A.G. Ulsoy

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.

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

Dynamics of a Radially Rotating Beam With Impact, Part 2: Experimental and Simulation Results

The experimental results are presented and they are compared with simulations using the momentum balance model described in Part. 1. Excellent agreement was found between the experiments and simulation. Sensitivity studies were employed to show that the model is applicable for a fairly wide range of parameter values

Perturbation Analysis of Spindle Speed Variation in Machine Tool Chatter

Spindle speed variation has been shown to be an effective method for chatter control. In this paper, a single-degree-of-freedom regenerative type chatter equation is treated using perturbation methods. Rather than using the time coordinate, the angle of revolution is taken as the independent coordinate for maintaining a constant delay in the equations. The spindle speed is taken to be harmonically varying about a constant mean speed. Approximate analytical solutions are sought using the method of strained parameters, a perturbation technique. The amplitude of speed fluctuations (ε) is assumed to be small, and solutions are constructed using this parameter as the perturbation parameter. The stability lobes for constant spindle speeds are calculated exactly. By using the approximate perturbation analysis, the gain in stability is calculated for variable spindle speeds. The analysis is valid for (ε) values up to 0.02 (i.e., 2% of the constant mean speed). Solutions are verified using numerical simulations of the original equation.

TURNING OF SLENDER WORKPIECES: MODELING AND EXPERIMENTS

This paper introduces a dynamic cutting force model for turning of slender workpieces, as well as experimental results related to the frequency response of the workpiece in turning. The model is based on a flexible workpiece and rigid machine tool, and a workpiece displacement dependent cutting force. The model is described and studied theoretically as well as experimentally. The experimental studies utilise both cutting force and workpiece vibration measurements in two orthogonal directions. This data is obtained for both cutting and non-cutting conditions, and analysed in the frequency domain. The model was found to be in partial agreement with the experimental results. The experimental procedure described here represents a new method for determining the cutting process damping ratio, based on differences in the measured workpiece natural frequencies with and without cutting.