In the above formula the integration operation · performed along profile circumference and Kij coefficients represent stiffnesses in 2D constitutive equations and are expressed by means of classical laminate theory stiffness quantities Aij, Bij, Dij. Again, for more details regarding the definitions of these coefficients please refer to paper [4].

Lead-lag displacement 2 0 with boundary conditions, |

0, |

•

3

For the purpose of numerical analysis a rectangular box beam is analyzed. As reported in the previous section, the lamination scheme exhibits the circumferentially asymmetric stiffness scheme (CAS). To get the strongest possible coupling between beam’s bending and twisting motion an analysis of a twist-bending coupling coefficient a37 has been performed. The generic data were used to provide versatile conclusions; the outcomes are given in Figure 3.

0

Cross-section rotation (considering shear effect) 0 with boundary conditions, |

0, |

•

0,

Twist angle 0, with boundary conditions, |

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0,

NUMERICAL STUDIES


Parameter 21 is expressed with respect to the mass moment of inertia of the hub Jh which is treated as a bifurcation parameter. We may notice that Eqs. (1) are coupled by inertia terms. Moreover the natural frequency of the whole system differs from that obtained for a separate composite beam. In order to show the hub influence we neglect in equations (1) damping, rotation of the hub and excitation terms. Then separating second derivatives we find the natural frequency of the hub-beam system assembly to be Figure 3. Stiffness coupling coefficient a37 with respect to composite fiber orientation (/2 corresponds to fibers along the beam length).

600

550

0

It is evident, that the increase of fiber orientation angle raises the magnitude of coupling coefficient until about 72°. Above that value the coupling decreases and at the /2 angle (fibers along the beam length) both modes fully decouple. Following this results the numerical simulations were performed for fibers orientation set to 70°. The material data of the graphite/epoxy composite material and geometry properties of the thin-walled beam used in the numerical analysis is listed in Table 1.

This frequency is a function of parameter 21 which depends on the hub’s mass moment of inertia. We can observe in Fig. 4 that the natural frequency approaches asymptotically the beam’s natural frequency while hub’s mass moment of inertia tends to infinity. Thus for a relatively light hub the dynamics may differ essentially from a heavier hub system.

Table 1. Material properties and geometric characteristics of a thin-walled beam featured by CAS lay-up. Material properties E1 = 206.751 × 109 Pa E2 = E3 = 5.17 × 109 Pa

500

450

400

G12 = G13 = 3.11 × 109 Pa G23 = 2.55 × 109 Pa

0.000

0.002

0.004

0.006

0.008

0.010

Jh

21 = 31 = 0.00625

Figure 4. Natural frequency of the beam-hub system against hub’s mass moment of inertia.

32 = 0.25  = 1528.15 km/m3 Geometry d = 0.0254 m h = 0.001 m l = 0.254 m

4

c = 0.00508 m

RESULTS

Equations of motion after one mode reduction take the form: (1) where ζ1 and ζh are accepted modal dampings of the beam and the hub system respectively. The external torque  is assumed to be a harmonic function t = e cost, where e and  are amplitude and frequency of excitation respectively. Coefficients ij are obtained after modal reduction and they take the following values: 11  154923, 12  0.89496, 13  0.9995,  21 

Figure 5. Resonance curves obtained for excited vibrations,  = 100, ζ1 is 0.01 of beams natural frequency, ζh = 0.1, aluminium hub: Jh = 1.7653 × 10-5 – black curve; steel hub Jh = 5.68819 × 10-5 – blue curve.

0.000713 0.0006769 J h

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On the resonance curves (Fig. 5) we see in fact that for a light hub made of aluminium (back curve) the resonance zone is shifted into higher frequency zone. For a heavier made of steel hub (blue curve) the resonance occurs earlier and with larger amplitudes. In Figure 6 we present time histories of the motion of the beam’s tip point for (a) transversal displacement w(l,t), (b) angle of rotation of the cross-section y(l,t) related to Timoshenko beam model, (c) angle of twist ϕ(l,t) and (d) angular velocity of the hub Ωh computed for data as given in the Figure 5. We can observe that in fact bending and twisting modes are coupled and the dynamics of the system strongly depends on hub mass moment of inertia. For the same excitation magnitude the response of a rotor assembly (hub and blade) composed of a heavier steel hub is much higher (red colour) than for a case with aluminium hub. 5

CONCLUSIONS

Dynamics of the thin walled composite beam attached to the rotating hub is analyzed in the paper. The mathematical model represented by partial differential equations has been reduced to the ordinary differential equations considering one mode reduction. The composite fibres orientation resulted in coupled bending and torsion vibrations. Therefore, it required solving a coupled eigenvalue problem which has been solved by the method of assumed modes. We have demonstrated that apart from the beam’s structural parameters also mass moment of inertia of the hub essentially influences the full beam-hub system dynamics. For the heavier hub made of steel the resonance zone has been shifted to lower frequencies and with the higher amplitudes comparing to the lighter one made of aluminium. ACKNOWLEDGMENTS The financial support of Structural Funds in the Operational Programme – Innovative Economy (IE OP) financed from the European Regional Development Fund – Project “Modern material technologies in aerospace industry”, Nr POIG.01.01.02-00-015/08-00 is gratefully acknowledged. REFERENCES [1] [2] [3] [4]

Figure 6. Time histories of the displacement w(l,t) (a), angle of cross-section rotation y(l,t) (b), angle of twist ϕ(l,t) (c) at the tip point, and angular velocity of the hub Ωh(t) (d) , computed for data as in Fig. 5. Aluminium hub – back colour, Steel hub – red colour.

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[5] [6]

Al-Bedoor B.O. Dynamic model of coupled shaft torsional and blade bending deformations in rotors. Computer Methods in Applied Mechanics and Engineering, 169(1-2): 177--190, 1999. Altenbach J., Altenbach H., Matzdorf V. A generalized Vlasov theory for thin-walled composite beam structures. Mechanics of Composite Materials, 30(1):43--54, 1994. Bauchau O.A., Loewy R.G., Bryan P.S. Approach to ideal twist distribution in tilt rotor VTOL blade designs. RTC Report No. D-86-2. Troy, NY: Rensselaer Polytechnic Institute; July 1986. Georgiades F., Latalski J., Warminski J.: Equations of motion of rotating composite beam with a nonconstant rotation speed and an arbitrary preset angle, Meccanica – paper after review & accepted for publication. Hodges D.H. Nonlinear composite beam theory. American Institute of Aeronautics and Astronautics. Progress in astronautics and aeronautics 213, Reston (VA), 2006. Huang S.C., Ho K.B. Coupled shaft-torsion and blade-bending vibrations of a rotating shaft-disk-blade unit. Journal of Engineering for Gas Turbines and Power-transactions of The ASME 01/1996; 118(1): 100-106.


[7]

[8] [9] [10]

[11] [12] [13] [14]

[15]

[16] [17] [18] [19]

[20] [21]

[22] [23]

Kaya M.O., Ozgumus O.O. Flexural-torsional-coupled vibration analysis of axially loaded closed-section composite Timoshenko beam by using DTM. Journal of Sound and Vibration, 306(3-5):495--506, 2007. Kovvali R.K., Hodges D.H. Verification of the Variational-Asymptotic Sectional Analysis for initially curved and twisted beams. Journal of Aircraft, 49(3):861--869, 2012. Kunz D. Survey and comparison of engineering beam theories for helicopter rotor blades. Journal of Aircraft, 31(3):473--479, 1994. Lake R.C., Nixon M.W., Wilbur M.L., Singleton J.D., Mirick R.H. A demonstration of passive blade twist control using extension–twist coupling. In: Paper AIAA-92-2468-CR, Structural Dynamics and Materials Conference, Dallas, Texas; 1992. Librescu, L; Song, O. Thin-walled composite beams. Theory and application. Dordrecht, the Netherlands; 2006 Springer. Low K.H. Vibration analysis of a tip-loaded beam attached to a rotating joint. Computers & Structures Volume 52, Issue 5, 3, 1994, Pages 955– 968. Oh S.-Y., Song O., Librescu L. Effects of pretwist and presetting on coupled bending vibrations of rotating thin-walled composite beams. International Journal of Solids and Structures 40 (2003) 1203–1224. Rehfield, L. W. and Atilgan, A. R. (1989) “Toward Understanding the Tailoring Mechanisms for Thin-Walled Composite Tubular Beams,” in Proceedings of the First USSR–U.S. Symposium on Mechanics of Composite Materials, 23–26 May, Riga, Latvia SSR, S. W. Tsai, J. M., Whitney, T. W. Chou and R. M. Jones (Eds.), ASME Publ. House, pp. 187–196. Sina S.A., Ashrafi M.J., Haddadpour H., Shadmehri F. Flexuraltorsional vibrations of rotating tapered thin-walled composite beams. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 225(4):387--402, 2011. Sina, S.A., Haddadpour, H. Axial-torsional vibrations of rotating pretwisted thin walled composite beams. International Journal of Mechanical Sciences 80 (2014) 93–101. Song O., Librescu L. Structural modeling and free vibration analysis of rotating composite thin-walled beams. Journal of the American Helicopter Society, 42(4):358--369, 1997. Song O., Librescu L. Free vibration of anisotropic composite wthinwalled beams of closed cross-section contour. Journal of Sound and Vibration, 167(1): 129-147, 1993. Stoykov S. Ribeiro P. Nonlinear forced vibrations and static deformations of 3D beams with rectangular cross section: The influence of warping, shear deformation and longitudinal displacements. International Journal of Mechanical Sciences, 52(11):1505-1521, 2010. Volovoi V.V., Hodges D.H., Cesnik C., Popescu B. Assessment of beam modeling methods for rotor blade applications. Mathematical and Computer Modelling, 33(10-11):1099--1112, 2001. Warminski J., Balthazar J.M. Nonlinear vibrations of a beam with a tip mass attached to a rotating hub. In Volume 1: 20th Biennial Conference on Mechanical Vibration and Noise. Long Beach, California, USA, 2005, technical publication, DETC2005-84518. Warminski J., Szmit Z., Latalski J.: Nonlinear dynamics and synchronisation of pendula attached to a rotating hub. European Physical Journal – paper after review & accepted for publication. Yu W., Volovoi V.V., Hodges D.H., Hong X. Validation of the Variational Asymptotic Beam Sectional Analysis (VABS). AIAA Journal, 40(10):2105--2113, 2002.

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