Nonsmooth Nonlinearities Research Articles

Adaptive regularization, discretization, and linearization for nonsmooth problems based on primal–dual gap estimators

We consider nonsmooth partial differential equations associated with a minimization of an energy functional. We adaptively regularize the nonsmooth nonlinearity so as to be able to apply the usual Newton linearization, which is not always possible otherwise. We apply the finite element method as a discretization. We focus on the choice of the regularization parameter and adjust it on the basis of an a posteriori error estimate for the difference of energies of the exact and approximate solutions. Importantly, our estimates distinguish the different error components, namely those of regularization, linearization, and discretization. This leads to an algorithm that steers the overall procedure by adaptive stopping criteria with parameters for the regularization, linearization, and discretization levels. We prove guaranteed upper bounds for the energy difference and discuss the robustness of the estimates with respect to the magnitude of the nonlinearity when the stopping criteria are satisfied. Numerical results illustrate the theoretical developments.

Sign-changing solution for a generalized Kirchhoff problem in the fractional Orlicz-Sobolev space with nonsmooth nonlinearity

In this paper, we study a nonlocal generalized Kirchhoff problem driven by the fractional Orlicz g-Laplace operator and involving a nonsmooth nonlinearity. Although this problem has a double nonlocality, we prove the existence of a least-energy nodal (or sign-changing) solution. Note that the tools used are minimization argument and quantitative deformation lemma.

Spiral bevel gears: Bifurcation and chaos analyses of pure torsional system

The goal of this paper is to provide a detailed analysis of the complex dynamic scenario of spiral bevel gears by considering the torsional shaft stiffness. The dynamic model takes into account: time-dependent stiffness and non-smooth nonlinearity due to the backlash, i.e., teeth contact loss. The time-varying meshing stiffness is evaluated by means of a nonlinear finite element model, which allowed an accurate evaluation of the global and local teeth deformation in both directions: forward and reverse motions. Due to intentional tooth profile errors, e.g., profile modification, manufacturing error, or assembly error, the loaded and unloaded tooth contact analyses were conducted. The dynamic model was validated by comparisons with a verified SDOF model in terms of linear natural frequencies and nonlinear dynamic response. The present study provides amplitude frequency diagrams and bifurcation diagrams; for specific regimes, periodic, quasiperiodic, and chaotic responses were found. The dynamic behavior of the systems was evaluated using various tools such as modal analysis, nonlinear time series analysis, spectra, 3D-phase diagrams, and Poincaré maps. This study proves that decreasing the DOFs of the system can lose some dynamics. Different phenomena have been found such as trapping and boom-and-bust cycle. The dynamic response of different cases is evaluated by estimating the largest Lyapunov exponent and the correlation dimension. It is found that the system undergoes complex dynamic phenomena including nT-periodic, trapping, and aperiodic motions, which are evidenced in 3D-bifurcation diagrams, spectra, Poincaré maps, and 3D-phase diagrams. The chaotic motion found in this study is an undesirable behavior from a practical point of view, which can be avoided by adjusting the designed parameters. The results provide a basis for parameter design and dynamic characteristic control of the spiral bevel gear drive system.

Method of Virtual Sensor Design for Faulty Physical Sensor Replacement

The paper considers the problem of virtual sensor design for nonlinear dynamic systems with non-smooth nonlinearities described by continuous-time models for faulty physical sensor replacement. The main purpose of virtual sensors is generating the estimates of the unmeasured components of the considered system to provide additional information for effective control and fault diagnosis. Besides, virtual sensors can be used for faulty physical sensor replacement. The methods of virtual sensor design for solving this problem differ from standard procedure since information from faulty physical sensor does not use to design the virtual sensor replacing this sensor. It is assumed that to solve the problem, the system is equipped by diagnostic system allowing detecting faulty sensor. For every such a sensor, the virtual sensor generating estimate replacing the faulty sensor is designed. To solve the problem, so-called logic-dynamic approach is used which does not guarantee optimal solution but uses only methods of linear algebra to solve the problem for systems with non-smooth nonlinearities. This approach contains three steps. Initially, the nonlinear term is removed from system and linear model is designed. Then, a possibility to estimate the faulty sensor and to insert in the model the transformed nonlinear term is checked. Finally, stability of sensor is provided. The virtual sensor can be designed in identification canonical form or Jordan canonical form. The advantage of the first form is a standard procedure of the virtual sensor design while Jordan form allows obtaining simpler solution. The relations allowing designing the virtual sensor as in identification canonical as Jordan canonical form are derived.

Model reduction of a periodically forced slow–fast continuous piecewise linear system

AbstractIn this paper, singular perturbation theory is exploited to obtain a reduced-order model of a slow–fast piecewise linear 2-DOF oscillator subjected to harmonic excitation. The nonsmooth nonlinearity of piecewise linear nature is studied in the case of bilinear damping as well as with bilinear stiffness characteristics. We propose a continuous matching of the locally invariant slow manifolds obtained in each subregion of the state space, which yields a reduced-order model of the same nature as the full dynamics. The frequency-response curves obtained from the full system and the reduced-order models suggest that the proposed reduction method can capture nonlinear behaviors such as super- and subharmonic resonances.

Tunable nonlinear metastructure with periodic bi-linear oscillators for broadband vibration suppression

A novel tunable nonlinear metastructure with periodically distributed bi-linear oscillators (BLOs) is proposed for broadband vibration suppression. The BLO with non-smooth nonlinearity is composed of a cantilever-beam resonator and two limiters, whose tunability for bi-linear stiffness can be achieved by adjusting the clearance size and limiting position. By simplifying the present nonlinear metastructure, we firstly establish a two degrees-of-freedom (2-DOF) dynamic model and analyze its nonlinear dynamic behaviors to reveal the underlying energy transfer mechanism. The governing equations of the nonlinear metastructure integrated with BLOs are derived using the Hamilton principle. The semi-analytical solutions of the nonlinear bandgap boundaries that depend on the amplitude-clearance ratio and the nonlinear stiffness ratio are derived to evaluate the bandgap behavior of the nonlinear metastructure, which is validated well by comparing with the frequency-response behavior of the metastructure using its analytical model and finite element model. The numerical simulation results show that the tunable-nonlinear-bandgap characteristic of the proposed metastructure can be achieved by modulating the clearance size and the nonlinear stiffness of the BLOs. In particular, a nonlinear bandgap possessing substantial amplitude reduction can be observed, which makes the nonlinear metastructure behave excellent broadband vibration suppression performance. Furthermore, there exist a minimum number of BLOs with a small additional mass which can form a dramatic vibration reduction for practical design with finite-size metastructure. The present work demonstrates that the designed tunable-BLOs-based metastructure with non-smooth nonlinearity can contribute to a breakthrough of the limitation of existing nonlinear metastructure and provide a novel and effective approach to tailor strong mechanical nonlinearity and to enhance broadband vibration attenuation performance.

Virtual Sensor Design for Replacement the Faulty Physical Sensors

Abstract: The paper considers the problem of virtual sensor design for nonlinear dynamic systems with non-smooth nonlinearities described by continuous-time models for faulty physical sensor replacement. It is assumed that to solve the problem, the system is equipped by diagnostic system allowing detecting and isolating the faulty sensor. For every such a sensor, the virtual sensor generating estimate replacing the faulty sensor is designed. To solve the problem, so-called logic-dynamic approach is used which does not guarantee optimal solution but uses only methods of linear algebra to solve the problem for systems with non-smooth nonlinearities. The virtual sensor can be designed in the identification canonical form or Jordan canonical form. The advantage of the first form is a standard procedure of the virtual sensor design while Jordan form allows obtaining a simpler solution. The relations allowing designing the virtual sensor both in identification and in Jordan canonical form are derived.

Ground State Sign-Changing Solution for a Large Class of $$ p(x) \& q(x)$$-Quasilinear Elliptic Equations with Nonsmooth Nonlinearity

Ground State Sign-Changing Solution for a Large Class of $$ p(x) \& q(x)$$-Quasilinear Elliptic Equations with Nonsmooth Nonlinearity

Evaluating route to impact convergence of the harmonic balance method for piecewise-smooth systems

In this work, we investigate the applicability of the harmonic balance method (HBM) to predict periodic solutions of a single degree-of-freedom forced Duffing oscillator with freeplay nonlinearity. By studying the route to impact, which refers to a parametric study as the contact stiffness increases from soft to hard, the convergence behavior of the HBM can be understood in terms of the strength of the non-smooth forcing term. HBM results are compared to time-integration results to facilitate an evaluation of the accuracy of nonlinear periodic responses. An additional contribution of this study is to perform convergence and stability analysis specifically for isolas generated by the non-smooth nonlinearity. Residual error analysis is used to determine the approximate number of harmonics required to get results accurate to a given error tolerance. Hill’s method and Floquet theory are employed to compute the stability of periodic solutions and identify the types of bifurcations in the system.

Virtual Sensors Design for Nonlinear Dynamic Systems

The objective of the paper is virtual sensors design, estimating prescribed components of the systems state vector to solve the tasks of fault diagnosis in nonlinear systems. To solve the problem, the method called logic-dynamic approach is used, which allows to solve the problem for systems with non-smooth nonlinearities subjected to external disturbance by methods of linear algebra. According to this method, the problem is solved in three steps: at the first step, the nonlinear term is removed from the system, and the linear model invariant with respect to the disturbance is designed; at the second step, a possibility to take into account the nonlinear term and to estimate the given variable is checked; finally, the transformed nonlinear term is added to the linear model. The relations allowing to design virtual sensor of minimal dimension estimating prescribed component of the state vector of the system are obtained. The main contribution of the present paper is that a procedure to design virtual sensors of minimal dimension for nonlinear systems estimating prescribed components of the state vector is developed. This allows to reduce complexity of the virtual sensors in comparison with known papers where such sensors of full dimension are constructed. Besides, the limitations imposed on the initial system are relaxed that allow to extend a class of systems for which the virtual sensors can be constructed.

Model updating of dynamic structures with strong nonlinearities using fixed frequency continuation tests

Model updating using multivalued responses is an important approach to construct strongly nonlinear models. During updating, swept frequency tests are usually used to measure the multivalued responses, but the measured responses only contain the stable branches. Accordingly, updating mainly uses these stable responses for the nonlinear model construction. Unstable responses are rarely tested or participate in the updating process, which may lead to errors in the constructed model with strong nonlinearities. Recently, fixed frequency tests using a continuation scheme have been used to measure fully multivalued responses, including both the stable and unstable branches, for structures with the strong nonlinearity. However, the measured responses were then transformed to equivalent swept frequency tests for the existing updating schemes, where this transformation may induce errors in the updating process. In this paper, a novel method for strong nonlinear model updating using the results of the fixed frequency test is proposed. Such updating is based on the directly measured data of the fixed frequency tests and can handle both the stable and unstable responses. Fixed frequency continuation using the Multi-Harmonic Balance Method (MHBM) is developed to predict the multivalued responses and their sensitivities. The multivalued responses of the prediction and measurement are directly matched in sensitivity-based updating. The proposed method is numerically verified on a 2 DOF nonlinear model. Using the proposed method, the nonlinear model is updated from linear to strongly nonlinear and the updated parameters all converge to the simulated values after 7 iterations. The proposed method is then experimentally validated on a 3 DOF nonlinear system with a non-smooth nonlinearity. A nonlinear model with a 7th order polynomial stiffness represents the equivalent system and is directly updated based on the fixed frequency test results. The updated responses are generally a good fit to the tested responses. Compared with updating using only the stable responses, the proposed updating method achieves higher precision and its updated responses are closer to the tested responses. These results show the validity and superiority of the proposed method.

Describing function method with pointwise balancing in two-dimensional regularized time domain for quasi-periodic responses

This paper proposes a new semi-analytical approach, called as describing function method with pointwise balancing (DFPB), to solve quasi-periodic (QP) responses of nonlinear dynamical systems especially those with non-smooth nonlinearities. This method is to describe the QP motion by generalized Fourier series with multiple regularized time scales. By discretizing the solution in the regularized time domain, the original equation is transformed into over-determined algebraic equations governing Fourier series coefficients via pointwise balancing. The coefficients are determined by solving the balance equations via the Newton–Raphson iteration algorithm. In addition, the Tikhonov regularization is utilized to enhance the convergence of the iteration scheme, so that the least squares solution of the over-determined equations can be obtained efficiently. On the one hand, the pointwise balancing avoids repeated transformations of non-smooth functions between time- to frequency-domain. On the other hand, the regularized time scales make it convenient to discretize the system equations regardless of unknown fundamental frequencies. With these procedures, the DFPB is well-designed to solve non-smooth systems for QP response with frequencies to be determined. In the numerical examples, the effectiveness and accuracy are elucidated in detail through both smooth and non-smooth models. The DFPB results are in excellent agreement with those attained by the Runge–Kutta method. In addition, it is worthy of pointing out the DFPB can track both stable and unstable QP responses.

Bifurcation Control for a Ship Maneuvering Model with Nonsmooth Nonlinearities

We consider a widely used form of models for ship maneuvering, whose nonlinearities entail continuous but nonsmooth second-order modulus terms. For such models bifurcations of straight motion are not amenable to standard center manifold reduction and normal forms. Based on a recently developed analytical approach, we nevertheless determine the character of local bifurcations when stabilizing the straight motion course with standard proportional control. For a specific model class we perform a detailed analysis of the linearization to determine the location of these bifurcations in the control parameter space and its dependence on selected design parameters. By computing the analytically derived characteristic parameters, we find that “safe” supercritical Andronov–Hopf bifurcations are typical. Through numerical continuation we provide a more global bifurcation analysis, which identifies the arrangement and relative location of stable and unstable equilibria and periodic orbits.

Existence, uniqueness and asymptotic behavior of solutions for a nonsmooth producer-grazer system with stoichiometric constraints

A reaction-diffusion-ODE system of stoichiometric producer-grazer type is considered in this paper. Since the system has nonsmooth nonlinearity, it is shown that the system has new dynamics different from the smooth case. We construct various types of discontinuous steady states and investigate their asymptotic behavior. In addition, the steady-state bifurcation near a constant solution is studied by treating the diffusion coefficient as a bifurcation parameter and the existence of Hopf bifurcation is derived. Our results cover the case where the number of positive equilibria of the kinetic system (i.e. without diffusion) changes from one to three in the spatial interval. Finally, some numerical simulations are given to illustrate the theoretical results.

Non-linear dynamic behaviour of spiral bevel gear by considering the torsional shaft stiffness

Spiral bevel gears (SBGs) play a crucial role in developing silent power transmissions for non-parallel shaft applications, offering advantages such as improved motor allocation flexibility and space reduction. While SBGs have been recognized for reducing vibration magnitude in high-speed gearboxes compared to straight bevel gears, complete vibration suppression remains elusive, leading to potential challenges such as teeth contact loss and complex dynamic scenarios. To construct the dynamical model of SBG, time-dependent mesh stiffness and non-smooth nonlinearity caused by backlash is considered. The employed dynamical system is a three-degree-of-freedom model, integrating rotational shaft stiffness, to investigate the dynamic behavior of SBGs. Through the utilization of various analysis tools such as bifurcation diagram, Fourier spectrum, 3D-phase diagram, Poincaré map, and amplitude-frequency diagram are generated, revealing the presence of periodic, quasiperiodic, and chaotic responses in specific regimes. This research provides an in-deep understanding of the dynamic behavior of SBG system, contributing to the characterization and prediction of nonlinear phenomena, which is vital for the optimization and design of gear mechanisms across various engineering applications.

Strong stationarity for a highly nonsmooth optimization problem with control constraints

<p style='text-indent:20px;'>This paper deals with a control constrained optimization problem governed by a nonsmooth elliptic PDE in the presence of a non-differentiable objective. The nonsmooth non-linearity in the state equation is locally Lipschitz continuous and directionally differentiable, while one of the nonsmooth terms appearing in the objective is convex. Since these mappings are not necessarily Gâteaux-differentiable, the application of standard adjoint calculus is excluded. Based on their limited differentiability properties, we derive a strong stationary optimality system, i.e., an optimality system which is equivalent to the purely primal optimality condition saying that the directional derivative of the reduced objective in feasible directions is nonnegative.</p>

On the subcritical period doubling of a non-smooth network system by incremental harmonic balance method

This paper presents an investigation on an unusual bifurcation, i.e., subcritical period doubling (PD), of the non-smooth network system of a multi-modular floating structure. It is cumbersome to deduce the incremental harmonic balance iteration scheme, as non-smooth nonlinearities are resulted from the connectors between adjacent modules. To solve this problem, a method of undetermined coefficient is implemented plus a difference scheme. Periodic solutions are obtained accurately and a series of PDs are tracked as the controlling parameter varies. Interestingly, an unusual case is found to the Period-2 (with 2 times the fundamental period) solution, that is, the Period-4 solution appearing at the bifurcation point grows as the controlling parameter decreases backward. According to this feature, this case is called as subcritical PD compared to a traditional one. At the bifurcation point, the Period-2 solution loses stability featured by the leading Floquet multiplier leaving the unit cycle at +1, which is just opposite to −1 in the traditional cases. Differently, the arising Period-4 solution is unstable and it finally gains stability through cyclic bifurcations.

Prescribed Performance Adaptive Control for a Nonlinear Aeroelastic System with Input Constraint

A constrained prescribed performance compensation controller is proposed for a nonlinear aeroelastic system in the presence of wind gust, system uncertainties, and input saturation. To deal with the effects of the nonsmooth saturation nonlinearity, an approximate saturation function is introduced into the controller design, which can smoothly approximate the real saturation with arbitrarily prescribed precision. Specifically, by designing the prescribed performance function, a fixed-time control framework is designed to ensure that the closed-loop system has the prescribed tracking performance. The designed control algorithm can not only compensate the adverse effect caused by disturbances and uncertainties but also restrain the excessive amplitude of control input. Finally, the stability analysis shows that all the signals in the closed-loop system are semiglobally uniformly ultimately bounded via the Lyapunov stability analysis method, and simulation results are presented to demonstrate the feasibility and effectiveness of the proposed method.

Numerical investigation into discontinuity-induced bifurcations in an aeroelastic system with coupled non-smooth nonlinearities

Numerical investigation into discontinuity-induced bifurcations in an aeroelastic system with coupled non-smooth nonlinearities

Detection of dead cores for reaction-diffusion equations with a non-smooth nonlinearity

This paper studies a class of nonlinear elliptic PDEs, arising for stationary reaction-diffusion models. The non-smooth nonlinearity gives rise to dead cores, that is, subdomains where the solution of the PDE vanishes. The paper gives a solid foundation for the numerical solution of the problem, including proper extensions of known maximum–minimum principles, Céa lemma, and convergence estimation of the FEM for locally Hölder continuous operators. Based on these, we finally detect dead cores numerically in various typical situations.