A MATHEMATICAL MODEL FOR KINEMATIC LINKAGES MADE OF DEFORMABLE COMPOSITE BARS BUILT BY USING THE HAMILTON S VARIATION PRINCIPLE

Transcription

1 Romanian Repors in Physics, Vol. 6, No., P. 5 5, 9 A MATHEMATICAL MODEL FOR KINEMATIC LINKAGE MADE OF DEFORMABLE COMPOITE BAR BUILT BY UING THE HAMILTON VARIATION PRINCIPLE ABIN RIZECU, MARCELA URACHE, ELENA TAINA AVRAMECU, DUMITRU BOLCU Universiy of Craiova, A.I. Cuza ree, Craiova, Romania (Received April 6, 7) Absrac. This paper consiss in building up a new perspecive concerning some analyical approach of modeling he behavior of kinemaic linkages made of linear-elasic deformable bars. We have been focused on case of composie orhoropic bars having elasic symmery. Key words: spaial linkages, deformable elemens, composie bars, elasic symmery, new HDT deformaion hypohesis.. INTRODUCTION Describing he elaso-dynamics of composie bars by using he Classic Theory of Elasiciy is pracically impossible due o he fac ha he Bernoulli hypohesis simply does no work in case of composie bars. Composie maerials are generally non-homogenous and non-isoropic [4, 5]. For such kind of maerials new mehods of sudy have been developed []. This paper proposes a new HDT deformaion heory [, ] which is o be used o kinemaic linkages made of deformable composie bars.. THE MECHANICAL WORK OF THE EXTERNAL CHARGE We shall consider ha on he bar axis are coninuously disribued he nex linear specific force: he linear specific couple: { p p i, () { m m i. ()

2 6 abin Rizescu, Marcela Ursache, Elena Taina Avramescu e al. Due o he fac ha he Gay condiions are fully saisfied, we can wrie he expression of he mechanical work of he exernal charges: L n { { { {. () Le p u m θ dx We shall noe he boundary poin in which is o be placed he origin O of he own reference sysem of bar, he second boundary poin will be obviously L, where L is he lengh of he bar. In our holonomic case we have: n L δ δ δ δθ Le d Le d { p { u { m { dx d. (4) Concenraed forces and concenraed couples are no o be considered.. THE MATHEMATICAL MODEL FOR THE VIBRATION OF EACH BAR OF A LINKAGE MADE OF DEFORMABLE COMPOITE BAR For he kineic energy which has been calculaed in [] we will furher apply he δ operaor in order o obain an even more simple form for he so-called differeniaed kineic energy. Considering all kinds of symmeries described in [] and [4] we finally obain: n L δ Td A u Ω x u { { { L n Ω u { u d dx d r Ω δ θ r r r r { Ω θ Ω Ω θ θ Q Ω { θ Q Ω Ω θ Q { Q δθ ddx d. (5)

3 Linkages made of deformable composie bars 7 The Hamilon s variaion principle is abou he exreme of he nex funcional: ( T Ld Le) d, (6) considered wih is own synchronic condiions: δ w ( ) δ w ( ). Le s pu (6) of [], (44) of [4] and (4) in (6) and aking ino accoun ha he synchronic condiions are o be saisfied which means ha { δ u and { δθ are o have arbirary values. In such a case he expression being in a posiion o muliply hem are o have zero value. o i means ha, for he expression relaed o { u δ : A u Ω x u Ω Ω {{ { u d { u E { I E Q E,,, θ { { I E Q E d p, And, for he expression relaed o { δθ : { { {. (7) r r r θ Ω θ Ω Ω θ r r θ Q Ω { θ Q Ω Ω Q Q d { r E { Q E,,, θ θ θ { {,,, Q E r d θ Q E Q d u E {

4 8 abin Rizescu, Marcela Ursache, Elena Taina Avramescu e al. 4 {,, { { {,, θ I E Q E I E Q E I d u E r E Q E Y Q θ d, { I E Q E I E Q E,, θ { { Y Q d u E θ r E Q E,,, {,, Y Q d I E Q E I θ E { { Q E Y Q d m,, Considering (9) of [] and considering ha:. (8) Ω Ω ; Ω Ω and Ω Ω, (9) ; () and for c., we have: Q Ω Q d [ ] The relaions (7) and (8) are acually parial derivae equaions having as unknowns { u and { θ. These unknowns depend only on he x coordinae and on he ime. In many cases he kinemaic linkages are made of composie orhoropic bars being mass homogenous (MEG). Le s noe B and H he dimensions of he ransversal secion of he bar. o, (7) and (8) are o become: P u P u P u P P u f 4 5 θ,, { { { { V V V V u V g where: 4 5 θ θ θ θ,, { { { { ; () ; ()

5 5 Linkages made of deformable composie bars 9 BH P BH ; () B H BH BH P BH BH ; (4) BH BH Ω is he ani-symmerical marix aached o he angular speed of T : ; (5) T wih respec o i i i { ; ; i ; (6) ( ) We have also he angular acceleraion of T wih respec ot : ε ε ( ) ( ) i i i ; ; i ; (7) where: Ω ; (8)

6 abin Rizescu, Marcela Ursache, Elena Taina Avramescu e al. 6 BH ( ) ( ) BH BH P BH BH ( ) ( ) BH BH BH BH ( ) ( ) (9) where: ( ) ( ) ( ) ( ) Ω ( ) ( ) ; () 4 P EBH ; () 9 EBH 9 EBH 5 P EBH EBH ; () p B H a ( ) ( ) x f p B H a x p B H a x { o ( ) o ( ) o ( ) ; ()

7 7 Linkages made of deformable composie bars BH ( B H ) 49 V BH H B ; (4) BH HB 6 6 BH V ; (5) 6 H B 6 V V V V V V V ; (6) V V V where: BH HB BH ; ( ) ( ) ( )( ) V B H V V BH ; H B V ; BH ;

8 abin Rizescu, Marcela Ursache, Elena Taina Avramescu e al. 8 ( ) ( ) V B H E B H E V ; V H B ; V ; H B ; ( ) ( ) V H B E B H E B H ; (7) 4 V EBH ; (8) 9 EBH 9 EBH EH B 49 V E B H E 4 H B ; (9) { g BH HB m BH m. () H B m We have o consider he siuaion which he boundary poins of bars are he places of he only exising couplings. In such a case he firs derivaive wih respec o ime of (7) of [] leads us o:

9 9 Linkages made of deformable composie bars k k k k k k A rk r k rk k k k k k k k k r k rk rk rk k { k r k k {. () Remember ha we are in he siuaion ha he origin of T is placed in he boundary poin L of he bar. In such a case we have: and { k k k { k { r ; () r are algebraic vecors having consan value wih respec o Tk, so, whaever derivaive of hem has o have zero value. o, () becomes: d d { k { k { () k k k k A r r r k d k d Obviously we have: r { a () () { ( ) { (). () a a A. (4) 4. CONCLUION An algorihm for esablishing he mahemaical model for he vibraions of each bar of he kinemaic linkage. We propose such an algorihm which is consisen wih all developmens we have been working on. We shall aach o each

10 4 abin Rizescu, Marcela Ursache, Elena Taina Avramescu e al. elemen of he kinemaic linkage is own reference sysem relaed o is nondeformed saus. We made consideraion on how is bes o choose such a sysem. We shall deermine he generalized coordinaes of he linkage, considering is bars as being rigid ones. k We shall deermine every marix. We shall deermine every marix: Ω. We shall esablish he posiion A,, which means we shall deermine he. vecors r and r We shall make a full-complee kinemaic and dynamic analysis for he linkage, considering bars ha being sill rigid. We shall calculae all marix of he mahemaical model according o relaions ().. (). We shall wrie he mahemaical model in is classic form (7) and (8). We shall wrie all possible relaions concerning he compaibiliy of displacemens in all kinemaic couplings. This kind of model can be improved in order o sudy more oher sors of bars like variable secion composie bars and composie bars having some complex geomerical form. More, his sor of HDT deformaion hypohesis can well be pu ogeher wih some sor of homogenizaion heory like his shown in [4, 5]. REFERENCE. Amirouché, F., Compuaional Mehods in Mulibody Dynamics, Prenice-Hall, 99.. Aia, O., El-Zafrany, A., A High Order hear Elemen for Nonlinear Vibraion Analysis of Composie Plaes and hells, Pergamon, In. J. Mech. ci., 4, 4 5, (999), Elsevier cience Ld.. Barrau, J.J., Laroze,., Calcul des srucures en maeriaux composies, 4, Ecole Naionale de l'aéronauique e de l'espace, Toulouse, Bolcu, D., ănescu, G., Ursache, M., Theoreical and experimenal sudy on deerminaion of he elasic properies of he composie maerials, Romanian Repors in Physics, 56,, (4). 5. Bolcu, D., ănescu, G., Ursache, M., Deerminaion of elasiciy modulus for he composie maerials based on resin-exile, Analele Universiăţii din Craiova, Physics AUC, 4, (4). 6. Gay, D., Maeriaux composies, Ediion Hermes, Paris, Gordaninead, F., Ghazavi, A., Chalhoub, N.G., Nonlinear Dynamic Modelling of a Revolue Prismaic Flexible Composie Maerial Robo Arm, Jour. Vibr. Acous.,, (99). 8. Librescu, L., aica şi dinamica srucurilor elasice, anizorope şi eerogene, Edi. Academiei Române, Librescu, L., Thangihan,., Parameric insabiliy laminaed composie shear-deformable fla panels subec o in-plane bads, Ind. J. Non-linear Mechanics, 5, 6 7 (99).

11 Linkages made of deformable composie bars 5. Reddy, J.N., Energy and Variaional Mehods in Applied Mechanics, John Wiley and ons, New York, Rizescu,., Bolcu, D., ănescu, G., A new deformaion hypohesis used o model he elasic behavior of he prismaic composie bars wih elasic symmery, Mashininelek, (4).. Rizescu,., Rinderu, P., Bolcu, D., Ecuaţii de mişcare penru un elemen cinemaic de ip bară compoziă, Conferinţa Naţională de Roboică, Craiova,.. Rizescu,., Ursache, M., The calculus of he kineic energy for a kinemaic linkage made of composie bars using an improved HDT-deformaion hypohesis, Analele Universiăţii din Craiova, seria Fizică, Physics AUC, 7 (Par II), 4 (7). 4. Rizescu., Ursache M., The calculus of he deformaion mechanical work for a kinemaic linkage made of composie bars using an improved HDT-deformaion hypohesis, Analele Universiăţii din Craiova, seria Fizică, Physics AUC, 7 (Par II), 9 (7). 5. aicu,., Aplicaţii ale calculului mariceal în mecanica solidelor, Edi. Academiei Române, 986.