This research work was supported by project SGS-2022-008 of the Czech Ministry of Education, Youth and Sport, and by the project 23-07280S of the Czech Science Foundatio

Bulín, R.

Byrtus, M.

Dyk, Š.

Hajžman, M.

Hrabačka, M.

Smolík, L.

DSpace at University of West Bohemia

Modeling and implementation of joints in 2-D multibodydynamics considering chosen imperfectionsM. Hrabačkaa, M. Hajžmana, M. Byrtusa, Š. Dykb, R. Bulı́nb, L. Smolı́kbaDepartment of Mechanics, Faculty of Applied Sciences, University of West Bohemia, Univerzitnı́ 8, 301 00 Plzeň, Czech RepublicbNTIS – New Technologies for the Information Society, Faculty of Applied Sciences, University of West Bohemia,Univerzitnı́ 8, 301 00 Plzeň, Czech RepublicSoftware for simulating multibody systems’ (MBS) dynamics allows engineers to study andinvestigate mechanical and mechatronic systems’ motion. It enables them to generate and solvevirtual 3D models to predict and visualise motion, coupling forces, or stresses [2].Because simulation of MBS dynamics is usually quite time demanding, MBS software mustbe efficient in calculations if it is to be employable in practice. This work focuses on joint for-mulations and their evaluation efficiency, as these expressions are evaluated many times duringsimulations. Moreover, the contribution investigates effects of joint imperfections (friction andclearance) on the resulting motion of a 2-D slider-crank mechanism.The mathematical basis of the in-house developed MBS software is formed by Lagrange’sequations of the first kind that represent a set of findings achieved by applying Hamilton’sprinciple. The dynamics of a spatial system of n interconnected rigid bodies can be describedaccording to [3] by a system of 6n+ r differential-algebraic equations (DAEs)[M CTqCq 0] [q̈λ]=[Qe +QvQd], (1)where M is the system mass matrix, q is the vector of generalised coordinates of all bodiesconsisting of generalised coordinates qi = [Ri,Φi]T of particular bodies (Ri are the absoluteCartesian coordinates and Φi are the orientation angles), λ is the vector of Lagrange multipli-ers, Qe is the vector of generalised applied forces, and Qv is the quadratic velocity vector. Theconstraint Jacobian matrix Cq and the vector Qd are products of time derivatives of kinematicconstraints (kinematic relationships describing mechanical joints or specified motion trajecto-ries) represented by an algebraic system of r constraint equationsC(q, t) = 0, (2)where C is the constraint vector. If system (2) is twice partially differentiated with respect totime t, the kinematic acceleration equations are obtained, forming the last r equations in (1),where the vector Qd can be expanded toQd = −Ctt − (Cqq̇)qq̇− 2Cqtq̇ . (3)Lower indices ·t and ·q indicate partial derivatives with respect to time and with respect togeneralised coordinates.One of many possible approaches to solve the problem of MBS dynamics is, for instance,to convert system (1) of DAEs of index 1 to the underlying system of ordinary differential78equations (ODEs) and subsequent usage of some standard ODE solver. The transformation of(1) to a system of ODEs can be done via the elimination of Lagrange multipliers, obtainingq̈ = M−1CTq (CqM−1CTq )−1[Qd −CqM−1(Qe +Qv)] +M−1(Qe +Qv). (4)System (4) of ODEs can be solved directly, and it is possible to improve numerical accuracy viastabilization techniques, e.g., Baumgarte’s stabilization, which can be found in [1].O0X0Y 0Ri RjjiOiOjuiJujJJriJ ≡ rjJ(a) Kinematic constraintrRFjinFjitji(b) Contact modelFig. 1. Two approaches to modeling a revolute jointThere are two basic ways how to define a connection (usually called a joint) between twobodies. First approach is to use kinematic constraints representing restrictions on the relativemotion between connected bodies. For instance, a revolute joint visualised in Fig. 1a eliminatesrelative translation between two bodies, or a prismatic joint eliminates relative rotation betweentwo bodies. There are several other joint constraints and some of them have realistic meaningonly in 3-D space such as a cylindrical joint or a spherical joint. All joints defined as kinematicconstraints are included in the constraint vector C. In Fig. 1a, a revolute joint constraint is visu-alised, where Ri,j are absolute coordinate vectors of linked bodies, ui,jJ are coordinate vectorsof the joint J in local coordinate systems with origins Oi,j , and ri,j are coordinate vectors of thejoint in absolute coordinate system with origin O0. The constraint equation is simple in case ofthe revolute joint: riJ − rjJ = 0.The idea of the second approach is in specifying contact forces that are generated whensurfaces of two particular bodies touch each other. Similarly to the kinematic approach, thereare different definitions of contact forces depending on the type of a joint. All joints that arerepresented by contact forces are simply included the vector Qe of generalised applied forces.Fig. 1b contains visualization of a revolute joint with contact forces Fjin and Fjit applied on thebody i (contact forces applied on the body j are not shown, as they can be trivially derived byapplying Newton’s third law), where R is the radius of the bearing, and r is the radius of thejournal.The advantage of the contact approach lies in the fact that it naturally contains parametersof joint imperfections (friction and clearance). In case of kinematic constraints, additionalequation defining friction force has to be included in the model, and it is obviously impossibleto include joint clearance. In this work, the first approach is used only in case of ideal jointswithout any imperfections. If it is needed to incorporate joint imperfection, the second approachis utilised.79During the implementation of any joint kinematic constraint to MBS software, it is neces-sary to have access not only to the constraint vector C but mainly to its derivatives Cq and Cqq,while the second expression is included in the modified form of (3). Equation (3) also containsderivatives with respect to time t, but since joint kinematic constraints are usually scleronomic,all time derivatives of C are zero. Nevertheless, evaluation of Cq and Cqq is generally quitetime consuming operation and represents a significant portion of total simulation time. Onthe other hand, evaluation of contact forces is not as time consuming as kinematic constraints,however, those forces bring important non-linearities to the mathematical model. This has to betaken into account when choosing and setting the numerical solver, and simulating non-linearsystem is generally more time consuming than solving the linear one.Fig. 2. A 2-D slider-crank mechanismThis work compares time demand for simulating the motion of a 2-D slider-crank mecha-nism, which is depicted in Fig. 2, without imperfections in two different model versions. Bothversions differ in the approach used for modeling the joint between the crank and the connectingrod. All other joints (revolute between the frame and the crank, revolute between connectingthe rod and the slider, prismatic between the slider and the frame) are defined as kinematicconstraints in both models.Furthermore, the effects of imperfections of the joint linking the crank and the connectingrod are investigated in a separate study. Specifically, the influence of the clearance sizeR−r andthe amount of friction are examined during various working conditions defined by the angularvelocity of the crank.AcknowledgementsThis research work was supported by project SGS-2022-008 of the Czech Ministry of Educa-tion, Youth and Sport, and by the project 23-07280S of the Czech Science Foundation.References[1] Hajžman, M., Polach, P., Application of stabilization techniques in the dynamic analysis of multi-body systems, Applied and Computational Mechanics 1 (2) (2007) 479-488.[2] Schiehlen, W., Advanced multibody system dynamics, Kluwer Academic Publishers, 1993.[3] Shabana, A. A., Computational dynamics, Third edition, John Wiley & Sons, 2010.80

Modeling and implementation of joints in 2-D multibody dynamics considering chosen imperfections

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Modeling and implementation of joints in 2-D multibody dynamics considering chosen imperfections

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