Detailed consideration is given to the modes and frequencies of a free rectangular Kirchoff plate subjected to in-plane stresses generated by prescribed non-uniform surface temperature distributions which are doubly symmetrical about the plate central axes. Physical understanding is sought of phenomena observed by previous investigators. Stress distributions corresponding to three different temperature distributions have first been studied and incorporated in a Rayleigh Ritz analysis to find natural frequencies and modes. All frequencies change as the temperature changes, some much more than others. All eventually vanish, one after the other, as the temperature reaches certain critical positive and negative values at which the plate goes into statically unstable buckling modes. Whether the frequencies rise or fall with rising temperature at the plate centre depends on the relative magnitudes of pairs of positive and negative critical temperatures. The modes of buckling at each pair of critical temperatures may differ greatly from one another and also from the vibration modes at zero temperature. The relationship between the square of the frequency and the temperature is then no longer approximately linear, although it is exactly so for certain simple in-plane stress distributions. Conditions have nevertheless been identified under which it is a very good approximation to the actual frequencies of the heated plate over wide temperature ranges.

This paper examines the natural frequencies and modes of transverse vibration of two simple redundant systems comprising straight uniform Euler–Bernoulli beams in which there are internal self-balancing axial loads (e.g., loads due to non-uniform thermal strains). The simplest system consists of two parallel beams joined at their ends and the other is a 6-beam rectangular plane frame. Symmetric mode vibration normal to the plane of the frame is studied. Transcendental frequency equations are established for the different systems. Computed frequencies and modes are presented which show the effect of (1) varying the axial loads over a wide range, up to and beyond the values which cause individual members to buckle (2) pinning or fixing the beam joints (3) varying the relative flexural stiffness of the component beams. When the internal axial loads first cause any one of the component beams to buckle, the fundamental frequency of the whole system vanishes. The critical axial loads required for this are determined. A simple criterion has been identified to predict whether a small increase from zero in the axial compressive load in any one member causes the natural frequencies of the whole system to rise or fall. It is shown that this depends on the relative flexural stiffnesses and buckling loads of the different members. Computed modes of vibration show that when the axial modes reach their critical values, the buckled beam(s) distort with large amplitudes while the unbuckled beam(s) move either as rigid bodies or with bending which decays rapidly from the ends to a near-rigid-body movement over the central part of the beam. The modes of the systems with fixed joints change very little (if at all) with changing axial load, except when the load is close to the value which maximizes or minimizes the frequency. In a narrow range around this load the mode changes rapidly. The results provide an explanation for some computed results (as yet unpublished) for the flexural modes and frequencies of flat plates with non-uniform thermal stress distributions.

A classification of the possible free wave motions in a three-layered composite thick cylinder is presented. The governing equations of motion for free wave motion for isotropic and orthotropic elastic media are given and a combined solution is developed for a thick cylinder made of three different materials, such as orthotropic-isotropic-orthotropic layers. The use of Bessel and special Frobenius series is required to obtain a correct, closed form solution for the propagating waves. Numerical results are given in the form of dispersion curves and these are discussed. A correct approach to the calculation of the cut-off frequencies is presented. It is shown that the wave speeds of decoupled longitudinal-shear motion in orthotropic shells is dependent upon the ratio of the different shear moduli,[formula]. The influence of a thin, soft rubbery material at the centre of a sandwich configuration is thoroughly analyzed. The effect of the isotropic core properties (its stiffness and thickness) on the dynamics of the wave motion is investigated.

A method is presented for studying harmonic wave propagation in thick circular cylinders, which are orthotropic. The analysis is carried out within the framework of the complete three-dimensional theory of elasticity. The displacements in the circumferential and longitudinal directions are taken in the form of trigonometric functions, while the radial displacement field is modelled by Frobenius power series developed through the thickness of the shell. It is shown that the Frobenius method is a powerful tool in solving for wave motions in anisotropic elastic continua. Dispersion curves are presented for axisymmetric and asymmetric waves in the case of transverse isotropy. The methodology breaks the problem down into four different tasks, which have then to be treated separately. They are as follows: the axisymmetric wave motion,n= 0; the flexural or beam-type wave motion,n= 1; the lobar wave motion,n= 2; finally the wave motion with higher circumferential wavenumbers,ngreater than 2. Exact solutions have been found for waves travelling in thick orthotropic shells. These solutions may serve as benchmark solutions for comparison with approximate treatments of similar problems.

The behavior of a feedforward active isolation system subjected to actuator output constraints is investigated. Distributed parameter models are developed to analyze the system response, and to produce a transfer matrix for the design of an integrated passive–active isolation system. Cost functions considered here comprise a combination of the vibration transmission energy and the sum of the squared control forces. The example system considered is a rigid body connected to a simply supported plate via two isolation mounts. The overall isolation performance is evaluated by numerical simulation. The results show that the control strategies which rely on unconstrained actuator outputs may give substantial power transmission reductions over a wide frequency range, but also require large control force amplitudes to control excited vibration modes of the system. Expected power transmission reductions for modified control strategies that incorporate constrained actuator outputs are considerably less than typical reductions with unconstrained actuator outputs. The active system with constrained control force outputs is shown to be more effective at the resonance frequencies of the supporting plate. However, in the frequency range in which rigid body modes are present, the control strategies employed using constrained actuator outputs can only achieve 5–10 dB power transmission reduction, while at off-resonance frequencies, little or no power transmission reduction can be obtained with realistic control forces. Analysis of the wave effects in the passive mounts is also presented.

The relationship between the free wave motion and the natural flexural modes of beams is re-examined. The reflection of both propagating and evanescent waves, incident upon a linearly constrained boundary, is first analyzed. Boundary conditions are identified which cause no evanescent wave to be reflected when a propagating wave is incident, and vice versa. The phase differences between the incident and reflected waves are analyzed for the case in which two waves (one propagating and one evanescent) are incident. The phase-closure principle is formally proved for single span uniform beams and is used to set up the exact frequency equation for a fully fixed beam. Whereas the principle has traditionally been applied to the propagating wave motion, it is applied in this paper also to the evanescent wave motion. Frequency equations having quite different forms are thereby obtained, but they have identical roots. A physical interpretation becomes obvious for the conventional frequency equation for the fully fixed beam.

An exact analysis is presented of the vibration response of an infinite beam on periodic supports, subjected to a transverse harmonic point force. The supports must all be the same and can be simply supported or be generally linear with elastic, inertial and dissipative properties. The total response is found as the sum of the flexural wave fields generated by the applied force and the infinite number of support reaction forces and moments. The concept of phased arrays of forces and moments is used to sum the support-generated wave fields. This utilizes the propagation constants of free-wave motion in the periodic beam. Equations for either four, six or eight of the unknown complex reactions (depending on the nature of the supports) are set up and solved numerically. This finite number is sufficient to permit the calculation of the beam displacement at any point and of all the other reactions. Some computed values of the beam direct receptance are presented to demonstrate its variation with forcing frequency, the effect of the location of the excitation force and the effect of changing the elastic properties of the supports.

An exact analytical method is presented for the vibration response of a finite, three-layered, rectangular sandwich plate with a visco-elastic core, subjected to a harmonic line force which varies sinusoidally across the plate. Uniform parallel stiffeners (which may all be different) span the plate between one pair of simply supported edges. The other pair of edges may have any degree or type of uniform constraint. In the analysis the known flexural wave motion in an infinite parallel unstiffened plate subjected to a single harmonic line force or moment is utilized. A matrix equation is set up for the reactions imposed on the plate by the stiffeners and for the amplitudes of wave motion reflected from the ends of a finite plate. The sandwich core may have large or small amounts of damping. Results computed from the theory are presented and are shown to compare well with experimental data. The influence of the stiffener and core properties on the plate harmonic response is readily determined.

Flat plates and cylindrical shells with identical stiffeners at regular intervals constitute spatially periodic structures, and specially convenient methods of analysis are available for the study of their vibrations. Some of the methods are suitable for the inclusion of the effects of fluid loading from adjacent acoustic media. This paper outlines the nature of the free wave motion that can occur in periodic structures that are stiffened either in one direction or in two orthogonal directions. It is shown how their responses to distributed sound fields can be determined by using displacement functions consisting of a series of space harmonics or of simple assumed polynomial modes. The sound that is reradiated or transmitted by the structure is also found. Methods that have been developed for analyzing the response and radiation due to line or point forces are reviewed. Recent developments in the analysis of periodically stiffened cylindrical shells are described. The hierarchical finite element method has been applied to determine flexural wave speeds in both flat reinforced plates and in reinforced cylinders. Symbolic computing has been used to set up the relevant stiffness and mass matrices. Some computed results are presented.

A wave approach is developed for the exact analysis of the harmonic response of uniform finite beams on multiple supports. The beam may be excited by single or multi-point harmonic forces or moments; its supports may have general linear characteristics which may include displacement-rotation coupling. Use is made of the harmonic response function for an infinite beam subjected to a single-point harmonic force or moment. The unknowns of the finite beam problem are the support reaction forces/moments and the magnitudes of four waves reflected from the ends of the beam. Equations are presented for the response of a single-bay beam with various support conditions and subjected to single-point harmonic excitation. The same equations, but with the simple addition of further straightforward terms on the right-hand side, are used for multi-point excitation. The effects of damping are easily incorporated. Equations for multi-supported beams are also presented together with illustrative computed frequency-response curves. Natural frequencies have been calculated by finding resonance frequencies of very lightly damped beams. These compare impeccably with the results of other investigators.

A thin cylindrical shell is considered, stiffened axially by equi-pitched, identical stringers and circumferentially by equi-pitched, identical frames. Generality of strigner and frame section is allowed. The structure is analyzed as a two-dimensional periodic structure by using wave propagation techniques in conjunction with the hierarchical finite element method. Results are presented in the form of phase-constant surfaces plotted against frequency. It is shown that free wave-motion can propagate in the infinite structure from zero frequency. A small frequency band has been identified in which predominantly flexural waves cannot propagate. Some experiments, which have been performed on a one-quarter scale fuselage model, confirm the main findings of the theoretical analysis.

A flat plate, reinforced by a regular orthogonal array of uniform beams, is analyzed by using techniques developed for studying wave propagation in two-dimensional periodic structures. A “plane-wave” type of motion is considered which may be characterized by different propagation phase constants in the x- and y-directions. The hierarchical finite element method is used to set up the governing equations of free wave motion, and these are then solved as an eigenvalue problem for the frequencies at which particular waves will propagate. Plots of phase constant surfaces vs. frequency are presented for a number of different plate-beam configurations. Excellent agreement is found between some of these and the results of earlier investigators. When the plate is supported by flexible beams in both directions, wave propagation is found to commerce at zero frequency. At higher frequencies alternating (and overlapping) attenuation and propagation bands occur. The nature and explanations of these are discussed. Wave speed surfaces vs. frequency are also presented and these give insight into the critical coincidence frequencies of the plate under acoustic excitation.

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.

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.

Wave propagation around a cylindrical shell which is reinforced at regular intervals by flexible stiffeners parallel to the shell generator is considered. The shell itself is restricted to a section between two circumferential frames on to which the shell is simply supported. The structure effectively constitutes a one-dimensional periodic system and is analyzed as such. Four degrees of freedom are allowed between each periodic element and equations are set up for the four pairs of propagation constants which characterize the possible wave motions. Symmetric or asymmetric stiffener sections may be accommodated in the analysis together with structural damping. Computed propagation constants are presented for two different stiffener cross-sections, each pitched at two different intervals around the shell. Natural frequencies are calculated for one of these.

The purpose of this paper is to show how a ceramic layer attached to a two-layered (elastic/viscoelastic) beam alters the wave propagation mechanism in the beam. The ceramic layer is assumed to possess mass but not longitudinal stiffness. Shear deformation, rotatory, longitudinal and transverse inertia forces are all included in the analysis. The equations of motion of the layered beam are derived by using the virtual work principle. Relevant dispersion curves for an infinite beam are presented and discussed, and are compared with dispersion curves obtained from a number of simplified theories. The influence of the inertia coupling between different wave types, caused by the ceramic layer, has been examined. The loss factors of the different waves are found to be critically dependent upon the inertia coupling and wave-number in particular wave-number regions. This occurs when uncoupled waves of different types have close wave-speeds. Under these circumstances a coupled wave can exist which has much less damping than any of its constituent uncoupled waves.

The method of receptance analysis is used to set up a frequency equation for the free vibration modes of a one-dimensional periodic lattice (mass-spring system) containing a disorder which is itself a one-dimensional periodic lattice. The concepts of the propagation constant and wave-receptance function are used to determine the receptances of the component systems, and these are used to set up a simple frequency equation. An accurate root-searching computer programme has been used to find the natural frequencies and corresponding modes of particle displacement. Some computed results are shown to demonstrate the capability of the method and programme. Special attention is given to modes which occur in the frequency “forbidden” zone, and receptance methods are used to derive formulae for the frequencies of systems with a single-mass disorder. The wider usefulness of the method is briefly discussed.

Periodic structure theory is used to study the interactions between flexural and longitudinal wave motion in a beam (representing a plate) to which offset spring-mounted masses (representing stiffeners) are attached at regular intervals. An equation for the propagation constants of the coupled waves is derived. The response of a semi-infinite periodic beam to a harmonic force or moment at the finite end is analyzed in terms of the characteristic free waves corresponding to these propagation constants. Computer results are presented which show how the propagation constants are affected by the coupling, and how the forced response varies with distance from the excitation point. The spring-mounted masses can provide very high attenuation of both longitudinal and flexural waves when no coupling is present, but when coupling is introduced the two waves combine to give very low (or zero) attenuation of the longitudinal wave. The influence of different damping levels on spatial attenuation is also studied.

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.

A description is given of an apparatus which enables the measurement of damping during vibration with non-zero mean strain. The loading condition is torsion on a pair of cylindrical specimens, which may be solid or hollow. Mean strain is applied by opposite pre-twist in the two specimens. Damping is determined from energy input during steady resonant vibration, although both frequency response and free decay techniques can also be used. Operating frequency is in the range 20-100 Hz and vibratory surface shear strain amplitude up to +or-0.003 can be applied, with mean surface shear strain up to large values (in principle, unlimited). An account is given of the apparatus development and operation, with particular reference to the use of a test specimen/low damping specimen combination which effectively halves the required power input and which allows a correction for extraneous losses to be applied.