1.8.4 Maxwell’s Reciprocity Theorem
The reciprocity theorem states that, for a linear elastic structure, the stiffness matrix
and the flexibility matrix (as defined in Sects. 1.8.2-3) are both symmetric. This can
ease the burden of computing stiffness- and flexibility-coefficients, since k ij = k ji and
a ij = a ji for (i, j) = 1, n. The theorem can be proved by calculating and equating the
work done by two forces f i and f j in different sequence (first j after i, then i after j).
1.9 Classification of Forces and Systems
Linear systems (or the linear part of nonlinear systems, if any) are sometimes
classified according to the kind of forces involved, such as instationary, dissipative,
(non-)circulatory, and gyroscopic. Here we just present a simple way to perform the
classification, referring to the Ziegler (1968) for a thorough treatment, and to
problems 4.7–4.10 that involves also system classification.
Consider a linear/linearized system of the form (1.14), i.e.:
M€ x þ C _
x þ Kx ¼ fðtÞ; xðtÞ 2 R
n
;
ð1:129Þ
which could model a multiple-DOF discrete mechanical system, or arise from using
mode shape expansion (Sect. 1.5.3) for a continuous system (then x would be the
modal amplitudes). Here the mass matrix M is assumed to be symmetric and
positive definite; this will normally be the case when the system is properly
restricted in space, and when symmetry has not been destroyed by manipulating
individual equations (e.g. multiplying any of the n equations in (1.129) by a constant would preserve model validity but destroy the symmetry of M and K). For this
rather general linear system we next describe the classification of forces, which is
then subsequently used as a basis for classifying systems, which in turn provides a
basis for stability considerations.
1.9.1 Force Classification
Any square matrix B can be split into symmetric and antisymmetric parts as
follows:
B ¼ B S þ B A ;
ð1:130Þ
where B’s symmetric part B S = B S
T and antisymmetric part B A = –B A
T can be calculated as:
42
1 Vibration Basics
The reciprocity theorem states that, for a linear elastic structure, the stiffness matrix
and the flexibility matrix (as defined in Sects. 1.8.2-3) are both symmetric. This can
ease the burden of computing stiffness- and flexibility-coefficients, since k ij = k ji and
a ij = a ji for (i, j) = 1, n. The theorem can be proved by calculating and equating the
work done by two forces f i and f j in different sequence (first j after i, then i after j).
1.9 Classification of Forces and Systems
Linear systems (or the linear part of nonlinear systems, if any) are sometimes
classified according to the kind of forces involved, such as instationary, dissipative,
(non-)circulatory, and gyroscopic. Here we just present a simple way to perform the
classification, referring to the Ziegler (1968) for a thorough treatment, and to
problems 4.7–4.10 that involves also system classification.
Consider a linear/linearized system of the form (1.14), i.e.:
M€ x þ C _
x þ Kx ¼ fðtÞ; xðtÞ 2 R
n
;
ð1:129Þ
which could model a multiple-DOF discrete mechanical system, or arise from using
mode shape expansion (Sect. 1.5.3) for a continuous system (then x would be the
modal amplitudes). Here the mass matrix M is assumed to be symmetric and
positive definite; this will normally be the case when the system is properly
restricted in space, and when symmetry has not been destroyed by manipulating
individual equations (e.g. multiplying any of the n equations in (1.129) by a constant would preserve model validity but destroy the symmetry of M and K). For this
rather general linear system we next describe the classification of forces, which is
then subsequently used as a basis for classifying systems, which in turn provides a
basis for stability considerations.
1.9.1 Force Classification
Any square matrix B can be split into symmetric and antisymmetric parts as
follows:
B ¼ B S þ B A ;
ð1:130Þ
where B’s symmetric part B S = B S
T and antisymmetric part B A = –B A
T can be calculated as:
42
1 Vibration Basics