Response Solutions for the Vibration of a Traveling String on an Elastic Foundation

1994 ◽  
Vol 116 (1) ◽  
pp. 137-139 ◽  
Author(s):  
J. A. Wickert
Keyword(s):  
Elastic Foundation ◽  
High Speed ◽  
Initial Conditions ◽  
Standard Form ◽  
Equation Of Motion ◽  
Air Bearings ◽  
Axial Translation ◽  
Axially Moving ◽  
Technical Interest ◽  
Acceleration Component

This Tech Brief presents solutions to the response problem for the vibration of an axially-moving string that is supported by an elastic foundation. This system is of technical interest in the area of flexible media which translates at a high speed, and which is guided by air bearings or similarly modeled distributed supports. The equation of motion is dispersive and contains a skew-symmetric “Coriolis” acceleration component which derives from axial translation of the string. The equation of motion is written in the standard form for a continuous gyroscopic system, so that the string’s stability and response can be analyzed within this broader context. Available modal analysis and Green’s function methods then provide closed form expressions for the response to arbitrary initial conditions and excitation.

Vibration and Guiding of Moving Media With Edge Weave Imperfections

2005 ◽  
Author(s):  
V. Kartik ◽  
J. A. Wickert
Keyword(s):  
Elastic Foundation ◽  
Moving Load ◽  
Lateral Position ◽  
Harmonic Excitation ◽  
Forced Response ◽  
Moving Medium ◽  
Web Handling ◽  
Moving Strip ◽  
Axially Moving ◽  
Technical Interest

This paper examines the steady-state forced vibration of a moving medium that is guided by a partial elastic foundation, and where geometric imperfections on the medium’s edge act as an excitation source. Such a system is of technical interest in the areas of web handling and magnetic tape transport where externally-pressurized air bearing guides are sometimes used to control lateral position. The axially-moving strip is modeled here as a string that is guided by elastic foundation segments and that is subjected to traveling wave excitation as the edge imperfections interact with the foundations. The equation of motion for this “moving medium and moving load” system incorporates a skew-symmetric Coriolis acceleration component that arises from convection. The governing equation is cast in the state-space form, with one symmetric and one skew-symmetric operator, as is characteristic of gyroscopic systems. Through modal analysis, the forced response of the system is obtained to the complex harmonic excitation associated with the interaction between the edge weave and the guides. Parameter studies are presented in the transport speed, foundation stiffness, guide placement, guide width, and imperfection wavelength. Of potential technological application, for a given wavelength of the edge imperfection, it is possible to reduce the medium’s vibration at a certain location by judiciously selecting the locations and spans of the foundation segments.

scholarly journals Effect of Changing Table Tennis Ball Material from Celluloid to Plastic on the Post-Collision Ball Trajectory

2017 ◽  
Vol 55 (1) ◽  
pp. 29-38 ◽  
Author(s):  
Yuki Inaba ◽  
Sho Tamaki ◽  
Haruhiko Ikebukuro ◽  
Koshi Yamada ◽  
Hiroki Ozaki ◽  
...  
Keyword(s):  
High Speed ◽  
Initial Conditions ◽  
Contact Point ◽  
Equation Of Motion ◽  
Tennis Ball ◽  
Table Tennis ◽  
Plastic Ball ◽  
Before And After ◽  
Ball Trajectory ◽  
Tennis Balls

AbstractThe official material used in table tennis balls was changed from celluloid to plastic, a material free of celluloid, in 2014. The purpose of this study was to understand the differences and similarities in the two types of ball materials by comparing their behavior upon collision with a table. The behavior of the balls before and after collision with a table, at various initial speeds ranging from 15 to 115 km/h, was captured using high-speed cameras. Velocities and spin rates before collision and velocities after collision were computed to calculate the coefficients of restitution and friction. Based on the computed variables, the post-collision trajectories of both balls were calculated by integrating the equation of motion of the ball for simulated service, smash and drive conditions with respect to time. The coefficients of restitution were higher for the plastic balls than the celluloid ones when the initial vertical velocities were higher. The coefficients of friction were higher for plastic balls when the initial horizontal contact point velocities were slower. Because of the differences in the material characteristics, the plastic ball trajectories of services with backspin and drives with great topspin were expected to be different from those of celluloid balls. Since the extent of differences between the two ball types varied depending on the initial conditions, testing at various initial conditions was suggested for comparing and understanding the characteristics of the balls.

Classical Vibration Analysis of Axially Moving Continua

1990 ◽  
Vol 57 (3) ◽  
pp. 738-744 ◽  
Author(s):  
J. A. Wickert ◽  
C. D. Mote
Keyword(s):  
High Speed ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Positive Definiteness ◽  
Canonical State ◽  
Divergence Instability ◽  
Traveling Beam ◽  
Axially Moving ◽  
Band Saw ◽  
Classical Vibration

Axially moving continua, such as high-speed magnetic tapes and band saw blades, experience a Coriolis acceleration component which renders such systems gyroscopic. The equations of motion for the traveling string and the traveling beam, the most common models of axially moving materials, are each cast in a canonical state space form defined by one symmetric and one skew-symmetric differential operator. When an equation of motion is represented in this form, the eigenfunctions are orthogonal with respect to each operator. Following this formulation, a classical vibration theory, comprised of a modal analysis and a Green’s function method, is derived for the class of axially moving continua. The analysis is applied to the representative traveling string and beam models, and exact closed-form expressions for their responses to arbitrary excitation and initial conditions result. In addition, the critical transport speed at which divergence instability occurs is determined explicitly from a sufficient condition for positive definiteness of the symmetric operator.

Particle motion with air resistance

2012 ◽  
Vol 8 (1) ◽  
pp. 1-15
Author(s):  
Gy. Sitkei
Keyword(s):  
Basic Equation ◽  
Initial Conditions ◽  
Equation Of Motion ◽  
Terminal Velocity ◽  
Dimensionless Form ◽  
Wood Industry ◽  
Acceptable Error ◽  
General Validity ◽  
Practical Applications ◽  
Air Resistance

Motion of particles with air resistance (e.g. horizontal and inclined throwing) plays an important role in many technological processes in agriculture, wood industry and several other fields. Although, the basic equation of motion of this problem is well known, however, the solutions for practical applications are not sufficient. In this article working diagrams were developed for quick estimation of the throwing distance and the terminal velocity. Approximate solution procedures are presented in closed form with acceptable error. The working diagrams provide with arbitrary initial conditions in dimensionless form of general validity.

High-Speed Imaging to Study an Auto-Oscillating Vocal Fold Replica for Different Initial Conditions

2017 ◽  
Vol 09 (05) ◽  
pp. 1750064 ◽  
Author(s):  
A. Van Hirtum ◽  
X. Pelorson
Keyword(s):  
Vocal Fold ◽  
High Speed ◽  
Initial Conditions ◽  
Vocal Folds ◽  
High Speed Imaging ◽  
Human Voice ◽  
Manual Intervention ◽  
Geometrical Features ◽  
Upstream Pressure

Experiments on mechanical deformable vocal folds replicas are important in physical studies of human voice production to understand the underlying fluid–structure interaction. At current date, most experiments are performed for constant initial conditions with respect to structural as well as geometrical features. Varying those conditions requires manual intervention, which might affect reproducibility and hence the quality of experimental results. In this work, a setup is described which allows setting elastic and geometrical initial conditions in an automated way for a deformable vocal fold replica. High-speed imaging is integrated in the setup in order to decorrelate elastic and geometrical features. This way, reproducible, accurate and systematic measurements can be performed for prescribed initial conditions of glottal area, mean upstream pressure and vocal fold elasticity. Moreover, quantification of geometrical features during auto-oscillation is shown to contribute to the experimental characterization and understanding.

Nonlinear Vibrations of a Hinged Beam Including Nonlinear Inertia Effects

1973 ◽  
Vol 40 (1) ◽  
pp. 121-126 ◽  
Author(s):  
S. Atluri
Keyword(s):  
Nonlinear Equation ◽  
Nonlinear Vibrations ◽  
Initial Conditions ◽  
Equation Of Motion ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Nonlinear Ordinary Differential Equations ◽  
Inertia Effects ◽  
Modal Expansion ◽  
General Response

This investigation treats the large amplitude transverse vibration of a hinged beam with no axial restraints and which has arbitrary initial conditions of motion. Nonlinear elasticity terms arising from moderately large curvatures, and nonlinear inertia terms arising from longitudinal and rotary inertia of the beam are included in the nonlinear equation of motion. Using a Galerkin variational method and a modal expansion, the problem is reduced to a system of coupled nonlinear ordinary differential equations which are solved for arbitrary initial conditions, using the perturbation procedure of multiple-time scales. The general response and frequency-amplitude relations are derived theoretically. Comparison with previously published results is made.

High-Performance Motion Control With Slosh: A Constrained Sliding Mode Approach

2008 ◽  
Author(s):  
Hanz Richter ◽  
Kedar B. Karnik
Keyword(s):  
High Speed ◽  
High Performance ◽  
Design Methodology ◽  
Closed Loop ◽  
Sliding Mode ◽  
Initial Conditions ◽  
Rectilinear Motion ◽  
Industrial Setting ◽  
Open Container ◽  
Nominal Performance

The problem of controlling the rectilinear motion of an open container without exceeding a prescribed liquid level and other constraints is considered using a recently-developed constrained sliding mode control design methodology based on invariant cylinders. A conventional sliding mode regulator is designed first to address nominal performance in the sliding mode. Then an robustly-invariant cylinder is constructed and used to describe the set of safe initial conditions from which the closed-loop controller can be operated without constraint violation. Simulations of a typical transfer illustrate the usefulness of the method in an industrial setting. Experimental results corresponding to a high-speed transfer validate the theory.

Free Vibration Analysis for Thin Wire of Modern Wiresaw Between Sliced Wafers in Wafer Manufacturing Processes

2000 ◽  
Author(s):  
Songbin Wei ◽  
Imin Kao
Keyword(s):  
High Speed ◽  
Free Vibration Analysis ◽  
Equation Of Motion ◽  
Manufacturing Processes ◽  
Thin Wire ◽  
Wire Vibration ◽  
The Wire ◽  
The Impact ◽  
Wafer Manufacturing ◽  
Impact Problem

Abstract In wiresaw manufacturing process where thin wire moving at high speed is pushed onto ingot to produce slices of wafer, the wire is constrained by two wafer walls as it slices into the ingot. In this paper, we investigate the vibration of such wire under the constraints of wafer walls. To address this problem, the model for wire vibration with impact to wafer walls is developed. The equation of motion is discretized using the Galerkin’s method. The principle of impulse and momentum is utilized to solve the impact problem. The results of analysis and simulation indicate that the response under a pointwise sinusoidal excitation is neither periodical nor symmetric with respect to the horizontal axis, due to the excitation from the impact. The wire vibration behavior is affected dramatically by the wafer wall constraints.

On the Vibration of a Rotating Disk

1972 ◽  
Vol 39 (4) ◽  
pp. 1143-1144 ◽  
Author(s):  
S. Barasch ◽  
Y. Chen
Keyword(s):  
Approximate Calculation ◽  
Initial Conditions ◽  
Rotating Disk ◽  
Equation Of Motion ◽  
Outer Radius ◽  
Order Equation ◽  
System Of Differential Equations ◽  
Fourth Order Equation ◽  
First Order ◽  
Inner Radius

The equation of motion of a rotating disk, clamped at the inner radius and free at the outer radius, is solved by reducing the fourth-order equation of motion to a set of four first-order equations subject to arbitrary initial conditions. A modified Adams’ method is used to numerically integrate the system of differential equations. Results show that Lamb-Southwell’s approximate calculation of the frequency is justified.

scholarly journals Multi-moving Loads Induced Vibration of FG Sandwich Beams Resting on Pasternak Elastic Foundation

2021 ◽  
Vol 18 (9) ◽  
Author(s):  
Wachirawit SONGSUWAN ◽  
Monsak PIMSARN ◽  
Nuttawit WATTANASAKULPONG
Keyword(s):  
Dynamic Response ◽  
Elastic Foundation ◽  
Excitation Frequency ◽  
Beam Theory ◽  
Equation Of Motion ◽  
Functionally Graded ◽  
Sandwich Beams ◽  
Moving Loads ◽  
Layer Thickness Ratio ◽  
Pasternak Elastic Foundation

The dynamic behavior of functionally graded (FG) sandwich beams resting on the Pasternak elastic foundation under an arbitrary number of harmonic moving loads is presented by using Timoshenko beam theory, including the significant effects of shear deformation and rotary inertia. The equation of motion governing the dynamic response of the beams is derived from Lagrange’s equations. The Ritz and Newmark methods are implemented to solve the equation of motion for obtaining free and forced vibration results of the beams with different boundary conditions. The influences of several parametric studies such as layer thickness ratio, boundary condition, spring constants, length to height ratio, velocity, excitation frequency, phase angle, etc., on the dynamic response of the beams are examined and discussed in detail. According to the present investigation, it is revealed that with an increase of the velocity of the moving loads, the dynamic deflection initially increases with fluctuations and then drops considerably after reaching the peak value at the critical velocity. Moreover, the distance between the loads is also one of the important parameters that affect the beams’ deflection results under a number of moving loads.

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