D.J. Mead

The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions

Abstract The sixth-order differential equation of motion is derived in terms of the transverse displacement, w, for a three-layer sandwich beam with a viscoelastic core. Mathematical expressions in terms of w are found for a variety of beam boundary conditions. The solution of the differential equation by the method of Di Taranto is shown to yield a special class of complex, forced modes of vibration which are completely uncoupled. These complex modes can only exist when the beam is externally excited by specific “damped normal loadings” which are also complex and which are proportional to the local transverse inertia loading on the beam. Use of these modes in the analysis of the forced vibration problem leads to a simple series form of solution. The orthogonality of these complex modes is briefly discussed and proved.

A general theory of harmonic wave propagation in linear periodic systems with multiple coupling

Abstract A general theory is presented of harmonic wave propagation in one-dimensional periodic systems with multiple coupling between adjacent periodic elements. The motion of each element is expressed in terms of a finite number of displacement coordinates. The nature and number of different wave propagation constants at any frequency are discussed, and the energy flow associated with waves having real, complex or imaginary propagation constants is investigated. Kinetic and potential energy functions are derived for the propagating waves and a generalized Rayleighs Quotient and Rayleighs Principle for the complex wave motion have been found. This is extended to yield a generalized Rayleigh-Ritz method of finding approximate, yet accurate, relationships between the frequencies and propagation constants of the propagating waves. The effect of damping is also considered, and a special class of “damped forced waves” is postulated for hysteretically damped periodic systems. An energy definition for the loss factor of these waves is found. Briefly considered is the two-dimensional multi-coupled periodic system in which a simple wave motion analogous to a plane wave propagates across the whole system.

Free wave propagation in periodically supported, infinite beams

The notion of propagation constants for the free harmonic motion of infinite beams on identical, equi-spaced supports is first reviewed. Expressions are then derived for the flexural propagation constants for beams on rigid supports which exert elastic rotational restraint, and also for beams on flexible supports. A beam on rigid supports has one propagation constant and a beam on flexible supports has two propagation constants for each frequency. Detailed consideration of the propagation constant leads to the conclusion that a freely propagating harmonic flexural wave in such a beam must be regarded as a wave group, having components of different wavelength, phase velocity and direction. The magnitudes of the components in some special cases are examined. The interaction between these groups and convected pressure fields is considered in qualitative terms. A mechanism is shown to exist whereby slow, subsonic convected pressure fields can generate flexural waves of supersonic velocity which can radiate sound.

Wave propagation and natural modes in periodic systems: I. Mono-coupled systems

In this paper the relationship is studied between the bounding frequencies of the propagation zones of mono-coupled periodic systems and the natural frequencies of the individual elements of which the system is composed. It is also shown how these relate to the natural frequencies of finite mono-coupled periodic systems. The concept of the characteristic receptance of a free wave in the periodic system is developed, and the manner in which this receptance varies with frequency is studied. It is used to examine the attenuation and phasechange undergone by a free wave when it impinges on and is reflected from a non-dissipative boundary. This leads on to a simple physical explanation for the occurrence of natural frequencies of finite systems in either the attenuation zones or propagation zones of the infinite system. Distinction is drawn between systems whose elements are symmetric or unsymmetric, positive-definite or semi-definite.

Wave propagation and natural modes in periodic systems: II. Multi-coupled systems, with and without damping

Free waves can propagate through periodic systems only in particular frequency zones. Equations for the bounding frequencies of these zones are obtained in terms of the receptance matrices of the elements of multi-coupled systems. The relationship between these frequencies and the natural frequencies of a single element of the system is considered, particular attention being given to elements which are symmetrical. The nature of the characteristic wave motions is studied, and a characteristic receptance matrix for a characteristic wave is defined. This is used to introduce the study of reflection of a characteristic wave from a boundary. The equations governing the reflection process are set up, and used to formulate the equations for the natural frequencies and modes of a finite periodic system with arbitrary boundaries. The modes are represented by superpositions of opposite-going pairs of characteristic waves, in terms of which a simple physical description of the natural wave motion is presented. Undamped systems are considered initially, but it is then shown that all the equations so derived for free wave motion are applicable to the damped forced motion of hysteretically damped multi-coupled systems. The bounding frequencies and loss factors of purely propagating damped forced wave motion are considered in relation to the resonant frequencies and loss factors of the damped forced normal modes of a single element. Finite periodic systems of damped multi-coupled elements are finally studied.

A new method of analyzing wave propagation in periodic structures; Applications to periodic timoshenko beams and stiffened plates

A response function is found for an infinite, uniform, one-dimensional structure which is subjected to an array of harmonic forces or moments, spaced equidistantly, and which have a constant phase or ratio between any adjacent pair. Receptance functions are derived for these “phased arrays”. They are used to set up a general determinantal equation for the propagation constants of the infinite structure when it is made periodic by the addition of an infinite set of regular constraints. They are also used to set up equations for the response of the structure to a convected harmonic pressure field. The method enables the equations for the propagation constants and for the response to convected loading to be set up with much greater facility than by earlier methods. It only requires a knowledge of the response function of the infinite uninterrupted structure under a single-point harmonic force or moment. The general equation for the propagation constants is used to study (a) a simply supported periodic Timoshenko beam, and (b) a parallel plate with periodic beam-type stiffeners. Some calculated propagation constants are presented and discussed. The periodic plate results are relevant to integrally stiffened skins of the type used in aeroplanes.

A comparison of some equations for the flexural vibration of damped sandwich beams

This paper compares the theories of flexural vibration of damped, three-layer sandwich beams as presented by Yan and Dowell, and by DiTaranto and Mead and Markus. Depending on the assumptions made about the internal shear stress distribution, the differential equation of transverse flexural displacement is either of fourth or sixth order. The inclusion of the effects of face-plate shear deformation and longitudinal inertia in the analysis yields a sixth order differential equation if the beam section is symmetric, and an eighth order equation if the section is unsymmetric. Flexural wave speeds and loss factors computed from the theories are presented and compared. The DiTaranto and Mead and Markus equations yield reliable values provided the flexural wavelength is greater than about four face-plate thicknesses. The Yan and Dowell equations yield reliable values only at much greater wavelengths or when the central layer in the sandwich is very thick.

Loss factors and resonant frequencies of encastré damped sandwich beams

Abstract The differential equation for the damped normal modes of a three-layer encastre sandwich beam is used, in conjunction with appropriate boundary conditions, to determine the characteristic equation for the resonant frequency, loss factor and modal roots. An iterative method of solving this equation is presented, and computed values for a wide range of parameter values are shown. The six modal roots are critically examined, and it is observed that all of them have both real and imaginary parts. The implication of this is that beams with different boundary conditions have different loss-factor/resonant frequency relationships. The loss factors of the encastre beams vary with the shear parameter, geometric parameter and core loss factor in a manner similar to those of simply-supported beams, but reach their maxima at values of the shear parameter which are greater than the simply-supported values. An approximate method of determining the resonant frequency and loss factor is presented.

Space-harmonic analysis of periodically supported beams: response to convected random loading

Abstract The solution for the response of stiffened beams due to a spatial and temporal harmonic pressure has been obtained in the form of a particular series of space harmonics, evolved from considerations of progressive wave propagation. The superiority of this method over the classical normal mode approach is indicated. It is applied to obtain the r.m.s. curvature at a point on a periodically supported beam excited by a random acoustic plane wave or boundary layer pressure fluctuation. The results obtained with different numbers of terms in the series are compared with known closed-form solutions. When seven terms are included, results bear good agreement with the exact solution and as few as three terms yield a solution of acceptable accuracy. The method of space harmonics can be adapted to the case of orthogonally stiffened plates which are excited by pressure fields convected across the plates at oblique angles to the direction of stiffeners. The general method should be applicable to the estimation of response of orthogonally stiffened cylindrical structures.

Free vibration of a thin cylindrical shell with periodic circumferential stiffeners

Abstract The theory is developed for obtaining the propagation constants of a thin uniform cylindrical shell, periodically stiffened by uniform circular frames of general cross-section. The free wave motion is analyzed and the stop and pass bands of free wave motion in the structure are located. Hysteretic damping is included. The natural frequencies of two stiffened finite cylindrical shells are deduced. The relative effects of the frame cross section and pitch on the free vibration characteristics of the whole structure are discussed.