Structural Dynamics: Heat Conduction PDF
Preview Structural Dynamics: Heat Conduction
INTERNATIONAL 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 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concemed specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. © 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:
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