*This preview shows
page 1. Sign up
to
view the full content.*

**Unformatted text preview: **the mathematical model would be grossly inconsistent with reality if the
symmetric matrix A in (7.6.2) were not positive deﬁnite. It turns out that A
is positive deﬁnite because there is a Cholesky factorization A = RT R with
⎞
⎛ r −1/r
1 R= ⎜
T⎜
⎜
mL ⎜
⎝ 1 r2 −1/r2
..
.
rn−1 .. .
−1/rn−1
rn ⎟
⎟
⎟
⎟
⎠ with rk = 2− k−1
,
k and thus we are insured that each λk > 0. In fact, since A is a tridiagonal
Toeplitz matrix, the results of Example 7.2.5 (p. 514) can be used to show that
λk = Copyright c 2000 SIAM 2T
mL 1 − cos kπ
n+1 = kπ
4T
sin2
mL
2(n + 1) (see Exercise 7.2.18). Buy online from SIAM
http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com
562
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540
Therefore,
√
√
⎧
⎫
= αk cos t λk + βk sin t λk ⎬
⎨ zk
√
zk (0) = ck
˜
˜
=⇒ zk = ck cos t λk , (7.6.3)
⎩
⎭
zk (0) = 0
and for P = x1 | x2 | · · · | xn ,
n It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] y = Pz = z1 x1 + z2 x2 + · · · + zn xn = √
cj cos t λk
˜ xj . (7.6.4) j =1 D
E This means that every possible mode of vibration is a combination of modes
determined by the eigenvectors xj . To understand this more clearly, suppose
that the beads are initially positioned according to the components of xj —i.e.,
˜
c = y(0) = xj . Then c = PT c = PT xj = ej , so (7.6.3) and (7.6.4) reduce to
√
if k = j =⇒ y = cos t√λ
zk = cos t λk
(7.6.5)
xj .
k
0
if k = j
th T
H In other words, when y(0) = xj , the j eigenpair (λj , xj ) completely determines the mode of vibration because the amplitudes are determined by xj , and
each bead vibrates with a common frequency f = λj /2π. This type of motion
(7.6.5) is called a normal mode of vibration. In these terms, equation (7.6.4)
translates to say that every possible mode of vibration is a combination of the
normal modes . For example, when n = 3, the matrix in (7.6.2) is
⎧
⎫
⎛
⎞
2 −1
0
⎨ λ1 = (T /mL)(2) ...

View
Full
Document