Dynamic Responses of Flexible-Link Mechanisms with Passive/Active Damping Treatment Jean-François Deü, Ana Cristina Galucio, Roger Ohayon

Dynamic responses of flexible-link mechanisms with passive/active damping treatment Jean-François Deü, Ana Cristina Galucio, Roger Ohayon

To cite this version:

Jean-François Deü, Ana Cristina Galucio, Roger Ohayon. Dynamic responses of flexible-link mechanisms with passive/active damping treatment. Computers and Structures, Elsevier, 2008, 86 (3-5), pp.258-265. 10.1016/j.compstruc.2007.01.028. hal-01519381

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Distributed under a Creative Commons Attribution| 4.0 International License Dynamic responses of ﬂexible-link mechanisms with passive/active damping treatment J.-F. Deü, A.C. Galucio, R. Ohayon

Conservatoire National des Arts et Me´tiers (CNAM), Structural Mechanics and Coupled Systems Laboratory, Case 353, 292 rue Saint Martin, 75141 Paris Cedex 03, France

This work presents a ﬁnite element formulation for non-linear transient dynamic analysis of adaptive beams. The main contribution of this work concerns the development of an original co-rotational sandwich beam element, which allows large displacements and rotations, and takes active/passive damping into account. This element is composed of a viscoelastic core and elastic/piezoelectric laminated faces. The latter are modeled using classical laminate theory, where the electromechanical coupling is considered by modifying the stiﬀ- ness of the piezoelectric layers. For the core, a four-parameter fractional derivative model is used to characterize its viscoelastic dissipative behavior. Equations of motion are solved using an incremental-iterative method based on the Newmark direct time integration scheme in conjunction with the Gru¨nwald approximation of fractional derivatives, and the Newton–Raphson algorithm.

Keywords: Piezoelectric material; Viscoelasticity; Fractional derivatives; Sandwich beam; Co-rotational formulation; Non-linear dynamics

1. Introduction ested reader can be referred to the review article [10] which compares these methods and summarizes the recent devel- In the context of linear dynamics, the active constrained opments in computational modeling of flexible multibody layer damping treatment is widely used due to its beneficial systems. In this paper, following the works of Crisfield performance in attenuating structural vibrations. The [2], a co-rotational approach is used to describe the large investigation of such a treatment in the geometric non-lin- displacement analysis of the sandwich beam. One of the ear case, more specifically for flexible multibody systems, is main difficulties in modeling such structures is the charac- very sparsely analyzed. This paper presents a non-linear terization of the damping properties of the viscoelastic finite element formulation for dynamic analysis of adaptive layer. In this study, a four-parameter fractional derivative sandwich beams composed of a viscoelastic core con- model [1] is used to describe the frequency-dependence of strained by elastic/piezoelectric laminated faces. It is the viscoelastic material. The advantage of employing frac- important to mention that flexible beams undergoing finite tional operators is related to the small number of parame- rotations have been investigated by several approaches ters required to represent the material damping over a namely the floating frame approach, the inertial frame broad range of frequencies. Concerning the active compo- approach or the co-rotational approach. Most of these nent, the electromechanical coupling is taken into account methods are described in standard textbooks and the inter- by modifying the stiffness matrix of the piezoelectric layers (sensor) and by applying a mechanical load written in terms of the applied tension (actuator). Consequently, in the final finite element formulation, no electrical degree- of-freedom explicitly appears [5]. Elastic/piezoelectric faces are modeled using classical laminate theory whereas a standard three-layer sandwich theory is used for the face/core/

1 face system [9]. In order to implement the fractional deriv- Let us introduce the mean and relative axial displace- ative constitutive equations, the Gru¨nwald formalism is ments given by u ¼ðua þ ubÞ=2 and u~ ¼ ua ub. Euler–Ber- 0 used in conjunction with a time discretization scheme based noulli hypotheses for the faces lead to hk ¼ w for k ¼ a; b on Newmark method. One of the particularities of the pro- with ðÞ0 ¼ oðÞ=ox. As all layers are supposed to be per- posed algorithm lies on the storage of displacement history fectly bonded, the displacement continuity conditions at only. This considerably reduces the numerical efforts interface layers can be written as uxa ¼ uxc at z ¼ hc=2 related to the non-locality of fractional operators. Finally, and uxb ¼ uxc at z ¼hc=2. Therefore, axial displacements numerical examples, including a flexible robot arm and a of the centerlines and rotations of each layer can be written slider-crank mechanism, are considered in order to show in terms of w0 and the above defined variables u and u~ as the effectiveness of the present co-rotational formulation. ~ u~ u~ h 0 ua ¼ u þ ; ub ¼ u ; uc ¼ u þ w ; 2. Active/passive damping treatment 2 2 4 0 0 u~ þ hw ha ¼ hb ¼ w ; hc ¼ ð2Þ Consider a sandwich beam composed of a viscoelastic hc core and elastic/piezoelectric laminated faces. Hypotheses ~ where h ¼ðha þ hbÞ=2 and h ¼ ha hb. proposed in this section are valid in a local coordinate sys- From (1) and (2), and taking the hypothesis of plane tem, which continuously rotates and translates with the stress state into account, we can write the axial strain of beam element. Furthermore, a linear strain definition is the ith layer e1i and the shear strain of the core e5c as used. The first part is addressed to the kinematical (for follows: the mechanical displacement) and electrostatic (for the electric potential) assumptions. In the second part, consti- e1i ¼ i þðz ziÞji; e5c ¼ cc ð3Þ tutive equations for the piezoelectric and viscoelastic mate- where i and ji are the membrane strain and curvature of rials are outlined. Finally, the internal virtual energy of the the ith layer, and cc the shear strain of the core. sandwich beam is derived. Concerning the electrostatic aspects, two main assumptions are taken under consideration. The first one concerns 2.1. Planar sandwich beam the electric potential, which is supposed to be linear within the thickness of each piezoelectric sublayer The sandwich beam is modeled using Euler–Bernoulli V ðx; tÞ assumptions for the faces and Timoshenko ones for the kj /kj ðx; z; tÞ¼/kj ðx; tÞþðz zkj Þ ð4Þ core. The elastic/piezoelectric faces are modeled using clas- hkj sical laminated theory with linear electric potential through þ þ where / / / =2andV / / . The quanti- kj ¼ð kj þ kj Þ kj ¼ kj kj the thickness of each piezoelectric sublayer. The mechani- þ ties / and / are the electrical boundary conditions at cal displacement field within the ith layer can be written as kj kj top ðz ¼ zkj þ hkj =2Þ and at bottom ðz ¼ zkj hkj =2Þ sur- uxiðx; z; tÞ¼uiðx; tÞðz ziÞhiðx; tÞð1aÞ faces. The local z-axis of the kjth face sublayer is situated at (for k ¼ aðþÞ; bðÞ) uziðx; z; tÞ¼wðx; tÞð1bÞ Xj1 where the subscript i a; b; c stands for upper (composed hkj þ hc ¼ z ¼ h kj 2 kr of N a sublayers), lower (composed of N b sublayers) and r¼1 middle layers, respectively; uxi and uzi are the axial and with hk being the thickness of the kjth layer. Secondly, the transverse displacements of each layer; ui and hi are the ax- j ial displacement of the center line and the fiber rotation of axial component of the electrical field can be neglected be- each layer; and w is the transverse displacement (Fig. 1). cause its contribution to the electromechanical energy is small if compared to that of the transverse one [5]. The two above assumptions imply a constant transverse electrical field within the kjth piezoelectric sublayer o /k V k E ¼ j ¼ j ð5Þ 3kj o z hkj

2.2. Fractional derivative viscoelastic model

The one-dimensional constitutive equation introduced by Bagley and Torvik [1] is adopted in this work to describe the viscoelastic behavior of the core a a a E1 a ri þ s D ri ¼ cii ei þ s D ei ð6Þ Fig. 1. Sandwich beam kinematics. E0

2 where ri and ei are axial ði ¼ 1Þ and shear ði ¼ 5Þ stress and For elastic faces, the piezoelectric constants vanish. More- 2 strain. Since the core is supposed to be isotropic, the elastic over, if the material is isotropic, c11 ¼ E=ð1 m Þ, where E 2 constants are defined by: c11 ¼ E0=ð1 m Þ and c55 ¼ and m are the elastic modulus and the Poisson’s ratio. E0=ð2 þ 2mÞ. Furthermore, E0 and E1 are the relaxed and non-relaxed elastic moduli, s is the relaxation time, a is 2.4. Virtual strain energy the fractional order of the time derivative, and Da denotes the operator of fractional derivation of ath order. The The virtual strain energy of the three-layer sandwich statements 0 < a < 1, s > 0 and E1 > E0 fulfill the second beam is the sum of the variations of the internal energy law of thermodynamics. of each layer: dU ¼ dU a þ dU b þ dU c. This will be used This four-parameter fractional derivative model has in the finite element formulation in order to evaluate the been shown to be an effective tool to describe the weak fre- internal force vector and stiffness matrix in the local coor- quency-dependence of most viscoelastic materials [1,7]. Its dinate system. behavior in frequency domain is described between two asymptotic values: the static modulus of elasticity E0 and 2.4.1. Viscoelastic core the high-frequency limit value of the dynamic modulus For the core, the variation of the strain energy is given E1. The Fourier transform of Eq. (6) leads to the calcula- by tion of the complex modulus of elasticity, which is given by Z a dU c ¼ ðr1cde1c þ r5cde5cÞdV ð11Þ E0 þ E ðixsÞ EðxÞ¼ 1 ð7Þ Xc 1 ixs a þð Þ In order to take into account the fractional derivative vis- a The fractional operator D , appearing in the constitutive coelastic behavior, let us introduce the internal variable eic Eq. (6), can be approximated by several methods. One of such that them is the Gru¨nwald definition, which is often adopted E r ¼ cc 1 ðe e Þð12Þ in literature since it is valid for all values of a and easy to ic ii E ic ic implement numerically. The finite difference approximation 0 of the Gru¨nwald definition is given by The constitutive equation (6) can then be rewritten as

XN t E1 E0 1 e þ saDae ¼ e ð13Þ ðDaf Þ A f ð8Þ ic ic E ic n Dta jþ1 nj 1 j¼0 This variable change implies that Eq. (13) contains only where Dt is the time step increment of the numerical scheme one fractional derivative term instead of two as in (6). (the function fn is approximated by f ðtnÞ,withtn ¼ nDt), N t Using the Gru¨nwald approximation (8) and noting that, is the truncation number of the series, and Ajþ1 represents by definition, A1 ¼ 1, relation (13) takes the following dis- the Gru¨nwald coefficients given either in terms of the gam- cretized form: ma function or by a recurrence formula XN t nþ1 E1 E0 nþ1 nþ1j Cðj aÞ j a 1 eic ¼ð1 caÞ eic ca Ajþ1eic ð14Þ A ¼ ¼ A E1 jþ1 CðaÞCðj þ 1Þ j j j¼1 where ca is a dimensionless constant given by c ¼ sa=ðsa þ DtaÞ. 2.3. Piezoelectric constitutive equations a Replacing this expression in Eq. (12), the axial and shear stresses in the core at time tnþ1 are given by The piezoelectric sublayers of the laminated faces are () poled in the thickness direction with an electrical field XN t nþ1 c E1 E0 nþ1 E1 nþ1j applied parallel to this polarization. Such a configuration ric ¼ cii 1 þ ca eic þ ca Ajþ1eic E0 E0 is characterized by the electromechanical coupling between j¼1 the axial strain e1 and the transverse electrical field E3. The ð15Þ three-dimensional constitutive equations can be reduced to The variation of the strain energy of the core is thus com- r1 ¼ c11e1 e31E3 ð9aÞ posed of two terms: the first one can be associated with a modified internal force vector or stiffness matrix, and the D ¼ e e þ d E ð9bÞ 3 31 1 33 3 second one, which depends on the anelastic deformation where r1 and D3 are the axial stress and the transverse elec- history, can be shifted to the right-hand side of the govern- trical displacement. Modified elastic, piezoelectric and ing equation modifying in this way the external force dielectric constants are respectively given by vector.

2 2 It should be stated that the Gru¨nwald coeﬃcients in Eq. c13 c13e33 e33 c11 ¼ c11 ; e31 ¼ e31 ; d33 ¼ d33 þ (14), which are strictly decreasing when j increases, describe c33 c33 c33 the fading memory phenomena. In other words, the behav- ð10Þ ior of the viscoelastic material at a given time step depends

3 more strongly on the recent time history than on the dis- describes the so-called inverse and direct piezoelectric tant one. effects. For an elastic material, since the constant ca is zero, An actuator configuration consists of applying an electri- using strain relations (3) and integrating over the cross sec- cal field to a piezoelectric ceramic in order to induce a tion, Eq. (11) gives mechanical deformation. This electrical field is imposed Z L here by means of an external force, which is written as a c c dU c ¼ c11ðAccdc þ I cjcdjcÞþc55Acccdcc dx ð16Þ function of the voltage applied to the piezoelectric patch. 0 Consequently, the variation of the difference of electrical 3 potential of the kjth lamina vanishes and the above men- where Ac ¼ bhc and Ic ¼ bhc =12 are the cross section area and moment of inertia of the core. Moreover, L and b tioned external force is extracted from Eq. (20a) (see [5] are the total length and width of the beam. for more details). The case where an electrical field is induced by a mechan- 2.4.2. Piezoelectric laminated faces ical deformation in the piezoelectric material corresponds For the piezoelectric/elastic laminated faces ðk ¼ a; bÞ, to a sensor configuration. The unknown is the difference of the virtual internal energy is electric potential of the kjth piezoelectric layer. Using Eqs. Z (20b) and (20c), it can be shown that the virtual electrome- XN k chanical internal energy of the kth lamina corresponds to an dU ¼ ðr de D dE ÞdX k 1kj 1k 3kj 3kj increase of the internal force vector or stiffness matrix of the j Xk ¼1 j beam due to the direct piezoelectric effect [5]. M ME EM E ¼ dU k þ dU k þ dU k þ dU k ð17Þ

M E 3. Co-rotational finite element formulation where dU k and dU k are the virtual mechanical and dielec- ME EM trical internal energies, and dU k and dU k comprise the virtual piezoelectric internal energy, i.e., the electrome- The co-rotational finite element formulation provides a chanical coupling. simple method for large displacement analysis. The key Using strain relations (3), piezoelectric constitutive idea is to separate the motion of each finite element in a equations (9), and electrical field expression (5), each term rigid body part and a deformational part, which is small in the right-hand side of Eq. (17) can be evaluated as relative to a local coordinate system [3,6]. The equations described below. of motion are defined in the global frame while strains The virtual mechanical internal energy can be written as are measured in the co-rotational coordinate system of Z the element. In this study, the linear sandwich beam theory L M k k k is used in a coordinate system, which rotates and translates dU k ¼ b A kdk þ B ðjkdk þ kdjkÞþD jkdjk dx 0 with the element but do not deform with it (Fig. 2). ð18Þ Stiffness and mass matrices of the individual elements are constructed in the element coordinate system, and then k k where A is the extensional stiffness, D is the bending stiff- transformed to the global coordinate system using k ness, and B is the bending-extensional coupling stiffness standard procedures [2]. The resulting equations of motion defined as are the same as those typically arising in non-linear N Z Xk zkj þhkj =2 k k k kj 2 ½A ; B ; D ¼ c11 ½1; ðz zkÞ; ðz zkÞ dz ð19Þ z zk hk =2 j¼1 j j ^ ^ x ^ u The three last terms in the Eq. (17) are related to the elec- z c2 tromechanical coupling and the dielectrical quantities in the virtual internal energy of the faces. These terms are de- θ^ c1 ^ fined as w'1 Z XN k L ME kj dU k ¼ b e31V kj dk þðzkj zkÞdjk dx ð20aÞ φ0 + φ j¼1 0 Z XN k L EM kj dU k ¼ b e31 k þðzkj zkÞjk dV kj dx ð20bÞ j¼1 0 Z XN k L k V k dU E ¼b d j j dV dx ð20cÞ φ 2 k 33 h kj 0 j¼1 0 kj 1 It is easy to see in Eqs. (20a) and (20b) that the electrome- x chanical coupling is taken into account by means of the kj piezoelectric constant e31. This electromechanical coupling Fig. 2. Co-rotational sandwich beam coordinate system.

4 structural dynamics. In this section, we first derive relations freedom vector. Hence, the transformation matrix T, which between the local and global virtual displacement vectors. connects local and global virtual displacement vectors Then we define the internal force vector as well as the tan- dq^ ¼ Tdq ð28Þ gent stiffness matrix in the global coordinate system. is given by Finally, the resolution method, based on the Newmark 2 3 2 2 direct integration scheme and the Newton–Raphson algo- h1s h1c ‘h1 þ h1s 0 h1sh1c h1s 0 6 7 6 sc‘ h s 0 s ‘chs 07 rithm, is briefly described. 6 1 1 7 6 7 16 h2s h2ch1h2s ‘ h2sh2c h1h2s 07 T ¼ 6 7 ‘ 6 ‘c h s ‘s h c ‘h c h2s 0 ‘c h s ‘s h c ‘h c 1 h2s 07 6 þ 1 1 1 þ 1 1 þ 1 1ð Þ 1 7 3.1. Virtual displacement vectors 4 5 sch1s 0 s c ‘ þ h1s 0 h s h chh s 0 h shc h h s 1 The linear finite element formulation of the sandwich 2 2 1 2 2 2 1 2 beam composed of elastic/piezoelectric laminated faces 3.2. Internal force vector and a viscoelastic core is thoroughly described in [5].We just recall that the displacements are discretized with linear The local and global internal force vectors are calculated (axial displacement) and cubic (deflection) shape functions. by using the internal virtual work in both local and global We propose in this section to determine relations between systems the local and global virtual displacement vectors. These T T b T T b relations will be used to compute the internal force vector dU ¼ dq F ¼ dq^ F ¼ dq T F ð29Þ and tangent stiffness matrix. The vectors of global and local displacements are respec- From the internal virtual energy in the local coordinate sys- tively defined by tem dU ¼ dU a þ dU b þ dU c and using linear and cubic shape functions, the local internal force vector u w u~ u w u~ T ð21Þ q ¼½ 1 1 b1 1 j 2 2 b2 2 b b b b e e T F ¼ Fa þ Fb þ Fc ¼ N 1 M 1 N 1 N 2 M 2 N 2 and ð30Þ q^ ¼½^ 0 ^ ^ 0 ^ T ð22Þ u1 w^ 1 u~1 j u2 w^ 2 u~2 can be evaluated analytically. It can be noted that the resul- e Using the notations defined in Fig. 2, the displacements tant forces N and N are associated with mean and relative and rotations for nodes 1 and 2 in the local coordinate sys- axial displacements. tem ðx^; ^zÞ are given, in terms of global quantities, by Since Eq. (29) must hold for any arbitrary dq, the global internal force vector is simply given by ^ 0 u^c ¼ 0; hc ¼ hc /; w^ ¼ b1 / 1 1 1 1 ð23Þ T b ^ 0 F ¼ T F ð31Þ u^c2 ¼ ‘ ‘0; hc2 ¼ hc2 /; w^2 ¼ b2 / 3.3. Tangent stiffness matrix where ‘0 and ‘ denote the initial and current lengths of the element. The global tangent stiffness matrix K is obtained by In these equations, local components of axial displace- T differentiation of Eq. (31) ments and rotations of the core can be written in terms T b T b of the local degrees of freedom (for node i ¼ 1; 2) dF ¼ T dF þ dT F ¼ KMdq þ KGdq ¼ KTdq ð32Þ u^ ¼ u^ þ h w^ 0 ð24Þ ci i 1 i where KM and KG are material and geometrical stiffness matrices. ^ 1 h2 0 hc ¼ u~î 1 w^ ð25Þ i i The global material stiffness matrix KM is computed by hc hc ~ T ^ where h1 ¼ h=4 and h2 ¼ hc þ h. KM ¼ T KT ð33Þ Using previous equations and geometrical consider- where the 6 · 6 local stiffness matrix is such that ations, it can be shown that the variations of ‘ and / are b given by dF ¼ K^dq^ ð34Þ Using previous definitions for q^ and Fb, this matrix can be d‘ ¼ rTdq ð26Þ evaluated analytically. For example, if the three layers 1 d/ ¼ zTdq ð27Þ are elastic homogeneous, we have ‘ 2 3 Kb Kb Kb Kb Kb Kb where the following notations are introduced: 6 11 12 13 11 12 13 7 6 b b b b b 7 T K K K K K c s h c cshc 6 22 23 12 25 23 7 r ¼½ 1 0 1 0 6 b b b b 7 T 6 K K K K 7 z s chs 0 sc h s 0 b 6 33 13 23 36 7 ¼ ½ 1 1 K ¼ 6 b b b 7 ð35Þ 6 K 11 K 12 K 13 7 with c ¼ cosð/ þ /0Þ and s ¼ sinð/ þ /0Þ. 6 b b 7 4 sym K 22 K 23 5 The virtual local displacements are obtained through b differentiation of the components of the local degrees of K 33

5 with the following components: cal axis passing through one end. This classical example was studied for example in [6,8] for a single layer beam. b 1 a b c K 11 ¼ ðc11Aa þc11Ab þc11AcÞ Here, the beam is composed of a viscoelastic core ‘0 b h1 c (ISD112 at 27 C) constrained by symmetrical elastic faces K 12 ¼ c11Ac ‘0 (Aluminum). Geometry data and material properties of the b 1 a b structure are shown in Fig. 3 and Table 1, respectively. K 13 ¼ ðc11Aa c11AbÞ 2‘0 Model parameters for the ISD112 at 27 C are identified 2 2 2 b 4 a b 4ðh2 hcÞ c 4h1 c 2h2‘0 c in [4], assuming that the Poisson’s ratio is frequency- K 22 ¼ ðc11Ia þc11IbÞþ 2 c11Ic þ c11Ac þ 2 c55Ac ‘0 ‘0hc ‘0 15hc independent. b h2 hc c h2‘0 c The robot arm is first repositioned to an angle of K 23 ¼ 2 c11Ic þ 2 c55Ac ‘0hc 12hc p=2 rad from its initial position. This is achieved by pre- 2 2 2 b 2 a b 2ðh2 hcÞ c 2h1 c h2‘0 c scribing the rotation angle wðtÞ as a linear function of time, K 25 ¼ ðc11Ia þc11IbÞþ 2 c11Ic þ c11Ac 2 c55Ac ‘0 ‘0hc ‘0 30hc as shown in Fig. 3. The beam is discretized by a regular b 1 a b 1 c ‘0 c mesh with 10 elements. The study is performed up to K 33 ¼ ðc11Aa þc11AbÞþ 2 c11Ic þ 2 c55Ac 4‘0 ‘0hc 3hc 200 ms with a fixed time step Dt ¼ 0:05 ms. The sequence b 1 a b 1 c ‘0 c of motion during the repositioning stage is depicted in K 36 ¼ ðc11Aa c11AbÞ 2 c11Ic þ 2 c55Ac 4‘0 ‘0hc 6hc Fig. 4a. Once wðtÞ¼p=2 rad for all time t P t1 ¼ 20 ms, the robot arm undergoes finite vibrations as shown in Moreover, the evaluation of dB gives the expression of the Fig. 4b. global geometrical tangent stiffness matrix Fig. 5a illustrates the attenuation of the tip elastic dis- 1 T 1 T T placements in x and z directions due to the constrained KG ¼ zz N 2 2 ðrz þ zr Þ ‘ hi‘ layer damping treatment. The damped oscillations for x component can also be observed in the phase-space dia- h ðN þ N ÞðM þ M Þþh ðNe þ Ne Þ ð36Þ 1 1 2 1 2 2 1 2 gram (Fig. 5b).

3.4. Resolution algorithm 4.2. Slider-crank mechanism An incremental-iterative method based on the Newmark direct integration scheme (with b ¼ 1=4 and c ¼ 1=2) and This second example corresponds to a slider-crank Newton–Raphson algorithm is employed here. Assuming mechanism with a rigid crank OA, a connecting sandwich the equilibrium conﬁguration at time tn is known, one beam AB and a slider block located at B. The sandwich nþ1 searches the solution q at time tnþ1 such that the follow- beam is composed by a viscoelastic core ðhc ¼ 0:2mmÞ ing non-linear equation of motion is satisﬁed:

nþ1 n 1 n 1 nþ1 e n 1 n 1 R ¼ Mq€ þ þ Fðq þ ÞG G þ G þ ¼ 0 ð37Þ ψ(t) π where R is the residual vector, M is the mass matrix, F the /2 rad OA = 200 mm internal force vector, G the mechanical external load, Ge the b = 0.1 mm electrical force associated with the actuator configuration ha = hb = 1.5 mm for the faces and G the dissipative force associated with hc = 0.2 mm the viscoelastic behavior of the core. All these quantities 20 ms t are defined in the global reference system using transforma- ψ(t) tion matrices and their local definitions given in [5] and thoroughly explained in [4]. O A It is well known that in a Newmark–Newton algorithm, ISD112 Aluminum an iteration loop on equilibrium is performed within the time loop. The non-linear equation (37) is then solved at Fig. 3. Flexible sandwich robot arm. each time step. Moreover, at each iteration, the displacement correction Dq is computed using the linearized equation Sðqnþ1ÞDq ¼Rnþ1, where the iteration matrix is Table 1 2 Mechanic and piezoelectric characteristics of the sandwich beam defined by S ¼ KT þ M=ðbDt Þ. Aluminum q ¼ 2690 kg=m3, m ¼ 0:345, E ¼ 70:3 GPa 4. Numerical examples ISD112 q ¼ 1600 kg=m3, m ¼ 0:5 E0 ¼ 1:5 MPa, E1 ¼ 69:95 MPa 2 4.1. Flexible robot arm a ¼ 0:7915, s ¼ 1:4052 10 ms 3 PZT5H q ¼ 7500 kg=m , c11 ¼ c33 ¼ 126 GPa 2 This simulation is concerned with the repositioning of a c13 ¼ 84:1 GPa, e31 ¼6:5C=m 2 8 sandwich flexible beam rotating horizontally about a verti- e33 ¼ 23:3C=m , d33 ¼ 1:3 10 F=m

6 t = 160 a t = 200 t = 100 b t = 60 t = 120 t = 18 t = 16 t = 140 t = 180 t = 20 t = 80 t = 14 t = 40

t = 12

t = 10

t = 8

t = 6 t = 4

Fig. 4. Deﬂected shapes of the sandwich robot arm: (a) repositioning sequence up to w ¼ p=2 rad and (b) free vibration about w ¼ p=2 rad.

a 300 b 30 200 20

100 x 10 z 0 0

–100 –10 velocity (mm/ms) Displacement (mm) –200 x –20

–300 –30

–400 –40 0 20 50 100 150 200 –350 –300 –250 –200 –150 –100 –50 0 Time (ms) x displacement (mm)

Fig. 5. (a) Time histories for x and z tip displacement components and (b) phase-space diagram for x component.

constrained by symmetrical elastic faces ðha1 ¼ hb1 ¼ The top piezoelectric patch works as an actuator and the

1mmÞ with piezoelectric patches ðha2 ¼ hb2 ¼ 0:5mmÞ. bottom one as a sensor. The time derivative of the voltage The dimensions and material properties of the slider-crank measured in the sensor is used for the feedback signal and mechanism are given in Fig. 6 and Table 1, respectively. then applied to the top piezoelectric patch with a constant _ The crank is driven by a constant angular speed w ¼ 0:15 gain Kd ¼100 ms, as V A ¼Kd V_ S. rad/ms. The points O, A and B are initially aligned with A between O and B. Moreover, the initial velocity and 0.04 acceleration of the mechanism are calculated by using kine- 0.02 matics of rigid mechanism. 0 The beam AB is discretized by a regular mesh of 10 ele- -0.02 ments. The study is performed up to 60 ms with a ﬁxed

Deflection (mm) -0.04 time step Dt ¼ 0:05 ms. The fractional operator is approx- -0.06 imated using the whole history in the Gru¨nwald series. 0 π/2 π 3π/2 2π 5π/2 3π 7π/2 4π Moreover, a sensor voltage feedback control law is used. Crank angle (rad)

10 A 5 t = 0 OA = 75 mm deflection 0 AB = 250 mm ψ(t) P AP = 100 mm -5 Velocity (mm/ms) Q PQ = 50 mm -10 O B -50 -40 -30 -20 -10 0 10 20 30 40 50 Deflection (mm) PZT5H ISD112 Aluminum Fig. 7. Deﬂection versus crank angle and phase-space diagram for the Fig. 6. Slider-crank mechanism. midpoint of the sandwich beam.

7 The transverse deflection of the midpoint of the connect- large displacements are proposed in order to test the co- ing beam AB is measured perpendicularly to the straight rotational active/passive sandwich beam element devel- line connecting points A and B. In Fig. 7, deflection versus oped in this work. The results illustrate the effectiveness crank angle and phase-space diagram are plotted in order of the proposed approach. An evaluation of the active con- to illustrate the attenuation of the elastic deformation strained layer damping treatment performances in the con- due to the active/passive damping treatment. text of non-linear dynamics will be investigated in a future work. Moreover, an experimental validation of the present 5. Conclusions formulation would be interesting particularly to demon- strate the effectiveness of the viscoelastic fractional deriva- A co-rotational approach for non-linear dynamic analy- tive model. sis of flexible linkage mechanisms with active constrained layer damping treatment is proposed in this work. An ori- References ginal co-rotational sandwich beam finite element, which allows large displacements and rotations, is developed for [1] Bagley RL, Torvik PJ. Fractional calculus – a different approach to the analysis of viscoelastically damped structures. AIAA J 1983;21: this purpose. The sandwich beam is composed of a visco- 741–8. elastic core, covered by elastic/piezoelectric laminated [2] Crisfield MA. Non-linear finite element analysis of solids and faces. No electrical degrees of freedom are required in the structures, Volume 1: essentials. Chichester: Wiley; 1991. finite element formulation, since the electromechanical cou- [3] Elkaranshawy HA, Dokainish MA. Corotational finite element pling is taken into account by means of an augmentation of analysis of planar flexible multibody systems. Comput Struct 1995;54:881–90. the stiffness of the piezoelectric layers in the sensor case. [4] A.C. Galucio, Atteńuation des re´ponses transitoires par traitement Concerning the viscoelastic damping, a four-parameter hybride pie´zoe´lectrique/viscoe´lastique en utilisant un mode`le a` fractional derivative model is used to describe the dissipa- de´riveés fractionnaires, PhD thesis; 2004 [in French]. tive behavior of the core. For the numerical examples, an [5] Galucio AC, Deu¨ J-F, Ohayon R. A fractional derivative viscoelastic incremental-iterative method based on the Newmark direct model for hybrid active/passive damping treatments in time domain – application to sandwich beams. J Intell Mater Syst Struct 2005;16: integration scheme and the Newton–Raphson algorithm is 33–45. employed. Two particularities are associated with the [6] Hsiao K-M, Jang J-Y. Dynamic analysis of planar flexible mecha- numerical resolution method. The first one concerns the nisms by corotational formulation. Comput Methods Appl Mech approximation of fractional operators by introducing an Eng 1991;87:1–14. external dissipative force which involves the storage of [7] Pritz T. Analysis of four-parameter fractional derivative model of real solid materials. J Sound Vib 1996;195:103–15. the displacement history. Obviously, this approach reduces [8] Simo JC, Vu-Quoc L. On the dynamics of flexible beams under large the numerical costs related to the non-locality of the viscoe- overall motions – the plane case: Parts I and II. J Appl Mech 1986;53: lactic behavior. The second one concerns the non linear 849–63. algorithm which requires the evaluation of the internal [9] Trindade MA, Benjeddou A, Ohayon R. Finite element modelling of force vector and tangent stiffness matrix associated with hybrid active–passive vibration damping of multilayer piezoelectric sandwich beams – part I: Formulation; part II: System analysis. Int J the sandwich element. As described in the paper, these Numer Methods Eng 2001;51:835–64. two quantities are calculated analytically in the global [10] Wasfy TM, Noor AK. Computational strategies for flexible multi- coordinate system. Finally, two numerical examples with body systems. Appl Mech Rev 2003;56:553–632.