E. Sanchez Palencia

Vibration and Coupling of Continuous Systems; Asymptotic Methods

Real problems concerning vibrations of elastic structures are among the most fascinating topics in mathematical and physical research as well as in applications in the engineering sciences. This book addresses the student familiar with the elementary mechanics of continua along with specialists. The authors start with an outline of the basic methods and lead the reader to research problems of current interest. An exposition of the method of spectra, asymptotic methods and perturbation is followed by applications to linear problems where elastic structures are coupled to fluids in bounded and unbounded domains, to radiation of immersed bodies, to local vibrations, to thermal effects and many more.

Integral Approach to Sensitive Singular Perturbations

The main purpose of this chapter is to give general ideas on a kind of singular perturbation arising in thin shell theory when the middle surface is elliptic and the shell is fixed on a part of the boundary and free on the rest, as well as an integral heuristic procedure reducing these problems to simpler ones. The system depends essentially on the parameter e equal to the relative thickness of the shell. It appears that the “limit problem” for e = 0 is highly ill posed. Indeed, the boundary conditions on the free boundary are not “adapted” to the system of equations; they do not satisfy the Shapiro–Lopatinskii (SL) condition. Roughly speaking, this amounts to some kind of “transparency” of the boundary conditions, which allows some kind of locally indeterminate oscillations along the boundary, exponentially decreasing inside the domain. This pathological behavior only occurs for e = 0. In fact, for e > 0 the problem is “classical” When e is positive but small, the “determinacy” of the oscillations only holds with the help of boundary conditions on other boundaries, as well as the small terms coming from e > 0.

Examples of Nonstandard Vibrations and Coupling

This chapter is devoted to the study of certain examples of mechanical systems which do not enter in the elementary framework of Chapter I, at least in a natural way. The system of equations of thermoelasticity (Section 1) may be considered as the coupling of a hyperbolic system (the elastic part) and a parabolic equation (the thermal part). As a consequence, the generator of the semigroup is neither self-adjoint nor skew-adjoint and furnishes a good example of the power of semigroup theory in proving existence and uniqueness of solutions of evolution problems. The second example (Section 2) is the system of viscoelasticity involving integro-differential terms. Sections 2 to 4 are devoted to problems where the imbedding of the spaces V ⊂ H is not compact. As a consequence, the operator A is not anticompact, and its spectrum may be much more involved than in Chapter I. The concept of essential spectrum is introduced and examples in mechanics are developed. Sections 6 to 8 contain examples of mechanical systems which are, in fact, in the context of Chapter I, but only after certain transformations. To some extent, the different sections may be read independently.

The Helmholtz Equation in Unbounded Domains

This chapter is devoted to the classical theory of the reduced wave equation in domains containing a neighborhood of infinity. The main difference with respect to the standard vibration problem of Chapter I is that the imbedding H1 ⊂ L2 is no longer compact. The spectrum of the Laplace operator (with standard boundary conditions) is purely continuous. The corresponding spectral family is considered in Section 5. The solution of the nonhomogeneous boundary value problems requires a radiation condition (Sections 3 and 4). These solutions describe the limit state as t → + ∞ of the corresponding time-dependent wave equation. The radiation of energy towards infinity implies some decay of solutions as t → +∞. There exist (complex) scattering frequencies ω s which are some sort of eigenvalues for the vibration (Sections 3, 4, and 8). The relation with the corresponding time-dependent vibration (which exhibits wave fronts) is explained in Section 2. Most of the material of this chapter is presented in the case of space dimension 3, but certain results hold true in the more difficult case of dimension 2.

Scattering Problems Depending on a Parameter. Elastic Structure-Fluid Interaction in Unbounded Domains

This chapter is devoted to some applications and generalizations of the material of Chapter VIII for problems of interest in mechanics. Moreover, we consider, in particular, problems depending on a parameter e which have a different structure for e > 0 and e = 0. Specifically, for e = 0 (unperturbed problem) we have a standard problem with discrete spectrum and real eigen-frequencies, which becomes, for e > 0, a problem with continuous spectrum. Moreover, some of the scattering frequencies ω(e) tends, as e ↘ 0, to the real eigenfrequencies of the unperturbed problem. As a consequence, lm{ω(e)} is small for small e.

Formal Perturbation Methods

The discussion in the preceding chapters, in particular spectral perturbation theory, was of a mathematically rigorous character. Unfortunately, many systems of the real world involve perturbations which do no fit within the framework of Chapter V. Typically, this occurs in boundary value problems involving a differential equation with a coefficient e tending to zero in front of the higher order derivatives. As e becomes 0, the order of the equation decreases and some boundary conditions are violated so that the perturbation exhibits singularities at the corresponding boundaries. Physical intuition led theorists and engineers concerned with mechanics to devise formal (heuristic) methods for studying certain types of such perturbations. One of these is the method of matched asymptotic expansions. This technique allows us to study problems where the perturbed solution has different structures in different regions. The typical example is the boundary layer generated by small viscosity; indeed, viscosity effects are only pronounced in the vicinity of a wall and the asymptotic expansions for small viscosity assume different forms in the inner and outer (to the boundary layer) regions. The compatibility of these two regions is expressed by suitable “matching” conditions. This technique provides formal asymptotic expansions in very general perturbation problems, but unfortunately it is mostly based on physical (or geometrical) intuition.

Some Classical Vibration Problems

In this chapter we present certain physical problems that are covered by the theory of Chapter I, and which serve as models in several branches of mechanics of continua. These two chapters constitute the first, elementary part of this book.

Elements of Operator Theory

In this chapter we present an outline of the theory of operators in Banach spaces which will be used freely in the sequel. The material is classical so we state the results without proof but, occasionally, with explanations. The proofs may be found in the classical treatises, for instance, in Brezis [1], Kolmogorov and Fomin [1], and Vulikh [1] (elementary exposition), or Dautray and Lions [1], Dunford and Schwartz [1], Reed and Simon [1], Riesz and Nagy [1], Smirnov [1, Vol. 5], and Yosida [1] (more complete theory). Special attention is paid to the theory of semigroups. In particular, we give a complete proof of the Lumer-Phillips theorem which is systematically used in the sequel. In this connection the reader is referred to Dautray and Lions [1], Kato [1], and Pazy [1]. Regularity theory for elliptic equations is often used throughout this book. Certain elements of the theory, including transmission problems for elliptic equations and systems, are presented in Section 9. The chapter concludes with a very brief account of the Lions-Magenes theory of very weak solutions (Section 10). Generally speaking, the presented material is valid for either real or complex spaces, and certain sections are slightly ambiguous with this respect. Nevertheless, spectral theory (in particular, for non-self-adjoint operators), only makes sense for complex spaces.

Perturbation of Vibrating Systems

In this chapter we apply the perturbation methods of Chapters V and VI to study asymptotic properties of vibrating systems involving a small parameter e. Most of the examples were considered, for fixed values of e, in Chapters II and IV. Several kinds of stiff problems in low frequency vibration are considered in Sections 1 to 4. The case of high frequencies and other problems involving spectral families are considered in Sections 5 and 6. Sections 7 and 8 are devoted to more classical problems involving boundary layers. An example of the splitting of an eigenvalue with infinite multiplicity appears in Section 9. The rest of the chapter (Sections 10 to 13) is devoted to problems with masses concentrated in small regions. To some extent, the different sections may be read independently.

Classical Theory of Vibration for Systems with Infinitely Many Degrees of Freedom

Chapters I and II constitute the first part of this book, which consists of an elementary, largely classical exposition of vibration theory of nondissipative systems with infinitely many degrees of freedom and discrete spectrum. The abstract theory is presented in this chapter, whereas physical examples are discussed in the following chapter.