Search

An experimental technique to predict the onset of aeroelastic flutter of an enclosed computer memory disk is presented. The aerodynamic pressure is modelled as the sum of dissipative and circulatory linear operators, which subsumes as a special case the pressure generated in a thin hydrodynamic film between the disk and the wall. It is shown that the aeroelastic model parameters can be extracted from the frequency response function of the disk spinning at subcritical speeds. The aeroelastic parameters for an acoustically excited single disk at different enclosure gaps are derived for the speed range 6,000–19,800 rpm. The flutter speed predicted is strongly influenced by the enclosure gap and the substrate damping. A flutter speed as low as 35,000 rpm has been predicted. The technique can also be extended to predict the flutter speed in other systems including DVD and CDROM drives.

A perturbation method is presented to analytically calculate eigensolutions of the two dimensional wave equation when asymmetric perturbations are present in the boundary conditions. The unique feature of the method is that the sequence of boundary value problems governing the eigensolution perturbations are solved exactly through fifth order perturbation. Two classes of asymmetry are considered: irregular domain shapes that cannot be treated by analytical means, and variation of the boundary condition along the boundary. The unperturbed eigensolutions are those for an annular domain with axisymmetric boundary conditions. Irregularly shaped domains are studied in detail to demonstrate the method and the accuracy of the results, which are compared with exact values for the elliptical and rectangular domain cases. The results show excellent agreement with these known solutions for large shape distortions, an achievement resulting from the extension to higher order perturbation. Fourier representation of the boundary asymmetries allows analysis of arbitrary distributions of asymmetry. Additionally, the exact perturbation solution retains the explicit parameter dependence of continuous system analysis, generates simple expressions for the perturbed eigensolutions, addresses all distinct and degenerate axisymmetric system eigensolutions, and requires minimal computation and programming.

Energy transfer and mode localization in a translating string, coupled to a stationary spring-mass-damper system, are analyzed using traveling waves. The string tension and the non-conservative centrifugal force at the constraint lead to energy transfer between the translating continuum and the stationary constraint. By calculating energy contained in a harmonic wave before and after interacting with the constraint, energy transferred at the constraint is quantified in terms of constraint parameters. At an undamped constraint, energy is transmitted when a downstream (forward) wave impinges on the constraint. At a damped constraint, energy dissipated by damping, as well as energy flux by the tension and centrifugal force, contribute to the energy variation at the coupling point. When the damping coefficient of the damped constraint equals twice the impedance of the string, vibration energy is maximally dissipated. Asymmetry caused by the constraint localizes vibration modes into a downstream or upstream region. The degree of localization in each vibration mode is described in terms of the reflection coefficient and the constraint location. The effect of the constraint parameters on the mode localization and veering of natural frequency loci are also examined.

The energetics of translating one-dimensional uniform strings and highly tensioned pipes with vanishing bending stiffness and flowing fluid are analyzed for fixed, free and damped boundary conditions. The interaction between the translating continua and the boundary supports causes energy transfer. At a fixed boundary, the transverse component of tension does work, and the Coriolis forces at a free-end cause energy flux into the second-order continuum. Under a symmetric boundary configuration, the total energy of free oscillation varies periodically at the fundamental natural frequency. Asymmetric boundary supports in the pipe-fluid system lead to damped or self-excited motions. At a viscously damped boundary, the condition for maximal energy dissipation, the destabilizing effect of dissipation and the stabilizing effect of negative damping are examined analytically using travelling wave solutions. The energies transferred at the different boundary supports are quantified by energy reflection coefficients which are determined completely by the boundary conditions. Numerical simulations verify the analytically predicted energy variations.

The sensitivity of eigenvalues to the location of bracing and/or boundary support is formulated by using a generalized variational principle. The constraints are incorporated through the use of Lagrange multipliers. A numerical example is provided to illustrate the proposed method and its accuracy in application.

The energetics of translating tensioned beams and fluid transporting pipes under fixed, simply supported and free boundary conditions are analyzed in a generalized manner. The conservative and non-conservative forces acting at the boundaries lead to energy transfer between the translating continua and the boundary supports. The forces and associated convective velocities are identified from the one-dimensional transport theorem. The group velocity and the wavenumber of propagating and evanescent waves in the dispersive continua are defined. The time variation of total energy is represented in terms of the impedances and the reflection coefficients of the propagating waves in the continuum and the dynamic stability of the translating continua is discussed based on the energy expressions. The critical fluid speed in a cantilevered pipe, at which the resultant energy flux into the pipe at the free end vanishes, is determined by the use of travelling wave solutions.

A model is proposed that describes the response of a smart hybrid bearing pair coupled to a flexible axially moving beam. Exact, explicit, closed form determination of the bearing force generated is derived. The maximum axial velocity of the beam for stable response of the coupled system, and the damping and stiffness forces created in the bearing are determined. The smart hybrid bearing can be tuned for vibration control by active and passive control of the rate of lubricant supply. The use of tuned hybrid bearings can increase the maximum stable axial velocity of the stable response of the beam.

A singular perturbation solution is developed for the transversely loaded, flexible rotating disk using the von Karman large deflection theory of thin plates. No solution exists for these equations in the limit of vanishing dimensionless bending stiffness, ϵ. This fact, along with the desire to have the terms describing non-linear membrane stretching appear in the leading order equations, motivates a perturbation expansion for displacement of the form ω ≈ ϵ −1/2 ω 0 + ω 1 + …. The dimensionless transverse loading term is retained in the leading order equations only if it is rescaled by ϵ 1/2 . This rescaling of transverse load magnitude constitutes a similitude prediction for unifying the equilibrium and stability measurements of different flexible, rotating disks. Experimental measurements of the stability of flexible rotating disks under transverse point loading support this prediction. In addition, the numerically computed stability of the perturbation solution agrees quantitatively with the experimental measurements.

A vibrating, translating string is coupled to a translating rigid surface through a fluid film at the interface. The coupling is modelled as a translating, distributed damping and investigated through the waves propagating in the fluid-string coupled system. An approximate method, similar to the method of slowly varying parameters, gives explicit predictions of the stability boundary in parameter space and of responses of the possibly unstable modes near the stability boundary. The translating damping destabilizes waves propagating in the direction of damping propagation when the mean film speed relative to the string exceeds the wave speed in the string without fluid coupling. Through the same excitation mechanism, the maximum stable speed of a shell rotating on a stationary shaft is shown to be twice the critical rotation speed of the shell in the absence of the fluid coupling.

The transverse response of a cable transport system, which is modelled as an ideal, constant tension string travelling at constant speed between two supports with a damped linear oscillator attached to it, is predicted for arbitrary initial conditions, external forces and boundary excitations. The exact formulation of the coupled system reduces to a single integral equation of Volterra type governing the interaction force between the string and the payload oscillator. The time history of the interaction force is discontinuous for non-vanishing damping of the oscillator. These discontinuities occur at the instants when transverse waves propagating along the string interact with the oscillator. The discontinuities are treated using the theory of distributions. Numerical algorithms for computing the integrals involving generalized functions and for solution of the delay-integral-differential equation are developed. Response analysis shows a discontinuous velocity history of the payload attachment point. Special conditions leading to absence of the discontinuities above are given.

The stability of a rotating disk coupled to the surrounding fluid is investigated analytically and experimentally. Dimensional analysis of the equations governing transverse motion of a centrifugally tensioned, Kirchhoff plate and irrotational flow of a compressible fluid identifies three dimensionless parameters that characterize the state of stability of the fluid-plate system. They are the ratio of the fluid and plate densities, Λ, the Mach number of the periphery of the disk, M, and the ratio of the stiffness of the disk in bending to that derived from centrifugal stresses, ϵ. Numerical analysis of these equations for subsonic speeds (M l 1. The experimental measurements show that instability is independent of ϵ for values of ϵ ranging over several orders of magnitude.

Residual membrane stresses are purposely induced in wide bandsaw plates through a localized cold rolling procedure, known as roll-tensioning, to improve the cutting performance of the blade. Despite the critical importance of this practice to efficient cutting operation with a bandsaw, quantitative understanding of roll-tensioning on bandsaw cutting performance remains unresolved. In this paper, the effect of roll-tensioning on bandsaw plate free vibration and plate stability are discussed. The residual stresses induced by roll-tensioning can substantially increase or decrease the transverse and torsional natural frequencies (or stiffnesses) of the bandsaw plate depending on the locations of the rolling tracks, the roller load and the number of rolling repetitions on each track. With roll-tensioning that results in a substantial increase of torsional natural frequencies (or stiffnesses) and moderate decrease of transverse natural frequencies (or stiffnesses), the bandsaw plate can resist a larger normal edge (cutting) force prior to buckling. Improper tensioning that decreases substantially the transverse natural frequencies, or both the transverse and torsional natural frequencies, can reduce the normal edge force at buckling.

Residual membrane stresses are induced in wide bandsaw plates by cold-rolling longitudinal tracks, a process known as roll-tensioning. When properly done, roll-tensioning substantially improves the cutting performance of the bandsaw. However, excessive rolling can buckle the plate causing rubbing of the plate against the work piece. The analysis in this paper shows that excessive roll-tensioning induces transverse or torsional buckling of the plate, depending on the location of the longitudinal tracks across the plate width and the magnitude of the uniform longitudinal (axial) tension applied to the plate during the cutting operation. The buckling modes are characterized by a number of waves along the length of the plate that are qualitatively predictable by theory.

The wide bandsaw plate is often initially stressed by "roll-tensioning" to improve cutting performance. Despite the critical importance of this process to practice, a quantitative understanding of the relationship between the "roll-tensioning" and the plate performance improvement is not available principally because the residual stresses generated in the roll-tensioning process are unknown. This paper predicts the approximate residual stress state induced in the bandsaw plate by the longitudinal plastic deformation generated by "roll-tensioning", given the number and location of each rolling track and the magnitude of longitudinal plastic strain in each rolling track. In the case of incremental indentation of the plate by rolling with spherical rollers, the longitudinal plastic deformation in each track, and consequently the residual stress state of the plate can be predicted by measuring the indentation profile of each rolling track.

Prediction of the natural frequencies of a rotating disk requires accurate representation of the disk membrane stresses caused by rotation. These stresses depend strongly on the membrane stress boundary condition at the clamping collar. This research evaluates experimentally the validity of the standard models of this boundary condition: vanishing radial stress, vanishing radial displacement, or finite stress at the center of an integral disk and collar assembly. It shows that they can lead to over 13% error in prediction of the natural frequencies of the disk spinning at high rotation speed. The error occurs because the idealized models do not include friction and stiction at the disk/collar interface and slippage of the disk between the collars. Stiction and slippage also cause hysteresis of the disk natural frequencies with start/stop cycles of rotation speed. Inclusion of an empirically determined friction traction at the collar diameter reduces the error in predicted frequencies to less than 1%.

This research describes aerodynamically excited supercritical disk vibration observed in experiments. Spectral analysis and moire photography show that for most fluid densities, a single supercritical backward traveling wave becomes unstable. At pre-flutter speeds, the fluid acts as a random excitation source, but at the flutter speed and at higher speeds the fluid and the disk are strongly coupled. Coupling becomes sufficiently strong that the post-flutter frequency becomes nearly independent of rotation speed, even though the disk stiffens centrifugally as speed increases. Disk flutter is subsonic and often occurs in incompressible flow. Reducing the fluid density decreases the vibration amplitude at all speeds and increases the flutter speed.

Eigensolutions, Green's functions, and steady state responses of three-dimensional, homogeneous, isotropic, finite linear elastic solids, containing small linear elastic imperfections, are presented in this paper. The inhomogeneities are treated as perturbations to otherwise homogeneous solids, the spectral eigensolutions of which are known. The exact eigensolutions of the imperfect solid, formulated through the Betti reciprocal theorem, satisfy a set of integral equations from which both analytic perturbation formulas and a numerical iteration scheme are developed. Special emphasis is given to analysis of perfect, homogeneous, elastic solids that are spectrally degenerate . Finally, the vibration of circular plates with evenly spaced radial inclusions, which cause splitting of some of the repeated vibration modes, is analyzed both analytically and numerically for the eigensolutions.

The classical moving load problem in which a mass traverses an elastic structure is extended to systems in which the guideway also has prescribed axial motion. A monocable ropeway, such as a tramway or ski lift, is an example of such a traveling structure-traveling load system, and it is modeled herein as an axially moving string that transports an attached discrete mass between two supports. The Green's function methods for the two constituent systems, the particle and the uniform traveling string, are combined to derive a Volterra integral equation with delay that governs the interaction force in the coupled system. The calculated response of the traveling string-particle system to arbitrary initial conditions and external excitation is compared to that predicted by the derived asymptotic solution for vanishing discrete mass. Repeated scattering by the particle of transverse waves in the string dominates the motion. Transient vibration as the particle is transported between the supports is maximized at a transport speed that can be substantially lower than the critical speed of divergence instability for the uniform traveling string. Agreement between responses measured in the laboratory and those predicted by the traveling structure-traveling load theory supports the analysis.

The response of a stationary, undamped, asymmetric, classical, circular plate subjected to a rotating spring-mass-dashpot system is formulated in a co-ordinate system fixed to the plate. Use of an eigenfunction expansion reduces the equation of motion to a set of coupled Hill's equations. Techniques for analytical solution of the Hill's equations are illustrated on axisymmetric plates. It is found that a spring rotating at supercritical speed will parametrically excite the plate to single mode and combination resonances. A rotating mass will destabilize the plate above a speed greater than the critical speed at which the inertia force of the rotating mass exceeds the elastic restoring force of the plate. A viscous dashpot destabilizes the plate when it rotates at supercritical speed by negatively damping the backward-travelling wave components of the plate eigenmodes.

A criterion guaranteeing asymptotic stability of non-conservative, linear gyroscopic systems is derived. The criterion applies to linear systems with internal damping, external damping, and circulatory forces as well as systems with negative definite stiffness matrix. A sufficient condition of flutter instability is also presented. The condition is shown applicable to both conservative and non-conservative gyroscopic systems. An application of the stability criteria explains why internal damping destabilizes a rotating system when operated above critical speed. It is also shown that the critical speed of a gyroscopic system is not an absolute stability measure; the critical speed depends on the frame of reference.

Initial stresses are often introduced deliberately into centrally clamped, peripherally free circular plates to increase the plate natural frequencies. Increased natural frequencies in the critical modes extend the range of stable rotation speed. Current axisymmetric initial stressing schemes increase the natural frequencies of the zero nodal circle, two and higher nodal diameter vibration modes, but the zero and one nodal diameter natural frequencies decrease. For sufficiently large initial stress, the zero nodal diameter natural frequency vanishes and the plate experiences divergence buckling, thereby limiting the initial stress that can be induced. In this paper a new initial stressing method is presented whereby the natural frequencies of all zero nodal circle modes increase simultaneously, thus avoiding buckling of the plate through excessive initial stress. The stress field is induced by applying uniform pressure to the boundaries of eccentric interior holes. Through proper location of the holes, the critical natural frequencies increase monotonically with increased pressure, and the magnitudes of the frequency shifts increase with the introduction of additional stressed holes. The initial stressing schemes can be directed towards optimal increases in specific vibration modes. Calculations predict natural frequency increases between 35% and 90% depending on the vibration mode. Experimental measurements are consistent with the model predictions.