Modelling of dynamics of mechanical systems with regard for constraint stabilization

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1 IOP Conference Series: Materials Science and Engineering PAPER OPEN ACCESS Modelling of dnamics of mechanical sstems with regard for constraint stabilization o cite this article: R G Muharlamov 018 IOP Conf. Ser.: Mater. Sci. Eng View the article online for updates and enhancements. his content was downloaded from IP address on 31/1/018 at 19:57

2 Fundamental and Applied Problems of Mechanics IOP Conf. Series: Materials Science and Engineering 468 (018) doi: / x/468/1/01041 Modelling of dnamics of mechanical sstems with regard for constraint stabilization R G Muharlamov Institute of Phsical Research and echnologies RUDN Universit Moscow Russia robgar@mail.ru Abstract. he main purpose of dnamical processes modelling is to formulate the motion equations of the sstem with regard for active forces and constraints restricted its movement. Desirable properties of sstem's motion which are provided b the influence of additional forces and b the variation of inertial sstem's properties can be specified b the constraint equations. Niola Zhuovsi studied two main problems on constructing motion equations: defining the force function that determines a set of motion trajectories and analsing its stabilit. he representation of constraint equations as partial integrals of motion equations allows to provide an asmptotic stabilit of the corresponding integral manifold and to solve the problem of constraint stabilization at numerical solution of dnamics equations. 1. Introduction he basis of the mathematical model of mechanical sstem s dnamics is the sstem of motion equations. If the set of constraint equations allows to uniquel represent the inematic state of a sstem using generalized coordinates and velocities then the sstem s behavior is determined with some particular level of accurac that depends on applicable methods. Lagrange multipliers can be applied for accounting the constraint influence in the case of impossibilit of generalized coordinates and velocities introduction. At the same time constraint reactions can be considered as control forces providing the realization of constraint equations. he problem of defining additional control forces which allow the motion of a mechanical sstem to have appropriate properties is related to the inverse dnamical problems [1]. So based on the properties of planetar motion sir Isaa Newton [] established the form of the gravit force and later it was discovered [3] that the motion of a material point on a conic section is the effect of a central force depending on the point's position [4] [5]. he problem of determining a force function corresponding to a holonomic sstem with given integrals was considered b Gavriil Suslov [6]. Also Niola Zhuovsi established the method of determining a force function based on a given set of trajectories of a material point on a curved surface and gave a solution to the problem of motion strength of a representation point [8] using indicators of the inetic energ sstem's force function. General Lapunov's theor of motion stabilit [9] allowed to formulate the stabilit criteria of a bunch of trajectories [10] and to develop some new methods of constructing the sstems of differential equations having a given stable integral manifold [11]. Determining the Lagrange multipliers constraint equations are usuall considered as the first integrals of motion equations so that the initial data corresponds to them. Deviations of initial data and an application of some approimation methods cause the disruption of constraint equations. he problem Content from this wor ma be used under the terms of the Creative Commons Attribution 3.0 licence. An further distribution of this wor must maintain attribution to the author(s) and the title of the wor journal citation and DOI. Published under licence b Ltd 1

3 Fundamental and Applied Problems of Mechanics IOP Conf. Series: Materials Science and Engineering 468 (018) doi: / x/468/1/01041 of a constraint stabilization was firstl mentioned b Joachim Baumgarte in his paper [1].An asmptotic stabilit of corresponding to the constraint equations integral manifold of a sstem of dnamics equations is a necessar condition for solving this problem. he solution of this problem can be obtained b introducing additional forces or b changing the inertial properties of a sstem [13] [14]. he problem of a constraint stabilization leads to the problem of constructing the sstem of differential equations admitting constraint equations as partial integrals and defining asmptotic stable invariant set [16] or integral manifold [17] of this sstem. It is quite possible to construct a sstem of motion equations with required accurac [18] of constraints deviations using a general approach of solving inverse dnamical problems. Some relevant dnamical analogies allow us to appl methods and equations of classical mechanics to solve problems of modeling and dnamical control of sstems of different nature.. Problem statement Let the state of a mechanical sstem be determined b generalized coordinates 1... n q q velocities 1... n i i v v v dq / v i 1... n Lagrange function L L t q q v and non-potential generalized forces interacting with a mechanical sstem Q Q Q Q Q t... 1 n i i q v. Let s consider that the sstem has both holonomic and nonholonomic constraints t 0 f 1 m f... f (1) f q m f q v t 0 1 s f... f f s n. () So the problem is to determine dnamical equations of this mechanical sstem that provide the constraint stabilization during the numerical integration. 3. Dnamical equations of an etended sstem n1 nm Let s introduce new variables: q n m 1 n s ( q... q ) q ( q... q n1 nm ) v ( v... v ) n m 1 n s v ( v... v n1 ) v v ns (... v ) such as q f q t 0 (3) v g q v t 0 (4) g g g f f g v g f. q t he sstem of equations that defines the virtual displacements can be constructed based on the equation (4) and it taes form g G q v G v. (5) A vector of virtual displacements q can be defined due to the sstem (5) considering the vector v to be arbitrar. If the columns of the matri G are linear independent then the solution of the sstem (5) q q q n consists of a general solution q l GC of the homogeneous equation and a partial solution q n G + v of the nonhomogeneous equation q l GC G v. (6) Here l is an arbitrar scalar value GC is a cross product of the vectors g 1... g n g 1... s composing the columns of the matri G and arbitrar vectors c 1... c n s 1... n 1 G G GG 1. c

4 Fundamental and Applied Problems of Mechanics IOP Conf. Series: Materials Science and Engineering 468 (018) doi: / x/468/1/01041 q v are non- Let s consider that L Lq v t is a Lagrange function Q Q... 1 Qn Qi Qi t potential generalized forces interacting with a mechanical sstem R R... 1 Rn i of the vector of constraint reaction. Let s define the functions L L q v q v t D D q v q v t that satisf the following conditions: L q v00 t L q v t D q v 0 0 t 0 and D q v q v t D q v 0 if q v D00 0. If R R R are components and do not go to zero simultaneousl and R... 1 n R i are components of the vector of constraint reaction then D Alembert s principle for the etended sstem with the Lagrange function L taes form Εq v Q R 0 (7) D Εq v 0 (8) v d L L q v q q q. v q We can define the sum of elementar wors b scalar multipling equations (7) and (8) b q and v correspondingl D Εq v Q R q Εq v v 0 (9) v then we can rewrite it with regard for (6): D Εq v Q R GCl Εq v Q R G Εq v v 0. (10) v he equalit (10) can be accomplished onl if the following conditions are satisfied Ε q v Q R GC 0 (11) D Εq v Q R G Εq v (1) v Let s choose the vector R so that the elementar wor of constraint reactions due to the displacements q will be equal to zero: R GC 0. his fact denotes the correspondence to the ideal constraints of the initial sstem λ... 1 s. From the identit (11) follows the equation describing the variation of generalized coordinates of the sstem. Identit (1) can be reduced to the equation of the constraints perturbations and as a result with regard for inematic equations identities (3) (4) and initial conditions we can obtain the following sstem of differentialalgebraic equations for q v q v λ : dq d L L v Q G λ (13) v q dq d L L D v (14) v q v q f q t q q q v gq v t (15) q t 0 q 0 q t0 f q0 t0 qt0 q 0 v t 0 v 0 v t0 g q0 v0 t0. (16) It is necessar to complete the right sides of differential equations (13) (14) to solve the sstems (13) (16). he values of the forces of constraint reactions are determined b defining the multipliers λ that ensure the equalities (15). If we assume that the values of the deviations from the solution of the sstem (13) (14) are defined with the help of etra variables q v then the solution q 0 v 0 of this R G λ 3

5 Fundamental and Applied Problems of Mechanics IOP Conf. Series: Materials Science and Engineering 468 (018) doi: / x/468/1/01041 sstem corresponds to the constraint equations and its stabilit depends on the choice of the Lagrange function L and dissipative function D.Let s consider that functions L and D tae form L P v Aq v v A q qv P q H qq D v Bqq v (17) Hq 0 H q. 0 0 he values of λ are determined from the equations (13) (15): 1 1 λ M q v t h q v t M GA qg 1 L da q q 1 daq L g g h A q q v Bq q v GA q v v q q q t q f q t v g q v t. Dnamical equations (13) with regard for the values L q v λ as functions of the variables q vt are reduced to the following sstem of differential equations: dq dv v 1 L da q A q v Qq v t G q v tλq v t (18) q L L q v f q t g q v t t that has the partial integrals: f q t 0 g q v t 0. (19) 4. Stabilit of integral manifold he constraint (1) () stabilization requires the asmptotic stabilit of an integral manifold of the sstem (18) given b the equalities (19). his stabilit of an integral manifold can be defined with the help of the following terms. An integral manifold of the sstem (18) given b the equalities (19) is stable if for an 0 there eists such 0 t v v t of the sstem corresponding to the initial so the solution q q conditions qt0 q 0 vt0 v 0 : f q 0t inequalities f q t t gqt v t t. g q v t for all t t0 will satisf the An integral manifold of the sstem (18) given b the equalities (19) is asmptoticall stable if it is stable and the following conditions are satisfied: f q t t lim 0 t lim g q t v t t 0. It is obvious that the stabilit of an integral manifold is determined b the corresponding stabilit of the trivial solution qt 0 v t 0 of the sstem of the perturbation constraint equations (14). With the regard for values of L D (17) the sstem (14) can be written in the form: dq dv 1 A q q v S q qq K q q v A q q v v q 1 1 da q q S q q A q qh q K q q A q q Bq q S q q K q q in a series in powers of q : or after epanding the matrices and S q t 0 K q0 S q q S q q S K q q K q q K q q 0 0 4

6 Fundamental and Applied Problems of Mechanics IOP Conf. Series: Materials Science and Engineering 468 (018) doi: / x/468/1/01041 it can tae the following form: dq dv v S q q K q v V (0) Sq S q0 K K q0 S q0 K q0 1 A qq V. q S q q K v A q q v v q q q Let s introduce the following notation: 0 E q v 0 W q Y -Sq -K q V and rewrite the sstem (0) in an abbreviated form: d. W q Y (1) If all of the roots of the characteristic polnomial of the matri W q at all of the possible values of the generalized coordinates q... 1 qn at their domain q have negative real parts then a trivial solution of the equation with primar approimation: d W q is asmptotic stable. he problem the holonomic constraint stabilization is studied in the paper [1] using the equations of perturbed constraints with the constant matri 0 E W. E E W W q v determined with An algorithm of solving the problem of stabilization with the matri the help of the matri G and its derivative is established in the paper [19]. In general the method of Lapunov s function is applied [11] to define the sufficient conditions of the stabilit of a trivial solution (1). If the constraints are scleronomic f q 0 gq v 0 then we can tae as a Lapunov s function a positive definite quadratic form with the constraint matri with coefficients V U. If the derivative of this function dv 3 UW q Y is negative definite then the trivial solution 0 of the equation (1) is asmptotic stable. 5. Numerical solution If the perturbation constraint equations have asmptotic stable trivial solution then we can limit our choice with the simplest numerical methods of solving the dnamical equations (18). So the application of the finite-difference scheme X 1 t t 1 t t 0 0 q v X v t F q v 1 L daq Fq v t A q v Qq v t G q v tλ q v t q with regard for (1) leads to the inequalit 5

7 Fundamental and Applied Problems of Mechanics IOP Conf. Series: Materials Science and Engineering 468 (018) doi: / x/468/1/01041 where 1 E W is a reminder. From inequalit (3) follows the estimation 1 if for all N the following conditions: E W 1 1 If solving equations (18) we use the finite difference scheme 1 1 ˆ X X X ˆ X t X N where const and for all N the following conditions are satisfied are satisfied. 1 dw Imr W W 1 then we have the estimation 1. he conditions of the constraint stabilization were obtained b the Runge-Kutta method in the paper [1]. 6. Eample Our goal is to determine the law of the variation of the force F providing the stable motion of the rocet on the trajector f 0. he rocet is considered as a material point that has coordinates and velocities d d q v v v v v and it interacts with the force of gravit mg directed verticall downwards. he deviation of the point from its trajector and its derivative are denoted as f f q f v v v. () Let s introduce Lagrange and dissipative functions L mv mg v cq D v (3) v c g const. v v From the equalities () follows the equation: f f v that determines virtual displacements of a point depending on the arbitrar values s and v : 1 f f f f f f f f s v s v. Using D'Alembert-Lagrange principle d L L d L L d L L D v 0 v v v q v let s construct the dnamical equations of the rocet d v d f d mv f mv mg and the perturbation constraint equations dq d v dv cq v. 1 (4) (5) 6

8 Fundamental and Applied Problems of Mechanics IOP Conf. Series: Materials Science and Engineering 468 (018) doi: / x/468/1/01041 Let s introduce as a velocit of a particle separation from the rocet and considering the equalit d d v m m dm v v v we can rewrite (4) in the following form: d v dv f m d v dv f m mg (6) v v v dm v v v dm. Right parts of the equations (6) contain the traction force motion trajector and the constraint reaction R f / f / directed along the tangent to the directed along the normal. hese two forces form the unnown force F. From the equations () (5) (6) we can determine the Lagrange multipliers m f f f f f f 1 f f dm v vv v cf v v g v v (7) N N v v f f N. he motion on the trajector will be asmptotic stable if the roots the characteristic equation of the sstem (5) c 0 have negative real parts. Numerical solution of the sstem (6) (7) t v v will satisf the equalit 1 v v 1 1 f 1 m q f v v 1 mg m for all K if the conditions q0 E W 1 are satisfied where W is a matri of the coefficients of the sstem (5) q f epansion. reminder of the function s 7. Conclusions he methods of solving inverse dnamical problems and the conditions of the stabilit of a set of trajectories based on the Zhuovsi s papers allow us to develop an algorithm od solving dnamical problems of mechanical sstems and problems of dnamical processes control in the sstems with different nature. his wor is supported b RFBR project А. Reference [1] Galiullin A S 1986 Methods of solving inverse dnamical problems (Moscow: Naua) [] Newton I 1989 Philosophiæ Naturalis Principia Mathematica. ranslation from Latin b A.N. Krlov (Moscow: Naua) [3] Bertrand M G 1873 Compterendus 77 pp [4] Imshenetsi V G 1879 Bulletin of Kharov Mathematical Societ pp 1 11 [5] Darbou M G 1877 Compterendus 84 pp [6] Suslov G K 1890 On force function with given integrals (Kiew: Kiew Universit) is a 7

9 Fundamental and Applied Problems of Mechanics IOP Conf. Series: Materials Science and Engineering 468 (018) doi: / x/468/1/01041 [7] Zhuovsi N E 1937 Collection of Papers 1 pp [8] Zhuovsi N E 1937 Collection of Papers 1 pp [9] Lapunov A M 1956 General problem of motion stabilit (Collection of papers vol ) (Moscow: Academ of Science USSR) [10] Zubov VI 1957 Methods of A.M. Lapunov and its application (Leningrad: LSU) [11] Muharlamov R G 1969 Differential Equations 5 pp [1] Baumgarte J 197 Comp. Math. Appl. Mech. Eng 1 pp 1 16 [13] Kozlov V V 015 Regul. Chaot. Dn.0 pp 05 4 [14] Kozlov V V 015 Regul. Chaot. Dn.0 pp [15] Erugin N P 1979 Boo for reading of general course of differential equations (Mins: Nauaiehnia) [16] Levi-Chivita and Amaldi U 195 Course of theoretical mechanics II (Moscow: IL) [17] Muharlamov R G 1971 Differential Equations 7 pp [18] Muharlamov R G 006 Applied Mathematics and Mechanics 70 pp [19] Ascher U M Chin H Petzold L R and Reich S 1995 Mechanics of Structures and Machines 3 pp [0] Muharlamov R G 015 Izvestia RAN 15 8 [1] Muharlamov R G and Assae W B 013 Bulletin of Peoples' Friendship Universit of Russia. Series: Mathematics Informatics and Phsics 3 pp