INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES EBERHARD BROMMUNDT TECHNICAL UNIVERSITY OF DARMSTADT VIBRATIONS OF CONTINUOUS SYSTEMS THEORY AND APPLICATIONS COURSE HELD AT THE DEPARTMENT FOR MECHANICS OF DEFORMABLE BODIES SEPTEMBER - OCTOBER 1969 UDINE 1969 CO U R S E SAN D L E C T U RES - N. 1 ISBN 978-3-211-81305-8 ISBN 978-3-7091-2918-0 (eBook) DOI 10.1007/978-3-7091-2918-0 Copyri~ht 1970 by Springer-Verlag Wien Originally published by Springer Vienna in 1970 First Reprint. PREFACE This booklet contains the notes of my lec tupes on vibpations of (solid) continuous systems de liveped at CISM in Fall of 1969. The lectures were presented to an auditory of engineers and physicists interested in vapious bpanches of mechanics. Stapting from vibpations of conservative, linear syst.ems I tried to give an introduction to some problems, methods of solution, and phenomena of non conservative and nonlineap systems. The examples cho sen to demonstpate the different notions and proce dures are very simple to avoid lengthy calculations which might hide the basic ideas. I would like to express my sincere thanks to the authorities of CISM, in particular to professors W.Olszak and L.Sobrero, for their kind invitation and continued interest. E.Brommundt Udine, October 1969. Introduction 5 o. Introduction Objectives of vibrational investigations are tech nical (physical, chemical), biological, economic, etc. systems. Purposes of such investigations: to "comprehend" phenomena observed (experimentally) in ac- tual systems; to "predict" the behavior, qualitatively as well as quantitative ly, of systems not yet (experimentally) tested,and of systems which are only projected as in engineering design. The procedures of these investigations are al ways similar, see Fig. O. 1. There is no way to compare mathematically the results obtained for the model with the behavior of the real system. In these lectures we shall restrict ourselves to (ct.p.8) 0' I-t a .., o g-o .... <"I-o I;j y e -' l t c n a e - o t n u th em he s o be con nstr tion wi he syst ice of t able s t sured; with i for c t o i a n a f n h ar e o Multiple inter environment Boundaries o vaguely know C v m ti nteraction with environ- ment partial- y suppressed I l l e d e h o implification) Well defined interaction with environment; well defined bound aries; list of the characteristicsof t various elements constituting the m s m e ® t s ) y el ealized s ical mod of state coor applica asic" titutive etc. solated, id model, phys Definition variables{ dinates), tion of "b laws, cons equations I ( 1:) "i o (') (1) §" "i (1) til o ...., I-< ~ (1) til ..... M- (JQ ?l ..... M-o ::s ..J s t n e s n t r I L:l'e res. ctoryt en nts " 0, (f) @ ~Meas perirnental tisfaJunsa <l agreem'", "" hnproverne ®) t @) (l)) x " " a E " " s " n of St-'lt "' o e is r ' r a , p I l l ~ a : . c D O i O e umer s) C1 th / FIG. odel on) "latiCS ical, n method sults ---1 __ / t mmatical ns of moti i rMathen® H(analyt iletc. ematical re solption s i I L_ - ctory ment ical model her calcu Mathe (equatio Math satisfa agree mathemats for furts en vo eri heat Tsl 8 Introduction CD mechanical system. We shall be concerned with the steps 0 and, mainly, of Fig. O. 1. Lit A. G. J. Macfarlane, Engineering systems analysis, G. G. Harrap & Co, London 1964 (German trans!. BI- Taschenbuch Nr. 81, Bibliogr. Institut, Mannheim 1967). R. H. Cannon, jr., Dynamics of physical systems, Mc Graw-Hill, New York 1967. Coo rdinate s 9 I. Continuous systems J. 1 Coordinates 1. 11 Reference coordinates v = Volume S = Surface Fig. 1. 1 The particle P is denoted by the Lagrangian coor- dinates which may be interpreted as its{curvilinear}coordinates in a certain reference configuration of the body. (Unique notation, continuously differentiable ---. no cracks, etc.) 1. 12 Position (spatial, Eulerian) coordinates t The position of the particle P at the time is given by 10 Continuous Systems (x(~.t),Y(K.t).z(~.t))= ( X1 (~ • t ) I XZ (~ • t ) I X3 (~ .l)) .. ~ (~. t) The systems of coordinates need not coincide. I. 2 State of the body 1. 21 Classical continuum The state is completely known far t ~ to if ! = ~(~. t) is known for all P E V and t ~ to 1.22 Modern extensions 1. 221 Thermodynamical state variables are taken in- to account. 1.222 Microstructure of the material (grains, cryst als, complex molecules) are taken into account by associating "directions" etc. with the particles ~ multipolar media. From a formal point of view both extensions mean (K.t) that the number of the state variables xi is increased, i = 1, ••• , N j N > 3 ; (x 1 (C. t) , ... , xN (~. t) )=~ (~ . t)