Biomedical Engineering Reference
In-Depth Information
In order to investigate these normal modes of vibration, I write the above equations in
matrix form, and then find the eigenvalues and eigenvectors as follows:
⎛
⎝
⎞
⎠
(
k
1
+
k
2
)
k
2
m
1
ξ
1
ξ
2
ω
2
ξ
1
ξ
2
−
m
1
=−
(4.3)
k
2
m
2
k
2
m
2
−
Matrix diagonalization gives the allowed values of ω
2
(the eigenvalues), and for each value
of
ω
2
we calculate the relevant combinations of the ξ values (the eigenvectors). The
eigenvectors of the matrix are called the
normal coordinates
.
−
4.2 Larger Systems of Balls on Springs
For a molecule comprising
N
atoms, there are 3
N
Cartesian coordinates. Of these, three
can be associated with the position of the centre of mass of the whole molecule and three
for the orientation of the molecule at the centre of mass (two for linear molecules). This
leaves 3
N
- 6 vibrational degrees of freedom (3
N
- 5 if the molecule is linear), and it is
appropriate to generalize some concepts at this point. I am going to use matrix notation, in
order to make the equations look friendlier.
The molecular potential energy
U
will depend on
p
6 (independent) variables.
For the minute, let me call them
q
1
,
q
2
, ...,
q
p
, and let me also write
q
1,e
,
q
2,e
,...,
q
p
,e
for
their 'equilibrium' values. These coordinates are often referred to as
internal coordinates
,
and they will be linear combinations of the Cartesian coordinates.
First of all, for the sake of neatness, I will collect all the
q
's into a column matrix
q
. I will
also collect together the 'equilibrium' values into a column matrix
q
e
and the extensions
into a column matrix ξ :
=
3
N
−
⎛
⎞
⎛
⎞
⎛
⎞
q
1
q
2
.
q
p
q
1,e
q
2,e
.
q
p
,e
q
1
−
q
1,e
⎝
⎠
⎝
⎠
⎝
⎠
q
2
−
q
2,e
q
=
,
q
e
=
,
ξ
=
(4.4)
.
q
p
−
q
p
,e
I will now write
U
(
q
) to indicate the dependence of
U
on these variables. If I use Taylor's
theorem to expand
U
(
q
) about point
q
e
then the one-dimensional equation
R
e
)
d
U
d
R
R
e
)
2
d
2
U
d
R
2
1
2
(
R
U
(
R
)
−
U
(
R
e
)
=
(
R
−
R
=
R
e
+
−
R
=
R
e
+···
(given previously in Chapter 3) has to be modified to take account of the larger number of
variables. First derivatives become partial first derivatives, and we have to take account of
the 'mixed' second-order derivatives:
ξ
i
∂
U
∂
q
i
ξ
i
ξ
j
∂
2
U
∂
q
i
∂
q
j
ξ
i
=
0,ξ
j
=
0
+···
p
p
p
1
2
U
(
q
)
−
U
(
q
e
)
=
0
+
(4.5)
ξ
i
=
i
=
1
i
=
1
j
=
1
