Page 412 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)
P. 412
Sec. 12.8 Transfer Matrices 399
for the symmetric and antisymmetric modes enables sufficient equations for the
solution.
12.8 TRANSFER MATRICES
The Holzer and the Myklestad methods can be recast in terms of transfer
matrices.^ The transfer matrix defines the geometric and dynamic relationships of
the element between the two stations and allows the state vector for the force and
displacement to be transferred from one station to the next station.
Torsional system. Signs are often a source of confusion in rotating
systems, and it is necessary to clearly define the sense of positive quantities. The
coordinate along the rotational axis is considered positive toward the right. If a cut
is made across the shaft, the face with the outward normal toward the positive
coordinate direction is called the positive face. Positive torques and positive
angular displacements are indicated on the positive face by arrows pointing
positively according to the right-hand screw rule, as shown in Fig. 12.8-1.
With the stations numbered from left to right, the nih element is represented
by the massless shaft of torsional stiffness and the mass of polar moment of
inertia J^, as shown in Fig. 12.8-2.
Separating the shaft from the rotating mass, we can write the following
equations and express them in matrix form. Superscripts L and R represent the
left and right sides of the members.
For the mass:
= e!; I 1
( 12.8-1)
—(O^j
For the shaft:
■ i e ( 12.8-2)
TV = TJ^-1 0 1
The matrix pertaining to the mass is called the point matrix and the matrix
associated with the shaft, the field matrix. The two can be combined to establish
the transfer matrix for the /ith element, which is
R p
( \
6 1
K
(12.8-3)
co^J
T -co^J
\ / n
’^E. C. Pestel and F. A. Leckie, ‘‘Matrix Methods in Elastomechanics,” (New York: McGraw-Hill,
1963).