Search results for: Galerkin

Authors: Y.Galerkin, O. Solovieva

Abstract:

Flow parameters are calculated in vaneless diffusers with relative width 0,014 – 0,10 constant along radii. Inlet flow angles and similarity criteria were varied. Information about flow structure is presented – meridian streamlines configuration, information on flow full development, flow separation. Polytrophic efficiency, loss and recovery coefficient are used to compare diffusers’ effectiveness. The sample of narrow diffuser optimization by conical walls application is presented. Three tampered variants of a wide diffuser are compared too. The work is made in the R&D laboratory “Gas dynamics of turbo machines” of the TU SPb.

Keywords: vaneless diffuser, relative width, flow angle, flow separation, loss coefficient, similarity criteria

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Authors: G. N. Sekhar, P. G. Siddheshwar, G. Jayalatha, R. Prakash

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The problem of thermal convection in temperature and magnetic field sensitive Newtonian ferromagnetic liquid is studied in the presence of uniform vertical magnetic field and throughflow. Using a combination of Galerkin and shooting techniques the critical eigenvalues are obtained for stationary mode. The effect of Prandtl number (Pr > 1) on onset is insignificant and nonlinearity of non-buoyancy magnetic parameter M3 is found to have no influence on the onset of ferroconvection. The magnetic buoyancy number, M1 and variable viscosity parameter, V have destabilizing influences on the system. The effect of throughflow Peclet number, Pe is to delay the onset of ferroconvection and this effect is independent of the direction of flow.

Keywords: ferroconvection, magnetic field dependent viscosity, temperature dependent viscosity, throughflow

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Authors: Y. Galerkin, K. Soldatova, A. Drozdov

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The 6th version of Universal modeling method for centrifugal compressor stage calculation is described. Identification of the new mathematical model was made. As a result of identification the uniform set of empirical coefficients is received. The efficiency definition error is 0,86 % at a design point. The efficiency definition error at five flow rate points (except a point of the maximum flow rate) is 1,22 %. Several variants of the stage with 3D impellers designed by 6th version program and quasi three-dimensional calculation programs were compared by their gas dynamic performances CFD (NUMECA FINE TURBO). Performance comparison demonstrated general principles of design validity and leads to some design recommendations.

Keywords: compressor design, loss model, performance prediction, test data, model stages, flow rate coefficient, work coefficient

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Authors: Y. Galerkin, A. Rekstin, K. Soldatova

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Universal modeling method well proven for industrial compressors was applied for design of the high flow rate supersonic stage. Results were checked by ANSYS CFX and NUMECA Fine Turbo calculations. The impeller appeared to be very effective at transonic flow velocities. Stator elements efficiency is acceptable at design Mach numbers too. Their loss coefficient versus inlet flow angle performances correlates well with Universal modeling prediction. The impeller demonstrated ability of satisfactory operation at design flow rate. Supersonic flow behavior in the impeller inducer at the shroud blade to blade surface Φdes deserves additional study.

Keywords: centrifugal compressor stage, supersonic impeller, inlet flow angle, loss coefficient, return channel, shock wave, vane diffuser

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Authors: K. Khanafer

Abstract:

The aim of this investigation was to validate the fluid-structure interaction (FSI) model of type B aortic dissection with our experimental results from a bench-top-model. Another objective was to study the relationship between the size of a septectomy that increases the outflow of the false lumen and its effect on the values of the differential of pressure between true lumen and false lumen. FSI analysis based on Galerkin’s formulation was used in this investigation to study flow pattern and hemodynamics within a flexible type B aortic dissection model using boundary conditions from our experimental data. The numerical results of our model were verified against the experimental data for various tear size and location. Thus, CFD tools have a potential role in evaluating different scenarios and aortic dissection configurations.

Keywords: aortic dissection, fluid-structure interaction, in vitro model, numerical

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Authors: Sendi Zohra, Belhadjsalah Hedi, Labergere Carl, Saanouni Khemais

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The modeling of mechanical problems including both material and geometric nonlinearities with Finite Element Method (FEM) remains challenging. Meshless methods offer special properties to get rid of well-known drawbacks of the FEM. The main objective of Meshless Methods is to eliminate the difficulty of meshing and remeshing the entire structure by simply insertion or deletion of nodes, and alleviate other problems associated with the FEM, such as element distortion, locking and others. In this study, a robust numerical implementation of an Element Free Galerkin Method for an elastoplastic coupled to damage problem is presented. Several results issued from the numerical simulations by a DynamicExplicit resolution scheme are analyzed and critically compared with Element Finite Method results. Finally, different numerical examples are carried out to demonstrate the efficiency of this method.

Keywords: damage, dynamic explicit, elastoplasticity, isotropic hardening, meshless

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Authors: Haci Mehmet Guzey, Levent Acar

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In this paper, a novel decentralized controller is developed for linear systems with nonlinear unknown interconnections. A model linear decoupled system is assigned for each system. By using the difference actual and model state dynamics, the problem is formulated as inverse problem. Then, the interconnected dynamics are approximated by using Galerkin’s expansion method for inverse problems. Two different sets of orthogonal basis functions are utilized to approximate the interconnected dynamics. Approximated interconnections are utilized in the controller to cancel the interconnections and decouple the systems. Subsequently, the interconnected systems behave as a collection of decoupled systems.

Keywords: decentralized control, inverse problems, large scale systems, nonlinear interconnections, basis functions, system identification

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Authors: Merbouha Barka, K. H. Benrahou, A. Fakrar, A. Tounsi, E. A. Adda Bedia

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This paper presents an analytical investigation on the buckling and postbuckling behaviors of thick functionally graded plates subjected to thermal load .Material properties are assumed to be temperature dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. The formulations are based on first order shear deformation plate theory taking into account Von Karman nonlinearity and initial geometrical imperfection. By applying Galerkin method, closed-form relations of postbuckling equilibrium paths for simply supported plates are determined. Analysis is carried out to show the effects of material and geometrical properties, in-plane boundary restraint, and imperfection on the buckling and postbuckling loading capacity of the plates.

Keywords: functionally graded materials, postbuckling, first order shear deformation theory, imperfection

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Authors: T. H. Young, Y. J. Tsai

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A method for determining the stress distribution of a rectangular plate subjected to two pairs of arbitrarily distributed in-plane edge shear loads is proposed, and the free vibration and buckling of such a rectangular plate are investigated in this work.  The method utilizes two stress functions to synthesize the stress-resultant field of the plate with each of the stress functions satisfying the biharmonic compatibility equation. The sum of stress-resultant fields due to these two stress functions satisfies the boundary conditions at the edges of the plate, from which these two stress functions are determined. Then, the free vibration and buckling of the rectangular plate are investigated by the Galerkin method. Numerical results obtained by this work are compared with those appeared in the literature, and good agreements are observed.

Keywords: stress analysis, free vibration, plate buckling, nonuniform in-plane edge shear

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Authors: Tai-Ping Chang

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In the present study, the effects on chaotic motion of single-walled carbon nanotube (SWCNT) due to the linear and nonlinear damping are investigated. By using the Hamilton’s principle, the nonlinear governing equation of the single-walled carbon nanotube embedded in a matrix is derived. The Galerkin’s method is adopted to simplify the integro-partial differential equation into a nonlinear dimensionless governing equation for the SWCNT, which turns out to be a forced Duffing equation. The variations of the Lyapunov exponents of the SWCNT with damping and harmonic forcing amplitudes are investigated. Based on the computations of the top Lyapunov exponent, it is concluded that the chaotic motion of the SWCNT occurs when the amplitude of the periodic excitation exceeds certain value, besides, the chaotic motion of the SWCNT occurs with small linear damping and tiny nonlinear damping.

Keywords: chaotic motion, damping, Lyapunov exponents, single-walled carbon nanotube

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Authors: M. J. Uddin, M. M. Rahman

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Transient natural convective flow dynamics of nanofluids in a horizontal homocentric annulus using nonhomogeneous dynamic model has been experimented numerically. The simulation is carried out for four different shapes of the inner wall, which is either cylindrical, elliptical, square or triangular. The outer surface of the annulus is maintained at constant low temperature while the inner wall is maintained at a uniform temperature; higher than the outer one. The enclosure is permeated by a uniform magnetic field having variable orientation. The Brownian motion and thermophoretic deposition phenomena of the nanoparticles are taken into account in model construction. The governing nonlinear momentum, energy, and concentration equations are solved numerically using Galerkin weighted residual finite element method. To find the best performer, the local Nusselt number is demonstrated for different shapes of the inner wall. The heat transfer enhancement for different nanofluids for four different shapes of the inner wall is exhibited.

Keywords: nanofluids, annulus, nonhomogeneous dynamic model, heat transfer

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Authors: Shangerganesh Lingeshwaran

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In this presentation, we have performed numerical simulations for a reaction-diffusion equation with various nonlinear density-dependent diffusion operators and proliferation functions. The mathematical model represented by parabolic partial differential equation is considered to study the invasion of gliomas (the most common type of brain tumors) and to describe the growth of cancer cells and response to their treatment. The unknown quantity of the given reaction-diffusion equation is the density of cancer cells and the mathematical model based on the proliferation and migration of glioma cells. A standard Galerkin finite element method is used to perform the numerical simulations of the given model. Finally, important observations on the each of nonlinear diffusion functions and proliferation functions are presented with the help of computational results.

Keywords: glioma invasion, nonlinear diffusion, reaction-diffusion, finite eleament method

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Authors: Korosh Khorshidi

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Due to the increasing use of different materials, such as auxetic structures, it is necessary to investigate mechanical phenomena, such as vibration, in structures made of these types of materials. This paper examines the forced vibrations of a three-layer cylindrical shell containing inviscid fluid under shock load. All three layers are made of aluminum, and the central layer is made of a re-entrant honeycomb cell structure. Using high-order shear deformation theories (HSDT) and Hamilton’s principle, the governing equations of the system have been extracted and solved by the Galerkin weighted residual method. The outputs of the Abaqus finite element software are used to validate the results. The system is investigated with both simple and clamped support conditions. Finally, this study investigates the influence of the geometrical parameters of the shell and the auxetic structure, as well as the type, intensity, duration, and location of the load, and the effect of the fluid on the dynamic and time responses.

Keywords: force vibration, cylindrical shell, auxetic structure, inviscid fluid

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Authors: Yu-Kai Ting, Jia-Ying Tu, Chung-Chun Hsiao

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New physical insights into the nonlinear Lorenz equations related to flow resistance is discussed in this work. The chaotic dynamics related to Lorenz equations has been studied in many papers, which is due to the sensitivity of Lorenz equations to initial conditions and parameter uncertainties. However, the physical implication arising from Lorenz equations about convectional motion attracts little attention in the relevant literature. Therefore, as a first step to understand the related fluid mechanics of convectional motion, this paper derives the Lorenz equations again with different forced conditions in the model. Simulation work of the modified Lorenz equations without the viscosity or buoyancy force is discussed. The time-domain simulation results may imply that the states of the Lorenz equations are related to certain flow speed and flow resistance. The flow speed of the underlying fluid system increases as the flow resistance reduces. This observation would be helpful to analyze the coupling effects of different fluid parameters in a convectional model in future work.

Keywords: Galerkin method, Lorenz equations, Navier-Stokes equations, convectional motion

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Authors: Y. Galerkin, E. Popova, K. Soldatova

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There is an evident trend to elevate pressure ratio of a single stage of a turbo compressors - axial compressors in particular. Whilst there was an opinion recently that a pressure ratio 1,9 was a reasonable limit, later appeared information on successful modeling tested of stages with pressure ratio up to 2,8. The Authors recon that lack of information on high pressure stages makes actual a study of rational choice of design parameters before high supersonic flow problems solving. The computer program of an engineering type was developed. Below is presented a sample of its application to study possible parameters of the impeller of the stage with pressure ratio π*=3,0. Influence of two main design parameters on expected efficiency, periphery blade speed and flow structure is demonstrated. The results had lead to choose a variant for further analysis and improvement by CFD methods.

Keywords: supersonic stage, impeller, efficiency, flow rate coefficient, work coefficient, loss coefficient, oblique shock, direct shock

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Authors: Youssef Abdelli, Rachid Nasri

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The large amplitude free vibration analysis of three-layered symmetric sandwich beams is carried out using two different approaches. The governing nonlinear partial differential equations of motion in free natural vibration are derived using Hamilton's principle. The formulation leads to two nonlinear partial differential equations that are coupled both in axial and binding deformations. In the first approach, the method of multiple scales is applied directly to the governing equation that is a nonlinear partial differential equation. In the second approach, we discretize the governing equation by using Galerkin's procedure and then apply the shooting method to the obtained ordinary differential equations. In order to check the validity of the solutions obtained by the two approaches, they are compared with the solutions obtained by two approaches; they are compared with the solutions obtained numerically by the finite difference method.

Keywords: finite difference method, large amplitude vibration, multiple scales, nonlinear vibration

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Authors: Win Ko Ko, A. N. Temnov

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The problem of nonlinear oscillations of a two-layer liquid completely filling a limited volume is considered. Using two basic asymmetric harmonics excited in two mutually perpendicular planes, ordinary differential equations of nonlinear oscillations of the interface of a two-layer liquid are investigated. In this paper, hydrodynamic coefficients of linear and nonlinear problems in integral relations were determined. As a result, the instability regions of forced oscillations of a two-layered liquid in a cylindrical tank occurring in the plane of action of the disturbing force are constructed, as well as the dynamic instability regions of the parametric resonance for different ratios of densities of the upper and lower liquids depending on the amplitudes of liquids from the excitations frequencies. Steady-state regimes of fluid motion were found in the regions of dynamic instability of the initial oscillation form. The Bubnov-Galerkin method is used to construct instability regions for approximate solution of nonlinear differential equations.

Keywords: nonlinear oscillations, two-layered liquid, instability region, hydrodynamic coefficients, resonance frequency

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Authors: M. J. Uddin, M. M. Rahman

Abstract:

The unsteady convective heat transfer flow of nanofluids in a half circular-annulus shape enclosure using nonhomogeneous dynamic model has been investigated numerically. The round upper wall of the enclosure is maintained at constant low temperature whereas the bottom wall is heated by three different thermal conditions. The enclosure is permeated by a uniform magnetic field having variable orientation. The Brownian motion and thermophoretic phenomena of the nanoparticles are taken into account in model construction. The governing nonlinear momentum, energy, and concentration equations are solved numerically using Galerkin weighted residual finite element method. To discover the best performer, the average Nusselt number is demonstrated for different types of nanofluids. The heat transfer rate for different flow parameters, positions of the annulus, thicknesses of the half circular-annulus and thermal conditions is also exhibited.

Keywords: nanofluid, convection, semicircular-annulus, nonhomogeneous dynamic model, finite element method

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Authors: Yashar Haghighatfar, Shahrzad Mirhosseini

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Pull-in instability is a nonlinear and crucial effect that is important for the design of microelectromechanical system devices. In this paper, the appropriate electrostatic voltage range is determined by measuring fluid flow pressure via micro pressure sensor based microbeam. The microbeam deflection contains two parts, the static and perturbation deflection of static. The second order equation regarding the equivalent stiffness, mass and damping matrices based on Galerkin method is introduced to predict pull-in instability due to the external voltage. Also the reduced order method is used for solving the second order nonlinear equation of motion. Furthermore, in the present study, the micro capacitive pressure sensor is designed for measuring special fluid flow pressure range. The results show that the measurable pressure range can be optimized, regarding damping field and external voltage.

Keywords: MEMS, pull-in instability, electrostatically actuated microbeam, reduced order method

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Authors: Easa Ali Abbasi, Akbar Allahverdizadeh, Reza Jahangiri, Behnam Dadashzadeh

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Electrical current measurement is a suitable method for the performance determination of electrical devices. There are two contact and noncontact methods in this measuring process. Contact method has some disadvantages like having direct connection with wire which may endamage the system. Thus, in this paper, a bimorph piezoelectric cantilever beam which has a permanent magnet on its free end is used to measure electrical current in a noncontact way. In mathematical modeling, based on Galerkin method, the governing equation of the cantilever beam is solved, and the equation presenting the relation between applied force and beam’s output voltage is presented. Magnetic force resulting from current carrying wire is considered as the external excitation force of the system. The results are compared with other references in order to demonstrate the accuracy of the mathematical model. Finally, the effects of geometric parameters on the output voltage and natural frequency are presented.

Keywords: cantilever beam, electrical current measurement, forced excitation, piezoelectric

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Authors: Ali Raheli, Rahim Habibifar, Behzad Mohammadi-Alasti, Mahdi Abbasgholipour

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This paper presents vibration behavior of a FGM micro-beam and its pull-in instability under a nonlinear electrostatic pressure. An exponential function has been applied to show the continuous gradation of the properties along thickness. Nonlinear integro-differential-electro-mechanical equation based on Euler–Bernoulli beam theory has been derived. The governing equation in the static analysis has been solved using Step-by-Step Linearization Method and Finite Difference Method. Fixed points or equilibrium positions and singular points have been shown in the state control space. In order to find the response to a step DC voltage, the nonlinear equation of motion has been solved using Galerkin-based reduced-order model and time histories and phase portrait for different applied voltages have been shown. The effects of electrostatic pressure on stability of FGM micro-beams having various amounts of the ceramic constituent have been investigated.

Keywords: FGM, MEMS, nonlinear vibration, electrical, dynamic pull-in voltage

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Authors: Khalil Khanafer

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This study emphasizes on analyzing the effect of flow conditions and the geometric variation of the microcantilever’s bluff body on the microcantilever detection capabilities within a fluidic device using a finite element fluid-structure interaction model. Such parameters include inlet velocity, flow direction, and height of the microcantilever’s supporting system within the fluidic cell. The transport equations are solved using a finite element formulation based on the Galerkin method of weighted residuals. For a flexible microcantilever, a fully coupled fluid-structure interaction (FSI) analysis is utilized and the fluid domain is described by an Arbitrary-Lagrangian–Eulerian (ALE) formulation that is fully coupled to the structure domain. The results of this study showed a profound effect on the magnitude and direction of the inlet velocity and the height of the bluff body on the deflection of the microcantilever. The vibration characteristics were also investigated in this study. This work paves the road for researchers to design efficient microcantilevers that display least errors in the measurements.

Keywords: fluidic cell, FSI, microcantilever, flow direction

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Authors: Norazuwin Najihah Mat Tahir, Fuziyah Ishak, Seripah Awang Kechil

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The onset of the convective instability in the horizontal through flow of a power-law nanofluid saturated by porous layer heated from below under the influences of magnetic field are investigated in this study. The linear stability theory is used for the transformation of the partial differential equations to system of ordinary differential equations through infinitesimal perturbations, scaling, linearization and method of normal modes with two-dimensional periodic waves. The system is solved analytically for the closed form solution of the Rayleigh number by using the Galerkin-type weighted residuals method to investigate the onset of both traveling wave and oscillatory convection. The effects of the power-law index, Lewis number and Peclet number on the stability of the system were investigated. The Lewis number stabilizes while the power-law index and Peclet number destabilize the nanofluid system. The system in the presence of magnetic field is more stable than the system in the absence of magnetic field.

Keywords: convection, instability, magnetic field, nanofluid, power-law

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Authors: Salma Parvin, M. A. Alim

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The convective and radiative heat transfer performance and entropy generation on forced convection through a direct absorption solar collector (DASC) is investigated numerically. Four different fluids, including Cu-water nanofluid, Al₂O₃-waternanofluid, TiO₂-waternanofluid, and pure water are used as the working fluid. Entropy production has been taken into account in addition to the collector efficiency and heat transfer enhancement. Penalty finite element method with Galerkin’s weighted residual technique is used to solve the governing non-linear partial differential equations. Numerical simulations are performed for the variation of mass flow rate. The outcomes are presented in the form of isotherms, average output temperature, the average Nusselt number, collector efficiency, average entropy generation, and Bejan number. The results present that the rate of heat transfer and collector efficiency enhance significantly for raising the values of m up to a certain range.

Keywords: DASC, forced convection, mass flow rate, nanofluid

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Authors: K. Soldatova, Y. Galerkin

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A loading factor performance is necessary for the modeling of centrifugal compressor gas dynamic performance curve. Measured loading factors are linear function of a flow coefficient at an impeller exit. The performance does not depend on the compressibility criterion. To simulate loading factor performances, the authors present two parameters: a loading factor at zero flow rate and an angle between an ordinate and performance line. The calculated loading factor performances of non-viscous are linear too and close to experimental performances. Loading factor performances of several dozens of impellers with different blade exit angles, blade thickness and number, ratio of blade exit/inlet height, and two different type of blade mean line configuration. There are some trends of influence, which are evident – comparatively small blade thickness influence, and influence of geometry parameters is more for impellers with bigger blade exit angles, etc. Approximating equations for both parameters are suggested. The next phase of work will be simulating of experimental performances with the suggested approximation equations as a base.

Keywords: loading factor performance, centrifugal compressor, impeller, modeling

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Authors: Ravi Chaithanya Mysa, Abouzar Kaboudian, Boo Cheong Khoo, Rajeev Kumar Jaiman

Abstract:

The phenomenon of vortex-induced vibration has applications in off-shore industry, power transmission, energy extraction, etc. Two cylinders in crossflow whose centers are displaced in transverse direction are considered in the present work. The effects of the gap distance between the cylinders on the vortex shedding are presented. The inline distance between the cylinder centers is kept at zero. Two setups are considered for the study: first, we assume the two cylinders vibrate as a single rigid body mounted on a spring, and in the other case, each cylinder is mounted on a separate spring with no rigid connection to the other cylinder. The study focuses on the effect of transverse gap on the fluid-structure coupled response of two setups mentioned and corresponding flow contours. Incompressible flow is assumed in the Eulerian framework. The cylinder movement is modeled by a single degree of freedom rigid body motion (translational motion) in the Lagrangian framework. The governing equations were numerically solved by standard Petrov-Galerkin second order finite element schemes.

Keywords: cross-flow, vortex-induced vibrations, cylinder, close proximity

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Authors: Yousef Bakhbakhi

Abstract:

Crystallization with supercritical fluids (SCFs), as a developed technology to produce particles of micron and sub-micron size with narrow size distribution, has found appreciable importance as an environmentally friendly technology. Particle synthesis using SCFs can be achieved employing a number of special processes involving solvent and antisolvent mechanisms. In this study, the compressed antisolvent (PCA) process is utilized as a model to analyze the theoretical complexity of crystallization with supercritical fluids. The population balance approach has proven to be an effectual technique to simulate and predict the particle size and size distribution. The nucleation and growth mechanisms of the particles formation in the PCA process is investigated using the population balance equation, which describes the evolution of the particle through coalescence and breakup levels with time. The employed mathematical population balance model contains a set of the partial differential equation with algebraic constraints, which demands a rigorous numerical approach. The combined Collocation and Galerkin finite element method are proposed as a high-resolution technique to solve the dynamics of the PCA process.

Keywords: particle formation, particle size and size distribution, PCA, supercritical carbon dioxide

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Authors: Shivani Saini

Abstract:

The effect of vertical throughflow on the onset of convection in Rivlin-Ericksen Elastico-Viscous nanofluid with convective boundary condition is investigated. The flow is stimulated with modified Darcy model under the assumption that the nanoparticle volume fraction is not actively managed on the boundaries. The heat conservation equation is formulated by introducing the convective term of nanoparticle flux. A linear stability analysis based upon normal mode is performed, and an approximate solution of eigenvalue problems is obtained using the Galerkin weighted residual method. Investigation of the dependence of the Rayleigh number on various viscous and nanofluid parameter is performed. It is found that through flow and nanofluid parameters hasten the convection while capacity ratio, kinematics viscoelasticity, and Vadasz number do not govern the stationary convection. Using the convective component of nanoparticle flux, critical wave number is the function of nanofluid parameters as well as the throughflow parameter. The obtained solution provides important physical insight into the behavior of this model.

Keywords: Darcy model, nanofluid, porous layer, throughflow

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Authors: S. Nandal, R. Bhargava

Abstract:

The paper contains a numerical study of the unsteady magneto-hydrodynamic natural convection flow of nanofluids within a symmetrical wavy walled trapezoidal enclosure. The length and height of enclosure are both considered equal to L. Two-phase nanofluid model is employed. The governing equations of nanofluid flow along with boundary conditions are non-dimensionalized and are solved using one of Meshfree technique (EFGM method). Meshfree numerical technique does not require a predefined mesh for discretization purpose. The bottom wavy wall of the enclosure is defined using a cosine function. Element free Galerkin method (EFGM) does not require the domain. The effects of various parameters namely time t, amplitude of bottom wavy wall a, Brownian motion parameter Nb and thermophoresis parameter Nt is examined on rate of heat and mass transfer to get a visualization of cooling and heating effects. Such problems have important applications in heat exchangers or solar collectors, as wavy walled enclosures enhance heat transfer in comparison to flat walled enclosures.

Keywords: heat transfer, meshfree methods, nanofluid, trapezoidal enclosure

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Authors: T. H. Young, S. J. Huang, Y. S. Chiu

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This paper investigates the parametric stability of an axially moving web subjected to nonuniform in-plane edge excitations on two opposite, simply-supported edges. The web is modeled as a viscoelastic plate whose constitutive relation obeys the Kelvin-Voigt model, and the in-plane edge excitations are expressed as the sum of a static tension and a periodical perturbation. Due to the in-plane edge excitations, the moving plate may bring about parametric instability under certain situations. First, the in-plane stresses of the plate due to the nonuniform edge excitations are determined by solving the in-plane forced vibration problem. Then, the dependence on the spatial coordinates in the equation of transverse motion is eliminated by the generalized Galerkin method, which results in a set of discretized system equations in time. Finally, the method of multiple scales is utilized to solve the set of system equations analytically if the periodical perturbation of the in-plane edge excitations is much smaller as compared with the static tension of the plate, from which the stability boundaries of the moving plate are obtained. Numerical results reveal that only combination resonances of the summed-type appear under the in-plane edge excitations considered in this work.

Keywords: axially moving viscoelastic plate, in-plane periodic excitation, nonuniformly distributed edge tension, dynamic stability

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