The Mathematical Model ad the Simulatio Modellig Algoritm of the Multitiered Mechaical System Demi Aatoliy, Kovalev Iva Dept. of Optical Digital Systems ad Techologies, The St. Petersburg Natioal Research Uiversity of Iformatio Techologies, Sait Petersburg, RUSSIAN FEDERATION (LOMO PLC) Moder mechaical systems are usually have complex spatial ad multitiered structure. The importat stage of the desig of such systems is to estimate its behavior i the dyamics which ca be obtaied by computer simulatio. Figure 1 below shows a multi-mechaical cotrol system of the supervisory system, icludig a mobile base (dec of a ship) ad the observatio device i gimbal suspesio. The feature of this system is that the mass ceters of compoets of the system are separated i space. Fig. 1 Multitiered mechaical cotrol system of the supervisory system. Asia Busiess Cosortium ABC-JAR Page 44
Figure 1 uses the followig otatios: the first three compoets K, T, R simulate the rotatio of the vessel relative to three orthogoal axes, so that, ω K the agular velocity of rotatio of the dec at the agle of heel, ω T the agular velocity of rotatio of the dec at the corer of yaw, ω R the agular velocity of rotatio of the dec at the corer of pitch attitude. The poles of these compoets are the same ad are i the ceter of the ship rollig. The secod three compoets G, V, C simulate the optical device rotatio so that, ω G the agular velocity of the optical device rotatio by horizotal agle, ω V the agular velocity of the rotatio of the optical device by vertical agle, ω C the agular velocity of the optical device rotatio by agle of compesatio. The simplest ad most effective way to set up the equatios of such systems with geometric costraits o the locatio of its poits (ot free holoomic system) is based o the represetatio of their relatively geeralized forces determied by the eergy of the system, ad geeralized coordiates. The equatios of motio i this case as the Lagrage equatios of the 2d id ca be represeted i vector-matrix form: Aq Q 0,5Dq Q 1 2 aiq q i 1 q where A the matrix of derivatives of the ietic eergy by coordiates; Q geeralized forces matrix; q geeralized coordiates matrix (i=1, 2,,); D partial derivatives matrix; the umber of degrees of freedom; a i coefficiets of iertia. The advatage of this represetatio is its geerality ad uiform of costructig the equatios. However, due to the fact that the presetatio of (1) is complex i terms of the spatial orietatio of the system ad aalysis of its costituet parts, appears difficulty i simulatio modelig. I order to simplify the procedure of buildig up the equatios, we tae the cocept of quasi-velocities, i.e. we ll carry out the liearizatio of the geeralized velocities. Such a simplificatio allows a mathematical model of motio of the mechaical part of the system i a form suitable for its partial aalysis. I this case the stage of the withdrawal of differetial equatios for the dyamics of the body ca be omitted. I this way we ca preset the mechaical multitiered cotrol system of the optical device (see fig.1) i form of the graphic-vector descriptio as show i Figure 2, where the followig otatios are used : О the ceter of the earth rotatio; V K the liear speed of the ship; K, T, R, G, V, C poles of the relevat parts of the system; R K,T,R,G,V,C radius vectors of the correspodig poles; R C radius vector of the mass ceter of the correspodig part to the coordiate system associated with this part; M K,T,R,G,V,C momets of the forces actig o the relevat body; ω O,K,T,R,G,V,C agular velocity vector of the relevat parts; D T,R,G,V,C the poles of the drive uits of the correspodig parts; R D radius vectors of drive uit poles to the coordiate system associated with the respective parts; ω D agular velocities of the respective drive uits; R CD radius vectors of the momets of the forces correspodig to the appropriate drive uits; M D momets of forces of correspodig drive uits; γ T,R,G,V,C agles of bodies rotatio uder the actio of the correspodig drive uits. (1) Asia Busiess Cosortium ABC-JAR Page 45
Fig.2 Graphic-vector descriptio of the multitiered mechaical system of optical device cotrol A mathematical model of motio of the mechaical multitiered system ca be made o the basis of: the law of coservatio of eergy; virtual wor i couctio with the priciple d'alembert; Newto's ad Euler's equatios; Lagrage equatios; Hamilto's priciple. Accordig to this ratio as the iitial set-ow equatios of the free body motio i a vector form: m 0 W 0 + ω 0 r 0 + ω 0 ω 0 r 0 = J 0 0 + 0 J 0 0 + m 0 r 0 W 0 = M (2 ) W 0 = V 0 + 0 V 0, where m 0 и J 0 mass ad iertia tesor give to the top of the coordiate system fixed to the body X 0Y 0Z 0; F 0 и M 0 the mai vectors of exteral forces ad momets of exteral forces applied to the body accordig to the coordiate system X 0Y 0Z 0; 0 и V 0 agular ad liear velocity of top of the coordiate system X 0Y 0Z 0; r 0 radius vector of the body's mass ceter; W 0 acceleratio vector of the coordiate system X 0Y 0Z 0. The first equatio expresses the theorem quatity of motio (equatio of motio of the mass ceter). The secod equatio expresses the theorem quatity of motio relative to the coordiate system X 0Y 0Z 0 (equatio of "rotatio"). Ay mechaical system ca be decomposed ito a fiite set of solid bodies S i(i=,,n; =0,1, ). The positio of each body S i will be determied relative to the body S i-1 of this set, ad the positio ad movemet of the body S 0 will be determied relative to the iertial coordiate system XYZ. We will also assume that the body S i ca move relatively to the body S i-1 oly uder the ifluece of the force vectors F i from the body Si-1. I case of motio of the system of solid bodies, the positio of the ceter of iertia C i does ot chage correspodig to the pole O i, associated with the body S i coordiate system X iy iz i, but varies i relatio to the poles of the coordiate system with the idex other tha the «i». The positio of the ceter of iertia С О of the body S 0 relative to the iertial coordiate system XYZ is defied by the Asia Busiess Cosortium ABC-JAR Page 46
radius vector R 00 (3а), the positio of the ceter of iertia C 1 of the body S 1 relative to the XYZ ad X 0Y 0Z 0 is determied by the radius vector R 01 (3б), ad i geeral case (3c): R 00 = R 0 + R C0, a) R 01 = R 00 + R 1 + R C1 = R 0 + R C0 + R 1 + R C1 b) (3) R оп = i=0 R сп = R оп R i + R ci 1 i=0 ; R i + R ci + R п. where R 0 radius vector of the top of the coordiate system X 0Y 0Z 0 relative to the top of the coordiate system XYZ; R C0 radius vector of the ceter of the mass of the body S 0 i the X 0Y 0Z 0 coordiate system. To determie the velocity vectors of the ceters of iertia of the body relative to the appropriate coordiate system, you must perform the operatio of differetiatio with respect to time of the correspodig proectios of the radius vectors at the axis of coordiate systems, as show i 4a ad 4b i geeral case: V 00 = R 00 = V 0 + 0 R co ; а) (4) V 01 = R 01 = V 0 + V 1 + 0 R 1 + 0 + 1 R c1 ; V оп = R оп. V i = R i = V + ω l ω =i R + ω R c, where i=0, 1, 2,, N; =, +1,, N; V = R velocity vector of pole O, associated with the body S of the coordiate system X Y Z, relative to the coordiate system X -1Y - 1Z -1, attached to the body S -1. Determiatio of the acceleratio vector of ceters of iertia of bodies Si respect to the correspodig coordiate systems is similar. We ca show this i geeral case we get: W i = V i = R i = V + 2 ω l ω =i + ω l ω R l R + R c ω l ω V + ω l ω + ω R R + ω l ω ω R c, where V acceleratio vector of the pole O, of the coordiate system X Y Z attached to the body S relative to the coordiate system X -1Y -1Z -1, attached to the body S -1 proected o the axis X Y Z. The process of buildig a mathematical model of multitiered mechaical system is a sequece of operatios preseted i the form of the followig algorithm (see Fig.3). c) b) ω (5) Asia Busiess Cosortium ABC-JAR Page 47
Fig. 3 Algorithm of buildig a mathematical model of the multitiered mechaical system CONCLUSION The techique of buildig а mathematical model of arbitrary multitiered mechaical system has bee examied i this article. The relatioship betwee the compoets of the system is determied usig a graphic-vector scheme. Locatio of items i space is determied by the equatios of the motio of a free body i vector form. The algorithm of buildig a mathematical model of the multitiered mechaical system has bee also formed. The method is demostrated o the example. CONCLUSION Ahmad, M., Alam, M., & Taluder, M. (2013). No-Associativity of Loretz Trasformatio ad Associative Reciprocal Symmetric Trasformatio. Asia Joural Of Applied Sciece Ad Egieerig, 2(1), 48-56. Ozada, N., & Saeed Madai, S. (2012). A Novel Musculoseletal Joit Modelig for Orthopaedic Applicatios. ABC Joural Of Advaced Research, 1(1), 35-46. Taluder, M., & Ahmad, M. (2013). Relativistic Rule of Multiplicatio of Velocities Cosistet with Loretz Eistei Law of Additio ad Derivatio of the Missig Equatios of Special Relativity. Asia Joural Of Applied Sciece Ad Egieerig, 2(1), 26-41. Asia Busiess Cosortium ABC-JAR Page 48