Transcript Materialy/01/Applied Mechanics-Lectures/Applied Mechanics

Slovak University of Technology
Faculty of Material Science and Technology in Trnava
APPLIED MECHANICS
Lecture 01
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.

Engineering Problems
TACOMA NARROWS
Bridge Collapse
Length of center span
Width
Start pf construction
Opened for traffic
Collapse of bridge
2800 ft
39 ft
Nov. 23, 1938
July 1, 1940
Nov. 7, 1940
Engineering Problems
Destruction of tank
Millenium bridge
Ear
MODELLING OF THE MECHANICAL SYSTEMS
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1. Problem identification
2. Assumptions
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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
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conservation of mass,
conservation of momentum,
conservation of energy,
second and third laws of thermodynamics,
MODELLING OF THE MECHANICAL SYSTEMS

4. Constitutive equations

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5. Geometric constraints

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provide information about the materials of which a system is
made
develop force-displacement relationships for mechanical
components
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

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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
FUNDAMENTALS OF RIGID-BODY DYNAMICS
Position vector
r  x(t )i  y (t ) j  z (t )k
Velocity vector
dr
v
 x (t )i  y (t ) j  z (t )k
dt

Acceleration vector
dv
a
 x(t )i  y(t ) j  z(t )k
dt

Angular velocity vector
  e
Angular acceleration vector
d
  e


dt

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
i, j, k, e - unit cartesian vectors
FUNDAMENTALS OF RIGID-BODY DYNAMICS

position, velocity and acceleration vectors of point B (Fig. 1)
z´
z
B
y´
rB
rBA
A
v B  v A  v BA  v A    rBA
rA
k
i
rB  rA  rBA
a B  a A  a BA  a A    v BA    (  rBA )
j
x´
x
Fig. 1
y
FUNDAMENTALS OF RIGID-BODY DYNAMICS
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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
ma   Fi  F,
i
I G ε   M Gi  M,
i
IG
Fi
- inertia moment of the rigid body about an axis through its
mass center G and parallel to the axis of rotation,
- forces acting on body.
M Gi - moments acting on a rigid body.
Method of FBD for rigid bodies
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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
Method of FBD for rigid bodies system

The equations of motion for j-th rigid body
mi ai  FiE   F jiI ,
j
E
I
I Gi ε i  M G

M
,

G
i
ji
j
ai , resp. εi - accelerati on, resp. angular accelerati on of i - th rigid body
mi , resp. I Gi - mass, resp. inertia moment of i - th rigid body
E
FiE , resp. MG
- external force, resp. external moment acting on i-th rigid body,
i
I
- internal force, resp. internal moment acting from j-th to i-th rigid body.
F jiI , resp. M G
ji
Method of FBD for rigid bodies system

The system of equations of motion is after formulating of
kinematical relation between connected rigid bodies, is
solved in the form
i , t )  0
fi (FAi , qi , q i , q
for
i  1 n
n
- number of DOF
FAi
- are action forces affecting on systems of rigid bodies
i - generalized displacement, velocity, and acceleration
q i , q i , q
of i-th rigid body
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.
Method of reduction of mass & force parameters

Basic conditions:
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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:
dEk
 PA
dt
where Ek is a kinetic energy, PA is a power of action forces.
Method of reduction of mass & force parameters

The vector position of any rigid body
ri (t )  ri (q(t ))
i  1 n
where q(t) is so-called generalized coordinate.

The vector of velocity of i-th rigid body
vi 
q
dri dri dq dri


q
dt
dq dt
dq
is so-called generalized velocity.
Method of reduction of mass & force parameters

The kinetic energy of system of rigid bodies
E k   E ki
i

 dri
1
1
2

  mi vi   mi 
2 i
 dq
i 2




2
 q 2

i 1  n
Power of action forces
PA   FA j
j

 v j   F A j
 j
 dr j
 
 dq

 q


j  1  nF
Method of reduction of mass & force parameters

The general equation of motion of reduced system
1 dm (q) 2
m (q)q 
q  Q(q)
2 dq

q
- so-called generalized acceleration
m (q) - reduced mass
Q (q )
- reduced force