Meccanica (2014) 49:1833–1858 DOI 10.1007/s11012-014-9926-9
NONLINEAR DYNAMICS AND CONTROL OF COMPOSITES FOR SMART ENGI DESIGN
Equations of motion of rotating composite beam with a nonconstant rotation speed and an arbitrary preset angle Fotios Georgiades • Jarosław Latalski Jerzy Warminski
•
Received: 2 August 2013 / Accepted: 10 March 2014 / Published online: 30 April 2014 The Author(s) 2014. This article is published with open access at Springerlink.com
Abstract In the presented paper the equations of motion of a rotating composite Timoshenko beam are derived by utilising the Hamilton principle. The nonclassical effects like material anisotropy, transverse shear and both primary and secondary cross-section warpings are taken into account in the analysis. As an extension of the other papers known to the authors a nonconstant rotating speed and an arbitrary beam’s preset (pitch) angle are considered. It is shown that the resulting general equations of motion are coupled together and form a nonlinear system of PDEs. Two cases of an open and closed box-beam cross-section made of symmetric laminate are analysed in details. It is shown that considering different pitch angles there is a strong effect in coupling of flapwise bending with chordwise bending motions due to a centrifugal force. Moreover, a consequence of terms related to nonconstant rotating speed is presented. Therefore it is shown that both the variable rotating speed and nonzero pitch angle have significant impact on systems dynamics
F. Georgiades School of Engineering, Faculty of Science, University of Lincoln, Lincoln, UK e-mail:
[email protected] J. Latalski (&) J. Warminski Department of Applied Mechanics, Lublin University of Technology, 20-618 Lublin, Poland e-mail:
[email protected] J. Warminski e-mail:
[email protected]
and need to be considered in modelling of rotating beams. Keywords Rotating structures Vibrations of composite beam Coupled modes Warping Timoshenko beam model
1 Introduction Rotating beams are important structures widely used in mechanical and aerospace engineering as turbine blades, various cooling fans, windmill blades, helicopter rotor blades, airplane propellers etc. Introduction of composite materials technology has significantly influenced their design and opened new research areas. Various elastic couplings, resulting from the directional-dependent properties of composites and ply-stacking sequences combinations might be exploited to enhance their response. A profound understanding of rotating composite beams dynamics is essential towards taking the full advantage of their design potential and avoiding failures. An interest of modelling composite rotating beams started in mid 80s and increasing number of research work led to several review papers, where different approaches to rotating composite beam modelling were discussed. In 1994 Kunz [17] and later Volovoi et al. [32] made an extended assessment of rotating beam modelling methods with special regard to helicopter rotor blades.
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Currently available approaches to analysis of composite beams fall in one of the three cases [13]: (a) theories in which some a’priori cross-sectional deformation is assumed leading to 1-D governing equations; (b) theories based on equations for the blade as a one-dimensional continuum, the crosssection properties of which are obtained from separate source e.g. [12]; (c) approach where the beam and cross-sectional governing equations are rigorously reduced from three-dimensional elasticity theory. Although ‘ad hock’ theories (a) have some drawbacks (see [36]) they still stay to be one of the most common ones. A very extensive work devoted to thin-walled composite beam analysis was done by Librescu and Song [21, 28, 29] and their co-workers e.g. [23, 24]. Authors developed and refined a theory of beams modelled as thin-walled structures of an arbitrary, closed or open cross-section. The work-out theory encompassed a number of non-classical effects such as the anisotropy and heterogeneity of constituent materials, transverse shear, primary and secondary warping phenomena (Vlasov effect) etc. under assumption of the cross-section to be rigid in its own plane. In the performed analysis of rotating systems [29] the effects of the centrifugal and Coriolis forces were taken into account. Furthermore, the importance of shear effects in composite material was emphasized. Kaya and Ozgumus in [15] performed an analysis of 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 the impact of the various couplings, as well as the slenderness ratio on the natural frequencies were investigated. Although only the cases of constant rotating speed and zero pitch angle were considered. Sina et al. [27] analysed a rotating tapered thinwalled composite Timoshenko beam in linear regime following Librescu’s approach. In the performed analysis centrifugal and Coriolis forces were taken into account; however no beam presetting angle nor variable rotating speed were considered in that paper. A discussion of taper and slenderness ratio impact on natural frequencies and mode shapes was furthermore presented. Jun et al. [14] studied the dynamics of axially loaded thin-walled beams of open, mono-symmetrical
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section. The studies considered the effect of warping stiffness, but were restricted to Euler–Bernoulli beam theory and isotropic material only. An impact of axial force and warping stiffness on the coupled bending torsional natural frequencies with various boundary conditions is examined. The analysis of a nonlinear rotating thin-walled composite beam was presented by e.g. Arvin and Bakhtiari-Nejad [2]. Authors developed a model for rotating composite Timoshenko beam considering centrifugal forces by means of von-Karmans strain– displacement relationships. The system was studied using the multiple time scales method and nonlinear normal modes theory. Again assumptions regarding zero pitch angle and constant rotation speed were made. Ghorasi [11] developed a nonlinear model of the rotating thin-walled composite Euler–Bernoulli beam using the previously mentioned variational asymptotic method elaborated by Cesnik and Hodges [6]. He considered a non-zero pitch angle but again only in parallel with constant rotating speed case. Avramov et al. [4] modelled rotating Euler–Bernoulli beams made by isotropic material with asymmetric cross-sections considering non-zero pitch angle but constant rotation speed. The study of flexural-flexuraltorsional nonlinear vibrations using nonlinear normal modes approach were presented. Effect of an initial beam pre-twist was examined by several researchers too. Chandiramani et al. [7] studied free linear vibrations of the rotating pretwisted thin-walled composite Timoshenko beams considering a non-zero pitch angle but a constant rotating speed. Also Oh et al. [23] modelled and studied the similar system, but again following an assumption of the constant rotation speed. Possible interactions and mode couplings observed in rotating ‘smart’ beams gained attention of several researchers in recent years too. Na et al. [22] addressed the problem of modelling and bending vibration control of tapered rotating blades represented as nonuniform thin-walled beams and incorporating adaptive capabilities. The blade model incorporated most non-classical features and their assessment on system dynamics including the taper characteristics was accomplished. Ren et al. [25] modelled a rotating thin-walled shape memory alloy composite Euler– Bernoulli beam, considering centrifugal and Coriolis forces with non-zero pitch angle but constant rotating speed. Free vibrations of the system were studied.
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Choi et al. [8] modelled rotating thin walled composite Timoshenko beams with macro fiber composite actuators and sensors using Librescu’s approach. Again zero pitch angle and constant rotating speed assumptions were made. The research on free vibrations of the rotating inclined cantilever beam was done by Lee and Sheu [20]. Influence of the beam setting angle and the inclination angle on the first two natural frequencies of the system were examined. Moreover different hub radius to beam length ratios were tested. This study was limited to isotropic materials, so only very little of possible internal couplings observed in the case of composite materials were recorded and commented. A nonlinear model of rotating isotropic blades based on the Cosserat theory of rods without restrictions on the geometry of deformation was presented in [18]. The roles of internal kinematic constraints such as unshearability of the slender blades, coupling between flapping, lagging, axial and torsional deformations were studied. The authors showed an influence of the angular velocity on the possible internal resonances which may occur in the considered rotating structure. The method of multiple time scales was applied directly to PDE of motion in order to determine the back-bone curves of the flapping modes out of internal resonance [3]. The hardening and softening effect was shown for selected angular velocities, and the importance of 2:1 internal resonance between flapping and axial modes leading to additional interactions and changing dynamics was demonstrated. As results from the paper the angular speed may be treated as the geometrical tuning mechanics leading to specific dynamic interactions between modes. As in the case of previously mentioned papers the assumption of constant rotating speed stayed in force. Case of non-constant rotating speed with respect to isotropic material blade was examined by Vyas and Rao [33]. The equations of motion of a Timoshenko beam mounted on a disk rotating with constant acceleration were given. In the derivation higher order effects due to Coriolis forces were included, but impact of non-zero presetting angle was ignored; moreover no numerical examples were given. Warmin´ski and Balthazar [34] modelled a rotating Euler– Bernoulli beam made of isotropic material with a tip mass. In the analysis geometrical nonlinearities and nonconstant rotating speed were taken into account, but an impact of pitch angle has been neglected. The
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authors showed transitions through the resonances for a reduced order discrete system. This topic was also investigated and continued afterwards by Fenili and Balthazar [9, 10]. Authors considered the non-ideal system’s power supply where the response of the excited beam affected the behaviour of the energy source. The performed simulations led to the observation, that this interaction caused a damping of the beam response. Apart from analytical based models composite thin walled beams are studied also by means of finite element method. Altenbach et al. [1] developed a generalized Vlasov theory for thin-walled composite beam and next an isoparametric finite element with arbitrary nodal degrees of freedom. The element was tested for multiple cases of open and closed crosssection cantilever designs, although only static cases were considered. The 3D model of a rotating beam with geometric nonlinearities was investigated in [30] 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. In the latest paper [31] the p-version of finite element was used for 3D Timoshenko beam model in which two types of nonlinearity were considered, a nonlinear strain-displacement relation and inertia forces due to rotation. The deformation in a longitudinal direction due to warping was taken into account. Dynamics was studied for constant and nonconstant rotation speed and for harmonic external loading. A very comprehensive theory of rotating slender beams has been developed since the nineties by Hodges and his co-workers. It was presented in a series of papers and later collected in a book by Hodges [13]. In order to derive the equations of motions they used Cosserat theory (a director theory) for the determination of generalized strains. The modelling was based on asymptotic procedures that exploited the smallness of system’s parameters such as strain and slenderness. The approach extracted 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 beeing the beam equations themselves.
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(a)
(b)
(c)
Fig. 1 Rotating thin-walled beam under consideration and reference frames
Authors developed a software called variational asymptotic beam section 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. [36], by Kovvali et al. [16]. The worked out theory is especially well suited for helicopter rotor blades modelling since it accounts for initial twist and geometrical nonlinearities (i.e. large displacements and rotations caused by deformation) preserving small strain condition. It allows for arbitrary cross-sectional geometry and material properties as well. In spite of the fact that authors have not assumed constant angular velocity there is a lack of explicit dependencies in the case when nonlinear couplings arise from nonconstant angular velocity. This phenomenon is essential in case of study of so called non-ideal system when energy source e.g. DC motor interacts with the power-driven structure. Nonlinear coupling terms may be observed by Sommerfeld effect [5, 34]. Although of evident practical importance, to the best of the authors knowledge, no studies on possible interactions occurring in composite beams rotating with nonconstant speed have been found in the specialised literature. Also the assumption of the arbitrary pitch angle in the performed analysis still leaves some space for discussion. To address this situation a derivation of equations of motion of the rotating thin-walled composite Timoshenko beam is presented. In the performed analysis most non-classical effects like material anisotropy, transverse shear and both primary and secondary
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cross-section warpings are taken into account, as well as arbitrary pitch angle and nonconstant rotating speed. The resulting general equations form a system of nonlinear PDEs that are coupled together. To illustrate the potential of this study a more detailed examination of two cases of open and closed box-beam cross-sections made of symmetric laminate is performed. It is shown that considering different pitch angles one observes a strong effect in coupling of flapwise bending and choordwise bending motions due to centrifugal force. Moreover, the consequence of terms related to nonconstant rotating speed is exemplified.
2 Theory 2.1 Problem formulation Let us consider a slender, straight and elastic composite thin-walled beam clamped at the rigid hub of radius R0 experiencing rotational motion as shown in Fig. 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 linearly elastic (Hookean) and its properties may vary in directions orthogonal to the middle surface. 2.1.1 Coordinate systems Four coordinate systems are defined to describe the motion of the beam:
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–
–
–
–
global and fixed in space Cartesian coordinate system ðX0 ; Y0 ; Z0 Þ attached at the center of the hub O (Fig. 1a), beam Cartesian coordinate system ðX1 ; Y1 ; Z1 Þ with origin O set at the center of the hub, rotating _ with arbitrary angular velocity wðtÞ about axis OZ1 ¼ OZ0 (Fig. 1a), beam Cartesian coordinate system ðx; y; zÞ located at the blade root and oriented with respect to plane of rotation (X1 Y1 ) at angle h denoting blade presetting (pitch) angle (Fig. 1b). Axis ox is directed along beam span and oz axis is normal to the beam chord. The origin o of the ðx; y; zÞ coordinates is set at the center of the beam crosssection, therefore axes ox and OX1 coincide, local, curvilinear coordinate system ðx; n; sÞ related to blade cross-section—see Fig. 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.
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(f)
In a special case of the prismatic beams made of planar segments this ratio is exactly 0, the stress in transverse normal (rnn Þ direction and the hoop stress resultant (Nss Þ are very small and can be neglected.
2.2 Beam model The equations of motion of the rotating beam are derived according to the extended Hamilton’s principle of the least action dJ ¼
Zt2
dT dU þ dWext dt ¼ 0;
ð1Þ
t1
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. One of the key steps of the derivation procedure is associated with the definition of a position vector and subsequently velocity and acceleration ones. 2.2.1 Position vector
2.1.2 Assumptions For the development of the equations of motion the following kinematic and static assumptions are postulated: (a)
the original shape of the cross-section is maintained in its plane, but is allowed to warp out of the plane, (b) the concept of the Saint-Venant torsional model is discarded in the favour of the non-uniform torsional one. Therefore the rate of beam twist u0 ¼ du=dx depends in general on the spanwise coordinate x, (c) in addition to the primary warping effects (related to the cross-section shape) the secondary warping related to the wall thickness is also considered, (d) the transverse beam shear deformations cxy , cxz are taken into account. These are assumed to be uniform over the beam cross-section, (e) the ratio of wall thickness to the radius of curvature at any point of the beam wall is negligibly small while compared to unity.
Let us consider any arbitrary point A located on the beam’s profile mid-line specified by its position vector r ¼ fx; y; zgT in the local beam’s coordinate system— Fig. 2. As the system rotates and A experiences displacement D ¼ fDx ; Dy ; Dz gT it occupies a new position A0 given by a position vector R defined in fixed inertial frame. To find the position vector R in the global coordinate system transformation matrices defining transitions to subsequent reference frames need to be defined. For this purpose the method of four dependent Euler parameters is used—see e.g. Shabana [26]. Transformation between X0 Y0 Z0 and X1 Y1 Z1 frames Transformation between X0 Y0 Z0 and X1 Y1 Z1 frames is related to the rotation through angle wðtÞ about OZ1 ¼ OZ0 axis. Therefore a transformation matrix is given as 2
AðIÞ
cos wðtÞ ¼ 4 sin wðtÞ 0
sin wðtÞ cos wðtÞ 0
3 0 05 1
ð2Þ
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(b)
(a)
Fig. 2 a Position vectors of point A in reference frames; b transformation of an arbitrary cross-section as rigid body motion; point P denotes cross-section pole and vP ; wP its transversal displacements while uðx; tÞ is cross-section rotation
Transformation between (xyz) and (X1 Y1 Z1 ) frames Transformation from beam’s local xyz frame to X1 Y1 Z1 one is related to the rotation about OX1 axis through angle h (see Fig. 1) corresponding to beam’s pitch angle and next the translation along the hub radius R0 . So the second rotation matrix is 2
AðIIÞ
1 ¼ 40 0
0 cos h sin h
3 0 sin h 5 cos h
ð3Þ
In the following discussion, to distinguish between coordinates associated with cross-section’s mid-line points and off-mid line ones lower (y, z) and capital (Y, Z) letters are used correspondingly. Following this notation setting (in beam’s local coordinate system) the initial position vector r of any arbitrary point of the cross-section r ¼ xi þ Yj þ Zk
ð4Þ
and applying subsequent transformations the position vector R in inertial (global) frame is R ¼ AðIÞ AðIIÞ ðr þ DÞ þ R0 ð5Þ Substituting (2), (3) and (4) into the above relation one obtains
where Dx , Dy and Dz are individual terms of deformation vector D in local coordinates as defined later in (9). 2.2.2 Velocity vector The velocity vector of an arbitrary point A of the elastic body in the inertial frame can be obtained by differentiating the position vector (6) with respect to time. This requires evaluating the time derivative of the transformation matrix. Next a skew symmetric matrix needs to be defined to formulate an angular velocity vector in a global coordinate system as described in e.g. [26]. After appropriate manipulations one arrives at R_ x ¼ ðDx þ x þ R0 ÞsinwðtÞ ðDy þ YÞcoshcoswðtÞ _ þ ðDz þ ZÞsinhcoswðtÞ wðtÞ þ D_ x coswðtÞ D_ y coshsinwðtÞ þ D_ z sinhsinwðtÞ R_ y ¼ ðDx þ x þ R0 ÞcoswðtÞ ðDy þ YÞcoshsinwðtÞ _ þ ðDz þ ZÞsinhsinwðtÞ wðtÞ þ D_ x sinwðtÞ þ D_ y coshcoswðtÞ D_ z sinhcoswðtÞ R_ z ¼D_ y sinh þ D_ z cosh ð7Þ where overdot means time derivative, so D_ x , D_ y and D_ z terms correspond to velocities of deformation.
9 8 > = < ðDx þ x þ R0 Þ cos wðtÞ ðDy þ YÞ cos h sin wðtÞ þ ðDz þ ZÞ sin h sin wðtÞ > R ¼ ðDx þ x þ R0 Þ sin wðtÞ þ ðDy þ YÞ cos h cos wðtÞ ðDz þ ZÞ sin h cos wðtÞ > > ; : ðDy þ YÞ sin h þ ðDz þ ZÞ cos h
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ð6Þ
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2.2.3 Acceleration vector Following the same approach as the above one an acceleration vector is obtained: R€x ¼ ðDx þxþR0 ÞsinwðtÞðDy þYÞcoshcoswðtÞ € þðDz þZÞsinhcoswðtÞ wðtÞ þ ðDx þxþR0 ÞcoswðtÞþðDy þYÞcoshsinwðtÞ 2 ðDz þZÞsinhsinwðtÞ w_ ðtÞþ2 D_ x sinwðtÞ _ D_ y coshcoswðtÞþ D_ z sinhcoswðtÞ wðtÞ €y coshsinwðtÞþ D €z sinhsinwðtÞ €x coswðtÞ D þD R€y ¼ ðDx þxþR0 ÞcoswðtÞðDy þYÞcoshsinwðtÞ € þðDz þZÞsinhsinwðtÞ wðtÞ þ ðDx þxþR0 ÞsinwðtÞðDy þYÞ 2 coshcoswðtÞþðDz þZÞsinhcoswðtÞ w_ ðtÞ þ2 D_ x coswðtÞ D_ y coshsinwðtÞ _ € x sinwðtÞ D þ D_ z sinhsinwðtÞ wðtÞþ €z sinhcoswðtÞ €y coshcoswðtÞ D þD €y sinhþ D € z cosh R€z ¼D ð8Þ Commenting on the velocity relation (7) and the acceleration formula (8) one can notice that supposing _ ¼ const: and constant angular velocity condition wðtÞ zero pitch angle h ¼ 0 these simplify exactly to the formulations given for cases examined by other researchers e.g. by Choi et al. [8], Librescu and Song [21] or Sina et al. [27]. 2.2.4 Displacement field Following the assumptions given in the section 2.1.2 the displacements of an arbitrary point of the crosssection are defined as (Librescu and Song [21]) dy dz Dx ¼ u0 ðx; tÞ þ #y ðx; tÞ z n þ #z ðx; tÞ y þ n ds ds Gðn; sÞu0 ðx; tÞ ¼ u0 ðx; tÞ þ #y ðx; tÞ Z þ #z ðx; tÞ Y 0
Gðn; sÞ u ðx; tÞ Dy ¼ vp ðx; tÞ ðY yp Þ 1 cos uðx; tÞ 1 ðZ zp Þ sin uðx; tÞ vp ðx; tÞ ðY yp Þ 2 2 ðuðx; tÞ ðZ zp Þuðx; tÞ
Dz ¼ wp ðx; tÞ þ ðY yp Þ sin uðx; tÞ ðZ zp Þ ð1 cos uðx; tÞ wp ðx; tÞ þ ðY yp Þuðx; tÞ 2 1 ðZ zp Þ uðx; tÞ ð9Þ 2 where Gðs; nÞ is a warping function formulated in Appendix 1, uðx; tÞ denotes rotation of the crosssection (twist angle) as seen in Fig. 2 and prime denotes differentiation with respect to the span coordinate (x). Angles #y ðx; tÞ ¼ cxz w0p and #z ðx; tÞ ¼ cxy v0p represent cross-sections’ rotations about respective axes y and z at pole P. The coordinates associated with off-mid-line points are denoted by capital letters (Y, Z), while mid-line ones are denoted by small y and z as already has been explained. Moreover, in the above formula the moderately large rotations are allowed by the approximation cos u 1 u2 =2. The later linearisation of the respective resulting equations is performed after incorporation of this approximation. 2.2.5 Strains We restrict our modelling to linear elastic range, therefore we consider only linear strains. The axial and circumferential-axial ones are split into mid-line terms (superscript ðÞð0Þ ) and off-mid-line ones (superscript ðÞð1Þ ) as proposed by Librescu and Song [21] ð1Þ exx ¼ eð0Þ xx þ nexx
ð10aÞ
ð1Þ cxs ¼ cð0Þ xs þ ncxs
ð10bÞ
cxn ¼ cð0Þ xn
ð10cÞ
It should be noted that considering linearised displacement field of equations (9) the following strains ðeyy ¼ ezz ¼ cyz Þ are identically zero. The respective parts are defined in terms of mid-line coordinates 0 0 0 ð0Þ 00 eð0Þ xx ¼ u0 þ z#y þ y#z G ðsÞu
ð11aÞ
dy 0 dz 0 # þ # Gð1Þ ðsÞu00 ds y ds z
ð11bÞ
eð1Þ xx ¼
ð0Þ 0 ð0Þ cð0Þ xs ¼ c xs þ g ðsÞu 0 cð0Þ xs ¼ ð#y þ wp Þ ð1Þ 0 cð1Þ xs ¼ g ðsÞu
dz dy þ ð#z þ v0p Þ ds ds
ð11cÞ ð11dÞ ð11eÞ
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Fig. 3 In-plane and transversal stress resultants and stress couples acting on a beam’s wall element
0 cð0Þ xn ¼ ð#y þ wp Þ
dy dz þ ð#z þ v0p Þ ; ds ds
ð11fÞ
and primary and secondary warping functions G ðsÞ, Gð1Þ ðsÞ and gð0Þ , gð1Þ functions formulas are given explicitly in the Appendix 1. ð0Þ
2.3 Energies Following the displacement field, the position vector formula and acceleration dependencies (see Sects. 2.2.1, 2.2.2 and 2.2.3) the formulas for the appropriate energies are derived in subsequent paragraphs.
The requested variations of strains given by formulas (11) are 0 0 0 ð0Þ 00 deð0Þ xx ¼ du0 þ zd#y þ yd#z G ðsÞdu
ð14aÞ
dy 0 dz 0 ð14bÞ d# þ d# Gð1Þ ðsÞdu00 ds y ds z dz dz dy dy d#y þ dw0p þ d#z þ dv0p þ gð0Þ ðsÞdu0 dcð0Þ xs ¼ ds ds ds ds ð14cÞ deð1Þ xx ¼
ð1Þ 0 dcð1Þ xs ¼ g ðsÞdu
ð14dÞ
dy dy dz dz d#y dw0p þ d#z þ dv0p ds ds ds ds
2.3.1 Potential energy
dcð0Þ xn ¼
The potential energy of the elastic system under consideration is given by
Integrating given in Appendix 2 formulas for stress resultants (95, 96) and stress couples (97) along the mid-line contour the following onedimensional stress measures are introduced
1 U¼ 2 1 2
Zl Z Zh=2
rxx exx þ rxn cxn þ rxs cxs dndsdx ¼
0 c h=2 Z lZ
0 c
Tx ¼
Nxx eð0Þ xx
þ
ð1Þ
þLxs cxs dsdx
Lxx eð1Þ xx
þ
Nxn cð0Þ xn
þ
Nxs cð0Þ xs
dU ¼
ð12Þ
þ
Lxs dcð1Þ xs
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Nxx ds
the axial force
Z dy dz Qy ¼ ds Nxs þ Nxn ds ds c
ð1Þ ð0Þ ð0Þ Nxx deð0Þ xx þ Lxx dexx þ Nxn dcxn þ Nxs dcxs
0 c
Z
the shear force in y-direction Z dz dy ds Nxs Nxn Qz ¼ ds ds c
ð13Þ
ð16Þ
ð17Þ
the shear force in z-direction Z gð0Þ ðsÞNxs þ gð1Þ ðsÞLxs ds Mx ¼ ð18Þ
c
dsdx:
ð15Þ
c
whereas formulations for stress resultants and stress couples (see Fig. 3) as defined in (95–97) were used. Integral’s subscript c denotes integration along the mid-line contour. The variation of the potential energy arising from Eq. (12) is Z lZ
ð14eÞ
the twisting moment about x-axis
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My ¼
Z dy ds zNxx Lxx ds c
the bending moment in y-axis Z dz Mz ¼ ds yNxx þ Lxx ds c
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ð19Þ
ð20Þ
Z
Nsn ðs; xÞ ¼ C45 cð0Þ xn
ð28Þ
and next substituted in (15–21). After unknowns reordering one arrives at final formulas for onedimensional stress measures þ a15 w0p þ a17 u0 a16 u00 ;
Gð0Þ ðsÞNxx þ Gð1Þ ðsÞLxx ds ð21Þ
c
ð27Þ
Tx ¼a11 u00 þ a15 #y þ a13 #0y þ a14 #z þ a12 #0z þ a14 v0p
the bending moment in z-axis Bw ¼
Nxn ðs; xÞ ¼ C44 cð0Þ xn
Qy ¼a14 u00 þ a45 #y þ a34 #0y þ a44 #z þ a24 #0z þ a44 v0p þ a45 w0p þ a47 u0 a46 u00 ;
the warping torque; bimoment Therefore considering equations (14a) and the above given stress measures, relation (13) after integration by parts and reordering takes the final form dU ¼
Zl n 0
ð22Þ Appearing in the above one-dimensional stress measures might be expressed in terms of basic unknown functions of the problem (or their spatial derivatives) i.e. u0 ðx;tÞ, vp ðx;tÞ, wp ðx;tÞ, #y ðx;tÞ, #z ðx;tÞ and uðx;tÞ. To do this the strain definitions (11) are substituted into stress resultants and stress couples definitions as given in (104) 0 ð1Þ ð0Þ Nxx ðs; xÞ ¼ K11 eð0Þ xx þ K12 c xs þ K13 u þ K14 exx
ð23Þ 0 ð1Þ ð0Þ Nxs ðs; xÞ ¼ K21 eð0Þ xx þ K22 c xs þ K23 u þ K24 exx
ð24Þ 0 ð1Þ ð0Þ Lxx ðs; xÞ ¼ K41 eð0Þ xx þ K42 c xs þ K43 u þ K44 exx
Mz ¼a12 u00 þ a25 #y þ a23 #0y þ a24 #z þ a22 #0z þ a24 v0p þ a25 w0p þ a27 u0 a26 u00 ; Bw ¼a16 u00 þ a56 #y þ a36 #0y þ a46 #z þ a26 #0z þ a46 v0p þ a56 w0p þ a67 u0 a66 u00 : ð29Þ where the coefficients aij (i; j ¼ 1; . . .7) are defined in Appendix 2. 2.3.2 Kinetic energy The kinetic energy of the system is defined Z 1 T¼ q R_ T R_ dV 2
0 ð1Þ þ K52 cð0Þ xs þ K53 u þ K54 exx
ð26Þ
ð30Þ
V
where designation q refers to average composite material density and V refers to volume element and dV ¼ dndsdx. From the above we obtain the variation dT ¼ ¼
ð25Þ Lxs ðs; xÞ ¼
þ a35 w0p þ a37 u0 a36 u00 ;
x¼0
K51 eð0Þ xx
Mx ¼a17 u00 þ a57 #y þ a37 #0y þ a47 #z þ a27 #0z þ a47 v0p My ¼a13 u00 þ a35 #y þ a33 #0y þ a34 #z þ a23 #0z þ a34 v0p
d#y Qy Mz0 d#z o Mx0 þB00w du dx x¼l x¼l x¼l þTx du0 þQy dvp þQz dwp x¼0 x¼0 x¼0 x¼l x¼l x¼l þMy d#y þMz d#z þ Mx þB0w du x¼0 x¼0 x¼0 x¼l Bw du0 Qz My0
þ a55 w0p þ a57 u0 a56 u00 ; þ a57 w0p þ a77 u0 a67 u00 ;
Tx0 du0 Q0y dvp Q0z dwp
Qz ¼a15 u00 þ a55 #y þ a35 #0y þ a45 #z þ a25 #0z þ a45 v0p
Z
_ q R_ T dRdV
V Z
q V
o _T R dR dV ot
Z
€ T dRdV qR
ð31Þ
V
Integration of the above relation over the time interval ht1 ; t2 i and keeping in mind that dRi vanishes at time
123
1842
Meccanica (2014) 49:1833–1858
t ¼ t1 and t ¼ t2 , yields the kinetic energy variation term for the Hamiltonian principle (1)
Wext;4 ¼ ns
Z l Zh=2
¼
8 > < 1 ns ¼ 1 > : 0
with
t1 V
Zt2Z
s¼s1
0 h=2
Zt2 Zt2Z €T dRdVdt dTdt ¼ qR t1
s¼s2 ^tx Dx þ ^ty Dy þ ^tz Dz dndx
q R€x dRx þ R€y dRy þ R€z dRz dVdt
s ¼ s1 s ¼ s2 elsewhere
Wext;5 ¼ Text;z wðtÞ
ð33Þ
t1 V
ð32Þ Cartesian coordinates of the position vector variation dR, as well as appropriate basic unknowns terms are given in Appendix 3.
whereas superposed bar indicates displacements of the mid-surface contour points (n ¼ 0). Therefore, the total work and its perturbation are as follows Wext ¼Wext;1 þ Wext;2 þ Wext;3 þ Wext;4 þ Wext;5 dWext ¼dWext;1 þ dWext;2 þ dWext;3 þ dWext;4 þ dWext;5
2.3.3 Work of external forces
ð34Þ
Consider the general case of beam loading expressed in terms of – – –
–
–
Fi —body forces per unit mass, with corresponding external work Wext;1 , qi —surface loads per unit area applied at the beam’s middle surface with external work Wext;2 , ti —traction loads per unit area applied at longitudinal (side) sections of open cross-section beams with resulting external work Wext;3 , ^ti —traction loads per unite area applied in beams with open cross-sections of at free edges of the cross-section with external work Wext;4 , Text;z —torque applied at hub for nonconstant rotational speed, Wext;5 .
The respective terms are as follows Z lZ Zh=2 Wext;1 ¼ q0 Fx Dx þ Fy Dy þ Fz Dz dndsdx 0 c h=2
Wext;2 ¼
Z lZ
Then the external works equations system (33) using also displacement relations (9) takes the form
Wext;1 ¼
Z l Z Zh=2 0 c h=2
Fx Gðn; sÞu0 þ Fy vp þ Fz wp
i þ ½Fz ðY yp Þ Fy ðZ zp Þu dndsdx Z lZ h Wext;2 ¼ qx u0 þ qx z#y þ qx y#z qx Gð0Þ ðsÞu0 0 c
i þ qy vp þ qz wp þ ½qz ðy yp Þ qy ðz zp Þu dsdx Z Zh=2 tx u0 þ tx Z#y þ tx Y#z Wext;3 ¼ nx c h=2
tx Gðn; sÞu0 þ ty vp þ tz wp
x¼l þ ½tz ðY yp Þ ty ðZ zp Þu dnds
qx Dx þ qy Dy þ qz Dz dsdx
x¼0
0 c
Wext;3 ¼ nx
Z Zh=2
x¼l tx Dx þ ty Dy þ tz Dz dnds
x¼0
c h=2
with
123
8 > < 1 nx ¼ 1 > : 0
x¼0 x¼l elsewhere
h q0 Fx u0 þ Fx Z#y þ Fx Y#z
Wext;4 ¼ d0 ns
Z l Zh=2
^tx u0 þ ^tx Z#y þ ^tx Y#z
0 h=2
^tx Gðn; sÞu0 þ ^ty vp þ ^tz wp
s¼s2 ^ ^ þ ½tz ðY yp Þ ty ðZ zp Þu dndx s¼s1
ð35Þ
Meccanica (2014) 49:1833–1858
1843
where d0 is a tag indicting case of open (d0 ¼ 1) or closed case (d0 ¼ 0). Formulas for the appropriate work terms in (35) are given in Appendix 4.
i € dx B3 sin h þ B2 cos h ðR0 þ xÞu Zl h _ 2B11 v_p wðtÞ _ sin h cos h 2B1 ðR0 þ xÞu_ 0 wðtÞ
2.4 Equations of motion
0
In this section the final form of the equations of motions of the system is derived. Starting from Eq. (1) and considering Eqs. (32), (13), and (121) the least action principle results in the relation Zt2 ( Z lZ Zh=2 € T dRdndsdx dJ ¼ qR t1
ð37Þ
m ð0Þ ð1Þ m ð0Þ m ð0Þ dexx þ Lm Nxx xx dexx þ Nxn dcxn þ Nxs dcxs
0 c
þ dWext;3 þ dWext;4 þ dWext;5 dt ¼ 0:
ð36Þ
dwðtÞ
€ B14 lwðtÞ € cos2 h B13 lwðtÞ € sin2 h B22 wðtÞ Zl h € € cos h sin h þ 2B15 lwðtÞ 2B1 ðR0 þ xÞu0 wðtÞ 0
€ þ 2 B12 cos h B11 sin h cos h vp wðtÞ € þ 2 B11 sin2 h B12 sin h cos h wp wðtÞ € þ 2B12 ðR0 þxÞ#z wðtÞ € þ 2B11 ðR0 þ xÞ#y wðtÞ € þ 2 B17 sin2 hB18 cos2 h uwðtÞ i € sin h cos h dx 2B7 ðR0 þxÞu0 þ 2 B16 B19 uwðtÞ
Zl h 0
•
du0 ,
_ cos h €0 þ 2B1 v_p wðtÞ B1 u€0 B12 #€z B11 #€y þ B7 u _ sin h 2B3 u_ wðtÞ _ cos h 2B1 w_ p wðtÞ _ sin h 2B2 u_ wðtÞ
Considering for variation of potential energy (22), variation of kinetic energy Eqs. (110)–(115) and variation of external work equation (122), the least action principle equation (36), after integration with respect to time leads to the following equations with respect to variation of problem’s independent variables
_ 2B7 ðR0 þ xÞu_ 0 wðtÞ _ þ 2B12 ðR0 þ xÞ#_z wðtÞ _ sin2 h 2B18 u_ wðtÞ _ cos2 h þ 2B17 u_ wðtÞ i _ sin h cos h dx þ Text;z ¼ 0 þ 2ðB16 B19 Þu_ wðtÞ
0 c h=2
Z lZ
ð1Þ þ Lm xs dcxs dsdx þ dWext;1 þ dWext;2 )
•
_ cos2 h 2B12 w_ p wðtÞ _ sin h cos h þ 2B12 v_p wðtÞ _ sin2 h þ 2B11 ðR0 þ xÞ#_y wðtÞ _ þ 2B11 w_ p wðtÞ
2
2 2 þ B1 ðR0 þ x þ u0 Þw_ ðtÞ þ B12 #z w_ ðtÞ 2 2 € sin h þ B11 #y w_ ðtÞ B7 u0 w_ ðtÞ B1 wp wðtÞ € sin h þ B1 vp wðtÞ € cos h € sin h B11 wðtÞ B3 uwðtÞ
€ cos h € cos h þ B12 wðtÞ B2 uwðtÞ 0 þ Tx þ Pext;x þ d0 P^ext;x ¼ 0 with boundary conditions, u0 x¼0 ¼ 0; •
Tx þ nx Qext;x x¼l ¼ 0
ð39Þ
dvp
_ cos h 2B12 #_z wðtÞ _ cos h € 2B1 u_ 0 wðtÞ B1 v€p þ B2 u _ cos h þ 2B7 u_ 0 wðtÞcos _ 2B11 #_y wðtÞ h 2 2 B1 wp w_ ðtÞsin h cos h B3 uw_ ðtÞsin h cos h 2 2 B11 w_ ðtÞsin h cos h þ B1 vp w_ ðtÞ cos2 h 2 2 B2 uw_ ðtÞcos2 h þ B12 w_ ðtÞ cos2 h B1 ðR0 þ x þ u0 Þ € cos h B11 #y wðtÞcos € € cos h B12 #z wðtÞ h wðtÞ
€ cos h þ Q0 þ Pext;y þ d0 P^ext;y ¼ 0 þ B7 u0 wðtÞ y
vp cos h B11 sin hB12 cos h u€0 þB1 ðR0 þ xÞ€
€p sin h þ B13 sin h B15 cos h #€y B1 ðR0 þ xÞw 0 € þ B15 sin h B14 cos h #€z þ B9 cos h B8 sin h u
ð38Þ
ð40Þ with boundary conditions, vp x¼0 ¼ 0;
Qy þ nx Qext;y x¼l ¼ 0
ð41Þ
123
1844
•
Meccanica (2014) 49:1833–1858
•
dwp
_ _ _ € €p B3 uþ2B B1 w 1 u_ 0 wðtÞsinhþ2B12 #z wðtÞsinh 2
2 _ _ _ _ 0 wðtÞsinhþB þ2B11 #_y wðtÞsinh2B 7u 1 wp w ðtÞsin h 2 2 þB3 uw_ ðtÞsin2 hþB11 w_ ðtÞsin2 h 2
2
B1 vp w_ ðtÞsinhcoshþB2 uw_ ðtÞsinhcosh
2
ð42Þ with boundary conditions,
•
Qz þ nx Qext;z x¼l ¼ 0
2
ð43Þ
2
sin h þ B11 ðR0 þ x þ u0 Þw_ ðtÞ þ B15 #z w_ ðtÞ 2 2 € sin h þ B15 #y w_ ðtÞ B8 u0 w_ ðtÞ B11 wp wðtÞ € cos h € sin h þ B11 vp wðtÞ B17 uwðtÞ
€ cos h B13 wðtÞ € sin h þ B15 wðtÞ € cos h B16 uwðtÞ 0 ^ ext;y ¼ 0 Qz þ My þ mext;y þ d0 m ð44Þ with boundary conditions,
•
2 2 þ B3 wp w_ ðtÞ sin2 h þ B4 uw_ ðtÞ cos2 h 2
_ cos h €0 þ 2B11 v_p wðtÞ B11 u€0 B13 #€y B15 #€z þ B8 u _ sin h 2B17 u_ wðtÞ _ cos h 2B11 w_ p wðtÞ _ 2B16 u_ wðtÞ
2
B3 vp w_ ðtÞ sin h cos h þ B2 wp w_ ðtÞ sin h cos h 2
B19 uw_ ðtÞ cos2 h þ B5 uw_ ðtÞ sin2 h
d#y
#y x¼0 ¼ 0;
€ sin h þ B3 ðR0 þ x þ u0 ÞwðtÞ € sin h B2 ðR0 þ xÞuwðtÞ € cos h þ B17 #y wðtÞ € sin h þ B18 #z wðtÞ € cos h þ B16 #y wðtÞ 2 2 B18 w_ ðtÞ cos2 h þ B17 w_ ðtÞ sin2 h 2 2 þ B16 B19 w_ ðtÞ cos h sin h B2 vp w_ ðtÞ cos2 h
0 € ^ B7 u0 wðtÞsinhþQ z þPext;z þd0 Pext;z ¼ 0
€ B3 w € € p B5 u B2 v€p B4 u € € cos h þ B3 ðR0 þ xÞuwðtÞ cos h þ B2 ðR0 þ x þ u0 ÞwðtÞ
€ sin h B20 u0 wðtÞ € cos h B21 u0 wðtÞ € sin h þ B19 #z wðtÞ
2 € B12 w_ ðtÞsinhcoshþB1 ðR0 þxþu0 ÞwðtÞsinh € € þB12 #z wðtÞsinhþB 11 #y wðtÞsinh
wp x¼0 ¼ 0;
du
My þ nx Mext;y x¼l ¼ 0
ð45Þ
d#z
2 2 B16 uw_ ðtÞ sin2 h þ B17 uw_ ðtÞ sin h cos h 2 _ cos h þ B18 uw_ ðtÞ cos h sin h þ 2B2 u_ 0 wðtÞ _ sin h þ 2B16 #_y wðtÞ _ cos h þ 2B3 u_ 0 wðtÞ
_ sin h þ 2B18 #_z wðtÞ _ cos h þ 2B17 #_y wðtÞ _ sin h þ 2B6 uw_ 2 ðtÞ sin h cos h þ 2B19 #_z wðtÞ _ cos h 2B21 u_ 0 wðtÞ _ sin h 2B20 u_ 0 wðtÞ 0 0 2 € cos h þ B7 ðR0 þ x þ u0 Þ w_ ðtÞ þ B7 vp wðtÞ 0 0 2 € sin h þ B8 #y w_ ðtÞ þ B9 #z 0 w_ 2 ðtÞ B7 wp wðtÞ 0 € cos h B21 u 0 wðtÞ € sin h B20 u wðtÞ 0 _ cos h 2 B7 w_ p 0 wðtÞ _ sin h þ 2 B7 v_p wðtÞ 0 0 _ cos h 2 B21 u_ wðtÞ _ sin h B8 #€y 0 2 B20 u_ wðtÞ 0 0 0 0 2 €0 B10 u0 w_ ðtÞ B9 #€z B7 u€0 þ B10 u ^ext;x þ d0 m ^0ext;w ¼ 0 þ Mx0 þ B00w þ mext;x þ m0ext;w þ d0 m ð48Þ
with boundary conditions, u0 x¼0 ¼ 0; ux¼0 ¼ 0; € € € B9 wðtÞcoshB 8 wðtÞsinhþB7 vp wðtÞcosh € € € 7 wp wðtÞsinhB21 wðtÞusinh x¼l _ sin h þ B12 ðR0 þ x þ u0 Þw_ 2 ðtÞ þ B14 #z w_ 2 ðtÞ B20 uwðtÞcoshB 2B19 u_ wðtÞ 2 2 2 2 2 þ ðR0 þxþu0 ÞB7 w_ ðtÞþB8 #y w_ ðtÞþB9 #z w_ ðtÞ € sin h þ B15 #y w_ ðtÞ B9 u0 w_ ðtÞ B12 wp wðtÞ 0 _2 _ v_p cosh þ 2B7 wðtÞ w u ðtÞ B 10 € € € x¼l B19 uwðtÞ sin h þ B12 vp wðtÞ cos h B18 uwðtÞ cos h _ ucosh2B _ _ p sinh _ 2B20 wðtÞ 7 wðtÞw € sin h þ B14 wðtÞ € cos h Qy þ M 0 þ mext;z B15 wðtÞ z _ usinh _ 2B21 wðtÞ x¼l ^ext;z ¼ 0 þ d0 m ð46Þ 0 € B7 u€0 B8 #€y B9 #€z þ B10 u with boundary conditions, x¼l ^ext;w þnx Mext;x x¼l ¼ 0 þ Mx þB0w mext;w m #z x¼0 ¼ 0; Mz þ nx Mext;z x¼l ¼ 0 ð47Þ ð49Þ Bw þnx Mext;w x¼l ¼ 0 _ cos h €0 þ 2B12 v_p wðtÞ B12 u€0 B15 #€y B14 #€z þ B9 u _ sin h _ cos h 2B12 w_ p wðtÞ 2B18 u_ wðtÞ
123
Meccanica (2014) 49:1833–1858
1845
Relations (37)–(49) form a nonlinear system of PDEs with all equations coupled together. It should be noted that in a case of the nonrotating beam and zero pitch angle the above system of equations corresponds exactly to the particular case given by Librescu and Song in [21]. The equations derived in this paper take into account nonconstant angular velocity and also arbitrary pitch angle of the beam. These two effects are essential for the study of dynamics of rotating blades.
•
du0
_ cos h 2B1 w_ 0 wðtÞ _ sin h B1 u€0 þ 2B1 v_0 wðtÞ 2 € sin h þ B1 ðR0 þ x þ u0 Þw_ ðtÞ B1 w0 wðtÞ € cos h þ a11 u0 0 þ a14 #z 0 þ a14 v0 0 þ B1 v0 wðtÞ 0
0
þ Pext;x þ d0 P^ext;x ¼ 0
ð52Þ
with boundary conditions, u0 x¼0 ¼ 0; a11 u00 þ a14 #z þ a14 v00 þ nx Qext;x x¼l ¼ 0;
3 Results
ð53Þ
In this section two cases of composite beams are examined in order to study the impact of the variable rotating speed and the pitch angle on system’s motion. The geometries of both cross-sections are presented in Fig. 4. To simplify the evaluation a symmetrically lay-up composite material is considered for both beams. Therefore the coupling stiffness matrix coefficients Bij [see definition (99)] vanish. Moreover it is assumed that the cross-section pole coincides with the spanwise axis x; therefore vp ! v0 , wp ! w0 . This results in simplification yp ¼ zp ¼ 0
•
2
_ cos h B1 w0 w_ ðtÞ sin h cos h B1 v€0 2B1 u_ 0 wðtÞ 2
€ cos h þ B1 v0 w_ ðtÞ cos2 h B1 ðR0 þxþu0 ÞwðtÞ 0 0 0 0 þ a14 u0 þ a44 ð#z þ v0 Þ þ Pext;y þ d0 P^ext;y ¼ 0 ð54Þ with boundary conditions, v0 x¼0 ¼ 0; a14 u00 þ a44 ð#z þ v00 Þ þ nx Qext;y x¼l ¼ 0 ð55Þ
ð50Þ •
3.1 Symmetric composite blade
dw
2 € sin h B1 v0 w_ ðtÞ sin h cos h þ B1 ðR0 þxþu0 ÞwðtÞ 0 þ a55 ð#y þ w00 Þ þ Pext;z þ d0 P^ext;z ¼ 0 ð56Þ
with boundary conditions, w0 x¼0 ¼ 0; a55 ð#y þ w00 Þ þ nx Qext;z x¼l ¼ 0 ð57Þ
€ B14 lwðtÞ € cos2 h B13 lwðtÞ € sin2 h B22 wðtÞ Zl € þ 2ðB16 B19 ÞuwðtÞ € 2B1 ðR0 þ xÞu0 wðtÞ
•
Zl
_ 2B1 ðR0 þ xÞu_ 0 wðtÞ
0
_ sin h cos h dx þ 2ðB16 B19 Þu_ wðtÞ Zl €0 sin h v0 cos h B1 ðR0 þxÞw B1 ðR0 þxÞ€
ð58Þ with boundary conditions, #y x¼0 ¼ 0; a33 #0y þ a37 u0 þ nx Mext;y x¼l ¼ 0;
0
þ B13 #€y sin h B14 #€z cos h dx þ Text;z ¼ 0
d#y
_ cos h þ B13 #y w_ 2 ðtÞ B13 #€y 2B16 u_ wðtÞ € sin h € cos h B13 wðtÞ B16 uwðtÞ 0 ^ext;y a55 #y a55 w00 þ a33 #0y þ mext;y þ d0 m 0 þ a37 u0 ¼ 0
0
sin h cos h dx
dw0
_ sin h þ B1 w0 w_ 2 ðtÞ sin2 h €0 þ 2B1 u_ 0 wðtÞ B1 w
A symmetric composite beam with rectangular open cross-section as indicated in Fig. 4a is examined. In this case, considering the inertia and stiffness coefficients given in Appendix 5, Tables 1 and 2 individual equations of motion are given by •
dv0
ð51Þ
ð59Þ
123
1846
Meccanica (2014) 49:1833–1858
(a)
(b)
Fig. 4 Beam cross-sections to be considered: a blade (open section) b closed box beam
•
3.2 Symmetric composite uniform box-beam
d#z
_ sin h þ B14 #z w_ 2 ðtÞ B14 #€z 2B19 u_ wðtÞ € cos h € sin h þ B14 wðtÞ B19 uwðtÞ ^ ext;z a14 u00 a44 #z a44 v00 þ mext;z þ d0 m 0 þ a22 #0z ¼ 0 with boundary conditions, 0 #z jx¼0 ¼ 0; a22 #z þnx Mext;z x¼l ¼ 0; •
ð60Þ
ð61Þ
In this section a composite symmetric beam with rectangular closed cross-section as indicated in Fig. 4b is examined; the circumferentially uniform stiffness (CUS) composite configuration is assumed, which is achievable via usual filament winding technology. In this case, considering the inertia and stiffness coefficients given in Appendix 5 Tables 1 and 2 respectively the equations of motion are given by •
du
€ cos h þ B19 #z wðtÞ € sin h € B5 u € þ B16 #y wðtÞ B4 u 2 2 þ B16 B19 w_ ðtÞ cos h sin h þ B4 uw_ ðtÞ cos2 h 2
€ B14 lwðtÞ € cos2 h B13 lwðtÞ € sin2 h I2 wðtÞ
Zl
€ 2B1 ðR0 þ xÞu0 wðtÞ
2
B19 uw_ ðtÞ cos2 h þ B5 uw_ ðtÞ sin2 h 2
_ cos h B16 uw_ ðtÞ sin2 h þ 2B16 #_y wðtÞ 0 0 2 _ sin h þ B10 u €0 B10 u0 w_ ðtÞ þ 2B19 #_z wðtÞ ^ext;x þ d0 m ^0ext;w þ mext;x þ m0ext;w þ d0 m 0 0 00 þ a37 #0y þ a77 u0 a66 u00 ¼ 0; ð62Þ with boundary conditions, ux¼0 ¼ 0; u0 x¼0 ¼ 0; a66 u00 þ nx Mext;w x¼l ¼ 0; 0 2 €0 þ a37 #0y þ a77 u0 a66 u00 B10 u0 w_ ðtÞ þ B10 u ^ ext;w þ nx Mext;x ¼ 0; mext;w m ð63Þ x¼l
It should be noted that in relations (52)–(63) in all equations there is a term which corresponds to nonconstant rotation. Therefore all equations are coupled together and form a nonlinear system of PDEs. Detailed analysis of the above equations is given together with a set arising for the box-beam case.
123
dw
0
€ sin h cos h dx þ 2ðB16 B19 ÞuwðtÞ
Zl
_ 2B1 ðR0 þ xÞu_ 0 wðtÞ
ð64Þ
0
_ sin h cos h dx þ 2ðB16 B19 Þu_ wðtÞ
Zl
€0 sin h B1 ðR0 þxÞ€ v0 cos h B1 ðR0 þxÞw
0
þ B13 #€y sin h B14 #€z cos h dx þ Text;z ¼ 0 •
du0 ,
_ cos h 2B1 w_ 0 wðtÞ _ sin h B1 u€0 þ 2B1 v_0 wðtÞ 2 € sin h þ B1 ðR0 þ x þ u0 Þw_ ðtÞ B1 w0 wðtÞ 0 € cos h þ a11 u0 þ a17 u0 0 þ Pext;x þ B1 v0 wðtÞ 0
þ d0 P^ext;x ¼ 0 ð65Þ
Meccanica (2014) 49:1833–1858 Table 1 General definitions of inertia coefficients Bi and formulas for beams made of symmetric laminate with rectangular cross-section
1847
Open section
Closed-section
.dnds
m0 d
2m0 ðc þ dÞ
Inertia coefficient Ii B1 ¼
R R h=2 c
h=2
B2 ¼
R R h=2
c h=2
.ðZ zp Þdnds
0
0
B3 ¼
R R h=2
c h=2
.ðY yp Þdnds
0
0
B4 ¼
R R h=2
c h=2
.ðZ zp Þ2 dnds
m2 d
þ 2m2 d
B5 ¼
R R h=2
m0 c2 ð3dþcÞ 6
c h=2
.ðY yp Þ dnds
1 3 12 m0 d
þ 2m2 c
B6 ¼
R R h=2
m0 d2 ð3cþdÞ 6
c h=2
.ðZ zp ÞðY yp Þdnds
0
0
B7 ¼
R R h=2
c h=2
.Gðn; sÞdnds
0
0
B8 ¼
R R h=2
c h=2
.ZGðn; sÞdnds
0
0
B9 ¼
R R h=2
c h=2
.YGðn; sÞdnds
0
0 c2 d2 ðcdÞ2 m0 24ðcþdÞ
2
B10 ¼
R R h=2
.G2 ðn; sÞdnds
1 3 12 m2 d
B11 ¼
R R h=2
c h=2
.Zdnds
0
0
B12 ¼
R R h=2
c h=2
.Ydnds
0
0
B13 ¼
R R h=2
c h=2
.Z 2 dnds
m2 d
þ 2m2 d
B14 ¼
R R h=2
m0 c2 ð3dþcÞ 6
.Y 2 dnds
1 3 12 m0 d
þ 2m2 c
B15 ¼
R R h=2
m0 d2 ð3cþdÞ 6
c h=2
.YZdnds
0
0
B16 ¼
R R h=2
c h=2
.ZðZ zp Þdnds
m2 d
B17 ¼
R R h=2
m0 c2 ð3dþcÞ 6
c h=2
.ZðY yp Þdnds
0
0
B18 ¼
R R h=2
c h=2
.YðZ zp Þdnds
0
0
B19 ¼
R R h=2
c h=2
.YðY yp Þdnds
1 3 12 m0 d
B20 ¼
R R h=2
m0 d2 ð3cþdÞ 6
c h=2
.ðZ zp ÞGðn; sÞdnds
0
0
B21 ¼
R R h=2
c h=2
.ðY yp ÞGðn; sÞdnds
0
B22 ¼
R l R R h=2
c h=2
c h=2
0 c h=2
.ðx2 þ R20 þ 2xR0 Þdndsdx
with boundary conditions, u0 x¼0 ¼ 0; a11 u00 þ a17 u0 þ nx Qext;x x¼l ¼ 0
b1
dv0
2 € cos h þ B1 v0 w_ ðtÞ cos2 h B1 ðR0 þxþu0 ÞwðtÞ 0 0 þ a44 ð#z þ v00 Þ þ a34 #0y þ Pext;y þ d0 P^ext;y ¼ 0
ð67Þ
Þ
þ 2m2 c
0 1
3l
3
þ R20 lþ R0 l
2
b1
l3 3
þ R20 l þ R0 l2
with boundary conditions, h i a44 ð#z þ v00 Þ þ a34 #0y þ nx Qext;y x¼l ¼ 0 v0 x¼0 ¼ 0; ð68Þ •
_ cos h B1 w0 w_ 2 ðtÞ sin h cos h B1 v€0 2B1 u_ 0 wðtÞ
3
þ 2m2 d
ð66Þ •
3
þ m2 ðc6þd
dw0
_ sin h þ B1 w0 w_ 2 ðtÞ sin2 h €0 þ 2B1 u_ 0 wðtÞ B1 w 2 € sin h B1 v0 w_ ðtÞ sin h cos h þ B1 ðR0 þxþu0 ÞwðtÞ 0 0 þ a25 #0z þ a55 ð#y þ w00 Þ þ Pext;z þ d0 P^ext;z ¼ 0
ð69Þ
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Table 2 Definitions of stiffness coefficients aij Stiffness coefficient aij
Open section
Closed-section
R a11 ¼ K11 ds (Extensional)
K11 d
2K11 ðc þ dÞ
0
0
(Extension-chordwise bending) R a13 ¼ c K11 z K14 dy ds ds
0
0
(Extension-flapwise bending) R a14 ¼ c K12 dy ds ds
K12 d
0
(Extension-choordwise transverse shear) R dz c K12 ds ds
0
0
a16 ¼
(Extension-flapwise transverse shear) R ð0Þ ð1Þ c K11 G ðsÞ þ K14 G ðsÞ ds
0
0
(Extension-warping) R c K13 ds (Extension-twist) dz 2 R a22 ¼ c K11 y2 þ 2K14 dz ds ds y þ K44 ds
0
2K12 cd
d3 K11 12
K11 d6 ð3cþdÞ þ 2D22 c
(Chordwise bending) R dz dz dy a23 ¼ c K11 yz K14 dy ds y þ K14 ds z K44 ds ds ds
0
0
(Chordwise bending-flapwise bending) R dy dz a24 ¼ c K12 dy ds y þ K24 ds ds ds
0
0
0
K12 cd
0
0
0
0
D22 d
K11 c6 ðcþ3dÞ þ 2D22 d
0
K12 cd
0
0
0
0
2D26 d
0
K22 d
2K22 d þ 2A44 c
(Chordwise transverse shear) R dy dz dz a45 ¼ c K22 dy ds ds A44 ds ds ds
0
0
(Chordwise transverse shear-flapwise transverse shear) R dy ð1Þ a46 ¼ c K12 Gð0Þ ðsÞ dy ds þ K24 G ðsÞ ds ds
0
0
(Chordwise transverse shear-warping) R a47 ¼ c K23 dy ds ds (Chordwise transverse shear-twist)
0
0
c
a12 ¼
R K11 y þ K14 dz ds ds c
a15 ¼
a17 ¼
(Chordwise bending-chordwise transverse shear) dz 2 R a25 ¼ c K12 dz ds ds y þ K24 ds (Chordwise bending-flapwise transverse shear) R a26 ¼ c K11 Gð0ÞðsÞyþK14 Gð1ÞðsÞyþK14 Gð0ÞðsÞ dz ds þK44 Gð1ÞðsÞ dz ds ds (Chordwise bending-warping) R (Chordwise bending-twist) a27 ¼ c K13 y þ K43 dz ds 2 R dy 2 a33 ¼ c K11 z 2K14 ds z þ K44 dy ds ds (Flapwise bending) dy 2 R dy ds c K12 ds z K24 ds
a34 ¼
(Flapwise bending-chordwise transverse shear) R dz dz dy c K12 ds z K24 ds ds ds
a35 ¼
(Flapwise bending-flapwise transverse shear) R dy ð0Þ ð1Þ ð0Þ c K11 G ðsÞzþK14 G ðsÞzK14 G ðsÞ ds K44 Gð1ÞðsÞ dy ds ds ( Flapwise bending-warping) R a37 ¼ c K13 z K43 dy ds ds (Flapwise bending-twist) 2 R 2 þ A44 dz a44 ¼ c K22 dy ds ds ds a36 ¼
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2
2
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Table 2 continued Stiffness coefficient aij a55 ¼
R c
K22
dz 2 ds
þ A44
dy 2 ds
ds
(Flapwise transverse shear) R dz ð1Þ a56 ¼ c K12 Gð0Þ ðsÞ dz ds þ K24 G ðsÞ ds ds (Flapwise transverse shear warping) R dz c K23 ds ds (Flapwise transverse shear-twist) 2 R a66 ¼ c K11 Gð0Þ ðsÞ þ 2K14 Gð0Þ ðsÞGð1Þ ðsÞ 2 þK44 Gð1Þ ðsÞ ds (Warping) R a67 ¼ c K13 Gð0ÞðsÞ þ K43 Gð1ÞðsÞ ds (Warping-twist) R a77 ¼ c K23 gð0ÞðsÞ þ K53 gð1ÞðsÞ ds a57 ¼
Open section
Closed-section
A44 d
2K22 c þ 2A44 d
0
0
0
0
d3 D22 12
d ðcdÞ K11 c 24ðcþdÞ
2
2 2
3
þD22 c þd 6
3
0
0
2D66 d
d 2 ccþd K22 þ 8ðc þ dÞD66
2 2
(Twist) Integrals calculated upon laminate symmetry assumption and circumferentially uniform stiffness (CUS) configuration for the closed cross-section
with boundary conditions, w0 x¼0 ¼ 0; a25 #0z þ a55 ð#y þ w00 Þ þ nx Qext;z x¼l ¼ 0 ð70Þ •
d#y
du
€ cos h þ B19 #z wðtÞ € sin h € B5 u € þ B16 #y wðtÞ B4 u 2 2 þ B16 B19 w_ ðtÞ cos h sin h þ ðB4 B19 Þuw_ ðtÞ 2 _ cos2 h þ ðB5 B16 Þuw_ ðtÞ sin2 h þ 2B16 #_y wðtÞ 0 _ sin h þ B10 u €0 cos h þ 2B19 #_z wðtÞ 0 2 ^ ext;x B10 u0 w_ ðtÞ þ mext;x þ m0ext;w þ d0 m 00 0 0 ^0ext;w þ a17 u00 þ a77 u0 a66 u00 ¼ 0; þ d0 m
_ cos h þ B13 #y w_ 2 ðtÞ B13 #€y 2B16 u_ wðtÞ € sin h € cos h B13 wðtÞ B16 uwðtÞ 0 0 a55 #y a25 #0z a55 w00 þ a33 #0y þ a34 v00 0 ^ext;y ¼ 0 þ a34 #z þ mext;y þ d0 m
ð75Þ ð71Þ
with boundary conditions, #y x¼0 ¼ 0; a33 #0y þ a34 v00 þ a34 #z þ nx Mext;y x¼l ¼ 0 ð72Þ •
•
d#z
with boundary conditions, ux¼0 ¼ 0; u0 x¼0 ¼ 0; a66 u00 þ nx Mext;w x¼l ¼ 0; 0 2 €0 þ a17 u00 þ a77 u0 a66 u00 B10 u0 w_ ðtÞ þ B10 u ^ext;w þ nx Mext;x ¼ 0; mext;w m x¼l
ð76Þ
2
_ sin h þ B14 #z w_ ðtÞ B14 #€z 2B19 u_ wðtÞ € cos h € sin h þ B14 wðtÞ B19 uwðtÞ 0 0 0 a44 #z a34 #y a44 v0 þ a25 #y 0 0 ^ ext;z ¼ 0 þ a25 w00 þ a22 #0z þ mext;z þ d0 m
Considering equations (64)–(76), similarly with open cross-section case, they form a nonlinear system of all equations coupled. ð73Þ
with boundary conditions, #z x¼0 ¼ 0; a25 #y þa25 w00 þa22 #0z þnx Mext;z x¼l ¼ 0 ð74Þ
4 Discussion Looking at relations (51)–(63) and (65)–(76) one observes in all these equations there is at least one term
123
1850
€ is multiplied by a where the angular acceleration wðtÞ certain problem variable (or it’s spatial derivative). Therefore all the resulting equations are coupled together and form a nonlinear system of PDEs. Similar terms have also arised in paper [34], where a case of Euler–Bernoulli beam made of isotropic material rotating with nonconstant velocity has been considered. It is worth to note here that assuming temporary the zero value for the pitch angle and the constant rotating speed case (and position vector restricted to the middle-line contour material points) the obtained system of equations (64)–(76) coincides exactly with the one reported by Librescu and Song in [21]. Analysing the derived axial displacement equation in both of the considered cross-section cases [see eqs. (52) and (65)] one can notice that u is coupled to transverse displacements (v and w DOFs) not only by Coriolis terms but by nonconstant rotation velocity € terms as well. In these two equations fixing the wðtÞ preset position of a structure at h ¼ 0 makes the terms corresponding to beam’s vertical movement (i.e. flapping w) to disappear. In the second limit case of h ¼ p=2 presetting the axial dynamics of the box-beam [Eq. (65)] is similar because all v originating terms disappear. Since transverse displacements v and w are defined in local coordinate system one can conclude, that in both of the two discussed limit cases (h ¼ 0; p=2) only the leadlag plane displacement enters the axial equation of a box-beam. For the symmetric blade and limit case of h ¼ p=2 presetting the axial dynamics is different. This is related to the presence of a coupling term ða14 v0p Þ0 in (52) which does not depend on presetting angle. It means that apart from the discussed special case of h ¼ 0 both lead-lag and flapping couplings are always present in (52). Also for the box-beam case coupling terms originating from both transverse displacements are present in axial equation (65) for the general presetting h 6¼ ð0; p=2Þ. It should be noted that the numerical modal analysis of the rotating symmetric composite beam with open section for 0 , 45 and 90 pitch angles has been presented by authors in paper [19]. In case of 0 and 90 presettings the observed free vibration bending modes correspond to either flapwise of choordwise bendings. However in case of analysed 45 pitch angle
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Meccanica (2014) 49:1833–1858
free vibration bending modes exhibit mixed flapwise and with chordwise displacements. Hence the impact of arbitrary presetting angle modelled in this article is in full agreement with the findings reported in [19]. Studying the transverse displacement equations (54)/(67) and (56)/(69) one observes that the nonconstant rotation velocity introduces additional inertia terms rendering coupling to the axial displacement u. Considering special cases of h ¼ 0; p=2 presetting one observes that these terms affect only the lead-lag displacements (i.e. parallel to the plane of rotation). This coupling scheme is also confirmed by the appropriate terms in torsion equation (62)/(75). Analysis of eqs. (58)/(71) and (60)/(73) reveals the considered nonconstant rotating speed to introduce additional coupling terms also between the rotatory DOFs. Discussion of the special cases h ¼ 0; p=2 yields that the additional coupling term involving torsion angle u impacts only flapping cross-section rotation (i.e. about horizontal axis) and is not present in lead-lag plane. This is also confirmed by the detailed analysis of torsion equation (62)/(75). Further analysis of the pitch angle impact on the systems dynamics reveals that its non-zero value introduces new additional coupling between transverse displacements. The terms incorporating sin h cos h product that are not present in case of zero or p=2 presetting appear in transverse displacement formulas (54)/(67) and (56)/(69) formulas. This renders the bendings in both planes to be coupled. The similar sin h cos h product appears also in the equation of torsion as an additional centrifugal term.
5 Conclusions The equations of motion of composite Timoshenko beam experiencing variable angular velocity were derived. In the performed analysis also the general case of non-zero pitch angle and arbitrary hub’s radius were taken into account. The equations of motion form a nonlinear system of PDEs coupled together. Detailed analysis of symmetric layup of the composite beam of rectangular open and closed cross-section cases showed similar characteristics considering nonconstant rotating speed and the effect of pitch angle. Considering nonconstant angular velocity in both cases the nonlinear system of PDEs with all equations
Meccanica (2014) 49:1833–1858
1851
coupled together arises. Non-zero pitch angle results in direct coupling of flapwise bending motion with choordwise bending motion. As it was shown the nonconstant rotating speed introduces additional couplings to the system of equations of motion, which are not observed in wðtÞ ¼ const: cases discussed in the literature. Therefore nonconstant angular velocity and non-zero pitch angle have to be considered in modelling of rotating beams and in examination of their dynamics. The equations derived in this paper are fundamental for further study of dynamics of rotating blade systems. The investigations will be developed in (a) the aspect of variable pitch angle which takes place in e.g. helicopter rotor dynamics and (b) the aspect of non-ideal systems where the structure interacts with the energy source [5, 35].
dz dy ðZ zp Þ ds ds dz dy rn ðsÞ ¼ ðy yp Þ ðz zp Þ ds ds
ð79Þ
Rn ðsÞ ¼ ðY yp Þ
ð80Þ
to represent distances from pole P to the tangent and to the normal of the mid-line beam contour as shown in Fig. 2. In case of open section one gets Gð0Þ o ðsÞ
¼
Zs
Gð1Þ o ðsÞ ¼ rt ðsÞ
rn ð sÞd s ¼ 2Xos
0
ð81Þ whereas X0s is the area enclosed by the center line and s is a dummy variable measured along centerline till ð0Þ
the point of interest. The torsional function go ðsÞ and ð1Þ
Acknowledgments The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007–2013), FP7–REGPOT–2009–1, under Grant Agreement No:245479. The support by the Polish Ministry of Science and Higher Education—Grant No 1471-1/ 7. PR UE/2010/7—is acknowledged by the second and third author. Authors would like also to thank Professor Giuseppe Rega for his comments and the valuable discussions while preparing the manuscript.
go one for this case are given by gð0Þ o ðsÞ ¼ 0
Gð0Þ c ðsÞ ¼
Zs
Zs H rn ðsÞds 1 H rn ð sÞd s d s 1 sÞGxs ð sÞ ðhðsÞGxs ðsÞÞds hð 0
2X ¼ 2Xos s b Gð1Þ c ðsÞ¼rt ðsÞ2
Zs 0
¼rt ðsÞ
Appendix 1 The primary and secondary warping functions and torsional functions for open and closed section beams are defined as given in [21]. The warping function Gðn; sÞ is the sum of primary Gð0Þ ðsÞ and secondary one nGð1Þ ðsÞ Gðn; sÞ ¼ Gð0Þ ðsÞ þ nGð1Þ ðsÞ
ð77Þ
To simplify further expressions the following quantities are defined (see also Fig. 2) rt ðsÞ ¼ Rt ðsÞ ¼ ðz zp Þ
dz dy þ ðy yp Þ ds ds
ð78Þ
ð82Þ
In case of closed sections both warping functions take the form
0
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
gð1Þ o ðsÞ ¼ 2
ð83Þ H
H
1 1 d s sÞGxs ð sÞ ðhðsÞG1xs ðsÞÞds hð ds
ð84Þ
where hðsÞ is the thickness, Gxs ðsÞ is the shear modulus of the cross-section in circumferential surface and b ð:Þ
denotes mid-line contour perimeter and the gc ðsÞ functions are given by, H sÞd s rn ð 1 2X H ¼ ðsÞ ¼ gð0Þ ð85Þ c d s hðsÞG b ðsÞ Þ ðhðsÞG xs sÞ xs ð H d s 1 ð1Þ gc ðsÞ ¼ 2 H ¼2 ð86Þ d s ðhðsÞGxs ðsÞÞ hðsÞGxs ðsÞ whereas the last equalities are true in case of uniform hðsÞGxs product with respect to s coordinate.
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Meccanica (2014) 49:1833–1858
Appendix 2
Define the following 2-D stress resultants:
The constitutive equations for orthotropic layer in reference principal axis have the form 2 ðkÞ ðkÞ ðkÞ 9 8 C11 C12 C13 r ss 6 ðkÞ > > > > ðkÞ ðkÞ 6C > > >r > > 6 12 C22 C23 xx > > > > > 6 > > < 6 CðkÞ CðkÞ C ðkÞ rnn = 6 23 33 ¼ 6 13 > > 6 r xn > > 0 0 > > 6 0 > > > > 6 > > r > > sn 6 > > 0 0 ; : 4 0 rxs ðkÞ ðkÞ ðkÞ ðkÞ C16 C26 C36 8 9 > > ess > > > > > > > exx > > > > > > = nn > cxn > > > > > > > > > > > c > sn > > ; : > cxs ðkÞ
the k th laminate system out of its 0
0
0
0
0
0
ðkÞ
C44
ðkÞ
ðkÞ
C45
ðkÞ
C45
C55
0
0
ðkÞ C16 ðkÞ C26 ðkÞ C36
3
7 7 7 7 7 7 7 0 7 7 7 0 7 5 ðkÞ C66
ðkÞ
C23
C36
C33
C33
C33
ð87Þ
e ðkÞ xx
c ðkÞ xs
ðkÞ
ðkÞ
rðkÞ ss ¼ Q11 ess þ Q12 exx þ Q16 cxs ðkÞ
ðkÞ
ðkÞ
rðkÞ xx ¼ Q12 ess þ Q22 exx þ Q26 cxs ðkÞ
rðkÞ xn ¼ Q44 cxn ðkÞ
rðkÞ xs ¼
ð89Þ ð90Þ
ð92Þ þ
ðkÞ Q26 exx
þ
ðkÞ Q66 cxs
ð93Þ
where ðkÞ
ðkÞ
ðkÞ
ðkÞ
Qij ¼ Qji ¼ Cij Q4i ¼ C4i
for
ðkÞ
Ci3 Cj3 ðkÞ
for
i; j ¼ 1; 2; 6;
C33
i ¼ 4; 5
are reduced stiffness coefficients for k-th laminate layer.
123
¼
Nsn –
Znk ( ðkÞ ) N X rxn ðkÞ
k¼1
nðk1Þ
rsn
dn
ð96Þ
ndn
ð97Þ
stress couples Lxx
¼
Znk ( ðkÞ ) N X rxx ðkÞ
k¼1
nðk1Þ
rxs
where Lss term is set to 0 due to shallow shell assumption [Sect. 2.1, item (e)]. According to the above Nss definition and also considering the (f) assumption and equations (10a, b), (89) one gets Znk N X Nss ¼ rðkÞ ss dn ¼ 0 () nðk1Þ
Znk N X k¼1
ðkÞ ðkÞ ðkÞ Q11 ess þ Q12 exx þ Q16 cxs dn ¼ 0 ()
nðk1Þ
A12 ð0Þ B12 ð1Þ A16 ð0Þ B16 ð1Þ e e c c ess ¼ A11 xx A11 xx A11 xs A11 xs whereas Znk N X ðkÞ Aij ¼ Qij dn k¼1
ð91Þ
rðkÞ sn ¼ Q45 cxn ðkÞ Q16 ess
Nxn
k¼1
ð88Þ
Setting the above into (87) and using the cross-section non-deformability condition cyz ¼ csn ¼ 0 [Sect. 2.1, item (a)] results in the system of reduced equations ðkÞ
ð95Þ
transverse
ðkÞ
ðkÞ
C13
e ðkÞ ss
–
Lxs
Considering the assumption of the normal stress rnn to be negligible (see Sect. 2.1, item (f)) the corresponding strain in wall thickness direction can be defined as enn ¼
– membraine 9 8 8 ðkÞ 9 > Znk > N = X = < Nss > < rss > ðkÞ Nxx ¼ rxx dn > > > ; k¼1 ; : : ðkÞ > nðk1Þ Nxs rxs
Dij ¼
N X k¼1
Znk N X k¼1
nðk1Þ
Znk
Bij ¼
ðkÞ
n2 Qij dn
ð98Þ
ðkÞ
nQij dn
nðk1Þ
ð99Þ
nðk1Þ
Therefore using relation (98) the constitutive equations (89)– (92) are:
ðkÞ ðkÞ Q12 A12 ð0Þ Q12 B12 ð1Þ ðkÞ ðkÞ rðkÞ ¼ Q þ nQ e exx 22 22 xx xx A11 A11
ðkÞ ðkÞ Q A16 ð0Þ Q B16 ð1Þ ðkÞ ðkÞ þ Q26 12 cxs þ nQ26 12 cxs A11 A11 ð100Þ
Meccanica (2014) 49:1833–1858
rðkÞ xn ¼
ðkÞ Q44 cð0Þ xn
ðkÞ
ð0Þ rðkÞ sn ¼ Q45 cxn
1853
ð101Þ
ð102Þ
ðkÞ ðkÞ ðkÞ Q16 A12 ðkÞ Q16 B12 ð0Þ Q26 exx þ nQ26 eð1Þ xx þ A11 A11
ðkÞ ðkÞ ðkÞ Q16 A16 ðkÞ Q16 B16 ð0Þ Q66 cxs þ nQ66 cð1Þ xs A11 A11
rðkÞ xs ¼
ð103Þ and subsequently after re-ordering and using (11d) the 2-D stress resultants and stress couples are 9 2 8 3 8 ð0Þ 9 K11 K12 K13 K14 0 > Nxx > > > > > > 7 6 > > > > exx > > Nxs > > > > > ð0Þ > 6 K21 K22 K23 K24 0 7 > > > > > > cxs > 7 > = 6 = < 7 6 K K K K 0 xx 41 42 43 44 0 7 ¼6 u 7 6 > > 6 K51 K52 K53 K54 0 7 > > > Lxs > > ð1Þ > > > > > > > exx > 7 > 6 > > > > 4 0 > > Nxn > > 5 > 0 0 0 A ; > : 44 > > ð0Þ ; : cxn Nsn 0 0 0 0 A45 ð104Þ where the beam effective stiffness coefficients Kij are given as follows
A12 A12 A12 A16 K11 ¼ A22 K12 ¼ A26 A11 A11
A12 B12 A16 A12 K14 ¼ B22 K21 ¼ A26 A11 A11
A16 A16 A16 B12 K22 ¼ A66 K24 ¼ B26 A11 A11
B12 A12 B12 A16 K41 ¼ B22 K42 ¼ B26 A11 A11
B12 B12 A12 B16 K44 ¼ D22 K51 ¼ B26 A11 A11
A16 B16 B12 B16 K52 ¼ B66 K54 ¼ D26 A11 A11 ð105Þ In the above relations symmetry is preserved for terms K12 ¼ K21 , K14 ¼ K41 and K24 ¼ K42 . The remaining Ki3 ; i ¼ 1; 2; 4; 5 terms depend whether the crosssection is open or closed one. General formulas have the form
A12 A16 A12 B16 þ gð1Þ B26 K13 ¼ gð0Þ A26 A11 A11
A A A 16 16 16 B16 K23 ¼ gð0Þ A66 þ gð1Þ B66 A11 A11
A B B 16 12 12 B16 ð0Þ ð1Þ K43 ¼ g B26 D26 þg A11 A11
A B B 16 16 16 B16 ð0Þ ð1Þ K53 ¼ g B66 D66 þg A11 A11 ð106Þ In case of open sections, according to (82) relation, the above expressions simplify
to
A B A16 B16 12 16 K13 ¼ gð1Þ B26 K23 ¼ gð1Þ B66 A11 A11
B B B 12 16 16 B16 ð1Þ ð1Þ K43 ¼ g D26 D66 K53 ¼ g A11 A11 ð107Þ Appendix 3 Following the position vector R as defined by (6) and displacement field (Sect. 2.2.4) the components of dR are dRx ¼ x þ R0 þ u0 ðx; tÞ þ #y ðx; tÞ Z þ #z ðx; tÞ Y Gðn; sÞ u0 ðx; tÞ sin wðtÞdwðtÞ u2 ðx; tÞ ðY yp Þ Y þ vp ðx; tÞ 2 ðZ zp Þuðx; tÞ cos h cos wðtÞdwðtÞ þ Z þ wp ðx; tÞ þ ðY yp Þuðx; tÞ u2 ðx; tÞ ðZ zp Þ sin h cos wðtÞdwðtÞ 2 þ cos wðtÞdu0 ðx; tÞ cos h sin wðtÞdvp ðx; tÞ þ sin h sin wðtÞdwp ðx; tÞ þ Z cos wðtÞd#y ðx; tÞ þ Ycos wðtÞd#z ðx; tÞ Gðn; sÞ cos wðtÞdu0 ðx; tÞ þ ðY yp Þuðx; tÞ cos h sin wðtÞ þ ðZ zp Þ cos h sin wðtÞ þ ðY yp Þ sin h sin wðtÞ
ðZ zp Þ sin h sin wðtÞuðx; tÞ duðx; tÞ dRy ¼ x þ R0 þ u0 ðx; tÞ þ #y ðx; tÞ Z þ #z ðx; tÞ Y Gðn; sÞ u0 ðx; tÞ cos wðtÞdwðtÞ
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u2 ðx; tÞ ðY yp Þ Y þ vp ðx; tÞ 2 ðZ zp Þuðx; tÞ cos h sin wðtÞdwðtÞ þ Z þ wp ðx; tÞ þ ðY yp Þuðx; tÞ u2 ðx; tÞ ðZ zp Þ sin h sin wðtÞdwðtÞ 2 þ sin wðtÞdu0 ðx; tÞ þ cos h cos wðtÞdvp ðx; tÞ
€p sin h þ B1 ðR0 þ xÞ€ vp cos h B1 ðR0 þ xÞw € þ B13 sin h B15 cos h #y þ B15 sin h B14 cos h #€z € B3 sin h þ B2 cos h ðR0 þ xÞu l Z h 0i _ € dx þ þ B9 cos h B8 sin h u 2B1 ðR0 þ xÞu_ 0 wðtÞ 0
_ þ 2 B12 cos2 h B11 sin h cos h v_p wðtÞ 2 _ þ 2 B11 sin h B12 sin h cos h w_ p wðtÞ _ þ 2B11 ðR0 þ xÞ#_y wðtÞ
sin h cos wðtÞdwp ðx; tÞ þ Z sin wðtÞd#y ðx; tÞ þ Ysin wðtÞd#z ðx; tÞ Gðn; sÞ sin wðtÞdu0 ðx; tÞ þ ðY yp Þuðx; tÞ cos h cos wðtÞ ðZ zp Þ cos h cos wðtÞ ðY yp Þ sin h cos wðtÞ
þ ðZ zp Þ sin h cos wðtÞuðx; tÞ duðx; tÞ dRz ¼ sin h dvp þ cos h dwp ðZ zp Þ sin h ðY yp Þ cos h duðx; tÞ ðZ zp Þ cos h þ ðY yp Þ sin h uðx; tÞ duðx; tÞ ð108Þ
_ 2B7 ðR0 þ xÞu_ 0 wðtÞ _ þ 2B12 ðR0 þ xÞ#_z wðtÞ 2 2 _ þ 2 B17 sin h B18 cos h u_ wðtÞ i o _ sin h cos h dx þ 2ðB16 B19 Þu_ wðtÞ
ð109Þ •
du0 term
Zt2 Z l h €0 dt B1 u€0 þ B12 #€z þ B11 #€y B7 u t1
0
_ cos h þ 2B2 u_ wðtÞ _ cos h 2B1 v_p wðtÞ _ sin h þ 2B3 u_ wðtÞ _ sin h þ 2B1 w_ p wðtÞ
Inserting the above terms (108) and accelerations (8) into energy variation time integral (32) and rearranging all the terms according to variations of respective basic unknowns of the problem and skipping higher order terms resulting from cosine approximation, yields •
2 2 B1 ðR0 þ x þ u0 Þw_ ðtÞ B12 #z w_ ðtÞ 2 2 € sin h B11 #y w_ ðtÞ þ B7 u0 w_ ðtÞ þ B1 wp wðtÞ € sin h € sin h þ B11 wðtÞ þ B3 uwðtÞ
dwðtÞ term
Zt2 n € þ B14 lwðtÞ € cos2 h dt B22 wðtÞ
€ cos h þ B2 uwðtÞ € cos h B1 vp wðtÞ i € cos h dx B12 wðtÞ
t1
€ sin2 h 2B15 lwðtÞ € cos h sin h þ B13 lwðtÞ þ
Zl h
€ 2B1 ðR0 þ xÞu0 wðtÞ
0
€ þ 2B11 ðR0 þ xÞ#y wðtÞ € 2B7 ðR0 þ xÞu0 wðtÞ € þ 2B12 vp wðtÞ € cos2 h þ 2B12 ðR0 þ xÞ#z wðtÞ € sin h cos h þ 2B11 wp wðtÞ € sin2 h 2B11 vp wðtÞ € sin h cos h 2B18 uwðtÞ € cos2 h 2B12 wp wðtÞ € sin2 h 2B19 uwðtÞ € sin h cos h þ 2B17 uwðtÞ i € sin h cos h dx þ 2B16 uwðtÞ þ
Zl h 0
123
B11 sin h B12 cos h u€0
•
ð110Þ
dvp term
Zt2 Z l h _ cos h € þ 2B1 u_ 0 wðtÞ dt B1 v€p B2 u t1
0
_ cos h þ 2B11 #_y wðtÞ _ cos h þ 2B12 #_z wðtÞ _ cos h þ B1 wp w_ 2 ðtÞ sin h cos h 2B7 u_ 0 wðtÞ 2 2 þ B3 uw_ ðtÞ sin h cos h þ B11 w_ ðtÞ sin h cos h 2
2
B1 vp w_ ðtÞ cos2 h þ B2 uw_ ðtÞ cos2 h 2 B12 w_ ðtÞ cos2 h þ B1 ðR0 þ x þ u0 Þ € cos h € cos h þ B12 #z wðtÞ wðtÞ i € cos h B7 u0 wðtÞ € cos h dx þ B11 #y wðtÞ
ð111Þ
Meccanica (2014) 49:1833–1858
•
1855
•
dwp term
Zt2 Z l h _ € 2B1 u_ 0 wðtÞsinh € p þ B3 u B1 w dt t1
du term
Zt2 Z l h € €p þ ðB4 þ B5 ÞuðtÞ B2 v€p þ B3 w dt t1
0
0
_ 2ðB2 cos h þ B3 sin hÞu_ 0 wðtÞ _ 2ðB16 cos h þ B17 sin hÞ#_y wðtÞ _ 2ðB18 cos h þ B19 sin hÞ#_z wðtÞ _ þ 2ðB20 cos h þ B21 sin hÞu_ 0 wðtÞ
_ _ 2B12 #_z wðtÞsinh 2B11 #_y wðtÞsinh 2 _ B1 wp w_ ðtÞsin2 h þ 2B7 u_ 0 wðtÞsinh 2 2 B3 uw_ ðtÞsin2 h B11 w_ ðtÞsin2 h 2
€ ðB3 cos h B2 sin hÞðR0 þxÞuwðtÞ 2 þ ðB2 cos2 h þ B3 sin h cos hÞvp w_ ðtÞ
2
þ B1 vp w_ ðtÞsinhcosh B2 uw_ ðtÞsinhcosh
2 ðB2 sin h cos h þ B3 sin2 hÞwp w_ ðtÞ ðB18 sin h cos h þ B17 sin h cos h 2 B16 sin2 h B19 cos2 hÞuw_ ðtÞ
2 þ B12 w_ ðtÞsinhcosh B1 ðR0 þxþu0 Þ
€ € wðtÞsinh B12 #z wðtÞsinh
i € € B11 #y wðtÞsinh þ B7 u0 wðtÞsinh dx
2 ð2B6 sin h cos h þ B5 sin2 h þ B4 cos2 hÞuw_ ðtÞ þ ðB18 cos2 h B17 sin2 h þ B9 sin h cos h 2 B16 sin h cos hÞw_ ðtÞ € ðB2 cos hþB3 sin hÞðR0 þxþu0 ÞwðtÞ € ðB16 cos h þ B17 sin hÞ#y wðtÞ € ðB18 cos hþB19 sin hÞ#z wðtÞ € þ ðB20 cos h þ B21 sin hÞu0 wðtÞ 0 0 € € €0 þ ðB7 u€0 Þ þðB8 #y Þ þ ðB9 #z Þ0 ðB10 uÞ 0 0 _ _ cos h þ ðB21 uÞ _ sin hwðtÞ þ2½ðB20 uÞ _ cos h þ 2ðB7 w_ p Þ0 wðtÞ _ sin h 2ðB7 v_p Þ0 wðtÞ
ð112Þ •
d#y term
Zt2 Z l h €0 dt B11 u€0 þ B13 #€y þ B15 #€z B8 u t1
0
_ cos h þ 2B16 u_ wðtÞcos _ 2B11 v_p wðtÞ h _ _ h þ 2B17 u_ wðtÞsin þ 2B11 w_ p wðtÞsin h
2 ½B7 ðR0 þ x þ u0 Þ0 w_ ðtÞ 2 2 ðB8 #y Þ0 w_ ðtÞ ðB9 #z Þ0 w_ ðtÞ
2 2 B11 ðR0 þ x þ u0 Þw_ ðtÞ B15 #z w_ ðtÞ 2 2 € B15 #y w_ ðtÞ þ B8 u0 w_ ðtÞ þ B11 wp wðtÞsin h
2
€ cos h þ ðB10 u0 Þ0 w_ ðtÞ ðB7 vp Þ0 wðtÞ 0€ € cos h þ ðB7 wp Þ wðtÞ sin h þ ðB20 uÞ0 wðtÞ i 0€ þ ðB21 uÞ wðtÞ sin h dx
€ € sin h B11 vp wðtÞcos h þ B17 uwðtÞ € sin h € cos h þ B13 wðtÞ þ B16 uwðtÞ i € h dx B15 wðtÞcos
ð113Þ
Zt2h
€0 B7 u€0 B8 #€y B9 #€z þ B10 u
t1
•
_ cos h 2B7 w_p wðtÞ _ sin h þ 2B7 v_p wðtÞ _ 2 B20 cos h þ B21 sin h u_ wðtÞ 2 2 þ B7 ðR0 þ x þ u0 Þw_ ðtÞ þ B8 #y w_ ðtÞ
d#z term
Zt2 Z l h dt B12 u€0 þB15 #€y þB14 #€z t1
2
0
_ _ €0 2B12 v_p wðtÞcoshþ2B _ wðtÞcosh B9 u 18 u _ _ _ wðtÞsinh þ2B12 w_ p wðtÞsinhþ2B 13 u 2 B12 ðR0 þxþu0 Þw_ ðtÞ
x¼0
2 2 2 B14 #z w_ ðtÞB15 #y w_ ðtÞþB9 u0 w_ ðtÞ € € þB12 wp wðtÞsinhþB 19 uwðtÞsinh
€ € B12 vp wðtÞcoshþB 18 uwðtÞcosh i € € þB15 wðtÞsinhB 14 wðtÞcosh dx
2
þ B9 #z w_ ðtÞ B10 u0 w_ ðtÞ € cos h B7 wp wðtÞ € sin h þ B7 vp wðtÞ € ðB20 cos h þ B21 sin hÞuwðtÞ x¼l i € þ ðB9 cos h B8 sin hÞwðtÞ dt du
ð114Þ
ð115Þ where, in order to simplify the notation, inertia Ii terms are given in Table 2 in 1. A reader may easily notice in the foregoing equations terms related to nonconstant angular velocity and terms corresponding to Coriolis and centrifugal accelerations.
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Meccanica (2014) 49:1833–1858
Appendix 4
•
According to the assumed general loading conditions the following external loads are defined •
c h=2
Shear forces
Pext;y ¼
Pext;z ¼
Zh=2
Z
q0 Fy dnds þ
c
Z
Qext;y ¼ nx
Z
^ ext;w m •
ty dnds Qext;z ¼ nx
Z
P^ext;y ¼ ns
Zh=2 tz dnds
s¼s2 ^ty dn P^ext;z ¼ ns s¼s1
h=2
s¼s2 ^tz dn s¼s1
h=2
ð116Þ Axial forces Z Zh=2
Pext;x ¼
q0 Fx dnds þ
Z
Z Zh=2 tx dnds
ð117Þ
s¼s2 Zh=2 ^tx dn ¼ ns
þ
½qz ðy yp Þ qy ðz zp Þds Z Zh=2 c h=2 Zh=2
^ext;x ¼ ns m
h=2
tx Y dnds
ð120Þ
s¼s1
þ Pext;y vp þ Pext;z wp þ mext;x u mext;w u0 dx h Wext;3 ¼ nx Qext;x u0 þ Mext;y #y þ Mext;z #z Mext;w u0
ix¼l þ Qext;y vp þ Qext;z wp þ Mext;x u
½tz ðY yp Þ ty ðZ zp Þdnds
s¼s2 ^tz ðY yp Þ ^ty ðZ zp Þ dn s¼s1
ð118Þ
123
Z Zh=2
s¼s2 ^tx Z dn ^ ext;y ¼ ns m s¼s1 Z h=2
s¼s h=2 2 ^tx Y dn ^ ext;z ¼ ns m
c
tx Z dnds
0
q0 ½Fz ðY yp Þ Fy ðZ zp Þdnds
h=2
Mext;x ¼ nx
Z Zh=2
Using equations (116)–(120) the work defining equations (35) take the form Z l Pext;x u0 þ mext;y #y þ mext;z #z Wext;1 þ Wext;2 ¼
Bending moments
Zc
Mext;y ¼ nx
qx y ds c
h=2
s¼s1
h=2
mext;x ¼
q0 Fx Y dnds þ
c h=2
c h=2 Z h=2
c h=2
Z Zh=2
c
Z
Mext;z ¼ nx
qx ds c
Qext;x ¼ nx
•
mext;z ¼
c h=2
Z Zh=2
c h=2
c h=2
P^ext;x
Bending moments Z Zh=2 Z mext;y ¼ q0 Fx Z dnds þ qx z ds
c h=2
Zh=2
ð119Þ
s¼s1
h=2
c h=2
Zh=2
s¼s2 Zh=2 ^tx Gðn; sÞ dn ¼ ns
qz ds c
Zh=2
tx Gðn; sÞ dnds c h=2
qy ds
Zh=2
q0 Fz dnds þ
Mext;w ¼ nx
Z
c h=2
Z
c
Z Zh=2
c h=2
•
Bi-moments Z Zh=2 Z q0 Fx Gðn; sÞdnds þ qx Gð0Þ ðsÞds mext;w ¼
x¼0
Wext;4
Z lh ^ext;y #y þ m ^ext;z #z ¼ P^ext;x u0 þ m 0
þ P^ext;y vp þ P^ext;z dwp i ^ext;x u m ^ext;w u0 dx þm
ð121Þ
Meccanica (2014) 49:1833–1858
1857
The variation of the total external work expressed by all its components takes the form dWext;1 þ dWext;2 ¼
Zl
4.
Pext;x du0 þ mext;y d#y þ mext;z d#z
0
þ Pext;y dvp þ Pext;z dwp þ mext;x þ m0ext;w du dx x¼l mext;w du ; x¼0 dWext;3 ¼ nx Qext;x du0 þ Mext;y d#y þ Mext;z d#z Mext;w du0
x¼l þ Qext;y dvp þ Qext;z dwp þ Mext;x du ; x¼0
dWext;4 ¼
Zl
^ ext;y d#y þ m ^ ext;z d#z P^ext;x du0 þ m
5.
6.
7.
8.
9.
0
þ P^ext;y dvp ^ ext;x þ m ^0ext;w du dx þ P^ext;z dwp þ m x¼l ^ ext;w du ; m
10.
11.
x¼0
dWext;5 ¼ Text;z dwðtÞ;
ð122Þ Appendix 5
13.
In order to provide a self-contained theory and a system of equations the inertia coefficients requested for kinetic energy (110)–(115) and stiffness coefficients of the system are given in explicit form. Setting the terms Zh=2 m0 ¼ .dn h=2
12.
Zh=2 m2 ¼ .n2 dn
14.
15.
ð123Þ
h=2
the definitions of inertia coefficients Bi are summarised in Table 1. The definition of introduced stiffness coefficients are given in Table 2.
16.
17. 18.
References 19. 1. Altenbach J, Altenbach H, Matzdorf V (1994) A generalized Vlasov theory for thin-walled composite beam structures. Mech Compos Mater 30(1):43–54 2. Arvin H, Bakhtiari-Nejad F (2013) Nonlinear free vibration analysis of rotating composite Timoshenko Beams. Compos Struct 96:29–43 3. Arvin H, Lacarbonara W, Bakhtiari-Nejad F (2012) A geometrically exact approach to the overall dynamics of
20. 21. 22.
elastic rotating blades–part 2: flapping nonlinear normal modes. Nonlinear Dyn 70(3):2279–2301 Avramov KV, Pierre C, Shyriaieva N (2007) Flexuralflexural-torsional nonlinear vibrations of pre-twisted rotating beams with asymmetric cross-sections. J Vib Control 13(4):329–364 Balthazar JM, Mook DT, Weber HI, Brasil RM, Fenili A, Belato D, Felix JLP (2003) An overview on non-ideal vibrations. Meccanica 38(6):613–621 Cesnik CES, Hodges DH (1997) VABS: a new concept for composite rotor blade cross-sectional modeling. J Am Helicopter Soc 42(1):27–38 Chandiramani NK, Shete CD, Librescu L (2003) Vibration of higher-order-shearable pretwisted rotating composite blades. Int J Mech Sci 45(12):2017–2041 Choi S-C, Park J-S, Kim J-H (2006) Active damping of rotating composite thin-walled beams using MFC actuators and PVDF sensors. Compos Struct 76(4):362–374 Fenili A, Balthazar JM (2005) Some remarks on nonlinear vibrations of ideal and nonideal slewing flexible structures. J Sound Vib 282(1–2):543–552 Fenili A, Balthazar JM, Brasil R (2003) Mathematical modelling of a beam-like flexible structure in slewing motion assuming non-linear curvature. J Sound Vib 268(4):825–838 Ghorashi M (2012) Nonlinear analysis of the dynamics of articulated composite rotor blades. Nonlinear Dyn 67(1): 227–249 Hodges DH (1990) A mixed variational formulation based on exact intrinsic equations for dynamics of moving beams. Int J Solids Struct 26(11):1253–1273 Hodges DH (2006) Nonlinear composite beam theory, volume 213 of Progress in astronautics and aeronautics. American Institute of Aeronautics and Astronautics, Reston, Va Jun L, Rongying S, Hongxing H, Xianding J (2004) Coupled bending and torsional vibration of axially loaded Bernoulli– Euler beams including warping effects. Appl Acoust 65(2): 153–170 Kaya MO, Ozdemir Ozgumus O (2007) Flexural-torsionalcoupled vibration analysis of axially loaded closed-section composite Timoshenko beam by using DTM. J Sound Vib 306(3–5):495–506 Kovvali RK, Hodges DH (2012) Verification of the variational-asymptotic sectional analysis for initially curved and twisted beams. J Aircr 49(3):861–869 Kunz D (1994) Survey and comparison of engineering beam theories for helicopter rotor blades. J Aircr 31(3):473–479 Lacarbonara W, Arvin H, Bakhtiari-Nejad F (2012) A geometrically exact approach to the overall dynamics of elastic rotating blades–part 1: linear modal properties. Nonlinear Dyn 70(1):659–675 Latalski J, Georgiades F, Warmin´ski J (2012) Rational placement of a macro fibre composite actuator in composite rotating beams. J Phys 382:012021 Lee SY, Sheu JJ (2007) Free vibrations of a rotating inclined beam. J Appl Mech 74(3):406–414 Librescu L, Song O (2006) Thin-walled composite beams: theory and application. Springer, Dordrecht Na S, Librescu L, Shim JK (2003) Modeling and bending vibration control of nonuniform thin-walled rotating beams
123
1858
23.
24.
25.
26. 27.
28.
29.
incorporating adaptive capabilities. Int J Mech Sci 45(8): 1347–1367 Oh SY, Song O, Librescu L (2003) Effects of pretwist and presetting on coupled bending vibrations of rotating thinwalled composite beams. Int J Solids Struct 40(5): 1203–1224 Qin Z, Librescu L (2002) On a shear-deformable theory of anisotropic thin-walled beams: further contribution and validations. Compos Struct 56(4):345–358 Ren Y, Yang S (2012) Modeling and free vibration behavior of rotating composite thin-walled closed-section beams with SMA fibers. Chin J Mech Eng 25(5):1029–1043 Shabana AA (2005) Dynamics of multibody systems, 3rd edn. Cambridge University Press, Cambridge Sina SA, Ashrafi MJ, Haddadpour H, Shadmehri F (2011) Flexural-torsional vibrations of rotating tapered thin-walled composite beams. Proc Inst Mech Eng 225(4):387–402 Song O, Librescu L (1993) Free vibration of anisotropic composite wthin-walled beams of closed cross-section contour. J Sound Vib 167(1):129–147 Song O, Librescu L (1997) Structural modeling and free vibration analysis of rotating composite thin-walled beams. J Am Helicopter Soc 42(4):358–369
123
Meccanica (2014) 49:1833–1858 30. Stoykov S, Ribeiro P (2010) Nonlinear forced vibrations and static deformations of 3D beams with rectangular cross section: the influence of warping, shear deformation and longitudinal displacements. Int J Mech Sci 52(11):1505–1521 31. Stoykov S, Ribeiro P (2013) Vibration analysis of rotating 3D beams by the p-version finite element method. Finite Elem Anal Des 65:76–88 32. Volovoi VV, Hodges DH, Cesnik CES, Popescu B (2001) Assessment of beam modeling methods for rotor blade applications. Math Comput Modell 33(10–11):1099–1112 33. Vyas NS, Rao JS (1992) Equation of motion of a blade rotating with variable angular velocity. J Sound Vib 156(2): 327–336 34. Warminski J, Balthazar JM (2005) 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, Parts A, B, and C, ASME, pp 1619–1624 35. Warminski J, Balthazar JM, Brasil RM (2001) Vibrations of a non-ideal parametrically and self-excited model. J Sound Vib 245(2):363–374 36. Wenbin Y, Volovoi VV, Hodges DH, Hong X (2002) Validation of the variational asymptotic beam sectional analysis (VABS). AIAA J 40(10):2105–2113
