Piecewise Linear Nonlinearity Research Articles

Towards digitally programmed nonlinear electroacoustic resonators for low amplitude sound pressure levels: Modeling and experiments

Passive acoustic foams have limitations in efficiently reducing low-frequency noise below 800 Hz due to the associated wavelengths for reasonable material thickness. To address this challenge, passive or semi-active resonators are commonly employed solutions. Linear resonators are very efficient within a narrow frequency bandwidth. However, nonlinear oscillators can broaden this bandwidth, but activation of their nonlinear responses typically requires high excitation amplitudes, beyond human hearing tolerance amplitudes. Furthermore, the specific type of nonlinearity in such devices is often predetermined by the inherent properties of the resonator. In this study, we employ a novel digital control algorithm, allowing to activate the nonlinear response of the electroacoustic resonator at low excitation amplitudes. This algorithm, which relies on real-time integration, facilitates the creation of nonlinear resonators featuring polynomial or diverse non-polynomial nonlinearities within the range of low amplitudes. The nonlinear control is carried out on a loudspeaker equipped with a microphone. Our research highlights the potential to create nonlinear resonators with different, versatile and programmable behaviors. Unprecedented non-polynomial nonlinear behaviors are experimentally exhibited, we consider cubic, piece-wise linear, and logarithmic nonlinearities. These behaviors are implemented and compared to a semi-analytic model to control an acoustic mode of a tube under conditions of low excitation amplitudes and frequencies.

Data-driven model order reduction for structures with piecewise linear nonlinearity using dynamic mode decomposition

Data-driven model order reduction for structures with piecewise linear nonlinearity using dynamic mode decomposition

Experimental observation of hidden Chua’s attractor

Eleven years have passed since the discovery of the first chaotic hidden attractor in Chua system with piecewise-linear nonlinearity. Experimental observation of such attractor is still a hard work as the attraction basin of such hidden attractor does not intersect with any system equilibrium point and is far away from the origin. The key technical difficulty is the accurate configuration of non-zero initial values for dynamic elements. In this Letter, offset control is applied to shift the attraction basin and mapping the non-zero initial values to zero ones. Consequently, the attraction basin is intersect with origin, which allows us to experimentally observe such hidden attractor via pre-discharge all dynamic elements.

Coexistence of Hidden Attractors in the Smooth Cubic Chua’s Circuit with Two Stable Equilibria

Since the invention of Chua’s circuit, numerous generalizations based on the substitution of the nonlinear function have been reported. One of the generalizations is obtained by substituting cubic nonlinearity for piece-wise linear (PWL) nonlinearity. Although hidden chaotic attractors with a PWL nonlinearity have been discovered in the classical Chua’s circuit, chaotic attractors with a smooth cubic nonlinearity have long been known as self-excited attractors. Through a systematically exhaustive computer search, this paper identifies coexisting hidden attractors in the smooth cubic Chua’s circuit. Either self-excited or coexisting hidden attractors are now possible in the smooth cubic Chua’s circuit with algebraically elegant values of both initial points and system parameters. The newly found coexisting attractors exhibit an inversion symmetry. Both initial points and system parameters are equally required to localize hidden attractors. Basins of attraction of individual equilibria are illustrated and clearly show critical areas of multistability where a tiny drift in an initial point potentially induces jumps among different basins of attraction and coexisting states. Such multistability poses potential threats to engineering applications. The dynamical regions of hidden and self-excited attractors, and areas of stability of equilibria, are illustrated against two parameter spaces. Both illustrations reveal that two nonzero equilibrium points of the smooth cubic Chua’s circuit have a transition from unstable to stable equilibrium points, leading to generations of self-excited and hidden attractors simultaneously.

Transition fronts and their universality classes.

Steadily moving transition (switching) fronts, associated with local transformation, symmetry breaking, or collapse, are among the most important dynamic coherent structures. The nonlinear mechanical waves of this type play a major role in many modern applications involving the transmission of mechanical information in systems ranging from crystal lattices and metamaterials to macroscopic civil engineering structures. While many different classes of such dynamic fronts are known, the interrelation between them remains obscure. Here we consider a minimal prototypical mechanical system, the Fermi-Pasta-Ulam (FPU) chain with piecewise linear nonlinearity, and show that there are exactly three distinct classes of switching fronts, which differ fundamentally in how (and whether) they produce and transport oscillations. The fact that all three types of fronts could be obtained as explicit Wiener-Hopf solutions of the same discrete FPU problem allows one to identify the exact mathematical origin of the particular features of each class. To make the underlying Hamiltonian dynamics analytically transparent, we construct a minimal quasicontinuum approximation of the FPU model that captures all three classes of the fronts and reveals interrelation between them. This approximation is of higher order than conventional ones (KdV, Boussinesq) and involves mixed space-time derivatives. The proposed framework unifies previous attempts to classify the mechanical transition fronts as radiative, dispersive, topological, or compressive and categorizes them instead as irreducible types of dynamic lattice defects.

Efficient Analysis of Stationary Dynamics of Piecewise-Linear Nonlinear Systems Modeled Using General State-Space Representations

AbstractIn this paper, a new technique is presented for parametrically studying the steady-state dynamics of piecewise-linear nonsmooth oscillators. This new method can be used as an efficient computational tool for analyzing the nonlinear behavior of dynamic systems with piecewise-linear nonlinearity. The new technique modifies and generalizes the bilinear amplitude approximation method, which was created for analyzing proportionally damped structural systems, to more general systems governed by state-space models; thus, the applicability of the method is expanded to many engineering disciplines. The new method utilizes the analytical solutions of the linear subsystems of the nonsmooth oscillators and uses a numerical optimization tool to construct the nonlinear periodic response of the oscillators. The method is validated both numerically and experimentally in this work. The proposed computational framework is demonstrated on a mechanical oscillator with contacting elements and an analog circuit with nonlinear resistance to show its broad applicability.

Data filtering-based multi-innovation forgetting gradient algorithms for input nonlinear FIR-MA systems with piecewise-linear characteristics

The piecewise-linear characteristics often appear in the nonlinear systems that operate in different ways in different input regions. This paper studies the identification issue of a class of block-oriented systems with piecewise-linear characteristics. The asymmetric piecewise-linear nonlinearity is expressed as a linear parametric representation through introducing an appropriate switching function, then the identification model of the system is derived by using the key term separation technique. On this model basis, a multi-innovation forgetting gradient algorithm is presented to estimate the unknown parameters. To further enhance the identification accuracy, the filtering identification model of the system is derived by changing the structure of the system without changing the relationship between the input and output. Further, a data filtering-based multi-innovation forgetting gradient algorithm is proposed through the use of the data filtering technique. A simulation example is employed to illustrate that the proposed approaches are effective for parameter estimation and the data filtering-based multi-innovation forgetting gradient algorithm has better estimation performance.

Modeling and simulating dynamics of lithium-ion batteries using block-oriented models with piecewise linear static nonlinearity

The article deals with the research of the efficiency of modelling the dynamics of voltage change in lithium-ion rechargeable batteries in charging/discharging modes using nonlinear block-oriented systems. Drawing on experimental data, a structural and parametric identification of the Hammerstein, Wiener and Hammerstein-Wiener models with a polynomial structure of the linear dynamic block and piecewise linear static nonlinearities was performed. It has been established that the best modelling accuracy was ensured by using the Hammerstein-Wiener system with a linear model having the 6th order of the numerator and denominator polynomials and an input delay of 3 samples. It showed 15.67% and 6.2% higher accuracy compared to the Wiener and Hammerstein systems, respectively. The application of those models in battery management systems will make it possible to improve the control quality for battery assemblies of solar and wind power plants in the context of the variable nature of the charging/discharging processes due to the variability of weather conditions and fluctuations in power consumption during a 24-hour period. This will ensure a wider introduction of renewable power generation into existing power systems, which is currently the leading way to ensure sustainable development of the energy sector.

Influence of friction and asymmetric freeplay on the limit cycle oscillation in aeroelastic system: An extended Hénon’s technique to temporal integration

Nonlinear effects such as friction and freeplay on the control surfaces can affect aeroelastic dynamics during flight. In particular, these nonlinearities can induce limit cycle oscillations (LCO), changing the system stability, and because of this it is essential to employ computational methods to predict this type of motion during the aircraft development cycle. In this context, the present article presents a matrix notation for describing the Hénon’s method used to reduce errors when considering piecewise linear nonlinearities in the numerical integration process. In addition, a new coordinate system is used to write the aeroelastic system of equations. The proposal defines a displacement vector with generalized and physical variables to simplify the computational implementation of the Hénon’s technique. Additionally, the article discusses the influence of asymmetric freeplay and friction on the LCO of an airfoil with control surface. The results show that the extended Hénon’s technique provides more accurate LCO predictions, that friction can change the frequency and amplitude of these motions, and the asymmetry of freeplay is important to determine the LCO behavior.

Method for controlling vibration by exploiting piecewise-linear nonlinearity in energy harvesters.

Vibration energy is becoming a significant alternative solution for energy generation. Recently, a great deal of research has been conducted on how to harvest energy from vibration sources ranging from ocean waves to human motion to microsystems. In this paper, a theoretical model of a piecewise-linear (PWL) nonlinear vibration harvester that has potential applications in variety of fields is proposed and numerically investigated. This new technique enables automatic frequency tunability in the energy harvester by controlling the gap size in the PWL oscillator so that it is able to adapt to changes in excitations. To optimize the performance of the proposed system, a control method combining the response prediction, signal measurement and gap adjustment mechanism is proposed in this paper. This new energy harvester not only overcomes the limitation of traditional linear energy harvesters that can only provide the maximum power generation efficiency over a narrow frequency range but also improves the performance of current nonlinear energy harvesters that are not as efficient as linear energy harvesters at resonance. The proposed system is demonstrated in several case studies to illustrate its effectiveness for a number of different excitations.

Nonlinear Energy Sinks With Piecewise-Linear Nonlinearities

Abstract An efficient nonlinear energy sink (NES) is developed here by employing a symmetric piecewise nonlinear coupling force. The proposed piecewise NES has a symmetric clearance about its equilibrium position in which zero stiffness is assumed. However, at the boundaries of the symmetric clearance, the NES is coupled to the linear structure by either linear or nonlinear stiffness elements. The damping is assumed to be continuous linear viscous damping during the oscillation of the NES mass inside and outside the clearance zone. This design is further modified by incorporating a negative coupling stiffness within the clearance zone. Therefore, the performance of the proposed piecewise NESs in rapid vibration suppression is numerically investigated with two-degree-of-freedom spring-mass system and compared with existing NESs in the literature. Accordingly, significant improvement in vibration suppression has been achieved by the proposed piecewise NES compared with other types of existing NESs. Moreover, the numerical simulation results have shown more robustness in the piecewise NES performance for a wide range of stiffness and damping variations than the linear absorber and other types of NESs in the literature.

Statistical Analysis of the Nonlinear Response of Bladed Disks with Mistuning and Cracks

In this paper, efficient techniques are presented for analyzing the dynamics of mistuned bladed disks with cracks. The analysis of the influence of cracks, coupled with the influence of mistuning on the dynamics of bladed disks, is computationally challenging for several reasons: 1) complex geometry of modern turbines results in very high-dimensional computational models; 2) mistuning in these structures breaks the cyclic symmetry in these systems; and 3) cracks further disrupt the symmetry and introduce a piecewise-linear nonlinearity into these systems. Recently, several approaches have been developed to handle these challenges individually. The component mode mistuning approach was developed to efficiently model small mistuning in bladed disks. The method was developed for generating reduced-order models of cyclically symmetric systems with cracks. More recently, the generalized bilinear amplitude approximation technique was created to efficiently approximate the nonlinear vibrational response of a class of piecewise-linear nonlinear systems. This paper modifies and combines these techniques, for the first time, to enable efficient modeling and statistical analysis of bladed disks with mistuning and cracks. The novel method is able to generate the reduced-order model of full-bladed disks using only single-sector models and approximate the nonlinear vibrational response of the system with significantly reduced computational effort. A high-dimensional finite element model of a mistuned bladed disk with a crack is studied using the proposed approach. The influence of mistuning patterns and cracks on the vibrational response of the bladed disk is discussed.

Analyzing Bilinear Systems Using a New Hybrid Symbolic-Numeric Computational Method

In this paper, an efficient and accurate computational method for determining responses of high-dimensional bilinear systems is developed. Predicting the dynamics of bilinear systems is computationally challenging since the piecewise-linear nonlinearity induced by contact eliminates the use of efficient linear analysis techniques. The new method, which is referred to as the hybrid symbolic-numeric computational (HSNC) method, is based on the idea that the entire nonlinear response of a bilinear system can be constructed by combining linear responses in each time interval where the system behaves linearly. The linear response in each time interval can be symbolically expressed in terms of the initial conditions. The transition time where the system switches from one linear state to the other and the displacement and velocity at the instant of transition are solved using a numerical scheme. The entire nonlinear response can then be obtained by joining each piece of the linear response together at the transition time points. The HSNC method is based on using linear features to obtain large computational savings. Both the transient and steady-state response of bilinear systems can be computed using the HSNC method. Thus, nonlinear characteristics, such as subharmonic motion, bifurcation, chaos, and multistability, can be efficiently analyzed using the HSNC method. The HSNC method is demonstrated on a single degree-of-freedom (DOF) system and a cracked cantilever beam model, and the nonlinear characteristics of these systems are examined.

Response of an oscillatory differential delay equation to a periodic stimulus.

Periodic hematological diseases such as cyclical neutropenia or cyclical thrombocytopenia, with their characteristic oscillations of circulating neutrophils or platelets, may pose grave problems for patients. Likewise, periodically administered chemotherapy has the unintended side effect of establishing periodic fluctuations in circulating white cells, red cell precursors and/or platelets. These fluctuations, either spontaneous or induced, often have serious consequences for the patient (e.g. neutropenia, anemia, or thrombocytopenia respectively) which exogenously administered cytokines can partially correct. The question of when and how to administer these drugs is a difficult one for clinicians and not easily answered. In this paper we use a simple model consisting of a delay differential equation with a piecewise linear nonlinearity, that has a periodic solution, to model the effect of a periodic disease or periodic chemotherapy. We then examine the response of this toy model to both single and periodic perturbations, meant to mimic the drug administration, as a function of the drug dose and the duration and frequency of its administration to best determine how to avoid side effects.

Fronts in subdiffusive FitzHugh–Nagumo systems

Front solutions of subdiffusive FitzHugh–Nagumo equations are studied for a piece-wise linear nonlinearity. Multiple solutions of the problem and the dependence of the propagation velocity on the parameters are discussed.

Generalized Bilinear Amplitude Approximation and X-Xr for Modeling Cyclically Symmetric Structures With Cracks

The analysis of the influence of cracks on the dynamics of bladed disks is critical for design, failure prognosis, and structural health monitoring. Predicting the dynamics of cracked bladed disks is computationally challenging for two reasons: (1) the model size is quite large and (2) the piecewise-linear nonlinearity caused by contact eliminates the use of linear analysis tools. Recently, a technique referred to as the X-Xr approach was developed to efficiently reduce the model size of the cracked bladed disks. The method employs relative coordinates to describe the motion of crack surfaces such that an effective model reduction can be achieved using single sector calculations. More recently, a method referred to as the generalized bilinear amplitude approximation (BAA) was developed to approximate the amplitude and frequency of piecewise-linear nonlinear systems. This paper modifies the generalized BAA method and combines it with the X-Xr approach to efficiently predict the dynamics of the cracked bladed disks. The combined method is able to construct the reduced-order model (ROM) of full disks using single-sector models only and estimate the amplitude and frequency with a significantly reduced computational effort. The proposed approach is demonstrated on a three degrees-of-freedom (DOF) spring–mass system and a cracked bladed disk.

Streaming electrification in the swinging plate system

The article presents the results of measurements and analysis of the flow electrification process. The research is presented with a new type of measurement instrument, in which one of the electrodes moves in an oscillating manner. The arrangement comprises two parallel plates and the liquid is relocated in one dimension. One of the electrodes is in the form of a copper wire. As a result, the liquid distribution is assumed as very simple and similar to the classic Stokes boundary layer. The electrification current measurements for different oscillation frequencies were performed in such an arrangement. The basic characteristics of the measured current and the determined process models were tested. The nonlinear approach and the linear approach were used. In the case of the nonlinear mode, the displacement of electrode was employed as an input. Hammerstein's mode with a piecewise linear input nonlinearity estimator was used for identification. In the case of the linear mode, the calculated velocity magnitude was used as an input into the system. The discrete time Output Error model and the Transfer Function model were used as a linear model. In each case, very good compliance was obtained between the simulation responses and the actual current measurements.

Response of an oscillatory differential delay equation to a single stimulus

Here we analytically examine the response of a limit cycle solution to a simple differential delay equation to a single pulse perturbation of the piecewise linear nonlinearity. We construct the unperturbed limit cycle analytically, and are able to completely characterize the perturbed response to a pulse of positive amplitude and duration with onset at different points in the limit cycle. We determine the perturbed minima and maxima and period of the limit cycle and show how the pulse modifies these from the unperturbed case.

Identification of Nonlinear Cascade Systems with Noninvertible Piecewise Linear Input and Backlash Output Nonlinearities

Abstract The paper deals with the parameter identification of cascade nonlinear dynamic systems with noninvertible piecewise linear input nonlinearities and backlash output nonlinearities. Application of the key term separation principle provides special expressions for the corresponding nonlinear model description that are linear in parameters. A least squares based iterative technique allows estimation of all the model parameters based on measured input/output data. Simulation studies illustrate the feasibility of proposed identification method.

Dynamic Characteristics of Fluid-Conveying Pipes with Piecewise Linear Support

For the analysis of dynamic characteristics of fluid-conveying pipes with piecewise linear support, a fluid–structure coupling dynamic model based on the finite element method is proposed. A user-defined pipe element based on Euler–Bernoulli beam is developed for modeling the pipes, considering the dynamic flow conditions. A nonlinear spring element is utilized to model the clamp between the pipe and the base. The dynamic responses of the system are obtained through the direct time integration. The stiffness of the clamp support is investigated by the analytical method and the experimental method, in which it is found that the clamp stiffness is piecewise linear. For different pipe geometries the user-defined element model, analytical model and measurement data are compared. The results show high quality of the element developed in this paper. Finally, the dynamic characteristics of the pipe system with piecewise linear support subjected to base harmonic excitation are calculated and the effects of the system parameters on pipe behaviors have also been studied. As a consequence, the model proposed in this paper can represent the piecewise linear nonlinearity of the clamp support and be used conveniently to investigate the effects of the fluid–structure coupling on the system behaviors.