Acta Polytechnica Vol. 52 No. 6/2012 Synthesis of Mechanisms by Methods of Nonlinear Dynamics Michael Valášek, Zbyněk Šika Department of Mechanics, Biomechanics and Mechatronics Faculty of Mechanical Engineering Czech Technical University in Prague Technická 4, 16607 Praha 6, Czech Republic Corresponding author: Michael.Valasek@fs.cvut.cz Abstract This paper deals with a new method for parametric kinematic synthesis of mechanisms. The traditional synthesis procedure based on collocation, correction and optimization suffers from the local minima of objective functions, usually due to the local unassembled configurations which must be overcome. The new method uses the time varying values of the synthesized dimensions of the mechanism as if the mechanism had elastic links and guidances. The time varying dimensions form the basis for an accompanying nonlinear dynamical dissipative system and the synthesis is transformed into the time evolution of this accompanying dynamical system. Its dissipativity guarantees the termination of the synthesis. The synthesis always covers the parametric kinematic synthesis, but it can be advantageously extended into the optimization of any further criteria. The main advantage of the method described here for dealing with mechanism synthesis is that it overcomes the unassembled configurations of the synthesized mechanisms and enables any further synthesis criteria to be introduced, and terminates due to dissipation of the accompanied dynamical system. Keywords: synthesis of mechanisms; time varying dimensions; evolution of dissipative systems; multi-objective optimiza- tion; dexterity; workspace; built-up space. 1 Introduction Like other engineering problems, the parametric kine- matic synthesis of mechanisms has profited from com- putational methods, e.g [1]. Traditional methods are described specifically for a particular type of mech- anisms [2–5]. General iterative procedures based on various optimization methods [1, 6–11] have been de- veloped recently. The currently used methods seem to be sufficiently powerful and able to find solutions for most problems in mechanism synthesis. However, all these methods suffer from two related problems. The first problem is that the dimensions of the mechanism that are being optimized do not allow the mechanism to be assembled in all the positions required for the desired motion. The second problem is that if some mechanism synthesis iteration fails for a certain pa- rameter because of a constraint and/or an assembly violation, the whole knowledge from this iteration is lost. A solution to the first problem has been pro- posed with the use of time-varying dimensions during the dimension iteration process for the mechanism [12]. By allowing the dimensions of the system that are treated as the design variables to vary during the motion of the mechanism, it is possible to guarantee that the system can be assembled in all configurations. This leads to a variation of each dimension during the cycle of the mechanism. The synthesis problem is then solved by attempting to minimize the deviation from the mean value for all the design variables during the cycle. A solution to the second problem is proposed in [13], where the non-assembly positions are used for the synthesis. However, this approach suffers from a slow iteration process with unclear termination. This problem is overcome by the new method described in this paper. This new approach has been described in [14–17]. Our paper formulates the method in a specialized way for a very large but restricted class of synthesis problems of mechanisms. This enables the description to be made in a more precise, systematic and algorithmic way. In addition, two interesting examples are included that have not previously been fully described [17]. 2 General formulation of the method 2.1 Traditional vector method The initial assumption is that the mechanism to be synthesized can be analyzed by the vector method (for 2D problems e.g. [18], for 3D problems e.g. [19]). This is the only reduction in generality compared to the formulation in [14–17]. Let us further describe the general procedure without loss of generality just for 2D. The mechanism to be synthesized is described by the vector method, which leads to a description of the mechanism by vector polygons with vector vertices Vi (i = 1, . . . ,n) (a simple case is shown in Fig. 1) and vectors bi (i = 1, . . . ,nV , nV ≥ n) with the parameters the lengths bi and the angles βi of the vectors (Fig. 2). 82 Acta Polytechnica Vol. 52 No. 6/2012 Figure 1: Simple vector polygon. These parameters include both the coordinates (variable parameters from bi and βi) and the dimen- sions (constant parameters from bi and βi) of the synthesized mechanism. Parameters bi and βi can therefore be split into variable coordinates sk and constant parameters pj (j = 1, · · · ,m). This is the traditional formulation of the mechanism synthesis where the dimensions being synthesized are constant values. The fundamental objective functions are typically constructed on the network of mechanism positions within the desired workspace. Let index r denote the general position of the mechanism, r = 1, 2, . . . ,N, where N is the number of such representative po- sitions, and the corresponding coordinates are sj,r. Some of the coordinates sj,r are prescribed for par- ticular positions r = 1, 2, . . . ,N. The traditional synthesis method is a search for constant parameters pj (j = 1, . . . ,m) such that the closure conditions of the vector polygons in all r (r = 1, 2, . . . ,N) positions are fulfilled [18–19]. 2.2 New method The new method is based on variation of the mecha- nism dimensions. They are no longer constant, and the time varying parameters pj,r can vary between the positions r (r = 1, 2, . . . ,N) of the mechanism. During the synthesis process it is therefore admitted p1,1 6= p1,2 6= · · · 6= p1,N, p2,1 6= p2,2 6= · · · 6= p2,N, ... pm,1 6= pm,2 6= · · · 6= pm,N (1) and the synthesis goal is to reach equality of all parameters at the end of the synthesis p1,1 ∼= p1,2 ∼= . . . ∼= p1,N, p2,1 ∼= p2,2 ∼= . . . ∼= p2,N, ... pm,1 ∼= pm,2 ∼= . . . ∼= pm,N (2) However, new coordinates are used by the new method. They are the Cartesian coordinates Figure 2: Vector parameters. xVi,r,yVi,r of the polygon vector vertices Vi (i = 1, . . . ,n), which are variable. The varying values of parameters pj,r that corre- spond to the dimensions being synthesized and that are constant in the traditional vector method can be determined from the positions of the vertices Vi in each position r. If the distance ViVi+1 corresponds to the constant length dimension of the synthesized mechanism, then its time varying value is computed in each position r and each time pi,r = bi,r = √ (xVi+1,r −xVi,r)2 + (yVi+1,r −yVi,r)2 (3) and if angle ViVi+1 with respect to the frame cor- responds to the constant length dimension of the synthesized mechanism then its time varying value is computed in each position r and each time pi,r = βi,r = atan yVi+1,r −yVi,r xVi+1,r −xVi,r . (4) The new coordinates xVi,r, yVi,r and the parame- ters pj,r are time varying and then constant at the synthesized mechanism after synthesis with the time varying values. The new coordinates xVi,r, yVi,r are the coordinates of the accompanying nonlinear dy- namical dissipative system, which is described by the Lagrange equations. Its kinetic energy is Ek = 1 2 n∑ k=1 N∑ r=1 mk(ẋ2k,r + ẏ 2 k,r), (5) where mk are artificially introduced masses, and its potential energy is Ep = 1 2 n∑ k=1 N∑ r=1 kk N∑ i=1 (pk,r −pk,i)2, (6) where kk are artificially introduced stiffnesses. The potential energy describes the excitation of the new dy- namic system whenever parameters pk,r are not equal to each other between the positions r = 1, . . . ,N. The dissipation is introduced by the Raleigh function D = 1 2 n∑ k=1 N∑ r=1 Bk(ẋ2k,r + ẏ 2 k,r), (7) 83 Acta Polytechnica Vol. 52 No. 6/2012 222 222 222 222 222 )()(e )()(d )()(c )()(b )()(a CiMiCiMi DiMiDiMi DiCiDiCi CiBiCiBi AiDiAiDi yyxx yyxx yyxx yyxx yyxx      (14). The optimization task is defined as follows     min 2 1 2 2 1 1       n i MiMi n i MiMi y'yq x'xqCF (15) A[xA,yA] B[xB,yB] D C M a c b d e β1 β2 β3 k β4 β5 Fig. 3: Four-bar mechanism and the set of constraints of the optimization task is 0; 1,1   i1i1i1i  (16), where the optimization parameters are: a, b, c, d, e, xA, yA, xB, yB and 1i, (i=1,2,…,n). The parameters qi again denote the penalization coefficients. Figure 3: Four-bar mechanism. where Bk are artificially introduced damping coef- ficients. Dissipation guarantees the removal of the energy from the new dynamic system, and thus brings the system into equilibrium. The synthesis process is now transformed into the evolution of the accompanying nonlinear dissipative dynamical system. The system has the coordinates xVi,r,yVi,r, the kinetic energy (5), the potential en- ergy (6) where the variables are described by formu- las (3)–(4) as functions of the coordinates xVi,r,yVi,r, the Raleigh function (7) and the initial conditions xVi,r(0),yVi,r(0) as estimations of the positions of the mechanism described by the vector polygon vertices i = 1, . . . ,n in particular positions r = 1, . . . ,N. It is supposed that this new accompanying system reaches its equilibrium given by Ek = 0, Ep = 0 (8) and Ek = 0 results from (5) into ẋk,r = 0, ẏk,r = 0 (9) and Ep = 0 results from (6) into pk,r = pk,i. (10) These final values pk = pk,r are the synthesized param- eters (dimensions) describing the synthesized mecha- nism [14–17]. 3 Extension of the method The method described here deals just with the para- metric positional synthesis of a mechanism. The syn- thesis is usually more complicated, and it can gener- ally be described as the minimization/ maximization of the set of further objective functions{ min CF`(s1,s2, . . . ,sn,p1,p2, . . . ,pm), ` = 1, 2, . . . ,nCF. (11) These objective functions are taken into consideration by extending the potential energy (6) by new terms Ep = · · · + 1 2 nCF∑ `=1 q`(CF` − CF D,`), (12) The associated dynamical dissipative system consists of n subsystems for the individual positions of point M (Fig. 4). The masses mA, mB, mD, mE are introduced in the points Ai, Bi, Di, Ei. The interactions between the subsystems are ensured by forces of a linear spring nature. A nonzero force acts into the relevant masses whenever the corresponding dimension differs between subsystems i and j (i,j = 1,2,….n). The stabilization of the whole system is ensured by damper elements between the masses and the inertial frame. The constraints of the associated system were changed into the form prescribedconsty prescribedconstx yyxx yyxx yyxx yyxx yyxx M M CiMiCiMi DiMiDiMi DiCiDiCi CiBiCiBi AiDiAiDi        i i 222 i 222 i 222 i 222 i 222 i )()(e )()(d )()(c )()(b )()(a (17). Fig. 4: Associated dissipative system of the four-bar me Di Fdi Mai FAyi FByi FBxi Fei Fei Fbi Fbi Fai Fci Fci Fai Ai Mi di ei Bi Ci ai bi ci x y FAxi Fdi bxA chanism Figure 4: Associated dissipative system of the four- bar mechanism. where q` are chosen positive constants and CF D,` are desired values of the objective function CF`. Then the equilibrium (8) of the new accompanying dynamic system also optimizes the objective functions (11) [15–17]. The artificially introduced parameters mk, kk, Bk and q` can be chosen as arbitrarily positive numbers, but their values influence the dynamics of the synthesis. 4 Planar example The main disadvantages of general optimization meth- ods combined with traditional kinematical synthesis are a high computational cost, extreme growth of the computational complexity with the number of opti- mized parameters, together with inability to find the solution even though it exists. The following example shows the main advantage of the evolution of the as- sociated dissipative system over general optimization methods, e.g. genetic algorithms. The comparison of the methods is focused on the synthesis of the kinematical system (a four-bar mecha- nism) from [12]. This mechanism was chosen because it is relatively simple and it simultaneously consists of 9 dimensional parameters. Moreover, the example can easily be extended with other optimized parameters, e.g. angles of the crank. The original classical optimization formulation of the example (Fig. 3) is as follows. The kinematical system has 9 + n dimensions to be synthesized, where n denotes the number of desired positions of the end effector. These dimensions are the length of the crank a, the length of the coupler c, the length of the follower b, and the lengths of the two rods d and e that are connected with the end effector. The other set of optimized dimensions are the x and y positions of the fixed part of the crank (xA,yA) and the follower (xB,yB). The last set of parameters to be optimized 84 Acta Polytechnica Vol. 52 No. 6/2012 Figure 5: Evolution of the coordinates of a four-bar mechanism. is the positive increments of the angles β1i of the crank β1i = β11 + i∑ j=2 β1j. (13) The point M of the mechanism should pass through the given positions Mi (i = 1, 2, . . . ,n) on the given trajectory. The coordinates of the mechanism are β1,β2 and β3. The system constraints are then a2 = (xDi −xAi) 2 + (yDi −yAi) 2, b2 = (xBi −xCi) 2 + (yBi −yCi) 2, c2 = (xCi −xDi) 2 + (yCi −yDi) 2, d2 = (xMi −xDi) 2 + (yMi −yDi) 2, e2 = (xMi −xCi) 2 + (yMi −yCi) 2. (14) The optimization task is defined as follows CF = q1 n∑ i=1 (xMi −x′Mi) 2 + q2 n∑ i=1 (yMi −y′Mi) 2 → min (15) and the set of constraints of the optimization task is β1(i+1) > β1i (16) where the optimization parameters are: a, b, c, d, e, xA, yA, xB, yB and β1i, (i = 1, 2, . . . ,n). The param- eters qi again denote the penalization coefficients. The associated dynamical dissipative system con- sists of n subsystems for the individual positions of point M (Fig. 4). The masses mA, mB, mD, mE are introduced in the points Ai, Bi, Di, Ei. The interactions between the subsystems are ensured by forces of a linear spring nature. A nonzero force acts into the relevant masses whenever the correspond- ing dimension differs between subsystems i and j (i,j = 1, 2, . . . ,n). The stabilization of the whole system is ensured by damper elements between the masses and the inertial frame. The constraints of the associated system were changed into the form a2i = (xDi −xAi) 2 + (yDi −yAi) 2, b2i = (xBi −xCi) 2 + (yBi −yCi) 2, c2i = (xCi −xDi) 2 + (yCi −yDi) 2, d2i = (xMi −xDi) 2 + (yMi −yDi) 2, (17) e2i = (xMi −xCi) 2 + (yMi −yCi) 2, xMi = const. (prescribed), yMi = const. (prescribed). The forces that act in the dynamical system are as follows FAxi = n∑ j=1 kxA(xAi −xAj), 85 Acta Polytechnica Vol. 52 No. 6/2012 Figure 6: Evolution of the dimensions of a four-bar mechanism. FAyi = n∑ j=1 kyA(yAi −yAj), FBxi = n∑ j=1 kxB(xBi −xBj), FByi = n∑ j=1 kyB(yBi −yBj), Fai = n∑ j=1 ka(ai −aj), (18) Fbi = n∑ j=1 kb(bi − bj), Fci = n∑ j=1 kc(ci − cj), Fdi = n∑ j=1 kd(di −dj), Fei = n∑ j=1 ke(ei −ej), Mai = kMi(βi−1 −βi), where for each mechanism position i the equations take into account all the other forces connecting with the other positions j of the mechanism. Therefore the system forms altogether 2n equations. The coef- ficient kMi takes the nonzero constant value only if β1(i+1) < β1i. The final dynamical equations for the mass particles in points B and the algebraic equations that must be fulfilled are as follows mAẍAi = − n∑ j=1 kxA(xAi −xAj) + n∑ j=1 ka(ai −aj) cos β1i − bxAẋAi, mAÿAi = − n∑ j=1 kyA(yAi −yAj) + n∑ j=1 ka(ai −aj) sin β1i − byAẏAi, mBẍBi = − n∑ j=1 kxB(xBi −xBj) − n∑ j=1 kb(bi − bj) cos β3i − bxBẋBi, 86 Acta Polytechnica Vol. 52 No. 6/2012 Figure 7: Evolution of the trajectories of a four-bar mechanism by the dissipative system. mBÿBi = − n∑ j=1 kyB(yBi −yBj) − n∑ j=1 kb(bi − bj) sin β3i − byBẏBi, mCẍCi = n∑ j=1 kb(bi − bj) cos β3i − n∑ j=1 kc(ci − cj) cos β2i + n∑ j=1 ke(ei −ej) cos β5i − bxCẋCi, (19) mCÿCi = n∑ j=1 kb(bi − bj) sin β3i − n∑ j=1 kc(ci − cj) sin β2i + n∑ j=1 ke(ei −ej) sin β5i − bxCẋCi, mDẍDi = − n∑ j=1 ka(ai −aj) cos β1i + n∑ j=1 kc(ci − cj) cos β2i + n∑ j=1 kd(di −dj) cos β4i − bxDẋDi, mDÿDi = − n∑ j=1 ka(ai −aj) sin β1i + n∑ j=1 kc(ci − cj) sin β2i + n∑ j=1 kd(di −dj) sin β4i − byDẏDi where the mechanism dimensions ai, bi, ci, di and ei are evaluated from the coordinates with the help of the constraint equations formulated in (17). The simulation started from some selected initial positions xA, yA, xB, yB, xC, yC, xD and yD, as was in general described in Section 2, above. The initial positions also determine the initial dimensions a, b, c, d and e of the mechanism. The results of the simulation are presented in the following figures. Fig. 5 and Fig. 6 show the history of the system coordinates. Fig. 7 presents the desired trajectories and the resulting trajectories. The system coordinates (xAi, yAi, xBi, yBi, ai, bi, ci, di, ei, i = 1, 2, . . . ,n) for all the subsystems (for all the positions) come to rest at the equilibrium values. The desired trajectory was reached using one reacti- vation of the dynamic process. All the coordinates for all the subsystems also come to rest at the equilibrium values. These equilibrium values can be interpreted as the searched parameters of the mechanism. The example was also simulated using genetic algo- rithms. The boundary conditions for each optimized 87 Acta Polytechnica Vol. 52 No. 6/2012 Figure 8: Evolution of the trajectories of a four-bar mechanism using a genetic algorithm. dimensional parameter were set according to the inter- val 〈desired, resulting〉 value of the simulation, which was done by the dissipative system. This interval was extended by 50 % on both sides, and was used as the boundary condition for the particular optimized parameter. The simulation result is presented in Fig. 8. It is shown that the desired trajectory was not found using this method. 5 Spatial example A further example is the synthesis of a 3D (RSSR) four-bar mechanism [17]. The original classical optimization formulation is presented in Fig. 9. It consists of two skew mechanism axes. The first mechanism axis is identical with the coordinate y axis. The second mechanism is shifted to level zA and rotated by angle βA round the z axis. This means that the vector of the axis of the second mechanisms still lies on the plane xy. There is a sleeve on the axis of each mechanism. There is a lever on each sleeve. There are spherical linkages on the other side of the levers. At the end, the spherical linkages are connected together by a pitman. The overall mechanism is naturally described by the height zA of point A, the length of the `A axis, the angle of rotation βA, the length of lever `AB, the length of pitman `BD, the length of lever `CD and the length of the axis yC = yD. Let us reformulate the description of the mechanism for conciseness, and in order to avoid the complicated equations in the following formulation. The mechanism can then be described by the posi- tions xA, yA, zA of the point A, by the lengths of the levers `AB and `CD, by the length of the pitman `BD and the length of the yD axis. Taking the relevant constraints into account, the representation described here is the minimal representation of the mechanism. The constraints that describe the mechanism are as follows. The first constraint ensures that points A and B reflect angle ψ for each subsystem. The following constraint ensures that points C and D reflect angle ϕ for each configuration of the mecha- nism. The third constraint describes the given length of the lever `AB, and ensures that it is constant across the configurations. The last constraint en- sures that vector [xA,yA, 0] is perpendicular to vector BA ([xB,yB,zB]−[xA,yA,zA]) for each configuration (subsystem). The constraints are cos ψ = zA −zB√ (xA −xB)2 + (yA −yB)2 + (zA −zB)2 , cos ϕ = xD x2D + z 2 D , (20) `AB = √ (xA −xB)2 + (yA −yB)2 + (zA −zB)2, 0 = xA(xB −xA) + yA(yB −yA). Synthesis of the transmission of the RSSR mech- anism is a very good example in the sense that in this case the described representation could at the same time mean the set of synthesized parameters. 88 Acta Polytechnica Vol. 52 No. 6/2012 The constraints that describe the mechanism are as follows. The first constraint ensures that points A and B reflect angle ψ for each subsystem. The following constraint ensures that points C and D reflect angle  for each configuration of the mechanism. The third constraint describes the given length of the lever lAB, and ensures that it is constant across the configurations. The last constraint ensures that vector [xA,yA,0] is perpendicular to vector BA ([xB,yB,zB]-[xA,yA,zA]) for each configuration (subsystem). The constraints are φi yC = yD lAB A [xA, yA, zA] C D B lD y x z zA lA lBD  ψi )()(0 )()()( )cos( )()()( )cos( 222 22 222 ABAABA ABABABAB DD D ABABAB AB yyyxxx zzyyxxl zx x zzyyxx zz          . (20) Synthesis of the transmission of the RSSR mechanism is a very good example in the sense that in this case the described representation could at the same time mean the set of synthesized parameters. Because it is a synthesis of transmission with two levers, a global solution of such an example exists and it consists in lengths of the levelers equal to zero. However, a solution of this kind is absolutely unimportant for the practical usability of the synthesized mechanism. Let us therefore choose the length of the lever lAB equal to a nonzero constant. The rest of the parameters can then be synthesized. They are once more: xA, yA, zA, lBD, lCD, yD. Figure 9: RSSR spatial mechanism. Because it is a synthesis of transmission with two levers, a global solution of such an example exists and it consists in lengths of the levelers equal to zero. However, a solution of this kind is absolutely unim- portant for the practical usability of the synthesized mechanism. Let us therefore choose the length of the lever `AB equal to a nonzero constant. The rest of the parameters can then be synthesized. They are once more: xA, yA, zA, `BD, `CD, yD. The task of transmission synthesis is to find dimen- sions of the mechanism that fulfill some transmission requirements. The requirement for this example is that the vector of the angles of lever ϕ should corre- spond with the vector of angles of lever ψ. This means that for ϕ = ϕ1 is ψ = ψ1, for ϕ = ϕ2 is ψ = ψ2, etc., for the constant dimensions of the mechanism. To sum up, the optimization task is as follows: F = n∑ i=1 (ϕi −ϕ′i) 2 → min, (21) where the optimization parameters are: xA, yA, zA, `BD, `CD and yD and where ψi is given. Parameters qi again denote the penalization coefficients. The associated dynamical dissipative system con- sists of n subsystems for the individual required po- sitions of the transmission mechanism. The masses mAi, mBi, mDi are introduced at points Ai, Bi, Di and act in the coordinates x, y, z. The interactions between the subsystems are ensured by forces of a linear spring nature. A nonzero force acts into the relevant masses when- ever the corresponding dimension differs between sub- systems i and j (i,j = 1, 2, . . . ,n). Stabilization of the whole system is ensured by the damper elements between the masses and the inertial frame, according to the sky-hook idea [20] or [21]. The idea of the transformation is presented in Fig. 10. In spite of the simple formulation of this exam- ple, only using three mass points, the 3D examples with difficult constraints drive us to formulate the dynamical equations by means of Lagrange equations of mixed type. The greatest difference that occurs in comparison with simple planar structures is the difficulty of for- mulating the dynamical system. With simple planar mechanisms, the dynamical equation in direct form can best be formulated using the Newton equations. The dynamical equations can be defined according to the formulation of the constraints (20). For the sake of conciseness, let us define the lengths of the lever `CD and the pitman `BD, and formulate the x, y, z projections of vectors [xD,yD,zD] − [xB,yB,zB] and [xD,yD,zD] − [0,yC, 0]. The lengths are: `DC = √ (xD − 0)2 + (yD −yC)2 + (zD − 0)2, (22) `BD = √ (xD −xB)2 + (yD −yB)2 + (zD −zB)2, The vectors are vecxDB = xB −xB√ (xD−xB)2 + (yD−yB)2 + (zD−zB)2 , vecyDB = yB −yB√ (xD−xB)2 + (yD−yB)2 + (zD−zB)2 , veczDB = zB −zB√ (xD−xB)2 + (yD−yB)2 + (zD−zB)2 , vecxDC = xD√ (xD − 0)2 + (yD −yC)2 + (zD − 0)2 , vecyDC = 0, (23) veczDC = zD√ (xD − 0)2 + (yD −yC)2 + (zD − 0)2 . 89 Acta Polytechnica Vol. 52 No. 6/2012 The task of transmission synthesis is to find dimensions of the mechanism that fulfill some transmission requirements. The requirement for this example is that the vector of the angles of lever  should correspond with the vector of angles of lever ψ. This means that for  = 1, is ψ = ψ1, for  = 2, ψ = ψ2, etc., for the constant dimensions of the mechanism. To sum up, the optimization task is as follows:   min 2 1   n i ii 'F  , (21) where the optimization parameters are: xA, yA, zA, lBD, lCD and yD and where ψi is given. Parameters qi again denote the penalization coefficients. The associated dynamical dissipative system consists of n subsystems for the individual required positions of the transmission mechanism. The masses mAi, mBi, mDi are introduced at points Ai, Bi, Di and act in the coordinates x,y,z. The interactions between the subsystems are ensured by forces of a linear spring nature. A nonzero force acts into the relevant masses whenever the corresponding dimension differs between subsystems i and j (i,j = 1,2,….n). Stabilization of the whole system is ensured by the damper elements between the masses and the inertial frame, according to the sky-hook idea [20] or [21]. The idea of the transformation is presented in Fig. 10. In spite of the simple formulation of this example, only using three mass points, the 3D examples with difficult constraints drive us to formulate the dynamical equations by means of Lagrange equations of mixed type. The greatest difference that occurs in comparison with simple planar structures is the difficulty of formulating the dynamical system. With simple planar mechanisms, the dynamical equation in direct form can best be formulated using the Newton equations. mAi ψi mBi C FlDCi mDi φi y x z FxAi FlDBi FyAi BzAi ByAi BxAi ByBi BxAi BzAi FlDBi BzDi FyDi ByDi BxDi Fig. 10: Associated system of the RSSR mechanism Figure 10: Associated system of the RSSR mechanism. The lengths (22) and the vectors (23) help in for- mulating the forces that act in the dynamical system. In addition, the constraints (20) have been changed into the form cos ψ = zB −zA√ (xB −xA)2 + (yB −yA)2 + (zB −zA)2 , cos ϕ = xD√ x2D + z 2 D , (24) `AB = √ (xB −xA)2 + (yB −yA)2 + (zB −zA)2, 0 = xA(xB −xA) + yA(yB −yA), ϕi = const., ψi = const. In order to describe the general transmission syn- thesis of this mechanism, let us take i = 1, 2, . . . ,n required transmissions and thus n required subsystems of the dynamical system. The dynamical equations and the constraints are identical for each subsystem. The forces that act in dynamical subsystem i are FxAi = n∑ j=1 k(xAi −xAj), FyAi = n∑ j=1 k(yAi −yAj), FzAi = n∑ j=1 k(zAi −zAj), F`DBi = n∑ j=1 k(`DBi − `DBj), (25) F`DCi = n∑ j=1 k(`DCi − `DCj), FyDi = n∑ j=1 k(yDi −yDj), BxAi =bẋAi, ByAi = bẏAi, BzAi = bżAi, BxBi =bẋBi, ByBi = bẏBi, BzBi = bżBi, BxCi =bẋCi, ByCi = bẏCi, BzCi = bżCi. This means that for each mechanism position i the equations take into account all the other forces con- necting with the other positions j of the mechanism. Therefore the system forms altogether 2n equations. The mechanism dimensions are evaluated from the coordinates as follows `DBi = √ (xDi−xBi)2 + (yDi−yBi)2 + (zDi−zBi)2, `Di = √ x2Di + z 2 Di. (26) The final dynamical equations for the mass particles in points A, B and D for dynamical subsystem i, together with the algebraic equations, are as follows mAẍAi = −FxAi −BxAi, mAÿAi = −FyAi −ByAi, mAz̈Ai = −FzAi −BzAi, mBẍBi = F`DBivecxDB −BxBi, mBÿBi = F`DBivecyDB −ByBi, mBz̈Bi = F`DBiveczDB −BzBi, (27) mDẍDi = −F`DBivecxDB −F`DCivecxDC −BxBi, mDÿDi = −F`DBivecyDB −FyDi −ByBi, mDz̈Di = −F`DBiveczDB −F`DCiveczDC −BzBi, ϕi = const., ψi = const., where the integrated coordinates are ẍAi, ÿAi, z̈Ai, ẍBi, ÿBi, z̈Bi, ẍCi, ÿCi, z̈Ci. The simulation started from some randomly gen- erated initial positions xA, yA, zA, xB, yB, zB, xD, 90 Acta Polytechnica Vol. 52 No. 6/2012 Figure 11: Dynamical response of the dimensions of the RSSR mechanism. yD, zD. The initial positions also determine the ini- tial dimensions xA, yA, zA, `BD, `DC, yD, of the mechanism. It is necessary to make a short note on the implementation. All the systems that have been simulated are easy in the context of the formulation. All the previous dynamical systems were thus for- mulated by Newton equations. Because this system is already quite complex, Lagrange equations of mixed type were chosen, instead of the Newton equations, as the optimal formulation tool. Thus the formulation was really simplified. However, the great disadvantage of this tool is the instability of the constraints. The results of the simulation are presented in the fol- lowing figures. The system coordinates xAi, yAi, zAi, `BDi, `DCi, yDi, (i = 1, 2, . . . , 8) for all the subsys- tems come to rest at the equilibrium values (Fig. 11). These equilibrium values can be interpreted as the searched parameters of the mechanism. The evolution of the constraints is presented in Fig. 12. The red lines show fulfilled constraints. The other lines represent the simulated values of the mech- anism. The upper part is dedicated to the given angles ψ and ϕ. The simulated values of the angles come to rest at the given equilibrium very soon after the system starts. The lower left picture shows the constraint that ensures the length of the lever `AB. It can be seen that the given value had been set equal to one. Here, too, the simulated value came to rest at the required value immediately after the system starts. The fourth history presents the perpendicularity of vectors [xA,yA, 0] and BA. This graph, too, shows how soon after the start the constraint condition is fulfilled. It is absolutely clear from all the pictures showing the evolution of the constraint conditions that the required constraints are perfectly fulfilled. The evolution of the whole structure of the space four-bar mechanism is presented in Fig. 13. The simu- lation started from the random positions of the struc- ture marked as the initial structure. The simulation finished in the final structure of the mechanism. The final image shows that the corresponding dimensions are equal. 6 Conclusion This paper has described a new method for solving the parametric kinematical synthesis of mechanisms. The robustness and the rapid synthesis procedure of the method have been proven. Especially the robustness is very valuable. Acknowledgements The authors appreciate kind support from the grant MSM6840770003 “Algorithms for computer simula- tion and application in engineering”. 91 Acta Polytechnica Vol. 52 No. 6/2012 Figure 12: Fulfilling the constraints of the RSSR mechanism. 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