Multi-parameter Perturbation Research Articles

Aeroelastic stability of a symmetric multi-body section model

The aeroelastic stability of a multi-body system is studied through a four-degree-of-freedom elastic model, which describes the linearized section dynamics of particular suspended bridges with doubly-symmetric cross-section, subject to a lateral stationary wind flow. A multi-parameter perturbation solution, applied to the modal problem in internal resonance conditions, allows a consistent reduction of the model dimensions. Focus is made on the particular parameter region corresponding to the triple internal resonance among a global torsional mode of the deck and two local modes of a pair of hanger cables. The bifurcation scenario described by a local stability analysis is featured by two dynamic instability boundaries, strongly interacting with each other. The parametric analysis of the critical wind velocity provides satisfying engineering results, pointing out how the presence of resonant light cables, as well as the addiction of dissipative couplings simulating passive viscous dampers acting on the cable-deck differential velocity, may have beneficial effects in preventing the aeroelastic instability of the full system.

Modal Estimation of Low-frequency Oscillation Using Perturbation Sensitivity Matrix with Multiple Parameters

A method based on multi-parameter second-order perturbation sensitivity is proposed to estimate the low-frequency oscillation modals in a power system, since the changing low-frequency oscillation modal is hard to estimate due to the variety of multiple parameters. First, the multi-parameter second-order perturbation sensitivity matrices of eigenvalues and eigenvectors are deduced, respectively. Second, second-order estimated values based on second-order perturbation sensitivity are deduced. On this basis, the changing oscillation modals under multiple parameters variation are then estimated. The simulation results of 4-generator, 16-generator, and China Southern Grid systems demonstrate that this method is not only able to estimate the oscillation modes but also the modals of the system with more accuracy in the case of multiple parameters of the system that change simultaneously. Then it can adjust an appropriate dispatching method accordingly to improve the damping of the dominant oscillation mode. Also, this method makes the solving process direct and explicit since it avoids the burdensome derivation calculation of second-order sensitivity, and it saves time by avoiding solving complicated high-dimensional state equations.

A parametric multi-body section model for modal interactions of cable-supported bridges

Abstract A parametric section model is formulated to synthetically describe the geometrically nonlinear dynamics of cable-stayed and suspended bridges through a planar elastic multi-body system. The four-degrees-of-freedom model accounts for both the flexo-torsional motion of the bridge deck and for the transversal motion of a pair of hangers or stay cables. After linearization around the pre-stressed static equilibrium configuration, the coupled equations of motion governing the global deck dynamics and the local cable motion are obtained. A multi-parameter perturbation method is employed to solve the modal problem of internally resonant systems. The perturbation-based modal solution furnishes, first, explicit formulae for the parameter combinations which realize the internal resonance conditions and, second, asymptotic approximations of the resonant frequencies and modes. Attention is focused on the triple internal resonance among a global torsional mode of the deck and two local modes of the cables, due to the relevant geometric coupling which maximizes the modal interaction. The asymptotic approximation of the modal solution is found to finely describe the multiple veering phenomenon which involves the three frequency loci under small variation of the most significant mechanical parameters, including terms of structural coupling or disorder. Moreover, the veering amplitude between any two of the three frequency loci can be expressed as an explicit parametric function. Finally, the disorder is recognized as the only parameter governing a complex phenomenon of triple modal hybridization involving all the resonant modes. The entire hybridization process is successfully described by an energy-based localization factor, presented in a new perturbation-based form, valid for internally resonant system.

Multi-parameter perturbation methods for the eigensolution sensitivity analysis of nearly-resonant non-defective multi-degree-of-freedom systems

The dynamic behavior of structural systems may be strongly characterized by the occurrence of multiple internal resonances for particular combinations of the mechanical parameters. The linear models governing these resonant or nearly-resonant systems tend to exhibit high sensitivity of the eigenvalues and eigenvectors to small parameter modifications. This pathological condition is recognized as a source of relevant phenomena, such as frequency veering and mode localization or hybridization. The paper presents the generalization of uniformly valid perturbation methods to perform eigensolution sensitivity analyses in multi-degree-of-freedom Hamiltonian systems with a generic number of close eigenvalues. The leading idea is to treat systematically nearly-resonant systems as multi-parameter perturbations of a perfectly-resonant, non-defective – though a priori unknown – reference system. Given a single nearly-resonant system, a multi-parameter perturbation method is presented to achieve a two-fold objective: first, identify a close resonant system suited to serve as a starting point for sensitivity analyses (inverse problem); second, to approximate asymptotically the eigensolution of all the nearly-resonant systems which may arise from its generic perturbation (direct problem). The conditions of existence and uniqueness of the inverse problem solution are discussed. The direct problem solution is analyzed with a focus on the eigensolution sensitivity to parameter perturbations with different physical meanings, such as a slight geometric disorder or weak elastic coupling in periodic structures. Finally, the procedure is verified on a prototypical structural system describing the section dynamics of a suspended bridge.

Mass Transfer Effects on Unsteady Hydromagnetic Convective Flow past a Vertical Porous Plate in a Porous Medium with Heat Source

The objective of this paper is to analyze the effect of mass transfer on unsteady hydromagnetic free convective flow of a viscous incompressible electrically conducting fluid past an infinite vertical porous plate in presence of constant suction and heat source. The governing equations of the flow field are solved using multi parameter perturbation technique and approximate solutions are obtained for velocity field, temperature field, concentration distribution, skin friction and the rate of heat transfer. The effects of the flow parameters such as Hartmann number M, Grashof number for heat and mass transfer Gr, Gc; permeability parameter Kp, Schmidt number Sc, heat source parameter S, Prandtl number Pr etc. on the flow field are analyzed with the help of figures and tables. It is observed that a growing Hartmann number or Schmidt number retards the mean velocity as well as the transient velocity of the flow field at all points. The effect of increasing Grashof number for heat and mass transfer or heat source parameter is to accelerate both mean and transient velocity of the flow field at all points. The mean velocity of the flow field increases with an increase in permeability parameter while the transient velocity increases for smaller values of Kp (≤1) and for higher values the effect reverses. A growing Hartmann number decreases the transient temperature of the flow field at all points while a growing permeability parameter or heat source parameter reverses the effect. The Prandtl number increases the transient temperature for small values of Pr (≤1) and for higher values the effect reverses. The effect of increasing Schmidt number is to reduce the concentration boundary layer thickness of the flow field at all points. The problem has some relevance in the geophysical and astrophysical studies.

Unsteady hydromagnetic convective flow in a vertical channel using Darcy–Brinkman–Forchheimer extended model with heat generation/absorption: Analysis with asymmetric heating/cooling of the channel walls

In the present paper transient as well as non-Darcian effects on laminar natural convection flow in a vertical channel embedded in a porous medium in the presence of uniform magnetic field applied normal to the flow region is studied. Forchheimer–Brinkman extended Darcy model is assumed to simulate momentum transfer within the porous medium. Approximate solutions are obtained using multi-parameter perturbation technique. Variations in the velocity field with Darcy number, Grashof number, kinematic viscosity ratio and variations in the temperature distribution are obtained and depicted graphically. Expression for the skin-friction and the rate of heat transfer at the channel walls are also derived and the numerical values for various physical parameters are presented in tabular form.

Effects of Unsteady Free Convective MHD Non-Newtonian Flow Through a Porous Medium Bounded by an Infinite Inclined Porous Plate

The problem of unsteady free convective MHD incompressible electrically conducting non-Newtonian fluid through porous medium bounded by on infinite inclined porous plate in the presence of constant suction and absorbing sinks is presented. Uniform magnetic field is applied normal to the plate. The equations governing the fluid flow have been solved using multi-parameter perturbation technique, subject to the relevant boundary conditions. It is noted that the velocity of the fluid and skin friction ore increased as permeability parameter and angle of inclination increases, whereas reverse phenomenon is observed in case of magnetic field strength and sink strength. Velocity and temperature are greater for mercury than that of electrolytic solution. Rate of heat transfer decreases with increase in the sink strength. The results ore discussed through graphs and tables.

Bifurcations of a cubic system perturbed by degree five

In this article, using multi-parameter perturbation theory and qualitative analysis, the authors studied a kind of cubic system perturbed by degree five and obtained the system that can have 17 limit cycles giving their two kinds of distributions (see Fig.5).

Global Bifurcation of a Perturbed Double-Homoclinic Loop*

Abstract This paper deals with a kind of fourth degree systems with perturbations. By using the method of multi-parameter perturbation theory and qualitative analysis, it is proved that the system can have six limit cycles.

The Global Bifurcation of a Cubic System

In this paper, we study the perturbation of certain of cubic system. By using the method of multi-parameter perturbation theory and qualitative analysis, we infer that the system under consideration can have five limit cycles.

The relative motion of the core and mantle of a planet in the gravitational field of a point mass

A model of a planet, consisting of two solid bodies – a core and a mantle – between which there is a spherical layer of a viscous incompressible liquid, is considered. The gravitational interaction between the core and the mantle is taken into account. The problem is investigated in a limited formulation, when the mass centre of the planet moves in a fixed elliptical orbit in the gravitational field of a point mass, while the mutual displacements of the core and the mantle are to be determined. The mutual displacements of the core and the mantle of the planet, and also the velocity field of the viscous liquid in the spherical layer, are obtained using multiparameter perturbation theory, where the Reynolds number, the orbit eccentricity and the ratio of the radius of the planet to the distance to its attracting centre are taken as small parameters. In addition, an approximate theory of gyroscopes is used to analyse the equations of motion. The results obtained are illustrated by the example of the motion of the Earth-Moon system.

Multiparameter Perturbation Analysis of Unsteady Oscillatory Magnetoconvection in Porous Media with Heat Source Effects

Unsteady natural convection flow of a viscous, incompressible, electrically conducting fluid past an infinite vertical porous plate embedded within a saturated, highly porous medium under the influence of a uniform magnetic field, Eckert heating, and a heat-absorbing sink is studied. It is assumed that the suction velocity is subjected to small-amplitude oscillations in time about the steady nonzero mean suction velocity. Approximate solutions for the velocity and temperature field are obtained using the multiparameter perturbation technique. Expressions for skin friction and rate of heat transfer are also derived. Mean temperatures are boosted by both magnetic field and heat-source parameter, but are lowered by a rise in the Prandtl number, Grashof number, and permeability parameter (k). The effects of all parameters on velocity, temperature, skin friction, and heat transfer rate (Nusselt number) are graphically presented and tabulated, and are discussed in detail. The model finds applications in nuclear heat transfer processes, metallurgy, aerospace/aval propulsion, and energy systems.

Nonlinear bending analysis of shear deformable functionally graded plates subjected to thermo-mechanical loads under various boundary conditions

Nonlinear bending analysis of shear deformable functionally graded plates subjected to thermo-mechanical loads and under various boundary conditions is presented. Theoretical formulations are based on Reddy's higher order shear deformation plate theory and include the thermal effects due to temperature rise. Material properties are assumed to be temperature-dependent and graded in the thickness direction according to a power law distribution in terms of the volume fractions of the constituents. The temperature field considered is assumed to be uniform in the XY plane and through the plate thickness. The plate may be clamped or simply supported on two opposite edges with the remaining others being either free, simply supported or clamped. A semi-numerical approach, which makes use of multi-parameter perturbation technique , one-dimensional differential quadrature approximation and Galerkin method, is employed to calculate the nonlinear bending of the plates. Numerical results for silicon nitride/stainless steel rectangular plates are given in dimensionless graphical forms . The roles played by the constituent volume fraction index, temperature rise, character of out-of-plane boundary conditions and in-plane constraints, width-to-thickness ratio as well as plate aspect ratio are studied.

Periodic behaviour in nonlinear biscrete-time systems and its application to a digital control system

In this paper, the behaviour of a two-dimensional nonlinear discrete-time model is considered with a view to understanding the effects of a varying sampling period in digital control systems. The analysis is carried out in the neighbourhood of a critical point which, on the unit circle, corresponds to a pair of complex-conjugate points with non-zero imaginary parts. It is demonstrated that, as the sampling period is varied from a critical sampling value, the branches of root-locus may cross the unit circle and secondary solutions bifurcate under certain conditions. The analysis here is facilitated by adapting the intrinsic method of harmonic balancing (a multi-parameter perturbation technique) to discrete-time systems

A Multiparameter Perturbation Approach for the Vibration of Plates Subject to Arbitrary, Independent, In-Plane Loads

A multiparameter perturbation approach for the solution of a class of differential equations is developed and is applied to the study of the vibration of rectangular plates under arbitrary in-plane loading fields describable by linear combinations of two or more independent loading parameters. The solution takes the form of a power series involving all the independent loading parameters and is valid over whole ranges of those parameters. Numerical results for fully clamped plates subject to uniform, combined, direct, and shear in-plane loads are presented in order to demonstrate the utility of the approach.