Nonlinear Stability Analysis Research Articles

Nonlinear Stability and Vibration Analyses of Functionally Graded Variable Thickness Toroidal Shell Segments Reinforced with Spiral Stiffeners

In this study, an analysis of nonlinear stability and vibration of functionally graded (FG) variable thickness toroidal shell segments (TSSs) reinforced with spiral stiffeners exposed to axial loading is presented using a combination of semi-analytical and analytical methods. Three types of variable thickness TSSs, including concave, convex, and cylindrical shells (CSs), are studied. Moreover, these structures are reinforced by external spiral stiffeners with various angles whose material properties are considered to be continuously graded along the thickness direction. In this regard, the smeared stiffeners technique is utilized to model the stiffeners, and the Donnell shell theory and the von Kármán equation are applied to derive the nonlinear governing equation for variable thickness TSSs reinforced with spiral stiffeners. Galerkin’s method is then used to obtain a discretized nonlinear governing equation to analyze the shells’ behavior. Also, the fourth-order P-T method is applied to analyze the nonlinear dynamic behaviors and the Budiansky–Roth criteria are used to examine the dynamic post-buckling (DPB) behavior. In this regard, it is noted that in terms of reliability and accuracy, the fourth-order P-T method has demonstrated advantages over the other numerical methods. Results are reported to evaluate the influences of stiffeners with different angles and input factors on the nonlinear vibration, dynamic and static post-buckling (SPB) behaviors of FG variable thickness TSSs reinforced with spiral stiffeners.

Nonlinear Modeling and Stability Analysis of Grid-Tied Paralleled-Converters Systems Based on the Proposed Dual-Iterative Equal Area Criterion

With the rapid development of distributed renewable energy generation and integration, stability issues of power converters systems have been widely studied in recent years. Many previous studies focus on the transient stability of a single converter connected to the weak grid. However, the nonlinear dynamics and stability mechanism of multiple converters systems remain unclear. In this paper, a unified nonlinear model of the paralleled-converters system is established, which reveals the interactive power and interactive damping among multiple converters and the weak grid. To analyze the impacts of the interactive power and damping terms on system stability, a dual-iterative equal area criterion (DITEAC) is proposed with a dual-layer iterative calculation of the accelerating area and decelerating area. Compared with the previous studies, the proposed DITEAC method could deal with the nonlinear interactive damping and reduce the conservatism caused by the inequality scaling, which greatly improves the calculation accuracy of stability boundaries. The expansion feasibility of the proposed unified nonlinear model and DITEAC method to the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> -converters system is fully discussed. Eventually, simulation and hardware-in-loop experiment results are presented to verify the effectiveness of the proposed method.

Stability analysis in a couple-stress fluid layer with variable viscosity heated from below: Different conducting boundaries

The Linear stability analysis technique and non-linear stability analysis technique are applied for couple stress fluid layer with having variable viscosity depending on temperature and pressure. The linear and nonlinear threshold that encapsulates the physics of the onset for convection is found to be same. Eigenvalue problems for free-free (FF), rigid-free (RF) and rigid-rigid (RR) conducting boundaries are solved with the help of a single term Galerkin technique. The couple stress is seen to have a stabilizing impact while the viscosity variation has destabilizing impact on the system.

Thermal Convection of Rivlin-Ericksen Fluid Governed by the Brownian Motion and Thermophoresis of Nanoparticles with Passive Behaviour of Nanoparticles at the Parallel Boundaries

Stability of Rivlin-Ericksen category of nanofluid saturated in a continuous medium bounded by infinite horizontal plates has been studied. Energy equation has been supplemented with the variables belonging to the Brownian motion and thermophoresis of nanoparticles. For the linear and the non-linear stability analyses, other than the specific boundary conditions appraised with the physical situation, the boundary conditions for the flux of nanoparticle mass, in analogy with the passive behaviour of temperature at the boundaries have been explored. The novelty of the paper is that the stationary convection exists for both positive as well as negative Rn (concentration Rayleigh number) and the convection sets in earlier in comparison to a porous medium. It is also shown that the non-existence of the oscillatory convection in a Newtonian nanofluid has been ruled out for Rivlin-Ericksen nanofluid, though it exists only for negative Rn, the situation when the density of the fluid is greater than the density of nanoparticle. The viscoelastic parameter of Rivlin-Ericksen nanofluid annihilates the instability of oscillatory convection. Under non-linear stability analysis, the truncated representation of Fourier series approach has been used and the parameters belonging to the heat and mass transfer have been evaluated. It is shown that corresponding to certain parameters, the rate of heat and mass transfer rises rapidly. The valuable results are shown graphically and verified numerically.

On nonlinear stability analysis of saturated embedded porous nanobeams

In this work, the nonlinear stability behaviors of the saturated porous nanobeams embedded in an elastic foundation are investigated. The restrained nanobeam is modeled using geometrical nonlinear equations in conjunction with the constitutive law of saturation. Three patterns for saturation along the thickness of the nanobeam are considered as porous/monotonous, porous/nonlinear symmetric, and porous/nonlinear non-symmetric distributions. The infinite terms of the power series are discretized thanks to Stokes' transformation and trigonometric series. Then, the effects of saturation and nonlinearity on buckling loads are studied by considering the restrained boundary conditions. Moreover, the nonlinear results are validated with the saturated nanobeam with the rigid supporting conditions.

Nonlinear vibration analysis and stability analysis of rotor systems supported on SFD by combining DQFEM, CMS and IHB methods

In this paper, the differential quadrature finite element method (DQFEM), component mode synthesis (CMS) method and incremental harmonic balance (IHB) method are combined to propose a methodology to perform vibration analysis and stability analysis of rotor systems supported on fluid-film bearings that exhibit localized nonlinearities such as squeeze-film damper (SFD). In multi-degree of freedom systems with SFD, a method for dynamic analysis of the system by reducing the degrees of freedom is first suggested in this paper. DQFEM are used to create the mass matrix, stiffness matrix, and gyroscopic matrix of the system. Then, the system is divided into linear system and nonlinear system, and the motion equation of the linear system reduces the degrees of freedom using the free interface CMS method, and the motion equation of the nonlinear system is written without reducing the degrees of freedom. By synthesizing the equations of motion of each component, the motion equation of the entire system is created. That is, the motion equation of the whole system uses hybrid coordinates. The motion equation of the system is solved using the modified IHB method. To evaluate the effectiveness of the suggested method, the computed solutions are compared with the results presented in the previous literature, and the results are very consistent. Also, stability analysis of the obtained solutions is performed by the Floquet theory. The method suggested in the paper can be effectively used to analyze the dynamic behavior of rotor systems exhibiting localized nonlinearities such as SFD.

The jamming transition of multi-lane lattice hydrodynamic model with passing effect

Multi-lane environments are not uncommon on the highway. In multi-lane scenarios, vehicles can not only switch freely between lanes but also overtake each other. However, as basic driving behavior, the passing effect under multi-lane environments has been neglected in traffic flow research. To fill this gap, we present a modified multi-lane lattice hydrodynamic model considering the passing effect. Based on the reduction perturbation method, the corresponding stability norm of the proposed model is obtained, which reveals that the total number of lanes is positively correlated with the stability of traffic flow. When the stability condition does not hold, to further examine the formation and transmission process of traffic jams near the neutral stability curve, we carry out a nonlinear stability analysis of the model. The modified Korteweg-de Vries (mKdV) equation and the existence of the above mKdV equation are deduced, respectively. When the passing ratio is sufficiently small, i.e., the above existence holds, we observe that the jamming transition emerges between uniform flow and kink jam; once the passing ratio exceeds the threshold, the jamming transition occurs among uniform flow and kink-Bando traffic wave through the chaotic phase. We also conduct numerical simulations to verify the theoretical derivation.

Spot–Ladder Selection of Dislocation Patterns in Metal Fatigue

Spontaneous pattern formation by a large number of dislocations is commonly observed during the initial stages of metal fatigue under cyclic straining. It was experimentally found that the geometry of the dislocation pattern undergoes a crossover from a 2D spot-scattered pattern to a 1D ladder-shaped pattern as the amplitude of external shear strain increases. However, the physical mechanism that causes the crossover between different dislocation patterns remains unclear. In this study, we theorized a bifurcation diagram that explains the crossover between the two dislocation patterns. The proposed theory is based on a weakly nonlinear stability analysis that considers the mutual interaction of dislocations as a nonlinearity. It was found that the selection rule among the two dislocation patterns, “spotted” and “ladder-shaped”, can be described by inequalities with respect to nonlinearity parameters contained in the governing equations.

Linear and nonlinear stability of Rayleigh–Bénard convection with zero-mean modulated heat flux

Linear and nonlinear stability analyses are performed to determine critical Rayleigh numbers ( ${Ra}_{cr}$ ) for a Rayleigh–Bénard convection configuration with an imposed bottom boundary heat flux that varies harmonically in time with zero mean. The ${Ra}_{cr}$ value depends on the non-dimensional frequency $\omega$ of the boundary heat-flux modulation. Floquet theory is used to find ${Ra}_{cr}$ for linear stability, and the energy method is used to find ${Ra}_{cr}$ for two different types of nonlinear stability: strong and asymptotic. The most unstable linear mode alternates between synchronous and subharmonic frequencies at low $\omega$ , with only the latter at large $\omega$ . For a given frequency, the linear stability ${Ra}_{cr}$ is generally higher than the nonlinear stability ${Ra}_{cr}$ , as expected. For large $\omega$ , ${Ra}_{cr} \omega ^{-2}$ approaches an $O(10)$ constant for linear stability but zero for nonlinear stability. Hence the domain for subcritical instability becomes increasingly large with increasing $\omega$ . The same conclusion is reached for decreasing Prandtl number. Changing temperature and/or velocity boundary conditions at the modulated or non-modulated plate leads to the same conclusions. These stability results are confirmed by selected direct numerical simulations of the initial value problem.

Size-dependent nonlinear thermomechanical in-plane stability of shallow arches at micro/nano-scale including nonlocal and couple stress tensors

In the present exploration, the unconventional tensors of nonlocal stress and couple stress are incorporated simultaneously in the in-plane nonlinear stability analysis of functionally graded (FG) multilayer shallow micro/nano-arches under thermomechanical loading conditions. For this purpose, the nonlocal couple stress (NCS) mechanics of continuum is implemented into the third-order shear flexible arch theory incorporating the von Karman kinematical nonlinearity. The nanocomposite material of FG multilayer shallow micro/nano-arches is reinforced with graphene nanofillers in accordance with different patterns of FG lamination. The generalized differential quadrature numerical strategy in conjunction with the pseudo arc-length continuation procedure are employed to deduce the roles of unconventional nonlocal and couple stress tensors in the NCS-based nonlinear stability paths of thermomechanical loaded FG multilayer shallow micro/nano-arches. It is found that for all patterns of the lamination, the contributions associated with the nonlocality and couple stress small scale effects on the value of the upper limit as well as the first bifurcation compressive loads are less than those on the lower limit and the second bifurcation ones. Also, it is demonstrated that by combining a temperature rise with the applied compressive lateral load, the upper limit lateral load increases, while the lower limit one decreases. Furthermore, it is seen that by applying a temperature rise, an initial lateral deflection is induced in the shallow micro/nano-arch before applying the compressive lateral load.

Directional crystallization with a mushy region. Part 2: nonlinear analysis of dynamic stability

Directional crystallization with a mushy region. Part 2: nonlinear analysis of dynamic stability

Nonlinear modeling and stability analysis of asymmetric hydro-turbine governing system

This paper aims to study the stability and nonlinear dynamics of hydro-turbine governing system with asymmetric water diversion system, i.e. asymmetric hydro turbine governing system, by using Hopf bifurcation theory. Firstly, the full nonlinear mathematical model of asymmetric hydro turbine governing system is established by all system components and nonlinear head loss. This model contains two units with different capacities which share a common pipeline. Based on the nonlinear mathematical model, the multi-stability characteristics of asymmetric hydro turbine governing system under load disturbance is studied by using stable domain and verified by numerical simulation. Moreover, the multi-time scale oscillation is revealed and its relationship to system multi-stability is investigated. Furthermore, the effect of system parameters and topological parameters on system stability is analyzed. Results indicate that: two stable domains are emerged under load disturbance, the overall stability of asymmetric hydro turbine governing system is determined by the intersection area of the two stable domains. In addition, the system parameters and topological parameters both have obvious effect on system stability, which can be significantly improved by reasonable tuning of system parameters and optimization of water diversion system layout.

Internal Heat Modulation on Darcy Convection in a Porous Media Saturated by Nanofluid

In this paper we investigate the effect of internal heat modulation over a nanofluid saturated porous medium. We consider a small variation in time dependant heat source and vary sinusoidally with slow time. An energy equation will be altered by adding time dependant internal heat source. This internal heat source has its time dependent and independent parts. Time dependent part shows that the internal heat modulation over a porous media and defines controls on heat/mass transfer in the layer. We have performed a nonlinear stability analysis to investigate heat/mass transfer in the system. The nonlinear system of partial differential equations are transformed into nonlinear ordinary differential equations under similarity transforms up to the second term. This system has different system parameters and they have been investigated on heat and mass transfer graphically. The dual nature, stabilize or destabilize is due to the significant effect of internal heating modulation of the system. Further, the effect of internal heating is to destabilize the system, as a consequence heat/mass transfer enhances. It is found that internal heating modulation can be used effectively to regulate heat/mass transfer in the system.

Nonlinear aeroelastic stability analysis of a folding wing by using geometrically exact fully intrinsic beam equations

In this paper, the aeroelastic instability of folding wings by using the geometrically exact fully intrinsic beam equations is investigated. The important advantages of these equations in comparison with other structural beam equations are complete modeling without simplifying assumptions in large deformations, low-order nonlinearities, and thus less complexity. For the first time, folding angles have been implemented in the geometrically exact fully intrinsic beam equations and hence this is the main novelty of this study. The applied aerodynamic loads in an incompressible flow regime are determined using Peter’s unsteady aerodynamic model. In order to check the stability of the system, first the resulting non-linear partial differential equations are discretized by employing the central finite difference method, and then linearized about the nonlinear steady-state condition. By obtaining the eigenvalues of the linearized system, the stability of the wing is evaluated. Furthermore, investigation of the effects of some important parameters such as stiffness ratio and length ratio on the flutter speed of the folding wing for various folding angles, is another achievement of this study. It is observed that the geometrically exact fully intrinsic beam equations can model the folding angles for the aeroelastic analysis more accurately and the capabilities of these equations became more specific.

Study on the layout methods of stiffening rings for steel cooling towers

Stiffening rings are commonly employed as strengthening measures in steel cooling towers. However, studies on reasonable layout methods are lacking. In this study, three layout methods are proposed: the equal-spacing method, the method based on the 1st-order eigenvalue, and the method based on the 1st-order buckled mode shape. Finite element models of steel cooling towers were established considering two tower types (hyperbolic-type and cylinder-frustum-type), three tower heights (120 m, 160 m, and 200 m), three geometric parameters, and two latticed shells (single-layer rectangular and double-layer square pyramid shells). A nonlinear stability analysis of steel cooling towers designed by three layout methods was performed, and the load-bearing capacities of the structures were compared to determine the superiority and inferiority of the three layout methods. The results indicated that double-layer latticed shell steel cooling towers can satisfy the requirements of stable bearing capacity without arranging stiffening rings. The layout method based on the 1st-order buckled mode shape is more suitable than the other two methods for single-layer latticed shell steel cooling towers involving various geometric parameters. It is recommended that this method be used to design the layout of the stiffening rings.

Linear and nonlinear stability analyses of double-diffusive convection in a porous layer due to magnetic field and throughflow effects

Linear and nonlinear stability analyses of double-diffusive convection in a porous layer due to magnetic field and throughflow effects

Global Instability of Rod Eutectic Growth in Directional Solidification

In our previous work, we obtained the uniformly valid asymptotic solution of a cylindrical rod eutectic. In order to further study the critical point of the stable growth of a rod eutectic, we have considered the unsteady growth of a rod eutectic on the basis of the steady solution of the rod eutectic. Based on the experimental system of rod eutectic growth, combined with solidification thermodynamics and kinetics, the unsteady mathematical model of the rod eutectic was established. We used the asymptotic analysis method to seek the analytical solution of the mathematical model and used the nonlinear stability analysis theory to analyze the analytical solution and establish the corresponding disturbance model. We obtained the analytic form of the global mode solution and the corresponding quantization conditions and find that there is a stable growth mode, namely the mode (ST-mode), for rod eutectic growth; when ε&lt;εST0, the rod eutectic growth is stable, when ε&gt;εST0, the rod eutectic growth is unstable and when ε=εST0, the rod eutectic growth is of a neutral stability. The critical eutectic spacing of succinonitrile(D)camphor (SCN-DC) predicted by us is smaller than that predicted by Jackson–Hunt, which is consistent with the corresponding experimental data. Finally, we found that the critical eutectic spacing and stable region of rod eutectic growth changed little with the temperature gradient.

Global stability analyses of Mack mode on the windward face of a hypersonic yawed cone

What is the natural transition scenario for the windward boundary layer over a hypersonic inclined cone? Based on comprehensive linear and nonlinear global stability analyses, this work shows that the K-type breakdown triggered by the Mack-mode instability is responsible for the windward boundary-layer transition.

Size-dependent nonlinear vibration analysis of cracked graphene-platelets-reinforced-composites (GPLRC) plate under parametric excitation

The response of a size-dependent randomly distributed graphene-platelet-reinforced-composite (GPLRC) plates, with an inclined part-through surface crack, resting on viscoelastic foundation under the action of periodic parametric excitation with quadratic and cubic nonlinearities are studied in the present study. The crack is centrally located at the center of the plate while inclined at an angle β concerning the X-axis. The angled crack’s mathematical formulation is based on a line-spring model. Indeed, the assumption of the problem is considered as the crack remains a continuous straight line. Nonlinearity is considered based on Berger’s formulation and Von-Karman’s geometric relations. Using classical plate theory, Eringen’s theory, micromechanical models, and equilibrium equation, the governing equation of motion of the problem have been derived. Then the Galerkin method was applied to transform the partial differential equation into the ordinary one. Eventually, the solution procedure of the cubic–quadratic nonlinear equation was followed by the multiple-scale perturbation method. Steady-state response and nonlinear softening or hardening treatment of the problem have been presented. Significant results of the study can be emphasized as GPLs, crack’s dimension and inclination, nonlocal parameter, damping, and detuning parameter effects on nonlinear behavior and stability of the structure. Dispersing GPLs as reinforcement material in the plate layer caused improvement in structural behavior, as the amplitude response of the system with the presence of a crack is decreased, comparing to a similar system without any reinforcement and crack. In the end, current work will help in the development of nonlinear behavior, stability analysis and critical bifurcation points of size-dependent cracked GPLRC plate exposed to external parametric loading.

Study the combined effect of electric field and variable viscosity on the stability of dielectric nanofluid (CuO + water) within a Hele–Shaw cell

In this research article, the combined effect of electric field and variable viscosity on the stability of dielectric nanofluid (CuO + water) within a Hele–Shaw cell is studied. The normal mode technique has been implemented for linear stability analysis and the truncated Fourier series method for nonlinear stability analysis. The effect of the Hele–Shaw number, thermorheological parameter, and alternating current electric Rayleigh number on the onset of convection and heat/mass transfer has been investigated analytically, graphically, and using tables. It is observed that the effect of the thermorheological parameter has a destabilizing effect, and the value of the parameter is always less than [Formula: see text] for this study. We also found that the Hele–Shaw number has stabilizing effect, and it is always greater or equal to [Formula: see text] for this study. The effect of alternating current electric Rayleigh number is destabilized. We observed that in the whole analysis, numerical results are better than analytical results.