A PPLIED M ECHANICS Lecture 01 Slovak University of Technology Faculty of Material Science and Technology in Trnava

INTRODUCTION Applied Mechanics:  Branch of the physical sciences and the practical application of mechanics.  Examines the response of bodies (solids & fluids) or systems of bodies to external forces.  Used in many fields of engineering, especially mechanical engineering  Useful in formulating new ideas and theories, discovering and interpreting phenomena, developing experimental and computational tools

INTRODUCTION Applied Mechanics:  As a scientific discipline - derives many of its principles and methods from the physical sciences mathematics and, increasingly, from computer science.  As a practical discipline - participates in major inventions throughout history, such as buildings, ships, automobiles, railways, engines, airplanes, nuclear reactors, composite materials, computers, medical implants. In such connections, the discipline is also known as engineering mechanics.

TACOMA NARROWS Bridge Collapse Length of center span2800 ft Width 39 ft Start pf constructionNov. 23, 1938 Opened for trafficJuly 1, 1940 Collapse of bridgeNov. 7, 1940 Engineering Problems

Destruction of tank Ear Millenium bridge

MODELLING OF THE MECHANICAL SYSTEMS 1. Problem identification 2. Assumptions  Physical properties - continuous functions of spatial variables.  The earth - inertial reference frame - allowing application of Newton´s laws.  Gravity is only external force field. Relativistic effects are ignored.  The systems considered are not subject to nuclear reactions, chemical reactions, external heat transfer, or any other source of thermal energy.  All materials are linear, isotropic, and homogeneous. 3. Basic laws of nature  conservation of mass,  conservation of momentum,  conservation of energy,  second and third laws of thermodynamics,

4. Constitutive equations  provide information about the materials of which a system is made  develop force-displacement relationships for mechanical components 5. Geometric constraints  kinematic relationships between displacement, velocity, and acceleration 6. Mathematical solution  many statics, dynamics, and mechanics of solids problems leads only to algebraic equations  vibrations problems leads to differential equations 7. Interpretation of results MODELLING OF THE MECHANICAL SYSTEMS

Mathematical modelling of a physical system requires the selection of a set of variables that describes the behaviour of the system  Independent variables – for example time  Dependent variables - variables describing the physical behaviour of system (functions of the independent variables ): dynamic problem - displacement of a system, fluid flow problem - velocity vector, heat transfer problem - temperature, a.o.  Degrees of freedom (DOF) -number of kinematically independent variables necessary to completely describe the motion of every particle in the system.  Generalized coordinates - set of n kinematically independent coordinates for a system with n DOF MODELLING OF THE MECHANICAL SYSTEMS

FUNDAMENTALS OF RIGID-BODY DYNAMICS Position vector Acceleration vector Velocity vector i, j, k, e - unit cartesian vectors Angular velocity vector Angular acceleration vector

position, velocity and acceleration vectors of point B (Fig. 1) FUNDAMENTALS OF RIGID-BODY DYNAMICS x z y rBrB rArA r BA i j k B A x´ y´ z´ Fig. 1

The principles governing rigid-body kinetics – based on application of Newton´s second law - vector dynamics. For rigid body in plane motion the equations of motion have the form FUNDAMENTALS OF RIGID-BODY DYNAMICS I G -inertia moment of the rigid body about an axis through its mass center G and parallel to the axis of rotation, F i -forces acting on body. -moments acting on a rigid body.

Method of FBD for rigid bodies universal method complete dynamical solution of system of rigid bodies is obtained bodies are released from systems of rigid bodies each released rigid body is loaded by appertain external forces and by internal forces which result from effects of other rigid bodies connected to the released rigid body for each released body, the equations of motion are formulated using Newton´s laws

The equations of motion for j-th rigid body Method of FBD for rigid bodies system - external force, resp. external moment acting on i-th rigid body, - internal force, resp. internal moment acting from j-th to i-th rigid body.

The system of equations of motion is after formulating of kinematical relation between connected rigid bodies, is solved in the form Method of FBD for rigid bodies system for n - number of DOF - are action forces affecting on systems of rigid bodies By solution of system of equations for defined initial condition, the motion and the dynamical properties of the system of rigid bodies are completely described. -generalized displacement, velocity, and acceleration of i-th rigid body

Method of reduction of mass & force parameters Basic conditions:  system of rigid bodies with one DOF  mass and force parameters are reduced on one of the rigid bodies of investigated system  this rigid body have to one DOF  only the relation between movement and action forces of system of rigid bodies can be determined using this method Method based on theorem of change of kinetic energy: where E k is a kinetic energy, P A is a power of action forces.

The vector position of any rigid body Method of reduction of mass & force parameters where q(t) is so-called generalized coordinate. The vector of velocity of i-th rigid body is so-called generalized velocity.

Power of action forces Method of reduction of mass & force parameters The kinetic energy of system of rigid bodies

Method of reduction of mass & force parameters - so-called generalized acceleration The general equation of motion of reduced system - reduced force - reduced mass