G. M. L. Gladwell

Branch mode analysis of vibrating systems

Abstract A method is presented for calculating the natural frequencies and principal modes of free undamped vibration of a system with many degrees of freedom. It is particularly suitable for the kind of system obtained by replacing each component of a complex continuous vibrating system by an appropriate lumped mass model. The method proceeds by dividing the complete problem into two stages. In the first, certain sets of constraints are imposed on the system and certain of the principal modes of the constrained systems are found. In the second stage these modes are used in a Rayleigh-Ritz analysis 0f the whole system. It is shown that the method is more economical than the straightforward analysis, and it is suggested that the errors introduced by its use can be made negligible compared to those incurred by the use of lumped-mass models. The method may easily be programmed for a computer.

Acoustic wave propagation in a sheared fluid contained in a duct

Abstract In assessing the propagation of an acoustic wave in a sheared fluid, first, the derivation of the linearized wave-equation for two-dimensional viscous flow is presented. Interactions with the acoustic wave of the mean flow, the shear and the viscosity of the fluid through shear have been included. This is followed by a numerical solution to the inviscid case, using a constant gradient and a turbulent velocity profile, for the first three symmetric modes. The “plane” mode is compared with results obtained by Pridmore-Brown using an analytical approach; there is good agreement for the case with constant gradient profile. Methods of solution are also presented for the problems of acoustic wave propagation in a flowing medium, both with and without shear, contained in ducts with finite wall admittance.

Discrete nodal domain theorems

We give a detailed proof for two discrete analogues of Courants Nodal Domain Theorem. (authors abstract)

An investigation into the theory of resonance testing

Many vibrating systems are subject to damping forces arising from the structure of the system itself, and in the simpler theories this damping is assumed to be linear. The first part of the paper (§§ 1 to 5) is devoted to a discussion of systems with n degrees of freedom subject to linear damping. In the remainder of the paper various resonance testing techniques are assessed in the light of this general theory.

The Vibration and Balancing of an Unbalanced Flexible Rotor

The motion of a flexible unbalanced rotating shaft is examined. The effects of balancing such a shaft (as a ‘rigid body’) in a conventional low-speed balancing machine are explained. The underlying...

A variational formulation of damped acousto structural vibration problems

Abstract The variational formulation of the equations governing the vibration of damped structural and acoustic systems is discussed. Following a suggestion of Morse and Feshbach, the problems are formulated using “adjoint” displacements, pressures, etc., in addition to the actual ones. A particular application of the method is the formulation of radiation problems, and it is shown that Sommerfelds radiation conditions may be obtained as natural boundary conditions of variational problems.

Natural frequencies of free finite-length circular cylinders

Sets of tables are given for the natural frequencies of the first five symmetric and first five antisymmetric modes of a hollow or solid cylinder for circumferential wave numbers n = 0, 1, 2. Contour graphs are given for the lowest frequencies as functions of (length)/(mean radius) and (thickness)/(mean radius).

The calculation of mechanical impedances relating to an indenter vibrating on the surface of a semi-infinite elastic body

Abstract The contact impedances for an area situated on the plane surface of a semi-infinite elastic medium have recently been used in the design of an ultrasonic hardness tester. The impedances relate to a displacement of the contact area in one of four ways: (i) rotation and (ii) translation in a direction normal to the plane face of the medium; (iii) rotation and (iv) translation in a direction parallel to the face. This paper presents a review of methods and results, within the theory of linear elasticity theory, which relate to the impedance for these problems and their static counterparts. The results presented for cases (i) and (ii) are known, some of those relating to cases (iii) and (iv) are new.

The inverse problem for the vibrating beam

The beam is modelled by a system of rigid rods joined together by rotational springs and with masses at the joints. The left end is clamped while the right end is subject to one of a number of conditions: it is free, pinned, sliding or clamped. It is shown that the resonant and antiresonant frequencies of the system may be almost completely ordered. Necessary and sufficient conditions are found that these frequencies must satisfy to correspond to an actual system, with positive parameters, and two stripping procedures are devised for the determination of these system parameters, from a knowledge of certain frequency spectra.

Finite element analysis of the axisymmetric vibrations of cylinders

Abstract A finite element analysis is developed for finite and infinite solid or hollow cylinders in axisymmetric vibration. The elements themselves are solid or annular cylinders, and have 16 degrees of freedom. Results are given for the propagation constants of solid and hollow infinite cylinders, and excellent agreement is found with those from the exact Pochhammer theory and Mindlin and McNivens three-mode theory. Frequency spectra are presented for the symmetric and antisymmetric modes of solid and hollow finite rods. Excellent agreement is found with experimental results, and this suggests that some of the results obtained from the three-mode theory by McNiven et al. , and in particular the frequency of the end mode, are in error by more than 10%. Details of the finite element inertia and stiffness matrices appear in an appendix.