V. Ramamurti

NATURAL FREQUENCIES OF TWISTED ROTATING PLATES

Abstract A detailed comparison is presented of the predicted eigenfrequencies of twisted rotating plates as obtained by using two different shape functions. Primarily, rotating twisted plates of two different aspect ratios and two different thickness ratios are considered. The effects of rotation are included by using a “stress smoothing” technique when calculating the augmented stiffness matrix. In addition, the effects of Coriolis acceleration, contributions from membrane behaviour, setting angle and sweep angle are considered. The effects of geometric non-linearity are briefly discussed. Finally, results of a brief study of cambered plates are presented.

Structural dynamic analysis of composite beams

In the treatment of the structural dynamic problem of composite materials, two alternate types of formulations, based on the elastic modulus and compliance quantities, exist in the literature. The definitions of the various rigidities are observed to differ in these two approaches. Following these two types of formulation, the structural dynamic characteristics of a composite beam are analyzed. The results of the analysis are compared with those available in the literature. Based on the comparison, the influence of the warping function in defining the coupling terms in the modulus approach and also on the natural frequencies of the beam has been identified. It is found from the analysis that, in certain cases, the difference between the results of the two approaches is appreciable. These differences may be attributed to the constraints imposed on the deformation and flexibility of the beam by the choice of the description of the warping behaviour. Finally, the influence of material properties on the structural dynamic characteristics of the beam is studied for different composites for various angles of orthotropy.

Dynamics of stockbridge dampers

The aim of the present investigation is to establish a theoretical analysis of the response characteristics of Stockbridge dampers and to verify the theoretical predictions experimentally. Expressions have been derived for the response of the Stockbridge damper, by treating it as a two-degree-of-freedom system. The dynamic stresses introduced on the stranded cable are also theoretically determined. The experimental findings have been reported and explanations for discrepancies, where they occur, have been made.

A parametric study of vibration of rotating pre-twisted and tapered low aspect ratio cantilever plates

A finite element technique has been used to determine the natural frequencies of a pre-twisted and tapered plate mounted on the periphery of a rotating disc. The pre-twisted plate has been idealized as an assemblage of three noded triangular shell elements with six degrees of freedom at each node. In the analysis the initial stress effect (geometric stiffness) and other rotational effects except the Coriolis acceleration effect have been included. The eigenvalues have been extracted by using a simultaneous iteration technique. Computation of frequencies has been carried out for plates of aspect ratios 1 and 2. Other parameters considered are pre-twist, taper, skew angle and disc radius. From the results of computations an extension to the existing empirical formulae derived by Dokainiah and Rawtani [1] has been suggested for computing natural frequencies of rotating pre-twisted and tapered cantilever plates.

Dynamic response of an annular disk to a moving concentrated, in-plane edge load

In-plane dynamic behaviour of a thin annular disk with a clamped inner boundary is analyzed. The frequencies of free in-plane vibration with a free outer boundary are first evaluated, by using Lame potentials, for various radius ratios ranging from 0.2 to 0.8. The steady state dynamic stresses induced by a concentrated load moving at a constant angular speed at the outer boundary are then evaluated through a Galilean transformation. Results are presented for a radius ratio of 0.5.

Free vibrations of circular cylindrical shells

The finite element method is used to predict the dynamic behaviour of circular cylindrical shells in free vibrations. A suitable shape function for the circumferential displacement distribution has been proposed. This reduces the three-dimensional character of the problem to a two-dimensional one. The simultaneous iteration method to determine the eigen-frequencies and eigenvectors is utilised for solving the eigenvalue problem. The accuracy of the method has been checked by verifying the results of known cases. Finally an experimental shell structure containing elastic rings welded at the ends has also been analysed and the experimental results compared with the theoretical ones.

Coupled free vibration characteristics of rotating tuned bladed disk systems

A detailed qualitative study is presented of the coupled free vibration characteristics of rotating turbomachinery bladed disk systems having useful ranges of geometric parameters. The analysis is carried out by using a three noded triangular shell element having six degrees of freedom per node together with the cyclic symmetric nature of the problem. The blade and disk attachment is established by a set of constraint equations obtained by the Love-Kirchhoff hypothesis. The effects of rotation are included by using a stress smoothing technique while computing the geometric stiffness matrix. The contributions from membrane behaviour and transverse shear are included. The severely coupled systems are identified and an exhaustive parametric study is carried out for these systems to predict the influence of stagger and twist on natural frequencies for stationary as well as rotating systems. An assessment of Southwells coefficients for these systems is made. Theoretical results agree closely with available experimental results.

Torsional vibrations of prismatic shells

Prismatic shells find wide application in mechanical, civil and aeronautical engineering. The object of this paper is to propose a method of analysing such shells having different internal ribbing for their torsional vibrations. The governing equations of motion are derived by using a finite difference method and solved by the simultaneous iteration technique. The accuracy of the method is ascertained by comparing the results for some wellknown boundary conditions with the results obtained by classical methods. Shells of square cross-section and rectangular shells of variable and constant cross-section have been analysed. The effects of introduction of diaphragms on the eignefrequencies have been analysed. The effects of variations in shell thickness and length on the dynamic behaviour also have been studied.

Natural frequencies of bladed disks by a combined cyclic symmetry and Rayleigh-Ritz method

A combined classical Rayleigh-Ritz and cyclic symmetry method is described for the free vibration analysis of rotating turbo-machinery bladed disk systems. It is shown that the first few lowest natural frequencies can be predicted accurately by taking only one repeatable substructure of the system. The formulation makes use of the shell theory for the blades so that no modes are missed. The accuracy is shown by means of a study of representative samples and comparison with available results.

Frequency analysis of centrifugal fan impellers

The geometric periodicity is utilized to determine the natural frequencies of centrifugal fan impellers. Triangular plate elements having six degrees of freedom are used [1,2] to model the impellers. Potters scheme of solving tridiagonal block matrices is extended to the present analysis. The simultaneous iteration scheme is modified to extract the natural frequencies from the resulting complex eigenvalue problem. Analytical results are compared with experimental results.