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.

On the Vibrations of an Axially Moving String With a Time-Dependent Velocity

2015 ◽  
Author(s):  
Rajab A. Malookani ◽  
Wim T. van Horssen
Keyword(s):  
Perturbation Method ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Axially Moving String ◽  
Fourier Sine Series ◽  
Axially Moving ◽  
Initial Boundary Value

The transverse vibrations of an axially moving string with a time-varying speed is studied in this paper. The governing equations of motion describing an axially moving string is analyzed using two different techniques. At first, the initial-boundary value problem is discretized using the Fourier sine series, and then the two timescales perturbation method is employed in search of infinite mode approximate solutions. Secondly, a new approach based on the two timescales perturbation method and the method of characteristics is used. It is found that there are infinitely many values of the velocity fluctuation frequency yielding infinitely many resonance conditions in the system. The response of the system with harmonically varying velocity function is computed for particular harmonic initial conditions.

scholarly journals Dynamic stability of an axially moving paper board with added subsystems

2018 ◽  
Vol 37 (1) ◽  
pp. 48-59 ◽  
Author(s):  
Yan Wang ◽  
Zhen Tian ◽  
Jimei Wu ◽  
Xuxia Guo ◽  
Mingyue Shao
Keyword(s):  
Dynamic Stability ◽  
High Speed ◽  
Equations Of Motion ◽  
Stable State ◽  
Element Free Galerkin ◽  
Axially Moving ◽  
Elastically Restrained Edges ◽  
Elastically Restrained ◽  
Element Free ◽  
The Relationship

Paper board with bending stiffness is usually used as the substrates on cigarette package printing industry. The vibration of these paper boards in high speed affects the printing precision. The dynamic characteristics and stability of moving paper board with finite interior elastic point supports and elastic edges restrained are investigated. First, the energy function of the system is established by using the extended Hamilton’s principle; second, the dimensionless equations of motion for the moving paper board are obtained using the element-free Galerkin method. The equations of motion and the eigenvalue equations of the system are established. The relationship between the first three complex frequencies of the system and the moving speed is then obtained by the numerical calculation. The effects of the elastic point supports, the elastically restrained edges, and the dimensionless speed of the motion on the dynamic stability of the paper board are analyzed. The critical speed when the paper board is in a stable state under different conditions is obtained. The results improve the dynamic stability of the paper board in printing process and provide the theoretical basis for the optimization of printing equipment.

A Sound Derivation of the Nonlinear Differential Equations of Motion of a Travelling String

1995 ◽  
Vol 23 (3) ◽  
pp. 215-228 ◽  
Author(s):  
Patrick Bar-Avi ◽  
Itzhak Porat
Keyword(s):  
Differential Equations ◽  
High Speed ◽  
Control Volume ◽  
Direct Method ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Plane Motion ◽  
Longitudinal Vibrations ◽  
Axially Moving ◽  
Volume Method

Axially moving materials, such as high-speed magnetic tapes, belts and band saws, have been discussed since 1897. In this paper the nonlinear differential equations, which describe the string's plane motion (lateral and longitudinal), are developed by two different methods: direct method (Newton's second law) and Hamilton's principle. The control volume method is presented briefly. The equations are stated in two different coordinates systems. Comparison between the equations developed by the different methods and coordinates systems shows that they are the same. The coupling between the lateral and longitudinal vibrations is of the second order, hence linearization (to the first order) leads to uncoupled equations.

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.

Stability and vibration analysis of an axially moving thin walled conical shell

2021 ◽  
pp. 107754632199760
Author(s):  
Hossein Abolhassanpour ◽  
Faramarz Ashenai Ghasemi ◽  
Majid Shahgholi ◽  
Arash Mohamadi
Keyword(s):  
Natural Frequency ◽  
Conical Shell ◽  
Shell Theory ◽  
Cone Angle ◽  
Theory Of Elasticity ◽  
Lower Velocity ◽  
Motion Equations ◽  
Conical Shells ◽  
Divergence Instability ◽  
Axially Moving

This article deals with the analysis of free vibration of an axially moving truncated conical shell. Based on the classical linear theory of elasticity, Donnell shell theory assumptions, Hamilton principle, and Galerkin method, the motion equations of axially moving truncated conical shells are derived. Then, the perturbation method is used to obtain the natural frequency of the system. One of the most important and controversial results in studies of axially moving structures is the velocity detection of critical points. Therefore, the effect of velocity on the creation of divergence instability is investigated. The other important goal in this study is to investigate the effect of the cone angle. As a novelty, our study found that increasing or decreasing the cone angle also affects the critical velocity of the structure in addition to changing the natural frequency, meaning that with increasing the cone angle, the instability occurs at a lower velocity. Also, the effect of other parameters such as aspect ratio and mechanical properties on the frequency and instability points is investigated.

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.

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.

Determination of Instability Boundaries for a Highly Flexible Prismatic-Jointed Robot Performing Repetitive Maneuvers

1991 ◽  
Author(s):  
Keith W. Buffinton
Keyword(s):  
Floquet Theory ◽  
Equations Of Motion ◽  
Perturbation Techniques ◽  
Axial Motion ◽  
The Third ◽  
Straightforward Application ◽  
Method Yield ◽  
Robotic Devices ◽  
Axially Moving

Abstract Presented in this work are the equations of motion governing the behavior of a simple, highly flexible, prismatic-jointed robotic manipulator performing repetitive maneuvers. The robot is modeled as a uniform cantilever beam that is subject to harmonic axial motions over a single bilateral support. To conveniently and accurately predict motions that lead to unstable behavior, three methods are investigated for determining the boundaries of unstable regions in the parameter space defined by the amplitude and frequency of axial motion. The first method is based on a straightforward application of Floquet theory; the second makes use of the results of a perturbation analysis; and the third employs Bolotin’s infinite determinate method. Results indicate that both perturbation techniques and Bolotin’s method yield acceptably accurate results for only very small amplitudes of axial motion and that a direct application of Floquet theory, while computational expensive, is the most reliable way to ensure that all instability boundaries are correctly represented. These results are particularly relevant to the study of prismatic-jointed robotic devices that experience amplitudes of periodic motion that are a significant percentage of the length of the axially moving member.

scholarly journals Dynamics of Space Particles and Spacecrafts Passing by the Atmosphere of the Earth

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Vivian Martins Gomes ◽  
Antonio Fernando Bertachini de Almeida Prado ◽  
Justyna Golebiewska
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
The Sun ◽  
Three Body Problem ◽  
The Earth ◽  
Before And After ◽  
Three Body ◽  
Body Energy ◽  
Spacecraft Position

The present research studies the motion of a particle or a spacecraft that comes from an orbit around the Sun, which can be elliptic or hyperbolic, and that makes a passage close enough to the Earth such that it crosses its atmosphere. The idea is to measure the Sun-particle two-body energy before and after this passage in order to verify its variation as a function of the periapsis distance, angle of approach, and velocity at the periapsis of the particle. The full system is formed by the Sun, the Earth, and the particle or the spacecraft. The Sun and the Earth are in circular orbits around their center of mass and the motion is planar for all the bodies involved. The equations of motion consider the restricted circular planar three-body problem with the addition of the atmospheric drag. The initial conditions of the particle or spacecraft (position and velocity) are given at the periapsis of its trajectory around the Earth.

scholarly journals Novel analysis methods of dynamic properties for vehicle pantographs

2018 ◽  
Vol 180 ◽  
pp. 01005 ◽  
Author(s):  
Andrzej Wilk
Keyword(s):  
Dynamic Analysis ◽  
Dynamic Simulation ◽  
High Speed ◽  
Dynamic Properties ◽  
Equations Of Motion ◽  
Computer Software ◽  
Fem Analysis ◽  
Electrical Energy ◽  
Modern Approach ◽  
Static And Dynamic Analysis

Transmission of electrical energy from a catenary system to traction units must be safe and reliable especially for high speed trains. Modern pantographs have to meet these requirements. Pantographs are subjected to several forces acting on their structural elements. These forces come from pantograph drive, inertia forces, aerodynamic effects, vibration of traction units etc. Modern approach to static and dynamic analysis should take into account: mass distribution of particular parts, physical properties of used materials, kinematic joints character at mechanical nodes, nonlinear parameters of kinematic joints, defining different parametric waveforms of forces and torques, and numerical dynamic simulation coupled with FEM calculations. In this work methods for the formulation of the governing equations of motion are presented. Some of these methods are more suitable for automated computer implementation. The novel computer methods recommended for static and dynamic analysis of pantographs are presented. Possibilities of dynamic analysis using CAD and CAE computer software are described. Original results are also presented. Conclusions related to dynamic properties of pantographs are included. Chapter 2 presents the methods used for formulation of the equation of pantograph motion. Chapter 3 is devoted to modelling of forces in multibody systems. In chapter 4 the selected computer tools for dynamic analysis are described. Chapter 5 shows the possibility of FEM analysis coupled with dynamic simulation. In chapter 6 the summary of this work is presented.

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