Nonlinear Conditions Research Articles

Nonlinear vibrations of temperature-dependent FGM cylindrical panels subjected to instantaneous surface heating

An analysis is performed in this research to deal with the large amplitude vibrations caused by rapid surface heating in an FGM cylindrical panel. It is assumed that the properties of the panel are distributed through the thickness in terms of volume fraction of constituents. Following the first-order shear deformation theory, von Karman type of kinematics, and Donnell kinematic assumptions, definition of kinetic and strain energies of the shell are established. These obtained equations are discretized via the method of Ritz, where the shape functions are estimated by the Chebyshev polynomials. Temperature distribution within the body is also obtained using the one-dimensional heat conduction equation. This equation is discretized using the finite different method and solved iteratively following the Picard and Crank-Nicolson methods. Thermally induced force and moment resultants are evaluated and inserted into the matrix representation of equations of motion. The temporal evolution of displacement is obtained using the iterative Picard method and Newmark time marching method. Results of this study are compared with the available data in the open literature and after that novel results are given to explore the effects of geometrical, mechanical, and thermal parameters. It is highlighted that power law index, shallowness, thickness, edge supports, temperature dependency, geometrical nonlinearity and thermal boundary conditions are all important factors on the response of the structure under thermal shock. It is highlighted that thermally induced vibrations indeed exists especially in thin class of shells. Besides, thermally induced deflections may be controlled through using a proper power law index.

Sensitivity of optimal double-layer grid designs to geometrical imperfections and geometric nonlinearity conditions in the analysis phase

Sensitivity of optimal double-layer grid designs to geometrical imperfections and geometric nonlinearity conditions in the analysis phase

A simplified and controllable model of mode coupling for addressing nonlinear phenomena in sound synthesis processes

This paper introduces a simplified and controllable model for mode coupling in the context of modal synthesis. The model employs efficient coupled filters for sound synthesis purposes, intended to emulate the generation of sounds radiated by sources under strongly nonlinear conditions. Such filters generate tonal components in an interdependent way and are intended to emulate realistic perceptually salient effects in musical instruments in an efficient manner. The control of energy transfer between the filters is realized through a coupling matrix. The generation of prototypical sounds corresponding to nonlinear sources with the filter bank is presented. In particular, examples are proposed to generate sounds corresponding to impacts on thin structures and to the perturbation of the vibration of objects when it collides with an other object. The sound examples presented in the paper and available for listening on the accompanying site illustrate that a simple control of the input parameters allows the generation of sounds whose evocation is coherent and that the addition of random processes yields a significant improvement to the realism of the generated sounds.

Solvability analysis for the Boussinesq model of heat transfer under the nonlinear Robin boundary condition for the temperature.

We consider the new boundary value problem for the generalized Boussinesq model of heat transfer under the inhomogeneous Dirichlet boundary condition for the velocity and under mixed boundary conditions for the temperature. It is assumed that the viscosity, thermal conductivity and buoyancy force in the model equations, as well as the heat exchange boundary coefficient, depend on the temperature. The mathematical apparatus for studying the inhomogeneous boundary value problem under study based on the variational method is being developed. Using this apparatus, we prove the main theorem on the global existence of a weak solution of the mentioned boundary value problem and establish sufficient conditions for the problem data ensuring the local uniqueness of the weak solution that has the additional property of smoothness with respect to temperature. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Thin inclusion at the junction of two elastic bodies: non-coercive case.

This article addresses an analysis of the non-coercive boundary value problem describing an equilibrium state of two contacting elastic bodies connected by a thin elastic inclusion. Nonlinear conditions of inequality type are imposed at the joint boundary of the bodies providing a mutual non-penetration. As for conditions at the external boundary, they are Neumann type and imply the non-coercivity of the problem. Assuming that external forces satisfy suitable conditions, a solution existence of the problem analysed is proved. Passages to limits are justified as the rigidity parameters of the inclusion and the elastic body tend to infinity.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

A Model of String System Deformations on a Star Graph with Nonlinear Condition at the Node

A Model of String System Deformations on a Star Graph with Nonlinear Condition at the Node

Radial diversification of pump and signal intensities in EDFA comprising LP01 mode dispersion compensation fibers in third order nonlinear condition

Radial diversification of pump and signal intensities in EDFA comprising LP01 mode dispersion compensation fibers in third order nonlinear condition

New Opportunities in Real-Time Diagnostics of Induction Machines

This manuscript addresses the critical challenges in achieving high-accuracy remote control of electromechanical systems, given their inherent nonlinearities and dynamic complexities. Traditional diagnostics often suffer from data inaccuracies and limitations in analytical techniques. The focus is on enhancing the dynamic model accuracy for remote induction motor control in both closed- and open-loop speed control systems, which is essential for real-time process monitoring. The proposed solution includes real-time measurements of input and output physical quantities to mitigate inaccuracies in traditional diagnostic methods. The manuscript discusses theoretical aspects of nonlinear torque formation in induction drives and introduces a dynamic model employing vector control and speed control schemes alongside standard frequency control methods. These approaches optimize frequency converter settings to enhance system performance under varying nonlinear conditions. Additionally, the manuscript explores methods to analyze dynamic, systematic errors arising from frequency converter inertial properties, thereby improving electromechanical equipment condition diagnostics. By addressing these challenges, the manuscript significantly advances the field, offering a promising future with enhanced dynamic model accuracy, real-time monitoring techniques, and advanced control methods to optimize system reliability and performance.

Performance of GRKM-method for solving classes of ordinary and partial differential equations of sixth-orders

Abstract A general quasilinear sixth-order ordinary differential equation (ODE) is an important class of ODEs. The primary objective of this study is to establish a numerical method for solving a general class of quasilinear sixth-order partial differential equations (PDEs) and ODEs. However, the Runge–Kutta method (RKM) approach for solving special classes of ODEs has been generalized as an effort to solve the general class of ODEs. Nonlinear algebraic order condition (OCs) equations have been obtained up to the tenth order using the Taylor-series expansion methodology which is used to derive the novel generalized Runge–Kutta method (GRKM). In this study, a GRKM integrator has been derived for solving a general class of quasilinear sixth-order ODEs and then this method is modified subsequently to solve a class of PDEs. Accordingly, the proposed GRKM is modified to solve a quasilinear sixth-order PDE by converting it to a system of sixth-ODEs using the method of lines. Nine problems have been implemented to prove the efficiency and accuracy of the proposed method. Simulation results of these problems showed that the proposed numerical GRKM is an accurate and efficient method. In contrast, by comparing the proposed GRKM numerical approach with the classical RK method, the numerical results demonstrate that the direct integrator outperforms the indirect classical RK method in terms of algorithm complexity and function evaluations, proving that the numerical GRKM is efficient.

Well-posedness and inverse problems for semilinear nonlocal wave equations

This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main challenge is due to the low regularity of the solutions of linear nonlocal wave equations. We then turn to an inverse problem of recovering the nonlinearity of the equation. More precisely, we show that the exterior Dirichlet-to-Neumann map uniquely determines homogeneous nonlinearities of the form f(x,u) under certain growth conditions. On the other hand, we also prove that initial data can be determined by using passive measurements under certain nonlinearity conditions. The main tools used for the inverse problem are the unique continuation principle of the fractional Laplacian and a Runge approximation property. The results hold for any spatial dimension n∈N.

On the stability of prestressed beams undergoing nonlinear flexural free oscillations

We study the nonlinear free undamped motions of a hinged-hinged beam exhibiting geometric stretching-induced nonlinearity and arbitrary initial conditions. We treat the governing integral-partial-differential equation of motion as an infinite dimensional Hamiltonian system. We analytically obtain a quantitative Birkhoff Normal Form via a nonlinear coordinate transformation that yields the reduced (modulation) equations describing the free oscillations to within a certain nonlinear order with an estimate of the reminder. The obtained solutions provide a very precise description of small amplitude oscillations over large time scales. The analytical optimization of the involved estimates yields time stability results obtained for plausible values of the physical quantities and of the perturbation parameter. The role played by internal resonances in determining the time stability of the solution is highlighted and discussed. We show that initial conditions with a finite number of eigenfunctions yield bounded solutions living on invariant subspaces of the involved modes at all times. Conversely, initial conditions comprising the full (infinite) spectrum of eigenfunctions provide solutions for which time stability for all times cannot be stated.

Error estimates for simplified Levenberg–Marquardt method for nonlinear ill-posed operator equations in Hilbert Spaces

Abstract In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form F ⁢ ( u ) = y {F(u)=y} , where F : 𝒟 ⁢ ( F ) ⊂ 𝖴 → 𝖸 {F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}} is a nonlinear operator between Hilbert spaces 𝖴 {\mathsf{U}} and 𝖸 {\mathsf{Y}} . The method is defined as follows: u n + 1 δ = u n δ - ( T 0 ∗ ⁢ T 0 + α n ⁢ I ) - 1 ⁢ T 0 ∗ ⁢ ( F ⁢ ( u n δ ) - y δ ) , u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{% \ast}(F(u_{n}^{\delta})-y^{\delta}), where T 0 = F ′ ⁢ ( u 0 ) {T_{0}=F^{\prime}(u_{0})} and T 0 ∗ = F ′ ⁢ ( u 0 ) ∗ {T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}} . Here F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} denotes the Frèchet derivative of F at an initial guess u 0 ∈ 𝒟 ⁢ ( F ) {u_{0}\in\mathcal{D}(F)} for the exact solution u † {u^{\dagger}} , F ′ ⁢ ( u 0 ) ∗ {F^{\prime}(u_{0})^{\ast}} is the adjoint of F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} and { α n } {\{\alpha_{n}\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F, we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element u 0 - u † {u_{0}-u^{\dagger}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions.

Formation and local stability of a two-dimensional Prandtl boundary layer system in fluid dynamics

By the linearized method, Oleǐnik-Samokhin had constructed solutions to distinguish between two cases of the behavior of the fluid at the initial stage of its motion past the surface in Oleinǐk and Samokhin (1999). In this paper, we consider the development of the boundary layer about a body that gradually starts to move in a resting fluid. In the analytical frame, the formation of the layer shows that, when the layer starts to move, the local solutions of the Prandtl boundary layer system is positive. By the Crocco transformation, the Prandtl boundary layer system reduces a degenerate parabolic equation with a nonlinear boundary value condition. Then by the reciprocal transformation, we can obtain a divergence type parabolic equation. We quote a new kind of BV entropy solution matching up with this divergence type parabolic equation when t≥t0, where t0 is a small enough positive constant. By imposing some close connections between the velocity of the outflow and the geometric characteristic of the spatial domain, using Kružkov’s bi-variables method, the local stability of BV entropy solutions is proved. Thus, by the Crocco inverse transformation, in the analytical frame, we obtain the existence and the uniqueness of the global solution of the Prandtl boundary layer system for a regular spatial domain.

Analysis of Non-Linear Static Procedure Based on Fema 356 to Evaluate Structural Performance in the Alton Apartment Building

Alton Apartment is a high-rise residential building. The city of Semarang, within a radius of 500 km, has a fault range with a strength of up to 8.3 magnitude. The purpose of this study is to calculate the performance level of the structure, find out the nonlinear static procedure conditions and whether the analysis results can be used, calculate the value of ductility R, check the boundaries between levels, check the influence of P-delta, adjust horizontal irregularities, and check the vertical irregularity conditions. This study uses the FEMA 356 method. New regulations are used in the analysis, such as peak acceleration maps and response spectra from PuSGeN in 2021, deaggregation maps of Indonesian earthquake hazards for planning and evaluation of earthquake-resistant infrastructure from the Director General of Cipta Karya in 2022, and earthquake resilience planning procedures for building and non-building structures from SNI 1726-2019. The result of the structure performance of the drift value is below 1% for the life safety (LS) performance level on the 250-year earthquake map, and the result of the drift value is below 2% for the collapse prevention (CP) performance level on the 100-year earthquake map. The results of the FEMA 356 condition one analysis show that the strength ratio (μstrength) value of less than μmax has been met for 250-year and 1000-year earthquake maps with a review of the x and y directions of the earthquake. The value of the R ductility in 250-year and 1000-year earthquakes is above seven according to the earthquake force holding system of special reinforced concrete sliding walls.

An Improvement Method for Improving the Surface Defect Detection of Industrial Products Based on Contour Matching Algorithms.

Aiming at the problems of the poor robustness and universality of traditional contour matching algorithms in engineering applications, a method for improving the surface defect detection of industrial products based on contour matching algorithms is detailed in this paper. Based on the image pyramid optimization method, a three-level matching method is designed, which can quickly obtain the candidate pose of the target contour at the top of the image pyramid, combining the integral graph and the integration graph acceleration strategy based on weak classification. It can quickly obtain the rough positioning and rough angle of the target contour, which greatly improves the performance of the algorithm. In addition, to solve the problem that a large number of duplicate candidate points will be generated when the target candidate points are expanded, a method to obtain the optimal candidate points in the neighborhood of the target candidate points is designed, which can guarantee the matching accuracy and greatly reduce the calculation amount. In order to verify the effectiveness of the algorithm, functional test experiments were designed for template building function and contour matching function, including uniform illumination condition, nonlinear condition and contour matching detection under different conditions. The results show that: (1) Under uniform illumination conditions, the detection accuracy can be maintained at about 93%. (2) Under nonlinear illumination conditions, the detection accuracy can be maintained at about 91.84%. (3) When there is an external interference source, there will be a false detection or no detection, and the overall defect detection rate remains above 94%. It is verified that the proposed method can meet the application requirements of common defect detection, and has good robustness and meets the expected functional requirements of the algorithm, providing a strong technical guarantee and data support for the design of embedded image sensors in the later stage.

Non-linear impedance boundary condition from linear piecewise B-H curve applied to induction heating systems

Purpose This paper aims to apply the non-linear impedance boundary condition (IBC) for a linear piecewise B–H curve in frequency domain simulations to find the equivalent impedance of a simple induction heating system model. Design/methodology/approach An electromagnetic description of the inductor system is performed to substitute the effects of the induction load, for a mathematical condition, the so-called IBC. This is suitable to be used in electromagnetic systems involving high conductive materials at medium frequencies, as it occurs in an induction heating system. Findings A reduction of the computational cost of electromagnetic simulation through the application of the IBC. The model based on linear piecewise B–H curve simplifies the electromagnetic description, and it can facilitate the identification of the induction load characteristics from experimental measurements. Practical implications This work is performed to assess the feasibility of using the non-linear boundary impedance condition of materials with linear piecewise B–H curve to simulate in the frequency domain with a reduced computational cost compared to time domain simulations. Originality/value In this paper, the use of the non-linear boundary impedance condition to describe materials with B–H curve by segments, which can approximate any dependence without hysteresis, has been studied. The results are compared with computationally more expensive time domain simulations.

Stabilised finite element method for Stokes problem with nonlinear slip condition

This work introduces a stabilised finite element formulation for the Stokes flow problem with a nonlinear slip boundary condition of friction type. The boundary condition is enforced with the help of an additional Lagrange multiplier representing boundary traction and the stabilised formulation is based on simultaneously stabilising both the pressure and the traction. We establish the stability and the a priori error analyses, and perform a numerical convergence study in order to verify the theory.

Stabilization and Blow-up in an Infinite Memory Wave Equation with Logarithmic Nonlinearity and Acoustic Boundary Conditions

Stabilization and Blow-up in an Infinite Memory Wave Equation with Logarithmic Nonlinearity and Acoustic Boundary Conditions

A Modified Variable Power Angle Control for Unified Power Quality Conditioner in a Distorted Utility Source

The distorted supply voltage degrades the control performance of a unified power quality conditioner (UPQC). This problem causes incorrect calculations in the harmonic identification and reference signal generation processes. This paper proposes a modified harmonic identification of the UPQC. The reference compensating current calculation for the shunt active power filter (shunt APF) is developed using the sliding window with the Fourier analysis (SWFA) method. In addition, the variable power angle control (PAC) is applied to operate the reference signal generation of the series APF and the shunt APF of the UPQC. Under the distorted voltage and nonlinear load conditions, the proposed approach can provide accurate reference compensating signals and successfully share the load reactive power compensation between the shunt APF and the series APF. In this work, a three-phase, three-wire power system with linear and nonlinear loads was implemented. The proposed method was validated using the processor-in-the-loop technique on an eZdsp™ F28335 board and the MATLAB/Simulink program. The testing results indicated that SWFA has excellent filtering performance and enhances harmonic identification compared to the operation without any filter or with low pass filters (LPF). With the proposed approach, the percentage of total harmonic distortion of voltage and current could be maintained within the IEEE519-2022 standard, and the magnitude of the RMS voltage across the load was in the recommended range specified by ANSI C84.1-2016.

Global existence of solutions for the drift–diffusion system with large initial data in [formula omitted] ([formula omitted])

In this paper, we study the Cauchy problem of the drift–diffusion system arising from semiconductor model. We prove that if a certain nonlinear function of the initial data is small enough, in a Besov type space, then there is a global solution to this drift–diffusion system. We also provide an example of initial data satisfying that nonlinear smallness condition, but whose norm be chosen arbitrarily large in Ḃ∞,∞−2(Rd).