ðN þ dNÞ þ qA gdx À N ¼ 0 ) N
0
¼ ÀqAg
ðT þ dTÞ À T ¼ 0 ) T
0
¼ 0
ðM þ dMÞ À M þ T dx À N dy ¼ 0 ) M
0
þ T À Ny
0
¼ 0:
ð2:14Þ
Solving the first equation with the boundary condition N(0) = −P, one finds that
N(x) = –qAgx − P. Solving the second equation with the boundary condition
T(0) = 0 yields T(x) = 0. Inserting into the third equation, and using Hooke’s law
(M = EIy′′), one obtains the differential equation for the EVP:
EIy
000
þ ðqAgx þ PÞy
0
¼ 0:
ð2:15Þ
The three boundary conditions required for this third-order equation are:
y
00
ð0Þ ¼ yðlÞ ¼ y
0
ðlÞ ¼ 0:
ð2:16Þ
On introducing the column rotation (assumed small) as a new variable,
hðxÞ ¼ y
0
ðxÞ;
ð2:17Þ
the differential equation is reduced to order two:
EIh
00
þ ðqAgx þ PÞh ¼ 0; h
0
ð0Þ ¼ hðlÞ ¼ 0:
ð2:18Þ
This equation allows a variety of EVPs to be posed. For example, one may
define the eigenvalue so as to describe a critical load P:
EIh
00
þ qAgxh ¼ Àkh; k P;
ð2:19Þ
or the eigenvalue could describe a critical column weight qAgl:
EIlh
00
þ Plh ¼ Àkxh ; k qAgl:
ð2:20Þ
What is to be noted is the appearance of a non-constant coefficient (qAgx or kx,
respectively), even though the beam is uniform.
2.5 Vibration-Related EVPs
In problems of vibrations EVPs arise primarily during the analysis of free,
undamped vibrations. The study of free vibrations in turn constitutes the basis of
many exact and approximate methods for calculating damped and/or forced
responses for linear as well as nonlinear problems. We consider two examples.
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2 Eigenvalue Problems of Vibrations and Stability