Dynamics of a system of rigid bodies

To introduce students to fundamentals of system of rigid bodies,(SRB). It is possible to solve direct and inverse kinematics and dynamics task of (SRB) using the classical approach as well as modern theory of finite rotation and quaternions. Determination of the (simulation) models SRB-differential equations of motion SRB which are important in practical problems of dynamics of SRB.

Attending the course students acquire the ability to analyze problems and synthesis solutions to the problem of rigid body dynamics system with the use of scientific methods and procedures as well as computer technology and equipment. Enabled him to connect knowledge of mechanics, mathematics, physics, the practical application of solving current problems of rigid body dynamics system.

Introduction to dynamics of system of rigid bodies (SRB). Fundamentals of kinematic chains.Orthogonal transformation of coordinates (OTC). Basic theory of finite rotations (FROT). FROT and spherical motion of rigid body. Quaternions. Hamilton-Rodriguez (HG) parameters. The transformation matrix in case of rotation in regard to (HG parameters and quaternions` notation), the application of the spherical motion SRB. Dynamics of spherical motion of the rigid body.The first integrals of differential equations (DIFE)of spherical motion of rigid body. Constraints of system, ideal and real constraints. The kinetic energy of the system of the rigid bodies.Metric tensor of system. Generalized forces and the principle of ideality RS- different cases -specially conservative case. The case of real constraints. (DIFE) of motion of the RS in (contra)covariant form,quaternion form. DIFE of motion of RS given in the form of kinematic chain with the structure of topological three; DIFE of motion of RS given in the form of closed-kinematic chain.Additional equations of contraints. Optimal motion of system of rigid bodies. Variational approach. Maximum-principle-application to real systems. Fundamentals of system dynamics of deformable bodies and contact mechanics.

Examples of determining the OTC. Determining the number of degrees of freedom for a given SRB. Application of Rodriguez matrix transformation-typical cases. An example of determining the configuration of a case of SRB-an industrial machine. Examples of application of the finite rotation and quaternions in spherical rigid body motion. Instances of the spherical rigid body motion-typical cases. Determination of the kinetic energy of the system of rigid bodies as well as the metric tensor of SRB. Application on a concrete example: a mechanical model of washing machines and robot Neuroarm as SRB. An instance of the formation of (contravariant)covariant forms of motion given SRB with 4-6 degrees of freedom. Synthesis of optimal control SRB.

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1.Wittenburg J.,Dynamics of Systems of Rigid Bodies,Teubner,Stuttgart,1977.(KSJ) 2.Čović M. V. Lazarević, Mechanics of Robot, MF Belgrade,2021.(Book) 3.Lazarević M. Exercises in mechanics of robot, MF Belgrade,2006.(ZZD) 4.Shabana A. Dynamics of Multibody Systems,2020.(KSJ) 5.Written abstracts from the lectures (Handouts) 6.Cyberbotics Webots - software package 7.Laboratory model of washing machine-4DOFs. 8.NeuroArm-laboratory robot-7 DOFs. 9.SimMechanics,GUI,(CSP)

Total assigned hours: 65

New material: 30
Elaboration and examples (recapitulation): 20

Auditory exercises: 0
Laboratory exercises: 0
Calculation tasks: 0
Seminar paper: 0
Project: 0
Consultations: 0
Discussion/workshop: 0
Research study work: 0

Review and grading of calculation tasks: 0
Review and grading of lab reports: 5
Review and grading of seminar papers: 5
Review and grading of the project: 0
Test: 0
Test: 0
Final exam: 5

Activity during lectures: 0
Test/test: 0
Laboratory practice: 0
Calculation tasks: 0
Seminar paper: 50
Project: 0
Final exam: 50
Requirement for taking the exam (required number of points): 30

Pfeiffer, F., Mechanical System Dynamics,Springer-Verlag Berlin Heidelberg,2008.; Ahmed A. Shabana,Computational Dynamics,John Wiley & Sons, Inc., 605 Third Avenue, New York, NY,2020.; Coutinho, M., Dynamic Simulation of Multibody Systems, Birkäuser, 2001.; Schielen,W. ed., Multibody Systems Handbook, Springer-Verlag, Berlin, 2012; Roberson, R.E., Schwertassek, R., Dynamics of Multibody Systems, Springer-Verlag, Berlin, 2012