Table Of ContentINTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
C 0 URS ES AN D L E C TU RES - No. 12G
B.M. FRAEI JS de VEUBEKE
UNIVERSITY OF LIEGE
M. GERADIN
BELGIAN NATIONAL SCIENCE FOUNDATION
A. HUCK
RESEARCH ENGINEER
M.A. HOGGE
UNIVERSITY OF LIEGE
STRUCTURAL DYNAMICS
HEAT CONDUCTION
COURSES HELD AT THE DEPARTMENT
OF MECHANICS OF SOLIDS
JULY Hn.2
UDINE 1972
SPRINGER-VERLAG WIEN GMBH
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© 1972 bySpringer-Verlag Wien
Originally published by Springer-Verlag Wien New York in 1972
ISBN 978-3-211-81201-3 ISBN 978-3-7091-2957-9 (eBook)
DOI 10.1007/978-3-7091-2957-9
STRUCTURAL DVNAMICS
M. GERADIN
A. HUCK
B. FRAEIJS de VEUBEKE
Laboratoire de Techniques Aeronautiques
et Spatiales
Un iversite de Liege
PREFACE
Beveral aspeets of Btruetural Dynamies
as they relate to the Finite Element Method are sur-
veyed.
In Beetion 2 the variational prineiples
of elastodynamies of both displaeement type and st-
ress type, on whieh rests the foundation of dual
mathematieal models of finite elements. In Beetion
3 the bases required for an eigenvalue analysis of
a strueture. In Beetion 4 the derivation of the
elassieal algebraie form of eigenvalue problern
using either displaeement or equilibrium finite
element models.
The presenee of kinematieal modes in
the strueture requires generalizations of some eon-
eepts, sueh as the one of flexibility matriees,
whieh are dealt with in Beetion 5.
Beetion 6 diseusses aeeuraey of the low
frequeney model analysis resulting from methods for
redueing the number of degrees of freedom; it also
deseribes bounding algorithms to assess this aeeu-
raey.
Beetion ? eontains numeriaal appliea-
tions to plate like struetures.
4 Preface
Beetions 1 to 7 were written by GERADIN
and FRAEYB de VEUBEKE.
In Beetion 8~ Huek eompares several
methods for the numerieal eomputations of transient
response. Partieular emphasis is put on the model
aeeeleration method that exploits a low frequeney
model analysis.
Udine~ July 1972
1. Introduction
Linear structural dynamics is one of the many
field problems of engineering that can receive a variational
formulation. The classical approach is the kinematical one, and
the discretization of the Hamiltonian variational principle in
finite elements results from a polynomial approximation of the
displacement field inside each separate region. Continuity will
be secured through identification of a suitable set of genera-
lized interface displacements, in wh~ch case the kinematical
elements are said tobe conforming. Integrating the kineticand
potential energies of the finite elements leads to lagrangian,
or so-called coherent, mass and stiffness matrices.
The first really satisfactory formulation of a
dual principle, in which the kinetic energy is transformed
through satisfaction of the dynamic equilibrium equations in
a functional expressed in terms of an impulse field, is due to
Toupin [38] . Similar approaches where followed by Crandall,
Yu Chen, Gladwell and Zimmermann [ 6 J , [41] , [24] . Some nu-
merical applications to beam and plate problems were presented
by Tabarrok, Sakaguchi and Karnopp [35] , [36] . Its use as an
efficient tool in finite element applications is however very
recent [19] , [21] and has stilltobe generalized tothe three-
dimensional linear elasticity.
6 Introduction
This presentation gives a logical derivation of
the dual dynamic principle through the canonical fonn in the
spirit of the Friedrichs transfonnation, Its discretization re-
sults from a polynomial approximation of the stress field with-
in each separate region. The diffusion of the boundary tractions
will be preserved if a suitable system of generalized boundary
loads can be chosen that may be defined uniquely in terms of the
parameters of the stress field inside each element.
This paper also discusses the general procedure
for assembling equilibrium, or statically admissible finite el-
ements, in order to implement the dual principle. It will appear
that, when starting from an assumed stress field, the expression
of the kinetic energy of an element involves only a small num-
ber of interior parameters; the equilibrium approach leadsthus
in a more natural way to an "eigenvalue economizer". Zero fre-
quency modes are associated \\'ith the other parameters, which
improve the representation of the strain energy without increas-
ing the order of the eigenvalue problems. Moreover, it is shown
that there is no advantages in using the dual principle tagether
with the requirement of forcing orthogonality with respect to
all zero frequency modes. On the contrary, experience proves
that, by ignoring this unnecessary requirement, the computed
eigenvalues generally converge to the exact values by lower
bounds, hence giving precious accuracy by comparison with the
displacement approach, which always gives upper bounds.
2. THE VARIATIONAL PRINCIPLES OF ELASTODYNAMICS
2.1 Hamilton's principle
Hamilton's principle, or displacement variation-
al principle, states that for time fixed end values of the dis-
placement field U· , the Lagrangian action
~
(2.1.1)
of a conservative system takes a stationary value on the true
trajectory of the motion.
T denotes the kinetic energy of the system:
T = ..
2L I 0r u.~ ü.~ d R (2.1.2)
R
and V , its potential energy, can be split into distinct parts:
The strain energy
V1 : jw(~) dR, (2.1.3)
R
results from the integration of the strain energy density W(t)
in the domain R . If we restriet oursel ves to the infini tesim-
al strains and rotations of a linear elastic material, W can
be written as a positive definite quadratic form
W(t.) = 2, c~" r. .. f.lll
•( L( n (2.1.4)
8 Variational principles of elastodynamics
of the symmetric strain tensor:
(2.1.5) t.
~..} = 12 (o.~ u.J + o.J u~. )
mn
The set of elastic moduli defining the tensor c. . verifie.s
~}
the symmetry conditions
mn mn nm
(2.1.6) C
". .} = CJ. .~ = C." .} = cm~ }n
The relation (2.1.5) show the strain energy tobe a functional
of the first order derivatives of the displacement field:
(2.1.7) vl = 1w ( o u) d R •
According to the symmetry properties (2.1.6), we obtain themore
explicit expression
(~.1.8) Vi = -12 1 C."m .J .n 0.• u.} 0m u n d R
R
Finally, we deduce from the strain energy density W the stress
tensor conjugated to the strain tensor through
the energy relation
mn mn
(2.1.9) - c .•.J- f, mn = c."-J 0 m u n '
under the condition that the formal distinction between
and t.. in the case i ~ i be kept.
~· d-
Another contribution to the potential energy, v2,
is a potential energy associated to the displacements on the
part 3 2 R of the boundary:
