R.A. Scott

The dynamic response of a rotating shaft subject to a moving load

The dynamic behavior of a rotating shaft subject to a constant velocity moving load is investigated. The Euler-Bernoulli, Rayleigh and Timoshenko beam theories are used to model the rotating shaft. The modal analysis method and an integral transformation method are employed in this study, for the case of a shaft with simply supported boundary conditions. The influence of parameters such as rotational speed of the shaft, the axial velocity of the load and the dimensions of the shaft are discussed for each shaft model. The results are presented and compared with the available solutions of a non-rotating beam subject to a moving load.

Flexural motion of a radially rotating beam attached to a rigid body

Flexural motion of a radially rotating beam attached to a rigid body is investigated. Fully coupled non-linear equations of motion are derived by using the extended Hamiltons Principle. Spatial dependence is suppressed by using the extended Galerkin method. A torque profile is used to drive the rotating body so that the rigid body motion is not known a priori . The effect of the coupling terms upon the vibration waveforms are investigated by using both a linearized analysis and numerical solution of the differential equations. It is found that for small values of the ratio of the flexible beam and rigid shaft inertia uncoupled equations can lead to substantially incorrect results, particularly with regard to frequencies.

Modal analysis of a distributed parameter rotating shaft

Forced response analysis of an undamped distributed parameter rotating shaft is investigated by using a modal analysis technique. The shaft model includes rotary inertia and gyroscopic effects, and various boundary conditions are allowed (not only the simply supported case). Presented here is a study of the resulting non-self-adjoint eigenvalue problem and its characteristics in the case of rotor dynamics. In addition to the modal analysis, Galerkins method is applied to analyze the forced response of an undamped gyroscopic system. Both methods are illustrated in a numerical example and the results are compared and discussed.

Dynamics of a Radially Rotating Beam With Impact, Part 1: Theoretical and Computational Model

The model uses a momentum balance method and a coefficient of restitution, and enables one to predict the rigid body motion as well as the elastic motion before and after impact

Dynamic Stability and Response of a Beam Subject to a Deflection Dependent Moving Load

The dynamic stability and transverse vibration of a beam subject to a deflection dependent moving load is considered. The deflection dependent moving load is motivated by the cutting forces in machining operations. Galerkin’s method is used to obtain a set of ordinary differential equations with periodic coefficients. It is found that parametric instability can be expected for a continuous sequence of moving loads. The response under the moving deflection dependent load is also calculated.

Spring-dashpot models for the dynamics of a radially rotating beam with impact

In a previous study, experimental results for the dynamics of a radially rotating beam with impact were found to be in excellent agreement with simulation results using the momentum balance method (for impact modeling). In this paper spring-dashpot models for impact modeling are compared to experiment for a radially rotating flexible beam. Excellent agreement is found between the simulation results using spring-dashpot models and the experiments. A sensitivity study is employed to investigate the issue of accurately determining the model parameters. The impact of a flexible radially rotating beam against a rigid impact surface is considered (see Figure 1). In an earlier study [l] experimental results were compared with simulation results obtained by using the momentum balance (coefficient of restitution) model for impact. Although that model is not intended for application to systems with flexible members, good agreement was found between the experiments and the simulation. Sensitivity studies were employed to show that the model is applicable over a fairly wide range of parameter values. Thus, the momentum balance method has been demonstrated to be capable of accurately predicting the dynamics of systems which consist of both rigid and elastic links undergoing impact. A second competing method for impact modeling, which is applicable to flexible systems, is the spring-dashpot model. In this paper the validity and utility of spring-dashpot models are investigated for the dynamics of a radially rotating beam with impact. A brief literature review on the spring-dashpot models for impact will now be given. Some of the energy losses during impact are associated with relative indentation and the damping mechanisms involved during this contact period. The first attempt to incorporate a theory of local indentations is an elastostatic one given by Hertz [2]. The deformation is assumed to be restricted to the vicinity of the contact area and to be given by static theory. Elastic wave motion in the impacting bodies is neglected, and the total mass of each of the bodies is assumed to be moving at any instant with the velocity of its center of mass. The impact, therefore, can be visualized as the collision of two rigid bodies restricted to move in the direction of impact with spring buffers; all deformations occur in the springs, the inertias of which are neglected [3]. The assumption that deformation is quasi-static can only be justified if the duration of impact is long enough to permit the stress waves to tranverse the length of the structure many times [4]. This criterion (Love’s criterion) does not apply in cases where an object impacts with another very large object, in which case no reflected wave returns to the point of impact [3]. Hunter [3] suggested as an alternative that the behavior of a large structure can be approximated by a dashpot in parallel with the spring to account for the energy radiated through the half-space by wave motion. If the time constant of the spring-dashpot system is short compared with

Non-planar, non-linear oscillations of a beam II. Free motions☆

Large amplitude whirling motions of a simply supported beam constrained to have a fixed length are investigated. Equations of motion taking into account bending in two planes and longitudinal deformations are developed. Using the method of harmonic balance, response curves for certain planar and non-planar steady state, forced motions are obtained. Another approximate scheme is used to study the stability of these motions. Stable regions corresponding to non-planar motions are found, thus confirming the existence of whirling motions. Numerical results are presented and discussed for several specific cases.

Non-linear vibrations of three-layer beams with viscoelastic cores I. Theory☆

Abstract Approximate equations of motion are developed for large amplitude motions of three-layer axially restrained unsymmetrical beams with viscoelastic cores. The external force consists of a constant plus an oscillatory term. The combination of this form of forcing and the large amplitude motions cause the beam to respond at multiples of the forcing frequency. This can lead to difficulties in the complex modulus approach to viscoelasticity. These are overcome here through use of hereditary integrals and their relationships with complex moduli. Theoretical results on the frequency response of clamped, symmetrical beams are compared with earlier experimental work. On the whole, reasonable agreement is found.

FORCED NON-LINEAR VIBRATIONS OF A DAMPED SANDWICH BEAM

Abstract Forced, damped, non-linear, low-frequency flexural motions of a clamped-clamped sandwich beam with thin face sheets and a soft viscoelastic core are examined experimentally and theoretically. The theory employed neglects the extensional rigidity of the core and treats the face sheets as membranes. The non-linearity stems from axial stretching of the face sheets. Damping is taken into account by modeling the core as a Kelvin solid, with the material parameters used being obtained experimentally as functions of frequency and temperature. Theoretical frequency-amplitude relations are obtained using Galerkins procedure and the method of harmonic balance. Results on fundamental natural frequencies, mode shapes, and stability are also presented. In the experiment, mechanical contact with the specimen was avoided by employing electromagnetic forcing and using a proximeter to measure displacements. Also, special attention was given to the interface bonds and to the reproduction, as close as possible, of clamped-clamped conditions. Agreement between the theoretical and experimental results is, in general, quite good.

Dynamic response of a rotating beam subjected to an accelerating distributed surface force

Abstract A method is given by which the response of a rotating or non-rotating timoshenko beam can be determined, subjected to an accelerating fixed direction distributed surface force. The beam model includes the gyroscopically induced displacement transverse to the direction of the load. The solution for pinned supports is set up using multi-integral transforms, and the inversion is expressed in terms of convulution integrals. These are numerically integrated for a uniformly distributed load having an exponentially varying velocity function. Results are presented for the displacement under the loads center as a function of position and for the displacement of every point on the beam at an instant in time. Comparisons are made between the beam response to a constant velocity load and its response to a load which accelerates to the same velocity.