Deflection of a Very Flexible Spinning Disk Due to a Stationary Transverse Load

1978 ◽  
Vol 45 (3) ◽  
pp. 636-642 ◽  
Author(s):  
R. C. Benson ◽  
D. B. Bogy
Keyword(s):  
Standing Waves ◽  
Outer Region ◽  
Transverse Load ◽  
Fourier Series Expansion ◽  
Stiffness Parameter ◽  
Membrane Theory ◽  
Concentrated Load ◽  
Transverse Loads ◽  
Spinning Disk ◽  
Load Location

This paper addresses the problem of steady deflection of a very flexible spinning disk due to transverse loads that are fixed in space. We first approach this problem within the context of membrane theory. The membrane differential operator is classified and shown to be hyperbolic in the outer region and elliptic in the inner portion. The characteristics are studied and qualitatively compared with some experimentally observed standing waves in the outer region of a membrane-like disk. The eigenvalue problem is examined and the membrane operator is found to be singular. We therefore conclude that the problem of interest cannot be solved within the context of membrane theory. Finally, the problem is formulated with bending stiffness retained. The concentrated load problem is solved by use of a Fourier series expansion in the angular direction in conjunction with a numerical solution for the radial modes. Graphical results are presented for various values of the stiffness parameter and load location.

Observations on the Steady-State Solution of an Extremely Flexible Spinning Disk With a Transverse Load

1983 ◽  
Vol 50 (3) ◽  
pp. 525-530 ◽  
Author(s):  
R. C. Benson
Keyword(s):  
Steady State ◽  
Hybrid System ◽  
Fourth Order ◽  
Steady State Solution ◽  
Transverse Load ◽  
System Of Differential Equations ◽  
Membrane Theory ◽  
Small Disk ◽  
Spinning Disk ◽  
Flexible Disk

The steady deflection of a transversely loaded, extremely flexible, spinning disk is studied. Membrane theory is used to predict the shapes and locations of waves that dominate the response. It is found that waves in disconnected regions are possible. Some results are presented to show how disk stiffness moderates the membrane waves, the most important result being an upper bound on the highest ordered wave of significant amplitude. A hybrid system of differential equations and boundary conditions is developed to replace the pure membrane formulation that is singular, and the full fourth-order plate formulation that is numerically sensitive. The hybrid formulation retains the salient features of the flexible disk response and facilitates calculations for very small disk stiffnesses.

Bending of Circular and Ring-Shaped Plates on an Elastic Foundation

1954 ◽  
Vol 21 (1) ◽  
pp. 45-51
Author(s):  
Herbert Reismann
Keyword(s):  
Circular Plate ◽  
Elastic Foundation ◽  
Series Solution ◽  
Circular Disk ◽  
Symmetric Problem ◽  
Concentrated Load ◽  
Transverse Loads ◽  
Infinite Extent ◽  
Theory Of Plates ◽  
Pure Moment

Abstract This paper develops a method for the evaluation of deflections, moments, shears, and stresses of a circular or ring-shaped plate on an elastic foundation under transverse loads. A series solution is derived for plates subjected to edge and/or concentrated loads and is given in terms of tabulated functions. It is exact within the assumptions underlying the classical theory of plates and includes, as a particular case, the known solution of the corresponding radially symmetric problem. Two examples displaying radial asymmetry are worked. A solution is given for (a) a circular plate resting on an elastic foundation, clamped at the boundary and subjected to an arbitrarily placed concentrated load, and (b) a plate of infinite extent, resting on an elastic foundation and clamped to the boundary of a rigid circular disk to which a pure moment is applied.

The Torsionless Bending of a Hollow Beam by a Transverse Load

1937 ◽  
Vol 4 (1) ◽  
pp. A25-A30
Author(s):  
W. L. Schwalbe
Keyword(s):  
Cross Sections ◽  
Equilateral Triangle ◽  
Bending Moment ◽  
Longitudinal Section ◽  
Transverse Load ◽  
Shearing Stress ◽  
Numerical Work ◽  
Hollow Beam ◽  
Transverse Loads ◽  
Hollow Beams

Abstract The author discusses the bending of hollow beams when subjected to transverse loads, and points out that shearing stresses and strains in the cross sections are necessary, and a particular longitudinal section remains plane only if the resultant of the shearing stress, and hence the plane of the applied bending moment, possesses a particular location. The author determines the location of this resultant shearing stress by applying a method based on St. Venant’s theory. Applications of the method are made to two hollow sections. One of the sections is that of an equilateral triangle which serves as a measure of accuracy for the numerical work presented by the author, since the location of the resultant of the shearing stresses is known by symmetry.

scholarly journals Simulation of the Vibratory Behavior of Slender Shafts Subject to Transverse Loads Moving in the Axial Direction

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
F. Bucchi ◽  
P. Forte
Keyword(s):  
Mathematical Model ◽  
Manufacturing Industry ◽  
Axial Direction ◽  
Transverse Load ◽  
Design Phase ◽  
Vibration Level ◽  
Machine Performance ◽  
Transverse Loads ◽  
Spatial Interval ◽  
Influential Parameters

In various machines of the manufacturing industry, and in particular in paper converting machinery, there are shafts operating under conditions similar to that of a slender beam subjected to a transverse load moving in the axial direction. This condition can lead to vibrations and consequent deterioration of the machine performance and of the product quality. The problem has been theoretically studied in the literature since the 1990s. While shaft mass and stiffness are universally considered among the most influential parameters on its vibratory behavior, less obvious and not investigated in the literature is the influence of the spatial interval between two successive loads, an aspect that should be considered in the shaft design phase. In fact, if that is less than the length of the shaft, i.e., if there is more than one transverse load on the shaft at a given time, the vibration level may decrease with respect to the single-load configuration. This work describes the development of a mathematical model of a slender shaft hinged at its ends, representing the rotor of a paper roll perforating unit, with the SW Mathematica. The effect of a load moving axially at a given speed followed by similar loads after given spatial intervals was simulated investigating the influence of speed and load interval on shaft vibrations and resonance. The results showed how reducing the load interval can lead to a reduction of the shaft vibration which is a useful indication on possible design corrective actions.

Response of a Viscoelastic Annulus to a Step Transverse Load

1970 ◽  
Vol 92 (3) ◽  
pp. 425-434
Author(s):  
S. R. Robertson
Keyword(s):  
Plane Strain ◽  
Poisson’S Ratio ◽  
Logarithmic Decrement ◽  
Poisson's Ratio ◽  
Transverse Load ◽  
Transverse Loads ◽  
Work Done

The problem of finding the response of a viscoelastic annulus in plane strain to step, transverse loads is solved. It is solved by employing Valanis’ method which assumes constant Poisson’s ratio. The resulting displacements are used to calculate the work done by the applied transverse load for various thicknesses of the annuli. A simple spring-dashpot model is then fitted to the work versus time curves so as to provide the logarithmic decrement for design.

Vibration of a Spinning Annular Disk With Coupled Rigid-Body Motion

1993 ◽  
Vol 115 (2) ◽  
pp. 159-164 ◽  
Author(s):  
Shih-Ming Yang
Keyword(s):  
Rigid Body ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Transverse Load ◽  
Elastic Displacement ◽  
Stable Operation ◽  
Coupling Effects ◽  
Annular Disk ◽  
Spinning Disk ◽  
Vibration Coupling

The vibration of a spinning annular disk with coupled translational and rotational rigid-body motion is analyzed. The spinning disk, with one linear spring as transverse load, is free to translate and rotate relative to the shaft axis. Modal functions of a stationary annular disk are used to describe the elastic displacement of the spinning disk. The governing equation includes the rigid disk translation, rotation, and the flexible disk vibration. Coupling effects between the rigid-body motion and the annular disk modal function are identified in the formulation. Because of the coupling effects, stable operation beyond divergence (critical speed) is achieved, the disk loses its stability to flutter. This stability prediction is different from that of a spinning disk without rigid-body motion where the disk is unstable at and right after divergence.

Flexural-Torsional Buckling Analysis of Steel Beams under Transverse Loads in High Temperature

2013 ◽  
Vol 641-642 ◽  
pp. 432-435
Author(s):  
Zhen Qing Wang ◽  
Jia Lei Li ◽  
Yu Lai Han ◽  
Zhu Ju
Keyword(s):  
High Temperature ◽  
Theoretical Analysis ◽  
Critical Stress ◽  
Transverse Load ◽  
Steel Beams ◽  
Beam Model ◽  
Torsional Buckling ◽  
Critical Moment ◽  
Transverse Loads ◽  
Flexural Torsional Buckling

Flexural-torsional buckling of steel beams in high temperature is studied by adopting the method of theoretical analysis. The flexural-torsional buckling critical moment equations of steel beams under concentrated transverse loads are driven in high temperature ranging from 20°C to 300°C. In order to verify the validity of the theoretical analysis equations, a steel beam model under concentrated transverse load is designed. The flexural-torsional buckling critical moments and critical stresses are calculated under different temperature conditions. The influence of load point on flexural-torsional buckling critical moment and critical stress are obtained. The variation tendency of flexural-torsional buckling critical moment and critical stress against temperature are also presented.

Analysis of continuous steel plates subjected to uniform transverse loads

1985 ◽  
Vol 12 (3) ◽  
pp. 685-699 ◽  
Author(s):  
K. P. Ratzlaff ◽  
D. J. L. Kennedy
Keyword(s):  
Cross Section ◽  
Reasonable Agreement ◽  
Steel Plates ◽  
Transverse Load ◽  
The Arctic ◽  
Continuous Steel ◽  
Flat Plates ◽  
Membrane Action ◽  
Transverse Loads ◽  
Analysis Design

The economic design of steel caissons for drilling and production platforms in the Arctic Ocean, formed from steel plates and supported by a rectangular grid of stiffeners, beams, and girders, requires that the full strength of the plates be mobilized to withstand extreme ice forces. A comprehensive method of analysis is needed to describe the behaviour of continuous steel plates into the inelastic range when they are acted upon by transverse loads. From this analysis, design procedures could then be developed.An extensive literature search has not revealed that satisfactory solutions exist for the load–deflection response of transversely loaded flat plates beyond the elastic limit when both flexural and membrane action are taken into account. Experimental data available in the inelastic range of behaviour are also limited.By considering various limiting simplified behavioural modes for the load–deflection response of uniformly loaded flat plates of zero aspect ratio, possible load–deflection domains are established. The limiting responses investigated are: elastic–inelastic flexural action, elastic membrane action, inelastic membrane action with increased stiffness resulting from increased Poisson's ratio in the inelastic range, elastic flexural membrane action, and action of a fully yielded cross section in flexure that gradually gives away to a fully yielded cross section in tension. Within the domain so established, a load–deflection behaviour is proposed that is in reasonable agreement with the results of the limited test data available. The results of a finite element analysis using the ADINA computer program are also in reasonable agreement with the proposed analysis. Design applications are discussed. Key words: deflection, elastic, elastoplastic, flexural resistance, membrane force, membrane resistance, plates, steel, strains, stresses, transverse load.

The Stability of a Spinning Elastic Disk With a Transverse Load System

1976 ◽  
Vol 43 (3) ◽  
pp. 485-490 ◽  
Author(s):  
W. D. Iwan ◽  
T. L. Moeller
Keyword(s):  
Equation Of Motion ◽  
Constant Angular Velocity ◽  
Transverse Load ◽  
Modal Interaction ◽  
Disk System ◽  
Linear Ordinary Differential Equations ◽  
Spinning Disk ◽  
Constant Coefficients ◽  
Elastic Disk ◽  
The Stability

This paper presents results of an investigation on the effect of a transverse load on the stability of a spinning elastic disk. The disk rotates at constant angular velocity and the load consists of a mass distributed over a small area of the disk, a spring, and a dashpot. The equation of motion for the transverse vibration of the disk is written as a system of linear ordinary differential equations with constant coefficients. The analysis indicates that the disk system is unstable for speeds in a region above the critical speeds of vibration of the spinning disk due to the effects of load stiffness. The mass and damping of the load system cause a terminal instability and other instabilities occur as a result of modal interaction.

Response of Annular Plates to Circumferentially and Radially Moving Loads

1993 ◽  
Vol 60 (3) ◽  
pp. 649-661 ◽  
Author(s):  
G. N. Weisensel ◽  
A. L. Schlack
Keyword(s):  
Plate Theory ◽  
Outer Boundary ◽  
Integral Form ◽  
Moving Loads ◽  
Classical Plate Theory ◽  
Concentrated Load ◽  
Annular Plates ◽  
Transverse Loads ◽  
Significant Extension ◽  
Loading Parameter

The forced dynamic response of annular plates to circumferentially and radially moving concentrated transverse loads is investigated utilizing classical plate theory, with damping included, and solved in integral form. The boundary conditions are that the inner boundary of the plate is clamped and the outer boundary is free. An analytical expression in Fourier-Bessel series form is obtained for the forced deflection response to an arbitrarily moving concentrated load. This study includes radially moving loads and is a significant extension of the understanding of circular and annular plate dynamics. This understanding of radially moving loads is used to examine the nature of resonance conditions and corresponding critical values of the load parameters. The shapes of deflection modes of plate vibration are also presented. Damping and loading parameter sensitivities are studied in detail.

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