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  • 1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME 400 CALCULATION OF THE UNDETERMINED STATIC REACTIONS FOR THE ARTICULATED PLAN QUADRILATERAL MECHANISM Jan-Cristian Grigore1 , Nicolae Pandrea2 1 (University of Piteşti, str. Targul din Vale nr.1, Romania) 2 (University of Piteşti, str. Targul din Vale nr.1, Romania) ABSTRACT Spatial mechanisms of the non-zero families constitute statically undetermined systems, the undetermination order is given by the number representing the family of the mechanism. The articulated plan quadrilateral mechanism, shown in this paper, is a third family mechanism, an undetermined static third order mechanism. This paper uses the relative displacement method and it establishes the mathematical model that allows the linear elastic calculation in order to determine the statically undetermined reactions. Keywords: coordinates pl ckeriene, matrix flexibility, stiffness matrix I. INTRODUCTION If the plane mechanisms are stressed by vector components forces perpendicular to the motion plane or by vector component moment placed in the plane of motion, they are statically undetermined systems. In these cases, in order to determine the components of the reaction forces perpendicular to the motion plane as well as the components of the reaction moments in the motion plane, the linear elastic calculation shall be used. This paper shows these components using the relative displacement method [3], [4] and the pl ckeriene coordinates. II. NOTATIONS, REFERENCE SYSTEMS, TRANSFORMATION RELATIONS Forces acting on a rigid point, the velocities of the points of a rigid, the small movements of the points of the rigid are systems reduced to a point O ( Fig.1)at a torsion vector consisting of mainly f and moment vector m . INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING AND TECHNOLOGY (IJMET) ISSN 0976 – 6340 (Print) ISSN 0976 – 6359 (Online) Volume 4, Issue 3, May - June (2013), pp. 400-408 © IAEME: www.iaeme.com/ijmet.asp Journal Impact Factor (2013): 5.7731 (Calculated by GISI) www.jifactor.com IJMET © I A E M E
  • 2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME 401 If the reduction is in point 0O , then equivalent torsion has components F , M satisfying the conditions fF = ; fxOOmM 0+= (1) Considering reference systems with origins in points 0,OO (Fig.1) and noting with ( )zyxzyx mmmfff ,,,,, , ( )zyxzyx MMMFFF ,,,,, vector projections ( )mf , , ( )MF, respective on the axis of the systems Oxyz , XYZO0 then these scalar components are pl ckeriene coordinates [4] of the torsion with representation by matrices column so; { } [ ]T zyxzyx mmmffff = ; { } [ ]T zMyMxMzFyFxFF = (2) For a system of forces, f is the resultant force vector and m is moment resulting in O , for rigid speeds f is the angular velocity of the rigid and m is the velocity of point O and for small displacements of rigid, f is small rotation vector, and m is small movement of the point O . Fig. 1. System of forces With notations: ( )000 ,, ZYX - point coordinates O ; 3,2,1,,, =iiii γβα , direction cosines of axes OzOyOx ,, ;[ ]G , [ ]R , [ ]T translation matrices, position respectively [ ]           − − − = 0 0 0 00 00 00 XY XZ YZ G ; [ ]           = 321 321 321 γγγ βββ ααα R ; [ ] [ ] [ ] [ ] [ ] [ ]      ⋅ = RRG R T 0 (3) Obtaining [4] transformation relations between the matrices column { }{ }fF , { } [ ] { } { } [ ] { }FTffTF ⋅=⋅= −1 ; (4) where [ ] [ ] [ ] [ ] [ ] [ ]       = − TTT T RGR T T 01 (5)
  • 3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME 402 III. RELATIONS BETWEEN RELATIVE MOVEMENTS AND EFFORTS AT THE ENDS OF A BAR Consider the straight bar AB (see Fig. 2), with length l , constant section, the area A , modules of elasticity GE, and either Axyz local reference system, AyAx, the central principal axes of inertia of the normal section A. Fig. 2. Straight bar AB , under the influence of efforts Noting with ( )AA mf , torsion effort from A ; with ( )BB mf , torsion effort from B ; with BBA mdd ,, vectors defined by the relations: BfxABBmBmABxABBBBdAAAd +=+′=′= * ;; θ (6) and noting the projections on the axes trihedral Axyz of vectors BdBAdABmBfAmAf ,,,,,,, θθ , respectively with ( )AzAyAx fff ,, , ( )AzAyAx mmm ,, , ( )BzByBx fff ,, , ( )BzByBx mmm ,, , ( )AzAyAx ddd ,, , ( )AzAyAx θθθ ,, , ( )BzByBx θθθ ,, , ( )BzByBx ddd ,, , we obtain column matrix of pl ckeriene coordinates in the local system Axyz { } [ ]T AzAyAxAzAyAxA mmmffff = ; { } [ ]T BzByBxBzByBxB mmmffff = (7) { } [ ]T AzAyAxAzAyAxA dddd θθθ= ; { } [ ]T BzByBxBzByBxB dddd θθθ= (8) and equality resulting from the equilibrium condition { } { } { }0=+ BA ff (9) Stiffness matrix [ ]ABk and matrix flexibility [ ] [ ] 1− = ABAB kh [4] expressed in the reference system Axyz are given by the equalities
  • 4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME 403 [ ]                     − − ⋅= 060 2 400 6000 2 40 00000 2 1200060 0120600 00 2 000 3 lzIlzI lyIlzI lzI E G zIlyI zIlzI Al l E ABk ; [ ]                                   − − ⋅= 0 3 0 2 2 00 3 000 2 2 0 00000 6 6 000 3 0 0 6 0 3 00 006000 6 yI l yI l zI l zI l A zIzI l yIyI l G E E l ABh (10) where zy II , are the principal central moments of inertia of normal areas on axis Ax and xI is defined by equality zyx III += (11) With these notations [4] to obtain equalities { } [ ] { } { } [ ] { }AABABABABA fkddkf ⋅=⋅= ; (12) where { }ABd is the relative displacement { } { } { }BAAB ddd −= (13) Switching to the reference OXYZ is done using relations (3), (4), (5) and these equalities are obtained { } [ ]{ } { } [ ]{ } { } { } { } { } [ ]{ } { } [ ]{ } { } [ ]{ }[ ] [ ] [ ]{ }[ ] 11 ; ; ;; −− == == −=== ABABABABABABABAB BABBAABA BAABBABBAABA TkTHTkTK fTFfTF DDDdTDdTD (14) { } [ ]{ } { } [ ]{ }AABABABABA FHDDkF == ; (15)
  • 5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME 404 IV. CALCULATION OF THE REACTIONS Considering articulated plan mechanism ABCD from Fig. 3 acted by external forces and moments [4] which are distributed in points DCB ,, and are marked by column matrix of pl ckeriene coordinate. Noting generally with { }E efforts at the ends of the bars, and with{ }R reactions by isolating bars and nodes DCB ,, obtain formal scheme from fig. 4, for which the following equations of equilibrium can be written Fig. 3. Articulated quadrilateral plan mechanism { } { } { } { } { } { } { } { } { } { } { } { } { } { } { }0; ;0;;0 3332 22211 =+=+ =+=+=+ DCCCC CBBBBBA EEPEE EEPEEEE (16) from which resulting { } { } { } { } { } { } { }CPBPAECEBPAEBE ++=+= 3;2 (17) { } { } { } { } { } { } { } { } { } { } { } { }AEDPCPBPDRAECPBPCRAEBPBR +++=++=+= ;; (18) and using relations (15) the following expressions are derived: { } [ ]{ } { }{ } { } { } [ ]{ } { }{ } { } { } { } [ ]{ } { }{ }33 221 ; DCCDCBA CBBCBABAABA DDKPPE DDKPEDDKE −=++ −=+−= (19) or { } { } [ ]{ } { } { } [ ]{ } { }{ } { } { } [ ]{ } { } { }{ }CBACDDC BABCCBAABBA PPEHDD PEHDDEHDD ++=− +=−=− 33 221 ; (20) where{ }iBD ,{ }iCD ,{ }iDD , 2,1=i , are the movements at the ends of indexed bars (Fig. 3) with indices 3,2,1 .
  • 6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME 405 Fig. 4. Formal mechanism quadrilateral scheme, the representations efforts and reaction forces Consider the insertion of A, { } { }( )0=AD in the linear elastic calculation and in this way the movements of the other sections relate to the section of A . Noting with { } { } { }DCB UUU ,, column matrices attached to kinematic couplings of rotation [6] to obtain expressions { } [ ] { } [ ] { } [ ]T DD CCC T BBB XU XYU XYU 00100 ,0100 ,0100 −= −= −= (21) and noting with DCB ξξξ ,, rotations of the joints, the following equalities are derived: { } { } { } { } { } { } { } { } { }DDD CCCC BBBB UD UDD UDD ξ ξ ξ += += += 3 23 12 0 ; ; (22) By adding relations (20), taking into account the equations (22) and using notations [ ] [ ] [ ] [ ] [ ] [ ] [ ] { } { } { }[ ] [ ] { } [ ]{ } [ ]{ } { }{ }CBCDBBC D C B DCBADAD CDBCABAD PPHPH UUUUHK HHHH ++=∆           === ++= − ~ ;; ; 1 ξ ξ ξ ξ (23) obtain the equation [ ] [ ][ ]{ } [ ]{ }∆−= ~ ADKUADKAE ξ (24) Using the notations
  • 7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME 406 { } [ ] { } [ ] { } [ ];1000 ~ ;1000 ~ ;0000 ~ DD T D CC T C BB T B XYU XYU XYU −= −= −= (25) { } { } { } { }[ ] ; ~~~~ T DCB UUUU = (26) knowing [6] that reactions satisfy the relations { } { } { } { } { } { } 0 ~ ;0 ~ ;0 ~ === D T DC T CB T B RURURU (27) taking into account the equality relations (18) and the notation { } { } { } { } { } { }{ } { } { } { } { }{ }            ++ +−= DCB T D CB T C B T B T PPPU PPU PU P ~ ~ ~ (28) obtain the expression [ ]{ } { }TA PEU = ~ (29) with that of (24) the matrix of rotations in the joints is deduced { } [ ] { } [ ][ ] { }{ }∆⋅+⋅⋅= − ~~~ 1 ADTAD KUPUKUξ (30) and then from (24) the reaction is deduced form { } { }AA ERA =, Relations (30), (29), (24) can be expressed in a simpler form if the following notations are made { } [ ] { } [ ]T zyxzyx T AzMAyMAxMAzEAyEAxEAE ∆∆∆=∆ = ~~~~~~~ θθθ (31) [ ] [ ] [ ]                                 === 565251 464241 363231)2( ; 656463 252423 151413)1( ; 065646300 560005251 460004241 360003231 025242300 015141300 KKK KKK KKK ADK KKK KKK KKK ADK KKK KKK KKK KKK KKK KKK ADK (32) [ ] [ ]           −−− =           − − − = DCB DCB DD CC BB XXX YYYB XY XY XY A 111 ; 1 1 1 (33) and then it follows { } [ ] [ ] [ ] { } [ ]           ∆ ∆+⋅⋅⋅= −−−− y x z TAD BPAKB ~ ~ ~ 111)1(1 θ ξ (34)
  • 8. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME 407 [ ] { } ( ) [ ]           ∆ ⋅−=           ⋅=           − z y x AD AY AX AZ T AZ AY AX K M M E PA M E E ~ ~ ~ ; 2 1 θ θ (35) Calculation of the other reactions { } { } { }DCB RRR ,, are made using the relations (18). V. CONCLUSION The matrix { }TP defined by the relation (28) depends only on the components of forces { } { }{ }DCB PPP ,, compatible with the movement of the mechanism, respectively on the components BZBYBX PPP ,, and the analogues ones. The statically determined components of the reaction { } { }AA ER = are given by the first relation (35) and as expected they depend on the components compatible with external forces movement and they do not depend on the stiffness of the elements of the mechanism. The matrix { }∆ ~ is the result matrix partitions { } [ ]T yxz ∆∆=∆ ~~~~ 1 θ ; { } [ ]T zyx ∆=∆ ~~~~ 2 θθ (36) The matrix { }1 ~ ∆ with components in the plane of motion and depending on the components compatible with movement BZBYBX MPP ,, and the analogues, and { }2 ~ ∆ with incompatible components with moving parts and is not compatible with motion-dependent BYBXBZ MMP ,, and analogues. It follows from this and from the second relation (35) that statically indeterminate reactions depend exclusively on the components of the external forces incompatible with moving parts. Concerning the matrix{ }ξ , movements of kinematic couplings resulting from the set and relation (35) it depends on the stiffness of elements as well as on the components of the external forces compatible with movement. Based on the relations established in this paper we can develop an algorithm and a program for numerical calculation of statically indeterminate components, an objective that will result in a subsequent paper. VI. ACKNOWLEDGEMENTS This paper is a continuation of research conducted under the grant "PD -683 / 2010", and we want to thanked the Romanian Government (UEFISCDI), which certainly those funding research.
  • 9. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME 408 REFERENCES [1] Bulac, I, Grigore, J.-C., (2012) Mathematical model for calculation of liniar elastic shaft, ANNALS of the ORADEA UNIVERSITY. Fascicle of Management and Technological Engineering, Volume XI (XXI), no. 2, p. 3.9-3.12. [2] Buculei, M., Băgnaru, D., Nanu, Gh., Marghitu, D.,(1994) Computational methods to analyze mechanisms with bars, Romanian Writing Publishing, Craiova. [3] Bulac, I, Grigore, J.-C., (2012) Mathematical model for calculation of liniar elastic shaft, ANNALS of the ORADEA UNIVERSITY. Fascicle of Management and Technological Engineering, Volume XI (XXI), no. 2, p. 3.9-3.12 [4] Pandrea, N.,(2000) Elements of solid mechanics coordinate pl ckeriene, Romanian Academy Publishing, București [5] Popa, D.,(1997) Contributions to the study of dynamic elastic-bar mechanisms. Thesis, Polytechnic Institute București. [6] Popa, D.,(1997) Contributions to the study of dynamic elastic-bar mechanisms. Thesis, Polytechnic Institute București. [7] Voinea, R., Pandrea, N.,(1973) Contributions to a general mathematical theory of kinematic couplings, SYROM, vol. B, București 1973 [8] Chikesh Ranjan and Dr R. P. Sharma, “Modeling, Simulation & Dynamic Analysis of Four-Bar Planar Mechanisms using Catia V5R21”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 4, Issue 2, 2013, pp. 444 - 452, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359. [9] Dr R. P. Sharma and Chikesh Ranjan, “Modeling and Simulation of Four-Bar Planar Mechanisms using Adams”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 4, Issue 2, 2013, pp. 429 - 435, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.