Free, Periodic, Nonlinear Oscillation of an Axially Moving Strip

1969 ◽  
Vol 36 (1) ◽  
pp. 83-91 ◽  
Author(s):  
A. L. Thurman ◽  
C. D. Mote
Keyword(s):  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Practical Interest ◽  
Limited Range ◽  
Fundamental Period ◽  
Transverse Motion ◽  
Longitudinal Motion ◽  
Moving Strip ◽  
Axially Moving ◽  
Linear Period

The free, nonlinear, fundamental period of transverse oscillation of axially moving strips (e.g., tapes, fibers, belts, and band saws) is determined by the approximate solution of two, coupled, nonlinear, partial differential equations. One equation describes longitudinal motion and the other transverse motion. A solution method is developed that permits accurate and efficient period calculations. The results indicate that the existence of the axial transport velocity reduces the fundamental period of oscillation and increases the relative importance of the nonlinear terms in the equations of motion. In many cases of practical interest the linear analysis is shown to be seriously in error and one may be led to erroneous conclusions because of its limited range of applicability. Curves are presented that assist one to estimate the accuracy of the linear period calculation.

Nonlinear Oscillation of a Cylinder Containing a Flowing Fluid

1969 ◽  
Vol 91 (4) ◽  
pp. 1147-1155 ◽  
Author(s):  
A. L. Thurman ◽  
C. D. Mote
Keyword(s):  
Approximate Solution ◽  
Fluid Transport ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Practical Interest ◽  
Longitudinal Motion ◽  
Flowing Fluid ◽  
Weakly Nonlinear ◽  
Efficient Calculation

The fundamental and second periods of transverse oscillation of a cylinder containing a flowing fluid are theoretically determined for the approximate solution of two, coupled, nonlinear partial differential equations describing the transverse and longitudinal motion. The calculations indicate that the existence of the fluid transport velocity reduces all cylinder natural periods of oscillation and increases the relative importance of non-linear terms in the equations of motion. Accordingly, in many cases of practical interest the linear analysis is shown to be severely limited in its applicability. Curves are presented that will assist one to estimate both the accuracy of the linear period and the approximate nonlinear period in selected examples. A new approximate solution method is utilized that permits accurate and efficient calculation of the nonlinear period. This method can be applied to the period determination of additional cylindrical models not examined herein; the method appears to be semi-generally applicable to the periodic solution of weakly nonlinear systems.

scholarly journals Nonlinear Vibration Analysis of Moving Strip with Inertial Boundary Condition

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Chong-yi Gao ◽  
Guo-jun Du ◽  
Yan Feng ◽  
Jian-xiong Li
Keyword(s):  
Nonlinear Vibration ◽  
Rolling Process ◽  
Vertical Vibration ◽  
Rotational Inertia ◽  
Longitudinal Motion ◽  
Euler Beam ◽  
Motion Equations ◽  
Moving Strip ◽  
Movement Mechanism ◽  
Axially Moving

According to the movement mechanism of strip and rollers in tandem mill, the strip between two stands was simplified to axially moving Euler beam and the rollers were simplified to the inertial component on the fixed axis rotation, namely, inertial boundary. Nonlinear vibration mechanical model of Euler beam with inertial boundary conditions was established. The transverse and longitudinal motion equations were derived based on Hamilton’s principle. Kantorovich averaging method was employed to discretize the motion equations and the inertial boundary equations, and the solutions were obtained using the modified iteration method. Depending on numerical calculation, the amplitude-frequency responses of Euler beam were determined. The axial velocity, tension, and rotational inertia have strong influences on the vibration characteristics. The results would provide an important theoretical reference to control and analyze the vertical vibration of moving strip in continuous rolling process.

Dynamic Properties of a Moving Thread Line

1976 ◽  
Vol 98 (3) ◽  
pp. 868-875 ◽  
Author(s):  
M. A. Moustafa ◽  
F. K. Salman
Keyword(s):  
Fundamental Frequency ◽  
Dynamic Properties ◽  
Unit Length ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Initial Tension ◽  
Tension Distribution ◽  
Elastic Strings ◽  
Axially Moving ◽  
Mathematical Relations

A mathematical model representing the transverse vibration of axially moving elastic strings is presented considering tension and mass variation. A suggested numerical scheme was successfully used to solve the nonlinear partial differential equations of motion. For axially nonmoving strings, the effect of initial amplitudes, and consequently the tension variation on the fundamental frequency is obtained. Also, the effect of the initial tension and the mass of the string per unit length on the fundamental frequency and their corresponding mathematical relations are presented. For axially moving strings, the effect of the axial velocity on the fundamental frequency as well as the tension distribution along the thread is given. Also the behavior of the string at velocities equal and greater than the wave speed is shown.

Stability and Bifurcations in Three-Dimensional Analysis of Axially Moving Beams

2013 ◽  
Author(s):  
Mergen H. Ghayesh ◽  
Marco Amabili ◽  
Hamed Farokhi
Keyword(s):  
Differential Equations ◽  
Fourier Transforms ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Time Integration ◽  
Three Dimensional ◽  
Axially Moving Beam ◽  
Bifurcation Diagrams ◽  
Response Curves ◽  
Axially Moving

The geometrically nonlinear dynamics of a three-dimensional axially moving beam is investigated numerically for both sub and supercritical regimes. Hamilton’s principle is employed to derive the equations of motion for in-plane and out-of plane displacements. The Galerkin scheme is applied to the nonlinear partial differential equations of motion yielding a set of second-order nonlinear ordinary differential equations with coupled terms. The pseudo-arclength continuation technique is employed to solve the discretized equations numerically so as to obtain the nonlinear resonant responses; direct time integration is conducted to obtain the bifurcation diagrams of the system. The results are presented in the form of the frequency-response curves, bifurcation diagrams, time histories, phase-plane portraits, and fast Fourier transforms for different sets of system parameters.

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.

Dynamics of Spinning Disc Parametrically Excited by Noise

1999 ◽  
Author(s):  
Narayanan Ramakrishnan ◽  
N. Sri Namachchivaya
Keyword(s):  
Equations Of Motion ◽  
Planck Equation ◽  
Stochastic Averaging ◽  
Transverse Motion ◽  
Probability Density Distribution ◽  
Integral Of Motion ◽  
Parametrically Excited ◽  
One Dimensional ◽  
Small Intensity ◽  
Spinning Disc

Abstract The nonlinear dynamics of a circular spinning disc parametrically excited by noise of small intensity is investigated. The governing PDEs are reduced using a Galerkin reduction procedure to a two-DOF system of ODEs which, govern the transverse motion of the disc. The dynamics is simplified by exploiting the S1 invariance of the equations of motion of the reduced system and further, reduced by performing stochastic averaging. The resulting one-dimensional Markov diffusive process is studied in detail. The stationary probability density distribution is obtained by solving the Fokker-Planck equation along with the appropriate boundary conditions. The boundary behaviour is studied using an asymptotic approach. Some aspects of dynamical and phenomenological bifurcations of the stationary solution are also investigated. The scheme of things presented here can be applied in principle to a four-dimensional Hamiltonian system possessing one integral of motion in addition to the hamiltonian and having one fixed point.

Investigation of the Stability of Axially Moving Beams With Discrete Masses

2021 ◽  
Author(s):  
Konstantina Ntarladima ◽  
Michael Pieber ◽  
Johannes Gerstmayr
Keyword(s):  
Large Deformations ◽  
Equations Of Motion ◽  
Arbitrary Lagrangian Eulerian ◽  
Axial Motion ◽  
Multibody Modeling ◽  
Conveyor Belts ◽  
Axially Moving Beams ◽  
Concentrated Masses ◽  
Axially Moving ◽  
The Stability

Abstract The present paper addresses axially moving beams with co-moving concentrated masses while undergoing large deformations. For the numerical modeling, a novel beam finite element is introduced, which is based on the absolute nodal coordinate formulation extended with an additional Eulerian coordinate to represent the axial motion. The resulting formulation is well known as Arbitrary Lagrangian Eulerian (ALE) method, which is often used for axially moving beams and pipes conveying fluids. As compared to previous formulations, the present formulation allows us to introduce the Eulerian part by an independent coordinate, which fully incorporates the dynamics of the axial motion, while the shape functions remain independent of the beam coordinates and are thus constant. The proposed approach, which is derived from an extended version of Lagrange’s equations of motion, allows for the investigation of the stability of axially moving beams for a certain axial velocity and stationary state of large deformation. A multibody modeling approach allows us to extend the beam formulation for co-moving discrete masses, which represent concentrated masses attached to the beam, e.g., gondolas in ropeway systems, or transported masses in conveyor belts. Within numerical investigations we show that a larger number of discrete masses behaves similarly as the case of (continuously) distributed mass along the beam.

Model Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes

1999 ◽  
Author(s):  
E. Pesheck ◽  
C. Pierre ◽  
S. W. Shaw
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Single Equation ◽  
Rotating Beam ◽  
Model Size ◽  
Nonlinear Normal

Abstract Equations of motion are developed for a rotating beam which is constrained to deform in the transverse (flapping) and axial directions. This process results in two coupled nonlinear partial differential equations which govern the attendant dynamics. These equations may be discretized through utilization of the classical normal modes of the nonrotating system in both the transverse and extensional directions. The resultant system may then be diagonalized to linear order and truncated to N nonlinear ordinary differential equations. Several methods are used to determine the model size necessary to ensure accuracy. Once the model size (N degrees of freedom) has been determined, nonlinear normal mode (NNM) theory is applied to reduce the system to a single equation, or a small set of equations, which accurately represent the dynamics of a mode, or set of modes, of interest. Results are presented which detail the convergence of the discretized model and compare its dynamics with those of the NNM-reduced model, as well as other reduced models. The results indicate a considerable improvement over other common reduction techniques, enabling the capture of many salient response features with the simulation of very few degrees of freedom.

Vibration Characteristics of Straight Blades of Asymmetrical Aerofoil Cross-Section

1969 ◽  
Vol 20 (2) ◽  
pp. 178-190 ◽  
Author(s):  
W. Carnegie ◽  
B. Dawson
Keyword(s):  
Analytical Solution ◽  
Cross Section ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Limited Range ◽  
First Order ◽  
Special Cases ◽  
Theoretical Results ◽  
Theoretical Procedure ◽  
Good Agreement

SummaryTheoretical and experimental natural frequencies and modal shapes up to the fifth mode of vibration are given for a straight blade of asymmetrical aerofoil cross-section. The theoretical procedure consists essentially of transforming the differential equations of motion into a set of simultaneous first-order equations and solving them by a step-by-step finite difference procedure. The natural frequency values are compared with results obtained by an analytical solution and with standard solutions for certain special cases. Good agreement is shown to exist between the theoretical results for the various methods presented. The equations of motion are dependent upon the coordinates of the axis of the centre of flexure of the beam relative to the centroidal axis. The effect of variations of the centre of flexure coordinates upon the frequencies and modal shapes is shown for a limited range of coordinate values. Comparison is made between the theoretical natural frequencies and modal shapes and corresponding results obtained by experiment.

Kinematic Aspects in Modeling Large-Amplitude Vibration of Axially Moving Beams

2019 ◽  
Vol 11 (02) ◽  
pp. 1950021 ◽  
Author(s):  
Yuanbin Wang ◽  
Hu Ding ◽  
Li-Qun Chen
Keyword(s):  
Large Amplitude ◽  
Model Equation ◽  
Classical Model ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Governing Equations ◽  
Amplitude Vibration ◽  
Large Amplitude Vibration ◽  
Axially Moving Beams ◽  
Axially Moving

This paper clarified kinematic aspects of motion of axially moving beams undergoing large-amplitude vibration. The kinematics was formulated in the mixed Eulerian–Lagrangian framework. Based on the kinematic analysis, the governing equations of nonlinear vibration were derived from the extended Hamilton principle and the higher-order shear beam theory. The derivation considered the effects of material parameters on the beam deformation. The proposed governing equations were compared with a few previous governing equations. The comparisons show that proposed equations are with higher precision. Besides, the proposed equations can be viewed as the asymptotic governing equations of Lagrange’s equations of motion for large displacement. Finally, the corresponding boundary conditions and the comparison between the presented model equation and classical model equation were provided.

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