Torsional vibrations of a shaft with non-uniform cross section

A solution is presented for the equation governing the propagation of torsional waves in shafts with variable cross sections. The solution permits one to treat several types of profiles, and in particular to compute the natural frequencies of an axisymmetrical shaft with various kinds of end conditions. Computed frequencies for three kinds of end conditions are compared with those of an equivalent cylindrical shaft, particularly with respect to the dependence on the geometrical characteristics.     

Forced vibrations of a rectangular plate with non-uniform thickness

The dependence on frequency of the maximum deflection and surface stresses of a simply supported rectangular plate subjected to a uniformly distributed sinusoidal excitation is discussed and simple formulae are proposed for estimating the deflection and surface stresses. The thickness of the plate varies linearly in one direction parallel to a side of the plate.     

Torsional vibration of multi-step non-uniform rods with various concentrated elements

An analytical approach and exact solutions for the torsional vibration of a multi-step non-uniform rod carrying an arbitrary number of concentrated elements such as rigid disks and with classical or non-classical boundary conditions is presented. The exact solutions for the free torsional vibration of non-uniform rods whose variations of cross-section are described by exponential functions and power functions are obtained. Then, the exact solutions for more general cases, non-uniform rods with arbitrary cross-section, are derived for the first time. In order to simplify the analysis for the title problem, the fundamental solutions and recurrence formulas are developed. The advantage of the proposed method is that the resulting frequency equation for torsional vibration of multi-step non-uniform rods with arbitrary number of concentrated elements can be conveniently determined from a homogeneous algebraic equation. As a consequence, the computational time required by the proposed method can be reduced significantly as compared with previously developed analytical procedures. A numerical example shows that the results obtained from the proposed method are in good agreement with those determined from the finite element method (FEM), but the proposed method takes less computational time than FEM, illustrating the present methods are efficient, convenient and accurate.     

Torsional lattice vibrations in molecular crystals

Atom-atom interaction potential energies derived from various sources have been used to calculate the torsional lattice vibration frequencies of crystalline benzene. The results are compared with available experimental data. The usefulness of this type of model potential is discussed.     

Large amplitude vibrations of rectangular plates with non-uniform elastic edge supports

Large amplitude vibrations of a rectangular plate with parabolically varying rotational constraints on two opposite edges are studied. The problem has been solved by both Berger's approach and the more rigorous von Kármán approach. Galerkin's method is used to obtain an ordinary differential equation in the modal function, the solution to which is given in terms of elliptic functions.     

Transverse vibrations of circular plates of varying thickness with non-uniform edge constraints

Free vibrations of circular plates varying in thickness and with flexible edge supports have been studied by several investigators for the restricted case when the supports are represented by uniformly distributed springs of constant stiffness.In the present study an approximate method is presented for dealing with supports possessing rotational flexibility which varies arbitrarily around the boundary.The method consists in representing the varying stiffness in terms of a Fourier expansion in the polar angle and approximately expressing the displacement function using a summation of polynomial co-ordinate functions which exactly satisfies only the essential boundary condition. The Ritz method is then applied in order to obtain the frequency determinant. The method can be easily extended to the forced vibrations case.     

Forced vibrations of a non-uniform thickness rectangular plate with two free sides

A forced vibration of a rectangular plate with thickness varying linearly in one direction is studied. The plate is simply supported along two opposite sides, and is free along the other sides. Approximate formulae are proposed for estimating the maximum deflection and surface stresses of a rectangular plate subject to a uniformly distributed harmonic lateral load. The accuracy of the formulae is discussed. The approximate formulae show the possibility that one can obtain solutions for the forced vibration of a plate with non-uniform thickness from those for the static bending of the same plate when the plate is subjected to a uniformly distributed load.     

Stability analysis of a nonlinear rotating blade with torsional vibrations

This paper discusses the stability of a spinning blade having periodically time varying coefficients for both linear model and geometric nonlinear model. To obtain a reduced nonlinear model from nodal space, a standard modal reduction procedure based on matrix operation is developed with essential geometric stiffening nonlinearities retained in the equation of motion. For the linear model, the stability chart with various spinning parameters of the blade is studied via the Bolotin method, and an efficient boundary tracing algorithm is developed to trace the stability boundary of the linear model. For the geometric nonlinear model, the method of multiple time scale is employed to study the steady state solutions, and their stability and bifurcations for the periodically time-varying rotating blade. The backbone curves of steady-state motions are achieved, and the parameter map for stability and bifurcation is developed.     

Coupled bending-torsion vibrations of rotating blades of asymmetric aerofoil cross section with allowance for shear deflection and rotary inertia by use of the Reissner method

Theoretical natural frequencies and modal shapes of the first five modes of vibration are presented for a rotating blade of asymmetric aerofoil cross section, with allowance for shear deflection and rotary inertia. Frequency equations for a rotating blade with asymmetry in one plane are developed by using the Ritz process, in two ways: namely, by proceeding according to the Reissner method and according to the classical potential energy method. In both cases shape functions for the bending moment, shearing force, twisting moment, bending slope, elastic twist and deflection are developed in series form. The results obtained are compared with those existing in the literature; it is found that the Reissner method approach yields more rapid convergence than does the classical potential energy method.     

Laser measurements of vibrations on rotating objects

Optical and Quantum Electronics -     

Scale effects in the bending vibrations of helicopter rotor blades

Scale effects and dynamic similarity in the bending vibrations of helicopter rotor blades are examined by expressing the first three modes of bending vibration of a uniform, conventional rotor blade by a series of Legendre polynomials as suggested by Wilde and others. The natural frequency ratios for these three modes are then determined as functions of a dynamic similarity parameter over the entire range of designs and operating conditions from very flexible, rapidly rotating blades to highly rigid, slowly turning conditions.