Motion of mechanical systems with non-Ideal constraints

Volume 35 Number 2 2 Vietnam Journal of Mechanics, VAST, Vol. 35, No. 2 (2013), pp. 157 – 167 MOTION OF MECHANICAL SYSTEMS WITH NON-IDEAL CONSTRAINTS Do Sanh1,∗, Dinh Van Phong1, Do Dang Khoa2 1Hanoi University of Science and Technology, Vietnam 2Medisend International, USA E-mail: ∗dosanhbka@gmail.com Abstract. In the paper a new method for mechanical systems with non-ideal constraints is presented. It is proved that a mechanical system subjected to physical non-ideal con- straints cannot be determined purely by theoretical analysis because the reaction forces depend on the physical parameters of interactive environment, which are identified only by measurement. The principle of compatibility is shown to be an effective tool in com- bination with experience to investigate such a problem. For illustration the dynamics of a digging machine is investigated. Keywords: The principle of compatibility, non-ideal constraints, physical constraints, interactive environment, method of transmission matrix. 1. INTRODUCTION The motion of mechanical systems under both physical and program constraints is paid more and more attention in technology and in automatic control industry. In the physical constraints, both the ideal and non-ideal constraints are attracted a lot of interests but only ideal constraints are considered. The non-ideal constraints are rarely discussed due to the fact that it is impossible to define the work (of virtual displacements) of the reaction forces corresponding to the non-ideal constraints. Therefore, the corresponding generalized forces cannot be determined. Up to now only the Coulomb friction constraints are considered as a particular case of non-ideal constraints which were mentioned in papers [4, 9-13]. However, non-ideal physical constraints are often met in practical applications, for example in machine tools (milling machine, lathe, planer, honing machine, etc.) or front loaders, and scoop loaders. Working surfaces created by machine tools or tunnels and ditches dug by loaders are not often considered as ideal constraints but non-ideal con- straints. The active forces (cutting or digging forces), which are not tangent to the con- straint hyperplanes need to meet the problem’s requirements. Thus, the constraints in those cases must be considered as non-ideal. For general non-ideal constraints such as the ones in metal fabricating machines, excavating machines, the active forces can only be measured by experiments and given in engineering notebooks [12]. 158 Do Sanh, Dinh Van Phong, Do Dang Khoa This paper presents a new method to analyse mechanical systems with non-ideal physical constraints and apply to the dynamics analysis of digging machines. Furthermore, this method can be extended for the cases of friction and program constraints. 2. EQUATIONS OF MOTION OF A MECHANICAL SYSTEM WITH CONSTRAINTS Let consider an unconstrained mechanical system of N particlesMk of mass mk. Let denote the position vector of the particle Mk and applied force acting to the particle Mk by ~rk(xk, yk, zk) and ~Fk( ~Fkx, ~Fky, ~Fkz), (k = 1, N), respectively. Suppose the constraints imposed on the considered system are of the form fα(xk, yk, zk) = 0;α = 1, r. (1) Let denote the reaction forces acting to the particle Mk by ~Rk(Xk, Yk, Zk). In the case of ideal constraints, the reaction forces are directed to the normal direction of the hypersurface (1) at the contact point. In the contrary case, i.e. the case of non-ideal con- straints, the reaction forces include the tangent components (the tangent and bi-tangent components), which are called the friction forces. The physical substance of these forces is very complicate, only one fact is known: they give negative works over virtual dis- placements. Let now consider the motion of the considered system in generalized coordi- nates. Suppose that the position of the system is defined by the redundant coordinates qj(j = 1, m). The constraint Eqs. (1) are of the form now fα(q1, q2, . . . , qm) = 0;α = 1, r. (2) Let the kinetic energy of the system be of the form T = 1 2 m∑ i,j=1 aij q˙iq˙j . In the matrix form, the kinetic energy is written as follows T = 1 2 q˙TAq˙T . (3) Where A = [aij ] is an (m×m) nonsingular symmetrical matrix, but q˙ = [ q˙1 q˙2 . . . q˙m ]T . The symbol T denotes the transpose of matrix. Analogously, the following matrix notation is introduced q¨ = [ q¨1 q¨2 . . . q¨m ]T , Q = [ Q1 Q2 . . . Qm ]T . The constraint Eqs. (2) in the matrix form can be written as follows f q¨+ f0= 0. (4) Motion of mechanical systems with non-ideal constraints 159 Where f and f0 are the r ×m and (r× 1) matrices respectively: f = [fαi] ; fαi = ∂fα ∂qi ; f0 =   m∑ i,j=1 ∂2fα ∂qi∂qj   ;α = 1, r; i = 1, m. (5) From now on a vector is treated as a matrix. By the Principle of Compatibility the motion equations of the constrained system are written in the form [3-7] Aq¨ = Q +Q 0 −Q∗+R, (6) where Qi = − ∂pi ∂qi + Q¯i; i = 1, m (7) pi - the potential energy of the system Q¯i - the generalized forces of non-potential forces, but Q0 = [ Q01 Q 0 2 . . . Q 0 m ] ;Q0i = 1 2 q˙T∂iAq˙;Q ∗ = m∑ i=1 ∂iAq˙ ∗ i ; ∂iA =   ∂a11 ∂qi ∂a12 ∂qi . . . ∂a1m ∂qi ∂a12 ∂qi ∂a22 ∂qi . . . ∂a2m ∂qi ... ... . . . ... ∂a1m ∂qi ∂a2m ∂qi . . . ∂amm ∂qi   ; q˙∗i =   q˙1q˙i q˙2q˙i ... q˙mq˙i   . (8) R - the generalized reaction forces corresponding to the i - generalized coordinate. By the Principle of Compatibility [3-7] the reaction R has to satisfy the equation FR+ F0 = 0. (9) Where F = fA−1;F0 = F(Q+Q0 −Q∗) + f0. (10) By such a way we obtain r algebraic equations containing m unknowns Ri(r < m). In order to determine the reactions Ri(r < m) it is necessary to fill up k = (m − r) equations containing only m unknowns, Ri(r < m) which together with r Eqs. (9) yield a complete set of equations of unknowns Ri(i = 1, m), where A −1 is the inverse matrix of the inertia matrix A. 3. THE IDEALITY AND NON-IDEALITY OF CONSTRAINTS 3.1. The ideality of constraints As known, the motion of the system with the constraints (4) is described by the Eqs. (6), where the reactions Ri(r < m) are determined by the Eqs. (9). However, the obtain system of equations is not complete yet. The problem will be solved in the case of the class of constraints, so-called ideal constraints, which can be defined by the axiom of ideality by Przeborski-Appell-Chetaev [4, 5]. 160 Do Sanh, Dinh Van Phong, Do Dang Khoa By this axiom we have [3-5] DR = 0. (11) Where the (k × m) matrix D consists of the elements, which are the coefficients in term of the expressions of generalized accelerations represented through independent generalized accelerations. In the other words, calculating the generalized accelerations q¨j(j = 1, m) from the Eqs. (2) we obtain q¨j = k=m−r∑ σ=1 dσj q¨σ + . . . ; (j = 1, m), where the non-written terms are the terms which do not include the generalized accelera- tions. The (k×m) matrix D D = [dσi] ; (i = 1, m; σ = 1, k = m− r), (12) can be determined by either analytical or by numerical algorithm [2]. We obtain a closed set of m algebraic equations containingm unknowns . By solving these equations we get the reaction forces Ri(i = 1, n), which are the functions of the generalized coordinates and velocities as R = R(q, q˙). (13) By integrating the Eqs. (6), where the reactions R determined from (9) and (11), we determine the motion of the considered system. Note that in this process the reaction force R = R(q, q˙) is calculated independently of integration of the differential Eqs. (6). In general, the motion of the system with ideal constraints is defined by means of 2m Eqs. (6), (9) and (11) and by means of m algebraic equations (9) with (11) and it is possible to compute separately the reactions Ri(qi, q˙i) with respect to the equations of motion (6). However, it is possible to define the motion of the system only by r Eqs. (2) and (m− r) differential equations DAq¨ = D(Q+Q0−Q∗). (14) In the other, the motion of the system with ideality constraints is determined by a set m differential algebraic Eqs. (2) and (14). By such a way the size of the problem is decreased by half. However, in the case of non-ideal constraints, the condition (11) is not satisfied. Thus, when the constraints are non-ideal, we have DR 6= 0. (15) 3.2. The motion of the system with non-ideal constraints Let consider the system with constraints, which are not to satisfy the condition (11). Such a constraint is called the non-ideal one and the system restricted by such constraints is called the system with non-ideal constraints. By the principle of compatibility [3-7] the equations of such a system are written in the form (6), in which the reactions of the constraints satisfy the Eqs. (9). In this case, the Motion of mechanical systems with non-ideal constraints 161 condition (11) is not occurred. Instead of that we have the condition (15). In the particular case of physical constraints, the reactions of constraints spend work, we have DR < 0. (16) This character will help to the investigation of motion of the systems with the non- ideal constraints. By such a way, we obtain (m+r) Eqs. (4) and (6) including 2m unknowns (qi, Ri); i = 1, m (2m > m+r). In other words, the motion of the considered system is not defined yet, which depends on the substance of the constraints restricted to the considered system. In the case of the physical constraints the condition (16) is arisen by acting one another between the system and constraints. It is important that these actions depend on the physic-mechanical characteristics of the environment. In other words, the reactions Ri(i = 1, n) are defined by experiments only. In connection with a non-ideal system, let write the Eqs. (4) and (6) in the form respectively Aq¨ = − ∂pi ∂q + Q¯+Q 0 −Q∗+R, (17) FQ¯ +F(− ∂pi ∂q +Q0−Q∗ +R) + f0 = 0, (18) where the reaction R can be determined by measuring machinery, based on the results of measured forces of environment acting on the system. For this purpose let us denote the components of the force acting from the constraints to the considered system at the contact pointMk(xk, yk, zk) by (Xk, Yk, Zk). Let introduce the following notations R (xyz) k = [ Xk Yk Zk ]T ; ∂irk = [ ∂xk ∂qi ∂yk ∂qi ∂zk ∂qi ] (19) The reaction R in the Eq. (17) is written as follows R = [ R1 R2 . . . Rm ]T ; Ri = N∑ k=1 ∂irkR (xyz) k ; i = 1, m (20) From the measuring result (Xk, Yk, Zk), the reaction R can be calculated by means of the formulas (20). Next, by the formula (18) it is possible to compute the drive force Q¯ for the system realizing the non-ideal constraints (2). After calculating the driven force Q¯, the motion of the system with non-ideal constraints (2) is defined by integrating the differential Eq. (17). The algorithm for solving the system of equation of motion is following [2]: – Determining the element Xk,Yk, Zk by measurement – Calculating Ri, i = 1, . . . , m by (20) – Determining Q¯ – Determining drive forces – Integration of equations of motion (17) – Repeating for next time step. 162 Do Sanh, Dinh Van Phong, Do Dang Khoa In the case the constraint is the surface or the curve, the reaction forces can be expressed by components in the axes of a moving trihedral composed of a tangent, a normal, and a bi-normal to the trajectory. Its origin moves along the trajectory. The components of the reaction in the coordinate system (Mxyz) can be expressed in term of its component in the (Mtnb) coordinate axes by means of the transmission matrix V [8], where Mt is oriented by the tangent direction, Mn - by the normal direction, but Mb - by the bi-normal direction. By such a way, we have R(Mxyz) = VR(Mtnb). (21) The Eq. (18) is written as follows FQ¯+ F(− ∂pi ∂q +Q0−Q∗ +VR(Mtnb)) + f0 = 0. (22) By this equation we can compute the driven force Q¯ corresponding to R(Mtn). It is noticed that from (21) we have R(Mtnb) = V−1R(Oxyz), (23) where V−1 is the inverse matrix of the transmission matrix V. 4. EXAMPLE Determine the drive moments M1,M2 of the servomotor acting on the links of the digging machine as Fig. 1. The length of the link OA and AB is equal l1 and l2 respectively and joined by the revolute joints. For simplicity let take l1 = l; l2 = βl The mass of the links are neglected, the ditching scoop of the mass m is treated as a particle with the body coordinate (a, 0) and the global coordinate (Bxy). The scoop is jointed to the end point B of the link AB. 0 A B 1 j 2 j a a X Y x y 1M 2 M Fig. 1. Close-loop two link model of the digging machine Suppose that the work trajectory of the scoop is of the form f ≡ x− y − a = 0. (24) Motion of mechanical systems with non-ideal constraints 163 Let choose the generalized coordinates by ϕ1, ϕ2, where ϕ1 is the angle of the link OA with respect to the fixed horizontal axis Oy, but the ϕ2 - the angles between links BA and OA. The transmission matrices are of the form t1 =   c1 −s1 0s1 c1 0 0 0 1   ; t11 =   −s1 −c1 0c1 −s1 0 0 0 0   ; t2 =   c2 −s2 ls2 c2 0 0 0 1   ; t21 =   −s2 −c2 0c2 −s2 0 0 0 0   ; r =   −βl0 1   ; t12 =   −c1 s1 0−s1 −c1 0 0 0 0   ; t22 =   −c2 s2 0−s2 −c2 0 0 0 0   . Where si ≡ sinϕi; ci ≡ cosϕi; ti1 is the matrix, its elements of which are the first derivatives of the elements of the matrix ti with respect to the variable ϕi, but ti2 - the second derivatives of the elements of the matrix [8]. The elements of the (2× 2) matrix of inertia of the manipulator A are as follows [8] a11 = mr T t2 T t11 T t11t2r = ml 2(1 + 2β cosϕ2 + β 2); a22 = mr T tT21t T 1 t1t21r = mβl 2; a12 = a21 = mr T tT21t T 1 t11t2r = mβl 2(β + cosϕ2). (25) The inverse matrix of the inertia matrix will be A−1 =   1 ml2 sin2 ϕ2 − (β + cosϕ2) mβl2 sin2 ϕ2 − (β + cosϕ2) mβl2 sin2 ϕ2 (1 + 2βcosϕ2 + β 2) mβl2 sin2 ϕ2   . (26) The potential energy is of the form pi = −mgl[ sinϕ1 − β sin(ϕ1 + ϕ2)] − ∂pi ∂ϕ1 = mgl[cosϕ1 − β cos(ϕ1 + ϕ2)]; − ∂pi ∂ϕ2 = −mglβ cos(ϕ1 + ϕ2). Therefore the potential generalized forces are calculated as follows Q(pi) = [ − ∂pi ∂ϕ1 − ∂pi ∂ϕ2 ]T = [ mgl[cosϕ1 − β cosϕ1 + ϕ2)] −mgβl cos(ϕ1 + ϕ2) ]T . (27) Suppose that the force from environment acting to the scoop of the manipulator in the axes of global coordinate is denoted by R(x, y) = [−X−Y]T , which are determined by the measure. The power of the force R will be W = vTBR = r T tT11t T 2Rϕ˙1 + r T tT1 t T 21Rϕ˙2. 164 Do Sanh, Dinh Van Phong, Do Dang Khoa Therefore the generalized forces R1, R2 corresponding to the generalized coordinates ϕ1, ϕ2 are of the form [8] R1 = r T tT11t T 2R = l [sinϕ1−β sin (ϕ1 + ϕ2)]+X + l [cosϕ1 − β cos (ϕ1 + ϕ2)]Y ; R2 = r T tT1 t T 21R = lβ [sin (ϕ1 + ϕ2)X − cos (ϕ1 + ϕ2)Y ] . (28) Where the quantities X, Y are obtained from the measure, but Q0 and Q∗ will be determined as follows By the matrix of inertia, we calculate ∂ϕ1A = 0; ∂ϕ2A = [ −2mβl2 sinϕ2 −mβl 2 sinϕ2 −mβl2 sinϕ2 0 ] ; Q01 = 1 2 q˙T1 ∂ϕ1Aq˙1 = 0; Q02 = 1 2 q˙T2 ∂ϕ2Aq˙ = −mβl 2 sinϕ2ϕ˙ 2 1 −mβl 2 sinϕ2ϕ˙1ϕ˙2; Q0 = [ Q01 Q0 2 ] = [ 0 −mβl2 sinϕ2(ϕ˙1 + ϕ˙2)ϕ˙1 ] . (29) By means of ∂ϕ1A = 0, in accordance to (8), we have Q∗1 = [ 0 0 ] , and Q∗2 = [ −2mβl2 sinϕ2 −mβl 2 sinϕ2 −mβl2 sinϕ2 0 ][ ϕ˙1ϕ˙2 ϕ˙22 ] = [ −mβl2 sinϕ2 (2ϕ˙1 + ϕ˙2) ϕ˙2 −mβl2 sinϕ2ϕ˙1ϕ˙2 ] . Hence, then Q∗= Q∗1+Q ∗ 2 = [ −mβl2 sinϕ2(2ϕ˙1 + ϕ˙2)ϕ˙2 −mβl2 sinϕ2ϕ˙1ϕ˙2 ] . (30) The acceleration of the scoop is calculated by the formula [8] 0aB = t11t2rϕ¨1 + t1t21rϕ¨2 + t12t2rϕ˙ 2 1 + t1t22rϕ˙ 2 2 + 2t11t21rϕ˙1ϕ˙2 [ 0x¨B 0y¨B ] =   −l [sinϕ1 + βl sin (ϕ1 + ϕ2)] ϕ¨1 − βl sin (ϕ1 + ϕ2) ϕ¨2 −l [cosϕ1 + β cos (ϕ1 + ϕ2)] ϕ˙ 2 1 + βl sin (ϕ1 + ϕ2) ϕ˙ 2 2 − 2βl cos (ϕ1 + ϕ2) ϕ˙1ϕ˙2 l [cosϕ1 + β cos (ϕ1 + ϕ2)] ϕ¨1 + βl cos (ϕ1 + ϕ2) ϕ¨2 −l [sinϕ1 + β sin (ϕ1 + ϕ2)] ϕ˙ 2 1 − βl cos (ϕ1 + ϕ2) ϕ˙ 2 2 − 2βl sin (ϕ1 + ϕ2) ϕ˙1ϕ˙2   where left superscript “0” denotes the acceleration in the global frame. Let write the constraint equation in the form x¨− y¨ = 0. Motion of mechanical systems with non-ideal constraints 165 We have f = [ f1 f2 ] ; f1 = l {[ sinϕ1 − cosϕ1] + β[sin(ϕ1 + ϕ2)−cos(ϕ1 + ϕ2)]} ; f2 = βl[cos(ϕ1 + ϕ2) + sin(ϕ1 + ϕ2)] f0 = {l(sinϕ1 − cosϕ1) + β[ sin(ϕ1 + ϕ2)− cos(ϕ1 + ϕ2)]} ϕ˙ 2 1 + βl[ sin(ϕ1 + ϕ2)− cos(ϕ1 + ϕ2)]ϕ˙ 2 2 + 2βl[ sin(ϕ1 + ϕ2)− cos(ϕ1 + ϕ2)]ϕ˙1ϕ˙2. (31) Let denote the matrix of the drive moments M1,M2 acting on the links of the manipulator by M M = [ M1 M2 ]T . (32) In order to determine the driven moments M1,M2 we write the Eq. (18) fA−1M+ fA−1(Q(pi) +Q0+R−Q∗)+f0= 0. (33) We obtain one equation of two unknownsM1, M2. The quantities Q (pi),Q0,Q∗ are computed by (27), (29), (30), but R is provided from the measurement result by means of X, Y , which are the reaction of the constraint (the environment) to the scoop of the manipulator. In result, we have one equation included two unknowns. Therefore we can choose arbitrarily one of two quantities M1,M2, for example, M1 ≡ 0 (corresponding to the link OA to be kept in fixed) or use the condition for optimizing some property of the scoop. Next, the motion of the manipulator is defined by means of integrating the differ- ential Eq. (17), where the matrix of inertia is computed by (26), the remaining quantities are in accordance with (27), (29), (30), (32). It is necessary to notice that the problem will be solved with the help of the numerical method. In the process of numerical integral the constraints will take the part of the criterion of verification. In some cases, it is necessary to know the normal and tangent reaction forces. For this aim, let introduce the natural moving coordinate axes Btn as shown as Fig. 2. The reaction forceR consists of a tangent and a normal component. By such a way, the reaction force R can be expressed in two forms R(Oxy) = [ X Y ]T ; R(Btn) = [ Ffric N ]T . Where (X, Y ) are the components of the reaction force in horizontal and vertical directions, but (N, Ffric) - the components in normal and tangent directions. In the other words, it is possible to establish the relation between these quantities by means of a transmission matrix. As known [8] the transmission matrix is of the form T =   cos θ − sin θ 0sin θ cos θ 0 0 0 1   . Where θ is the position angle between two coordinate axes 166 Do Sanh, Dinh Van Phong, Do Dang Khoa -y 0 R N fricF Y X t q a B n x a a Fig. 2. Components of reaction in the natural moving frame It is easy to obtain R(xy) ≡ [ X Y ] = [ cos θ − sin θ sin θ cos θ ][ −Ffric N ] → R(xy) = VR(t,n); V = [ cos θ − sin θ sin θ cos θ ] = [ sinα − cosα cosα sinα ] . (34) The Eq. (33) is written as follows fA−1M+ fA−1(Q(pi) +Q0+VR(Btnb)−Q∗)+f0= 0. (35) This equation allows to calculate the reaction forces in normal and tangent direc- tions. Based on these components it is possible to find out the relation between the friction force and the normal force. Note : The components of the reaction force in the (Btn) and rectangular (Oxy) coordinates R(Btn) = V−1R(Oxy); V−1 = [ sinα cosα − cosα sinα ] . (36) Where V−1 denotes the inverse matrix of the matrix V. 5. CONCLUSION This paper presents a new general method to analyze mechanical systems with non-ideal constraints. The program constraints are applied to the mechanical system by using the Principle of Compatibility. The paper pointed out that if a mechanical system is subject to non-ideal physical constraints, its motion depends on the interaction between the constraints and the system through mechanics-physics parameters. In other words, if a mechanical system is subject to non-ideal physical constraints, its motion cannot be determined only by theoretical analysis but also experimental measurements of the interaction between the system and the environment. It is necessary to emphasize that Motion of mechanical systems with non-ideal constraints 167 with the help of the method of transmission matrix and measured reaction forces the generalized forces corresponding to non-ideal constraints are defined. Accordingly, the paper contributed a method to analyze the motion of mechanical systems in the interactive relation between the system and the environment. Although only the physical constraints were analyzed in the paper, this method can be extended to problems with other types of constraints such as program constraints. Of course, the presented method can be applied for the case of the non-ideal constraints of Coulomb friction type [5,9-14] and the program constraints [4-6]. REFERENCES [1] Amirouch F., Determination of the generalized constraint forces in multi-body systems using kane’s equations, Journal of Theory and Applied Mechanics, 7(1), (1988), pp. 3-20. [2] Dinh Van Phong, Principle of compatibility and criteria of ideality in study of constrained me- chanical system, Strojnicky casopis 47 (Journal of Czech and Slovak mechanical engineering, Bratislava), 47(1), (1996), pp. 2-11. [3] Do Sanh, On the principle of compatibility and equations of motion of constrained mechanical systems, ZAMM 60, Berlin, (1980), pp. 210-212. 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[10] Rumantxep V.V., On the motion of the systems with non-ideal constraints, Vestnik MGU, No.5, (in Russian), (1961). [11] Rumantxep V.V., On the systems with friction, No 6, (in Russian), (1961). [12] Shart I., et al., Identification of friction coefficient for constrained robotic tasks, Journal of Dynamic Systems, Measurement ,and Control, December, 124, (2002). [13] Udwadia F.E., A new perspective on the tracking control of nonlinear structural and mechanical systems, Proc.R. Soc.A, (2003). [14] Udwadia Firdaus, Funcdamental principle of lagrange dynamics: Mechanical systems with nonideal holonomic and nonholonomic constraints, Journal of Mathematical Analysics and Applications, (2000), pp. 341-355. [15] Wojtura M., Joint reaction forces in multi-body systems with redundant constraints, Multibody System Dynamics, 14, (2005). Received August 10, 2012 VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY VIETNAM JOURNAL OF MECHANICS VOLUME 35, N. 2, 2013 CONTENTS Pages 1. Dang The Ba, Numerical simulation of a wave energy converter using linear generator. 103 2. Buntara S. Gan, Kien Nguyen-Dinh, Mitsuharu Kurata, Eiji Nouchi, Dynamic reduction method for frame structures. 113 3. Nguyen Viet Khoa, Monitoring breathing cracks of a beam-like bridge subjected to moving vehicle using wavelet spectrum. 131 4. Chu Anh My, Vuong Xuan Hai, Generalized pseudo inverse kinematics at singularities for developing five-axes CNC machine tool postprocessor. 147 5. Do Sanh, Dinh Van Phong, Do Dang Khoa, Motion of mechanical systems with non-ideal constraints. 157 6. N. D. Anh, Weighted Dual approach to the problem of equivalent replacement. 169