W.G. Price

Coupled bending and twisting of a timoshenko beam

Abstract Allowance is made for shear deflection and for rotary inertia of a non-uniform beam that executes coupled bending and twisting vibration. Principal modes are found, orthogonality conditions established and modal equations of forced motion derived.

On the relationship between “dry modes” and “wet modes” in the theory of ship response

Abstract It has been suggested that ship strength might be estimated on the basis of linear modal analysis Two distinct approaches have been proposed, employing “wet modes” and “dry modes”. The relationship between these two alternatives is examined, and some possible simplifications of the theory are noted.

The Uses of Functional Analysis in Ship Dynamics

Generally speaking, linear theory is unsatisfactory for the analysis of ship manoeuvres. The reason for this probably lies in the representation of fluid forces and moments, which are normally specified (on the basis of quasi-steady flow) by ‘slow motion derivatives’. Fluid forces and moments are at best represented only very crudely even though they play a dominant part in ship motions. It is known that time history effects must exist, in that the flow conditions at some instant cannot uniquely determine the flow forces and moments at that instant. Failure to allow for this is probably the major potential source of weakness in most techniques of analysing ship motion. A method is shown by which the effects of time history may be allowed for when specifying fluid force or moment. The specification, which is in terms of a Volterra series, has an approximate linear form. The standing of slow motion derivatives is examined in the light of this improved linear specification. There is no evident major obstacle to the application of the new approach to any of the branches of hydrodynamics conventionally studied in naval architecture, e. g. resistance in waves, use of the planar motion mechanism for model testing, directional stability and control (in calm water or in waves and on the surface or submerged).

On modal analysis of ship strength

An approach is formulated by which structural dynamics of ships may be analysed in a linear modal form. By employing the principal modes of the ship in vacuo, simple orthogonality relations can be retained without dependence on the necessarily approximate techniques used to estimate fluid forces. It is also possible to identify modal contributions to mass, damping and stiffness for the hull and for the hydrodynamic actions separately. Those contributions of hydrodynamic origin may depend significantly on time history effects which can be measured by means of a model test; these effects can be admitted into the ship strength analysis, it is believed for the first time. It is shown how existing modal theories of ship strength and theories of seakeeping (i. e. of ‘rigid body’ motion in a seaway) fit into this more general analysis.

Application of functional analysis to oscillatory ship model testing

The need for data relating to fluid forces and moments has led to general acceptance of oscillatory testing of ship models. Although the technique is well established, certain problems still attend the interpretation of results. The nature of the difficulties is explained and they are elucidated by making allowance for time history effects, using functional analysis. Allowance for these effects in this way also establishes that certain results which have hitherto been assumed to require nonlinear representations are in fact capable of very accurate linear specification.

The fifth annual fairey lecture: On the linear representation of fluid forces and moments in unsteady flow

When a rigid body departs from a uniform motion in a fluid, it is usual to assume that the forces and moments exerted on the body at any instant by the fluid are determined by the motion at that instant. This is the assumption of “quasi-steady flow” and it is on this basis that the fluid forces and moments are represented. Although this approach often suffices for aircraft it is known to be quite inadequate for ships, with the result that complicated and diverse non-linear fluid representations have been devised. It has recently been found, however [1, 2], that an unambiguous representation of the fluid forces and moments on a ship can be formulated in terms of Volterra series. This theory allows the assumption of quasi-steady flow to be relaxed and also has a simple approximate form. The latter does not destroy the linearity of the representation with which fluid forces and moments may be specified whilst the ship performs a non-steady motion. This fresh linear theory, which may well supplant the semi-empirical non-linear theories in current use, is explained and developed in simple terms. In particular, compatible definitions of “fluid derivatives” and “oscillatory coefficients” are presented.

AN INTRODUCTION TO SHIP HYDROELASTICITY

Abstract The linear theory of ship hydroelasticity is not yet familiar in naval architecture, yet it provides the most powerful techniques of investigation available today. The background of that theory is explained in very simple terms, by using the concept of a “uniform ship”—that is, of a uniform floating beam. So rudimentary is this idealization of a ship that no claim can be made that numerical results have much practical significance; nevertheless, the underlying ideas are those employed in practical studies and some typical results are given for an actual ship (an old destroyer).

The vibration characteristics of a beam with an axial force

Abstract Equations of motion are found for a non-uniform damped Timoshenko beam with a distributed axial force. Principal modes may be extracted by numerical means when the boundary conditions are specified, and the appropriate orthogonality conditions are given. The theory of linear forced vibration can thus be derived. It is an implicit requirement that all axial forces are conservative. That is to say, tangential, follower and partial follower axial forces (whether applied at an extremity or distributed along the beam) are excluded.

A note on structural damping of ship hulls

Abstract Existing information on the structural damping of ships is far from satisfactory. It cannot be calculated and it can only be measured in the presence of hydrodynamic damping, whose nature and magnitude are also somewhat obscure. Yet it is very important. Symmetric responses to wave excitation can be estimated on the basis of existing hydrodynamic theories, with use of rough estimates of hull damping; our limited knowledge of structural damping is then only likely to be a handicap with heavy slender ships and/or fast ships. Much less is known about antisymmetric responses to waves, either as regards the means of estimating them or the appropriate levels of hull damping. Vibration at higher frequencies, due to excitation by machinery (notably propellers), is limited by structural damping to a much greater extent that it is by the fluid actions of the sea. Damping measurements at these frequencies therefore give more accurate estimates of hull damping. Even so, the estimation of ship vibration responses to excitation by machinery remains a matter of considerable difficulty.

On the nature of slow motion derivatives

It has been shown that functional representation of non-steady fluid actions is more general than representation by “derivatives”, in the sense that the latter can be deduced from the former. Methods are examined by which the derivatives may be found from the appropriate functionals. It is shown that, while they lead to the same results in general, they do so only in a restricted sense; the techniques discussed shed some light on the (necessarily somewhat arbitrary) definition of a derivative.