B.1.2 Eigenvalue Problem, Natural Frequencies and Mode Shapes
a) Define on basis of (B.1) an eigenvalue problem (EVP) for the determination of
the undamped linear natural frequencies x j and associated mode shapes u j (x),
j = 1, 2, … for the shaft (Q 1 = Q 2 = me = c = 0, but Q 0 6 ¼ 0).
b) Determine whether the EVP is self-adjoint and completely definite, and supply
an interval for Q 0 ensuring the eigenvalues to be real and positive.
c) Compute the Rayleigh quotients R[u j ] for the set of orthogonal test functions
u j = sin(jpx), j = 1, 2, ….
d) Show that these Rayleigh quotients and test functions are, respectively, eigenvalues and eigenfunctions for the EVP, and hence that
x
2
j ¼
EI
qA
jp
L
4
þ
Q 0
qA
jp
L
2
; j ¼ 1; 2; Á Á Á ;
u j ðxÞ ¼ sin jpx; 0 x 1 ;
ðB:2Þ
are (squared) natural frequencies and mode shapes for the shaft. Which physical
condition corresponds to the case x 1
2 < 0?
B.1.3 Discretizations, Choice of Control Law
a) Assume that solutions u(x, t) for the nonlinear PDE in (B.1) can be expanded in
terms of the n lowest mode shapes u j , j = 1, n of the undamped system, that is:
uðx; tÞ ¼
X n
j¼1
y j ðtÞu j ðxÞ; u j ðxÞ ¼ sin jpx;
ðB:3Þ
and show that the modal amplitudes y i (t) are solutions of the following set of
nonlinear ODEs:
€ y i þ 2b_ y i þ x
2
i þ i
2 q 1 ðtÞ
À
Á
y i þ
X n
j¼1
q
ij
2 y j ¼ p i X
2 cos Xt; i ¼ 1; n;
ðB:4Þ
where x i denote the linear natural frequencies given by (B.2), and:
q 1 ðtÞ ¼
Q 1 ðtÞ
qA
p
L
2 ; p i ¼
2me
qAL 2 sin
ip
2
;
q
ij
2 ¼
À2
qAL 2
Z 1
0
Q 2 ð
X n
k¼1
y k u k ;
X n
k¼1
_
y k u k Þu i u
00
j dx :
ðB:5Þ
Appendix B: Major Exercises
459