Lateral Vibration of Two Axially Translating Beams Interconnected by a Winkler Foundation

2006 ◽  
Vol 129 (3) ◽  
pp. 380-385 ◽  
Author(s):  
Mohamed Gaith ◽  
Sinan Müftü
Keyword(s):  
Natural Frequencies ◽  
Flexural Rigidity ◽  
Mode Shapes ◽  
Winkler Foundation ◽  
Axial Tension ◽  
Winkler Elastic Foundation ◽  
Critical Tension ◽  
Divergence Instability ◽  
Axially Moving Beams ◽  
Axially Moving

Transverse vibration of two axially moving beams connected by a Winkler elastic foundation is analyzed analytically. The two beams are tensioned, translating axially with a common constant velocity, simply supported at their ends, and of different materials and geometry. The natural frequencies and associated mode shapes are obtained. The natural frequencies of the system are composed of two infinite sets describing in-phase and out-of-phase vibrations. In case the beams are identical, these modes become synchronous and asynchronous, respectively. Divergence instability occurs at a critical velocity and a critical tension; and, divergence and flutter instabilities coexist at postcritical speeds, and divergence instability takes place precritical tensions. The effects of the mass, flexural rigidity, and axial tension ratios of the two beams are presented.

Transverse Vibration of Two Axially Moving Beams Connected by an Elastic Foundation

2005 ◽  
Author(s):  
Mohamed Gaith ◽  
Sinan Mu¨ftu¨
Keyword(s):  
Elastic Foundation ◽  
Transverse Vibration ◽  
Natural Frequencies ◽  
Flexural Rigidity ◽  
Beam Theory ◽  
Mode Shapes ◽  
Winkler Elastic Foundation ◽  
Canonical State ◽  
Axially Moving Beams ◽  
Axially Moving

Transverse vibration of two axially moving beams connected by a Winkler elastic foundation is analyzed analytically. The system is a model of paper and paper-cloth (wire-screen) used in paper making. The two beams are tensioned, translating axially with a common constant velocity, simply supported at their ends, and of different materials and geometry. Due to the effect of translation, the dynamics of the system displays gyroscopic motion. The Euler-Bernoulli beam theory is used to model the deflections, and the governing equations are expressed in the canonical state form. The natural frequencies and associated mode shapes are obtained. It is found that the natural frequencies of the system are composed of two infinite sets describing in-phase and out-of-phase vibrations. In case the beams are identical, these modes become synchronous and asynchronous, respectively. Divergence instability occurs at the critical velocity; and, the frequency-velocity relationship is similar to that of a single traveling beam. The effects of the mass, flexural rigidity, and axial tension ratios of the two beams, as well as the effects of the elastic foundation stiffness are investigated.

Computing Natural Frequencies and Mode Shapes of an Axially Moving Non-Uniform Beam

2020 ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Cross Sections ◽  
Critical Speed ◽  
Natural Frequencies ◽  
Linear Time ◽  
Mode Shapes ◽  
Axially Moving Beam ◽  
Time Varying Systems ◽  
Axially Moving ◽  
The Galerkin Method ◽  
Spatially Varying

Abstract The partial differential equation of motion of an axially moving beam with spatially varying geometric, mass and material properties has been derived. Using the theory of linear time-varying systems, a general algorithm has been developed to compute natural frequencies, mode shapes, and the critical speed for stability. Numerical results from the new method are presented for beams with spatially varying rectangular cross sections with sinusoidal variation in thickness and sine-squared variation in width. They are also compared to those from the Galerkin method. It has been found that critical speed of the beam can be significantly reduced by non-uniformity in a beam’s cross section.

Free vibration analysis of partially connected parallel beams with elastically restrained ends

2016 ◽  
Vol 230 (16) ◽  
pp. 2851-2864 ◽  
Author(s):  
Alborz Mirzabeigy ◽  
Reza Madoliat
Keyword(s):  
Free Vibration ◽  
Natural Frequencies ◽  
Flexural Rigidity ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Computational Stability ◽  
Transform Method ◽  
Practical Engineering ◽  
Elastically Restrained ◽  
Connection Length

In the present paper, the problem of transverse free vibration of two parallel beams partially connected to each other by a Winkler-type elastic layer is investigated. Euler–Bernoulli beam hypothesis has been applied, and translational and rotational elastic springs in each end considered as support. The motion of the system is described by coupled, piece-wise differential equations. The differential transform method (DTM) is employed to derive natural frequencies and mode shapes. DTM is a semi-analytical approach based on Taylor expansion series which does not require any admissible functions and yields rapid convergence and computational stability. After validation of the DTM results with results reported by well-known references and finite elements solution, the influences of the inner layer connection length, boundary conditions, the coefficient of elastic inner layer and ratio of beam’s flexural rigidity on natural frequencies as well as influences of the inner layer connection length on mode shapes are discussed. This problem is treated for the first time, and results are completely new which candidate them to being considered for practical engineering applications.

scholarly journals Vibration frequency of prestress slender beams resting on Winkler elastic foundation

2006 ◽  
Vol 28 (4) ◽  
pp. 241-251
Author(s):  
Nguyen Dinh Kien
Keyword(s):  
Axial Force ◽  
Vibration Frequency ◽  
Strain Energy ◽  
Natural Frequencies ◽  
Winkler Foundation ◽  
The Finite Element Method ◽  
Various Boundary Conditions ◽  
Winkler Elastic Foundation ◽  
Mass Matrices ◽  
Slender Beams

The present paper investigates the vibration frequency of slender beams prestressing by axial force and resting on an elastic Winkler foundation by the finite element method. A beam element taking the effects of both the prestress and foundation support into account is formulated using the expression of strain energy. Using the developed element, the natural frequencies of beams having various boundary conditions are computed for different values of the axial force and foundation stiffness. The influence of the axial force and the foundation stiffness on the frequency of the beams is investigated. The effect of partial support by the foundation and the type of mass matrices on the vibration frequency of the beam is also studied and highlighted.

scholarly journals Parametric Vibrations of Axially Moving Beams with Multiple Edge Cracks

2019 ◽  
Vol 24 (2) ◽  
pp. 241-252 ◽  
Author(s):  
Murat Sarıgül
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Crack Depth ◽  
Multiple Cracks ◽  
Mean Velocity ◽  
Beam System ◽  
Crack Tips ◽  
Axially Moving Beams ◽  
Edge Cracks ◽  
Axially Moving

Nonlinear transverse vibrations of axially moving beams with multiple cracks is handled studied. Assuming that the beam moves with mean velocity having harmonically variation, influence of the edge crack on the moving continua are investigated in this study. Due to existence of the crack in the transverse direction, the healthily beam is divided into parts. The translational and rotational springs are replaced between these parts so that high stressed regions around the crack tips are redefined with the springs' energies. Thus, the problem is converted to an axially moving spring-beam system. The equations of motion and its corresponding conditions are obtained by means of the Hamilton Principle. In numerical analysis, the natural frequencies and responses of the spring-beam system are investigated for principal parametric resonance in detail. Some important results are obtained; the natural frequencies decreases with increasing crack depth. In case of the beam travelling with high velocities, the effects of crack's depth on natural frequencies seems to be vanished.

On the Vibration of Coupled Traveling String and Air Bearing Systems

1996 ◽  
Vol 118 (3) ◽  
pp. 398-405 ◽  
Author(s):  
A. V. Lakshmikumaran ◽  
J. A. Wickert
Keyword(s):  
High Speed ◽  
Natural Frequencies ◽  
Nodal Point ◽  
Fold Increase ◽  
Computational Effort ◽  
Air Pressure ◽  
Mode Shapes ◽  
Air Bearing ◽  
Design Variables ◽  
Axially Moving

Air bearings are used to position and guide such axially-moving materials as high speed magnetic tapes, paper sheets, and webs. In each case, vibration of the moving medium couples with the air bearing’s dynamics, and techniques are developed here to reduce the computational effort that is required to predict the natural frequencies, damping ratios, and vibration modes of the prototypical traveling string and self-pressurized air bearing model. Automatic nodal point allocation reduces the number of nonlinear equations that arise in finding the equilibrium string displacement and air pressure, and in subsequent vibration analysis, the response is obtained in closed form by using the Green’s function for the traveling string. Global discretization of the air pressure alone then yields a matrix eigenvalue problem which is simpler than that obtained through previous methods which required discretization of both displacement and pressure. Overall, essentially a five-fold increase in computational speed is achieved, thus facilitating design and parameter studies. Changes in the natural frequencies, damping ratios, and coupled displacement-pressure mode shapes with respect to several design variables are discussed and compared with experiments.

Computing Natural Frequencies and Mode Shapes of an Axially Moving Non-Uniform Beam

2021 ◽  
Author(s):  
Alok Sinha
Keyword(s):  
Cross Sections ◽  
Critical Speed ◽  
Natural Frequencies ◽  
Linear Time ◽  
Mode Shapes ◽  
Axially Moving Beam ◽  
Time Varying Systems ◽  
Axially Moving ◽  
The Galerkin Method ◽  
Spatially Varying

Abstract The partial differential equation of motion of an axially moving beam with spatially varying geometric, mass and material properties has been derived. Using the theory of linear time-varying systems and numerical optimization, a general algorithm has been developed to compute complex eigenvalues/natural frequencies, mode shapes, and the critical speed for stability. Numerical results from the new method are presented for beams with spatially varying rectangular cross sections with sinusoidal variation in thickness and sine-squared variation in width. They are also compared to those from the Galerkin method. It has been found that critical speed of the beam can be significantly reduced by non-uniformity in a beam's cross section.

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