Biomedical Engineering Reference
In-Depth Information
the consequence of making a Euclidean space model of an object is that it permits the
powerful computational machinery of classical geometry and the integral and differ-
ential calculus to be used to calculate quantities of physical or biological interest.
Since the representation of physical phenomena must be independent of the
observer, it is necessary to express physical quantities in ways that are independent
of coordinate systems. This is because different observers may select different
coordinate systems. It therefore becomes a requirement that physical quantities be
invariant of the coordinate system selected to express them. On the other hand, in
order to work with these physical quantities and evaluate their magnitudes, it
is necessary to refer physical quantities to coordinate systems as illustrated in
Fig.
1.1
. The resolution of this conflict is to express physical quantities as tensors;
vectors are tensors of order one and scalars are tensors of order zero. Thus it is no
surprise that in classical mechanics the essential concepts of force, velocity, and
acceleration are all vectors; hence the mathematical language of classical mechan-
ics is that of vectors. In the mechanics of rigid objects the concepts of position,
velocity, and acceleration are all vectors and moments of inertia are second-order
tensors. In the mechanics of deformable media the essential concepts of stress,
strain, rate of deformation, etc., are all second order tensors; thus, by analogy, one
can expect to deal quite frequently with second-order tensors in this branch of
mechanics. The reason for this widespread use of tensors is that they satisfy the
requirement of invariance of a particular coordinate system on one hand, and yet
permit the use of coordinate systems on the other hand. Thus a vector
u
represents a
quantity that is independent of coordinate system, i.e., the displacement of a point
on an object, yet it can be expressed relative to the three-dimensional Cartesian
coordinate system with base vectors
e
a
,
u
III
e
III
and also expressed relative to another three-dimensional Cartesian coordinate
system with base vectors
e
i
,
i
a ¼
I, II, III, as
u
¼
u
I
e
I
þ
u
II
e
II
þ
u
3
e
3
. The vector
u
and the two coordinate systems are illustrated in Fig. 2.2. These two
representations of the components of the vector
u
are different and both are
correct because a rule, based on the relationship between the two vector bases
e
a
,
a ¼
I, II, III, and
e
i
,
i
¼
1, 2, 3, by
u
¼
u
1
e
1
þ
u
2
e
2
þ
¼
1, 2, 3, can be derived for calculating one set of
components in terms of the other. Thus the vector
u
has a physical significance
independent of any coordinate system, yet it may be expressed in component form
relative to any coordinate system. The property of vectors is shared by all tensors.
This is the reason that tensors, as well as vectors and scalars, play a leading role in
modeling mechanics phenomena.
1.4 The Types of Models Used in Mechanics
It is possible to divide the discipline of mechanics according to the predominant
type of motion an object is considered to be undergoing. The three types of motion
are translational, rotational, and deformational. In translational motion all the
points of the moving object have the same velocity vector at any instant of time.
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