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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
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