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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME AND TECHNOLOGY (IJMET) ISSN 0976 – 6340 (Print) ISSN 0976 – 6359 (Online) IJMET 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) ©IAEME www.jifactor.com 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 . 400 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME If the reduction is in point O0 , then equivalent torsion has components F , M satisfying the conditions F = f ; M = m + O0 O x f (1) Considering reference systems with origins in points O , O 0 (Fig.1) and noting with (f x , f y , f z , m x , m y , m z ), (F , F , F , M x y z x ,M y,M z ) vector projections ( f , m ), (F, M ) respective on the axis of the systems Oxyz , O0 XYZ then these scalar components are pl ckeriene coordinates [4] of the torsion with representation by matrices column so; { f } = [ f x f y f z m x m y m z ]T ; {F } = Fx Fy Fz M x M y M z T [ (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: ( X 0 ,Y0 , Z 0 ) - point coordinates O ; α i , β i , γ i , i = 1, 2, 3 , direction cosines of axes Ox, Oy , Oz ; [G ] , [R ], [T ] translation matrices, position respectively 0 − Z0 Y0 α1 α 2 α3 [R ] [0] [G ] = Z 0 0 − X0 ; [R ] = β β β 3 ; [T ] = (3) 1 2 [G ]⋅ [R ] [R ] − Y0 X0 0 γ 1 γ 2 γ3 Obtaining [4] transformation relations between the matrices column {F }{ f } , {F } = [T ]⋅ { f } ;{ f } = [T ]−1 ⋅ {F } (4) where [T ]−1 = [T ] T [0]T T (5) [R ] [G ] [R ] T 401 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME 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 E , G and either Axyz local reference system, Ax , Ay the central principal axes of inertia of the normal section A . Fig. 2. Straight bar AB , under the influence of efforts Noting with (f A , m A ) torsion effort from A ; with ( f B , m B ) torsion effort from B ; with d A , d B , m B vectors defined by the relations: * d A = AA′ ; d B = BB ′ + AB x θ AB ; m B = m B + AB x f B (6) and noting the projections on the axes trihedral Axyz of vectors f A , m A , f B , m B ,θ A , d A , θ B ,d B , respectively with (fAx , f Ay , f Az ) , (m Ax , m Ay , m Az ) , (f Bx , f By , f Bz ) , (m Bx , m By , m Bz ), (d Ax , d Ay , d Az ) , (θ Ax ,θ Ay ,θ Az ) , (θ Bx ,θ By ,θ Bz ) , (d Bx , d By , d Bz ) , we obtain column matrix of pl ckeriene coordinates in the local system Axyz { f A } = [ f Ax f Ay f Az m Ax m Ay m Az ]T ; { f B } = [ f Bx f By f Bz mBx mBy mBz ]T (7) {d A } = [θ Ax θ Ay θ Az d Ax d Ay d Az ]T ; {d B } = [θ Bx θ By θ Bz d Bx d By d Bz ]T (8) and equality resulting from the equilibrium condition { f A }+ { f B } = {0} (9) Stiffness matrix [k AB ] and matrix flexibility [h AB ] = [k AB ] −1 [4] expressed in the reference system Axyz are given by the equalities 402 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME 0 0 0 Al 2 0 0 0 0 6I z l 0 12 I z 0 E 0 − 6I y l 0 0 0 12 I z [k AB ] = 3 ⋅ G 2 0 ; l I zl 0 0 0 0 E 2 0 4I z l 0 0 0 − 6I y l 2 0 0 0 4I z l 0 6I z l E 0 0 0 6 G 0 0 3l 6 0 0 0 0 (10) Iy Iy 3l 6 0 − 0 0 0 I Iz [h AB ] = l ⋅ 6 z 6E 0 0 0 0 0 A 2l 2 3l 0 0 0 0 − Iz Iz 2l 2 3l 0 0 0 0 Iy Iy where I y , I z are the principal central moments of inertia of normal areas on axis Ax and I x is defined by equality Ix = I y + Iz (11) With these notations [4] to obtain equalities { f A } = [k AB ] ⋅ {d AB } ; {d AB } = [k AB ] ⋅ { f A } (12) where {d AB } is the relative displacement {d AB }= {d A }− {d B } (13) Switching to the reference OXYZ is done using relations (3), (4), (5) and these equalities are obtained {D A }= [TAB ]{d A } ; {DB }= [TAB ]{d B } ; {D AB }= {D A }− {DB } {FA }= [TAB ]{ f A } ; {FB }= [TAB ]{ f B } (14) {K AB }= [TAB ]{k AB }[TAB ] ; [H AB ] = [TAB ]{k AB }[TAB ] −1 −1 {FA }= [k AB ]{D AB } ; {D AB }= [H AB ]{FA } (15) 403 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME 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 B, C , D 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 B, C , D obtain formal scheme from fig. 4, for which the following equations of equilibrium can be written Fig. 3. Articulated quadrilateral plan mechanism {E A }+ {E B } = {0} ; {E B }+ {E B } = {PB } ; {E B }+ {EC } = {0} ; 1 1 2 2 2 (16) {EC }+ {EC }= {PC } ; {EC }+ {E D }= {0} 2 3 3 3 from which resulting {E B 2 }= {E A }+ {PB }; {EC 3 }= {E A }+ {PB }+ {PC } (17) {RB }= {PB }+ {E A }; {RC }= {PB }+ {PC }+ {E A }; {RD }= {PB }+ {PC }+ {PD }+ {E A } (18) and using relations (15) the following expressions are derived: {E A }= [K AB ]{{D A }− {DB }}; {E A }+ {PB } = [K BC ]{{DB }− {DC }} 1 2 2 (19) {E A }+ {PB }+ {PC } = [K CD ]{{DC }− {DD }} 3 3 or {D A }− {DB }= [H AB ]{E A } ; {D B }− {D C }= [H BC ]{{E A }+ {PB }} 1 2 2 (20) {D C }− {D D }= [H CD ]{{E A }+ {PB }+ {PC }} 3 3 { } { }{ } where D Bi , D Ci , D Di , i = 1, 2 , are the movements at the ends of indexed bars (Fig. 3) with indices 1, 2, 3 . 404 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME Fig. 4. Formal mechanism quadrilateral scheme, the representations efforts and reaction forces Consider the insertion of A , ({D A } = {0}) in the linear elastic calculation and in this way the movements of the other sections relate to the section of A . Noting with {U B },{U C }, {U D } column matrices attached to kinematic couplings of rotation [6] to obtain expressions {U B } = [0 0 1 YB − X B 0] , T {U C } = [0 0 1 YC − X C 0] , (21) {U D } = [0 0 1 0 − X D 0] T and noting with ξ B , ξ C , ξ D rotations of the joints, the following equalities are derived: {D }= {D }+ ξ {U } ; B2 B1 B B {D }= {D }+ ξ {U } ; C3 C2 C C (22) {0} = {D }+ ξ {U } D3 D D By adding relations (20), taking into account the equations (22) and using notations [H AD ] = [H AB ] + [H BC ] + [H CD ] ; ξ B [K AD ] = [H AD ] ; [U ] = [{U B } {U C } −1 {U D }]; [ξ ] = ξ C (23) ξ D ~ {∆}= [H ]{P }+ [H ]{{P }+ {P }} BC B CD B C obtain the equation [E A ] = [K AD ][U ]{ξ } − [K AD ]{∆} ~ (24) Using the notations 405 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME ~ {U } = [Y − X 0 0 0 0]; B T B B ~ {U } = [Y − X 0 0 0 1]; C T C C (25) ~ {U } = [Y − X 0 0 0 1]; D T D D {U }= [{U } {U } {U }] ; ~ ~ ~ ~ B C D T (26) knowing [6] that reactions satisfy the relations ~ ~ ~ {U } {R } = 0 ; {U } {R } = 0 ; {U } {R } = 0 B T B C T C D T D (27) taking into account the equality relations (18) and the notation ~ T { } U B {PB } (28) ~ { } T {PT } = − U C {{PB }+ {PC }} U T {{P }+ {P }+ {P } ~ { } D D } B C obtain the expression [U ]{E ~ A } = {P } T (29) with that of (24) the matrix of rotations in the joints is deduced {ξ } = [U ⋅ K AD ⋅ U ]−1 {{PT }+ [U ][K AD ]⋅ {∆}} ~ ~ ~ (30) and then from (24) the reaction is deduced form A , {R A } = {E A } Relations (30), (29), (24) can be expressed in a simpler form if the following notations are made {E A } = [E Ax E Ay E Az M Ax M Ay M Az ]T (31) {∆} = [θ x ~ ~ ~ θy ~ θz ~ ∆x ~ ∆y ] ~ T ∆z 0 0 K13 K14 K15 0 0 0 K 23 K 24 K 25 0 K13 K14 K15 K 31 K 32 K 36 K 41 K 42 0 0 0 K 46 1) [ ] [K AD ] = K 31 K 32 0 0 0 K 36 ; K (AD = K 23 K 24 K 25 ; K (AD = K 41 K 42 K 46 2) [ ] (32) K 51 K 52 0 0 0 K 56 K 63 K 64 K 65 K 51 K 52 K 56 0 0 K 63 K 64 K 65 0 YB − X B 1 1 1 1 [A] = YC − XC 1 ; [B ] = Y B YC YD (33) YD − X D 1 − X B − XC − XD and then it follows ~ θz −1 ~ {ξ } = [B]−1 ⋅ [K AD ] (1) −1 ⋅ [A] ⋅ {PT } + [B ] ∆ x −1 (34) ~ ∆ y 406 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME E AX E = [A]−1 ⋅ {P } ; AY T M AZ ~ (35) E AZ θ x M = − K (2 ) ⋅ θ [ ~ ] AX AD y ~ ∆ z M AY Calculation of the other reactions {R B }, {RC }, {RD } are made using the relations (18). V. CONCLUSION The matrix {PT } defined by the relation (28) depends only on the components of forces {PB }, {PC }, {PD } compatible with the movement of the mechanism, respectively on the components PBX , PBY , PBZ and the analogues ones. The statically determined components of the reaction {R A } = {E A } 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 {∆ }= [θ~ ~ 1 z ~ ∆x ~ ∆y ] ; {∆ }= [θ~ T ~ 2 x ~ θy ~ ∆z ] T (36) ~ { } The matrix ∆ 1 with components in the plane of motion and depending on the ~ components compatible with movement PBX , PBY , M BZ and the analogues, and ∆ 2 with { } incompatible components with moving parts and is not compatible with motion-dependent PBZ , M BX , M BY 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. 407 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME 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. 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