Showing papers on "Rotary inertia published in 1968"

Rotary Inertia and Shear in Beam Vibration Treated by the Ritz Method

Abstract: The natural frequencies of vibration of a cantilever beam allowing for rotary inertia and shear deformation are obtained by the approximate Ritz method. The workability of the method is dependent upon the approximating functions chosen for the dynamic displacement curves. A series of characteristic functions representing the normal modes of vibration of cantilever beams in simple flexure is used as the approximating functions for both deflections due to flexure and shear deformation. Good agreement is shown between frequencies obtained by the Ritz method and those resulting from an analytical solution. The effect upon the natural frequencies of allowing for rotary inertia alone is shown and it is seen to increase rapidly with mode number.

Calculation of Added Virtual Mass and Added Virtual Mass Moment of Inertia of Ship Hull Vibration by the Finite Element Method

Abstract: The added virtual mass and the added virtual mass moment of inertia induced by the vibration of a ship are defined as the equivalent mass and the equivalent mass moment of inertia deduced from the effects of water surrounding a vibrating ship. These equivalent mass and the equivalent mass moment of inertia can be estimated either from the inertia force (or moment) concept or from the kinetic energy concept as long as the ideal fluid is concerned, and the solution of the Laplace's differential equation is required in both cases.This paper, dealing with the problem by the use of the inertia force (or moment) concept, shows that the so-called “finite element method” can be applied, with the aid of a high speed digital computer, to obtain the approximate solution of the Laplace's differential equation with any arbitrary boundary condition. In order to examine the accuracy of this method, the inertia coefficients Cv, CH and CT of vertical, horizontal and rotational vibrations of bars of circular and rectangular cross section are computed first by this method. Then, as an application of this method, the effects on the inertia coefficients of the restricted water and of the bilgekeel are shown numerically.

Vibration characteristics of tapered cantilever beams

Abstract: This thesis is concerned with the vibration characteristic of tapered rectangular cross-section cantilever beams with and without pretwist. A number of conventional methods of analysis of vibration problems are investigated and a finite difference method of solution of the differential equation of motion of a beam of uniform cross-section is presented. In this method the equation of motion is reduced to a set of linear simultaneous algebraic equations by replacing the derivatives in the equation of motion by their corresponding central difference relationships using a finite step length. The resulting simultaneous equations are expressed as a matrix equation and the natural frequencies and their corresponding mode shapes are obtained by an iteration technique. Successive closer approximations to the actual results are obtained by using successive smaller step lengths thus increasing the size of matrix. An extrapolation formula for obtaining a very close approximation to the true result from successive approximation is developed. The results are compared to the work of other investigators using other methods of analysis and good agreement is obtained. The method is extended to solve the equation of motion of tapered and pretwisted tapered beams in bending vibration, with and without including the shear and rotary inertia effects. The method is also extended to solve the equation of motion of torsional vibration of tapered and pretwisted tapered beams with and without warping effect. The effect of width and depth tapers on the bending vibration of a cantilever beam is studied by considering a beam of rectangular cross-section, thus eliminating the coupling due to non-coincident mass and elastic axis. The theoretical results obtained by the finite difference method for the frequencies and mode shapes of the first five modes for various width and depth tapers are presented. Experimental results obtained showed very good agreement with the theoretical results. The effect of depth and width tapers on the coupled bending-bending vibration of a beam is studied by considering a beam of square cross-section at the root, thus eliminating the coupling purely due to width to depth ratio of the beam cross-section. Theoretical results obtained showed good agreement with the experimental results for the first six modes presented. The effect of the root width to depth ratio on the frequency of vibration of a pretwisted tapered beam is investigated and the results are presented for tapered pretwisted beam for a range of root width to depth ratio. The effects of shear deformation and rotary inertia on the frequency of vibration of tapered and tapered pretwisted beams are studied and the theoretical results obtained are presented for a few selected taper ratios. Results obtained by applying the finite difference method in the analysis of torsional vibration of uniform cross-section beams are compared with the exact solution available in order to establish the accuracy of the method. Using this method the characteristics of tapered beams in torsional vibration are obtained. The frequencies and mode shapes of the first five modes are presented. Experimental results obtained showed good agreement with the theoretical results. The increase in torsional frequency of beams due to pretwist is investigated and results for uniform and some tapered beams are presented. Experimental results showed some agreement with the theoretical results. The effect of warping of the cross-sections in torsion when considered for short beams shows an increase in frequency of vibration as the length of the beam decreased. The theoretical and experimental results obtained for a uniform cross-section beam show some agreement. The theoretical results obtained for tapered beams are presented for the first few modes.