APPLIED MECHANICS

Lecture 01

Slovak University of TechnologyFaculty 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 NARROWSTACOMA NARROWSBridge Collapse

Length of center span 2800 ftWidth 39 ftStart pf construction Nov. 23, 1938Opened for traffic July 1, 1940Collapse of bridge Nov. 7, 1940

Engineering Problems

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 kjir )()()( tztytx

Acceleration vector

Velocity vector kjir

v )()()( tztytxdt

d

kjiv

a )()()( tztytxdt

d

i, j, k, e - unit cartesian vectors

Angular velocity vector e

Angular acceleration vector e

dt

d

position, velocity and acceleration vectors of point B (Fig. 1)

FUNDAMENTALS OF RIGID-BODY DYNAMICS

x

z

y

rB

rA

rBA

i j

k

B

A

x´

y´

z´

)( BABAABAAB

BAABAAB

BAAB

rvaaaa

rvvvv

rrr

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

,

,

MMε

FFa

iGG

ii

iI

m

IG - inertia moment of the rigid body about an axis through

its mass center G and parallel to the axis of rotation, Fi - forces acting on body.

iGM - 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

,

,

j

IG

EGiG

j

Iji

Eiii

jiiiI

m

MMε

FFa

body rigid th of onaccelerati angular resp. on,accelerati -resp. -i , ii εa

EG

Ei i

MF resp. , - external force, resp. external moment acting on i-th rigid body,

IG

Iji ji

MF resp. , - internal force, resp. internal moment acting from j-th to i-th rigid body.

body rigid th of moment inertia resp. mass, -resp. -i , iGi Im

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

0),,,,( tf iiiAi iqqqF for ni 1

n - number of DOF

iAF - are action forces affecting on systems of rigid bodies

iii qqq ,,

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:

Ak Pt

E

d

d

where Ek is a kinetic energy, PA is a power of action forces.

The vector position of any rigid body

Method of reduction of mass & force parameters

))(()( tqt ii rr ni 1

where q(t) is so-called generalized coordinate.

The vector of velocity of i-th rigid body

qqt

q

qtiii

i d

d

d

d

d

d

d

d rrrv

q is so-called generalized velocity.

Power of action forces

Method of reduction of mass & force parameters

22

2

d

d

2

1

2

1q

qmvmEE

i

ii

iii

ikik

r

qq

Pj

jAj

jAA jj

d

drFvF

Fnj 1

The kinetic energy of system of rigid bodies

ni 1

Method of reduction of mass & force parameters

)(d

)(d

2

1)( 2 qQq

q

qmqqm

q - so-called generalized acceleration

The general equation of motion of reduced system

- reduced force

- reduced mass )(qm

)(qQ