Information Technology Reference
In-Depth Information
beam is simply a polynomial of limited order (3). The transfer matrix of Eq. (4.91) is now
fully defined.
A multitude of methods for computing transfer matrices are treated in the references.
The solution
e
A
can be represented as a matrix polynomial using the Cayley
3
-Hamilton
4
theorem (Pestel and Leckie, 1963) i.e., the minimal polynomial, which requires knowledge
of the eigenvalues of
A
. For a large matrix
A
, these can be difficult to obtain. Pade
5
approx-
imations can be useful in reducing the number of terms needed in an expansion of
e
A
.
For a nonconstant
A
, methods such as the Picard
6
iteration (Pestel and Leckie, 1963)
are available. Also important are the numerical integration techniques, such as Runge
7
-
Kutta (1867-1944) (Problem 4.23) that can be employed to solve differential equations. State
space control methods also often involve the solution of a system of first order differential
equations. Hence the relevant control theory literature is a fruitful source of information
on the calculation of transfer matrices.
Two General Analytical Techniques
Two procedures are to be presented which are suitable for finding the transfer matrices
for general forms of the governing equations of motion (Pilkey, 1994). These techniques,
based on the Cayley-Hamilton theorem mentioned earlier and on the Laplace transform,
apply to any set of governing equations for which an analytical solution can be obtained.
A beam will be used to demonstrate the method.
The first order differential equations for the static response of a beam with axial load
N,
displacement foundation
k
w
=
k,
and rotary foundation
k
∗
,
are derived from the relations
of Example 4.4 as
d
dx
=−
θ
+
V
k
s
GA
d
dx
=
M
EI
(4.94)
dV
dx
=
dM
dx
=
k
∗
−
k
w
−
p
z
V
+
(
N
)θ
Thus, Eq. (4.88), when expanded to include a compressive axial force
N
(Chapter 11),
shear deformation effects, and elastic foundations, becomes
z
=
Az
+
P
(4.95)
3
Arthur Cayley (1821-1895) was a prolific (almost one thousand papers) English mathematician and Cambridge
University professor. He was elected a fellow of Trinity College, Cambridge, the year (1842) he graduated from
Cambridge. He also practiced law for several years.
4
Sir William Rowan Hamilton (1805-1865) was a brilliant Irish mathematician who discovered the theory of
quaternions and made major contributions to optics and dynamics. He was elected to the chair of Astronomy at
Trinity College, Dublin, while he was still an undergraduate. Although appointed astronomer royal at Dunsink
Observatory, he was not considered to be a successful practical astronomer.
5
Henri Eugene Pade (1863-1953) was a French mathematician who received some of his higher education in
Leipzig and G ottingen, Germany. His PhD thesis was written with Charles Hermite as the advisor and Picard on
his committee and is the best known of his writings. It dealt with approximations to functions.
6
Charles Emile Picard (1856-1941) was a Frenchman who turned to mathematics at the end of his secondary school
tenure. After an interview with Pasteur, he attended the Ecole Normale Superieure, where he was permitted to
devote himself exclusively to research. He had a brilliant career in mathematical analysis and algebraic geometry,
including two theorems that bear his name. He married the daughter of his mentor Charles Hermite. He was a
mountain climber whose family life was marred by tragedy. His daughter and two sons died in World War I and
his grandsons were wounded and taken prisoner in World War II. The occupation of France clouded his final two
years.
7
Carl David Tolme Runge (1856-1927) was a German professor of mathematics in Hannover and G ottingen. He
performed research on spectroscopy, the theory of functions, and the numerical solution of differential equations.
Search WWH ::
Custom Search