Damped Nonlinear Oscillator Research Articles

Pipe roughness calibration approach for water distribution network models using a nonlinear state observer

This paper introduces an online approach based on a nonlinear state observer (NSO) to calibrate the roughness of each pipe within a water distribution network (WDN) or a sector thereof. The NSO is designed to obtain the estimations of pipes' friction factors, which are then used to estimate the roughness. The core of the NSO is a dynamic WDN model formulated through a structured set of ordinary differential equations derived from fundamental physical principles and taking advantage of both graph theory and rigid water column theory. By applying a coordinate transformation, the WDN model is represented as a fully connected network of damped nonlinear oscillators, with each oscillator formulated as a Liénard system. This representation allows for estimating the friction factors for each pipe using only flow rate information. The proposed approach facilitates a continuous calibration when hydrodynamic data are readily accessible, which is a capability that empowers engineers to enhance, concurrently or proactively, the day-to-day operations of water distribution networks, such as control or diagnose tasks, whenever data are available. The results of numerical simulations are presented to illustrate the practical utility of the proposed method.

KCC theory for a type of nonlinear damped SD oscillators

Smooth and discontinuous (SD) oscillators are the nonlinear models that will exhibit SD dynamics due to continuous variation of the smooth parameter. Kosambi–Cartan–Chern (KCC) theory is a differential geometric theory that describes the deviations from the entire trajectory of the variational equation to the nearby trajectory. This paper presents a completely new dynamical analysis of a type of SD oscillators with nonlinear damping based on the KCC theory. First, the paper gives five KCC geometrical invariants of SD oscillators in the smooth and non-smooth cases, respectively. The results show that the geometric quantities in the non-smooth case cannot be derived directly from the geometric quantities in the smooth case. Second, from these geometric quantities, KCC stability of SD oscillators trajectory at any point except [Formula: see text] is analyzed. Lastly, numerical results show that in some regions that deviate from the equilibrium point of system, the system will exhibit complex and variable dynamical behavior due to small changes in parameters. This paper shows that the KCC theory is also a useful tool in the dynamical analysis of non-smooth systems.

Contact acoustic nonlinearity and local damage resonance for the detection of kissing bonds in structural adhesive joints

Adhesively bonded joints are susceptible to contamination of surfaces during manufacture and environmental deterioration in real operating conditions. These may cause the generation of so-called “kissing bonds” that can dramatically alter the strength of the joint leading to premature failure. Nonlinear acousto-ultrasonic (AU) techniques have shown great potential for monitoring kissing bonds with piezoelectric sensors permanently installed on the structural joint, thus enabling online and in-situ inspection. This paper investigated the combined contact acoustic nonlinearity (CAN) and local damage resonance (LDR) effects in the presence of kissing bonds in adhesive joints. Experimental nonlinear AU tests showed the formation of LDR frequency down-shifts and “jumps” of the fundamental damage resonance in adhesively bonded aluminium joints, with the kissing bond located internally to the overlapping region between the two adherends. These results were supported by a theoretical model based on the solution of the nonlinear Duffing's equation, under the assumption that the debonded region is a damped nonlinear harmonic oscillator subject to harmonic forcing. The Harmonic Balance Method was used to solve the nonlinear differential problem, showing the generation of frequency down-shifts via the dependence of the ratio between the excitation and LDR frequencies with the amplitude of the fundamental damage resonance. Additionally, two-dimensional finite element simulations using a reduced order model based on the Craig-Bampton technique were carried out to support experimental AU tests for the identification of the LDR frequency and the generation of nonlinear resonance effects. Good agreement between analytical, numerical, and experimental results revealed that a monitoring approach combining CAN and LDR is an extremely efficient and sensitive tool for ensuring integrity and safety of structural adhesive joints.

A new modified Lindstedt–Poincare method for nonlinear damped forced oscillations

Nonlinear damped forced oscillators are very importance in the fields of mechanical, electrical and other dynamical systems. Small oscillations are being investigated by classical perturbations. Earlier Cheung et al. presented a modified Lindstedt–Poincare method for large amplitude of oscillation; but its mathematical manipulation is tremendously difficult. Moreover, it fails to determine the solutions when linear restoring force vanishes. In this article, a modification of the same method is made which greatly reduces mathematical calculations. On the other hand, it provides more accurate results and is used whether linear restoring force is present or not.

A modified Krylov–Bogoliubov–Mitropolskii method for solving damped nonlinear oscillators with large oscillation

A modification of the extended Krylov–Bogoliubov–Mitropolskii​ method (by Popov et al.) has been presented to investigate damping effects of strongly nonlinear oscillators when the amplitude of oscillation is large. The modification is similar to the Lindstedt-Poincare method, which was early presented by Cheung et al. for handling un-damped large oscillations. The determination of solution is quite different from earlier all techniques and it is valid whether linear restring force is present or not. Moreover, the solution approaches toward modified Lindstedt-Poincare’s solution when damped force vanishes.

Analysis of the Vibro-Impact Nonlinear Damped and Forced Oscillator in the Dynamics of the Electromagnetic Actuation

In this work, the effect of vibro-impact nonlinear, forced, and damped oscillator on the dynamics of the electromagnetic actuation (EA) near primary resonance is studied. The vibro-impact regime is given by the presence of the Hertzian contact. The EA is supplied by a constant current generating a static force and by an actuation generating a fast alternative force. The deformations between the solids in contact are supposed to be elastic and the contact is maintained. In this study, a single degree of freedom nonlinear damped oscillator under a static normal load is considered. An analytical approximate solution of this problem is obtained using the Optimal Auxiliary Functions Method (OAFM). By means of some auxiliary functions and introducing so-called convergence-control parameters, a very accurate approximate solution of the governing equation can be obtained. We need only the first iteration for this technique, applying a rigorous mathematical procedure in finding the optimal values of the convergence-control parameters. Local stability by means of the Routh-Hurwitz criteria and global stability using the Lyapunov function are also studied. It should be emphasized that the amplitude of AC excitation voltage is not considered much lower than bias voltage (in contrast to other studies). Also, the Hertzian contact coupled with EA is analytically studied for the first time in the present work. The approximate analytical solution is determined with a high accuracy on two domains. Local stability is established in five cases with some cases depending on the trace of the Jacobian matrix and of the discriminant of the characteristic equation. In the study of global stability, the estimate parameters which are components of the Lyapunov function are given in a closed form and a graphical form and therefore the Lyapunov function is well-determined.

Dynamical analysis of a damped harmonic forced duffing oscillator with time delay

This paper is concerned with a time-delayed controller of a damped nonlinear excited Duffing oscillator (DO). Since time-delayed techniques have recently been the focus of numerous studies, the topic of this investigation is quite contemporary. Therefore, time delays of position and velocity are utilized to reduce the nonlinear oscillation of the model under consideration. A much supplementary precise approximate solution is achieved using an advanced Homotopy perturbation method (HPM). The temporal variation of this solution is graphed for different amounts of the employed factors. The organization of the model is verified through a comparison between the plots of the estimated solution and the numerical one which is obtained utilizing the fourth order Runge–Kutta technique (RK4). The outcomes show that the improved HPM is appropriate for a variety of damped nonlinear oscillators since it minimizes the error of the solution while increasing the validation variety. Furthermore, it presents a potential model that deals with a diversity of nonlinear problems. The multiple scales homotopy technique is used to achieve an estimated formula for the suggested time-delayed structure. The controlling nonlinear algebraic equation for the amplitude oscillation at the steady state is gained. The effectiveness of the proposed controller, the time delays impact, controller gains, and feedback gains have been investigated. The realized outcomes show that the controller performance is influenced by the total of the product of the control and feedback gains, in addition to the time delays in the control loop. The analytical and numerical calculations reveal that for certain amounts of the control and feedback signal improvement, the suggested controller could completely reduce the system vibrations. The obtained outcomes are considered novel, in which the used methods are applied on the DO with time-delay. The increase of the time delay parameter leads to a stable case for the DO, which is in harmony with the influence of this parameter. This drawn curves show that the system reaches a stable fixed point which assert the presented discussion.

Approximate solutions of the fractional damped nonlinear oscillator subject to Van der Pol system

This paper deals with fractal Van der Pol damped nonlinear oscillators equation having nonlinearity. By combining the techniques of the Laplace transform and the variational iteration method, we establish approximate periodic solutions for the fractal damped nonlinear systems. In this simple way, nonlinear differential equations can be easily converted into linear differential equations. Illustrative examples including the Van der Pol damped nonlinear oscillator reveal that this method is very effective and convenient for solving fractal nonlinear differential equations. Finally, comparison of the obtained results with those of the other achieved method, also reveals that this coupling method not only suggests an easier method due to the Lagrange multiplier but also can be easily extended to other nonlinear systems.

A heuristic approach to the prediction of a periodic solution for a damping nonlinear oscillator with the non-perturbative technique

The present work attracts attention to obtaining a new result of the periodic solution of a damped nonlinear Duffing oscillator and a damped Klein–Gordon equation. It is known that the frequency response equation in the Duffing equation can be derived from the homotopy analysis method only in the absence of the damping force. We suggest a suitable new scheme successfully to produce a periodic solution without losing the damping coefficient. The novel strategy is centered on establishing an alternate equation apart from any difficulty in handling the influence of the linear damped term. This alternative equation was obtained with the rank upgrading technique. The periodic solution of the problem is presented using the non-perturbative method and validated by the modified homotopy perturbation technique. This technique is successful in obtaining new results toward a periodic solution, frequency equation, and the corresponding stability conditions. This methodology yields a more effective outcome of the damped nonlinear oscillators. With the help of this procedure, one can analyze many problems in the domain of physical engineering that involve oscillators and a linear damping influence. Moreover, this method can help all interested plasma authors for modeling different nonlinear acoustic oscillations in plasma.

Analytical and Numerical Approximations to Some Coupled Forced Damped Duffing Oscillators

In this investigation, two different models for two coupled asymmetrical oscillators, known as, coupled forced damped Duffing oscillators (FDDOs) are reported. The first model of coupled FDDOs consists of a nonlinear forced damped Duffing oscillator (FDDO) with a linear oscillator, while the second model is composed of two nonlinear FDDOs. The Krylov–Bogoliubov–Mitropolsky (KBM) method, is carried out for analyzing the coupled FDDOs for any model. To do that, the coupled FDDOs are reduced to a decoupled system of two individual FDDOs using a suitable linear transformation. After that, the KBM method is implemented to find some approximations for both unforced and forced damped Duffing oscillators (DDOs). Furthermore, the KBM analytical approximations are compared with the fourth-order Runge–Kutta (RK4) numerical approximations to check the accuracy of all obtained approximations. Moreover, the RK4 numerical approximations to both coupling and decoupling systems of FDDOs are compared with each other.

Estimated the frequencies of a coupled damped nonlinear oscillator with the non-Perturbative method

The current article is concerned with a comprehensive investigation of achieving the simplest solution of non-conservative coupled nonlinear forced oscillators. The article mainly depends on a non-perturbative method. It depends on yielding an equivalent linear system. The advantage of this linear system is that its coefficients are easily computed and that they include the effects of the original nonlinear coefficients. This linearization approach allows achieving quasi-exact solutions even in the presence of periodic forces. The relationships of frequencies with amplitudes are easily established. The analytical solution so yielded may serve as a basis for a qualitative understanding of the actual behavior of the coupled nonlinear oscillators. Additionally, numerical calculations are carried out graphically to address the validation of the new approach and to further illustrate the effectiveness and convenience of the method. The results are compared with the exact numerical solutions which show perfect accuracy. The method can be easily extended to other nonlinear systems and can therefore be exceedingly applicable in engineering and other sciences.

Nonlinear oscillations of a one-dimensional dielectric elastomer generator system

The nonlinear oscillations of a dielectric elastomer generator (DEG) consisting of an oscillator and two hyperelastic membranes are investigated in this paper. Due to the finite deformations of the membranes and the Maxwell stress induced by external electric fields, the DEG can be considered as a damped nonlinear oscillator. We then develop a theoretical model for the DEG system to describe the nonlinear oscillatory behavior, which can be used to elucidate the effects of the pre-stretch, membrane size, electric charge, and forcing amplitude on the DEG. From our theoretical analysis, it is found that the Maxwell stress plays a significant role in stabilizing the oscillations of the system. In practical applications, nonlinear elastic DE films exhibit strong oscillations possibly associated with a jumping phenomenon which may be harmful to the system. To describe the jumping phenomenon, an asymptotic solution on the primary resonance is derived based on the averaging method, and the obtained asymptotic solutions are further validated by the corresponding numerical results. The evolutions of the jumping phenomena with respect to the relevant factors are further analyzed. It is found that by tuning the membrane size, electric charge amount, or forcing amplitude, the jumping phenomenon can be avoided. Finally, the onsets of the jumping phenomenon are acquired from the graphic representation of the evolutions of the system resonance characteristics. It is anticipated that the current analysis could provide useful insights into designs of DE generators.

Parametrically Excited Vibrations in a Nonlinear Damped Triple-Well Oscillator with Resonant Frequency

Parametrically Excited Vibrations in a Nonlinear Damped Triple-Well Oscillator with Resonant Frequency

A quasi-periodic route to chaos in a parametrically driven nonlinear medium

Small-sized systems exhibit a finite number of routes to chaos. However, in extended systems, not all routes to complex spatiotemporal behavior have been fully explored. Starting from the sine-Gordon model of parametrically driven chain of damped nonlinear oscillators, we investigate a route to spatiotemporal chaos emerging from standing waves. The route from the stationary to the chaotic state proceeds through quasiperiodic dynamics. The standing wave undergoes the onset of oscillatory instability, which subsequently exhibits a different critical frequency, from which the complexity originates. A suitable amplitude equation, valid close to the parametric resonance, makes it possible to produce universe results. The respective phase-space structure and bifurcation diagrams are produced in a numerical form. We characterize the relevant dynamical regimes by means of the largest Lyapunov exponent, the power spectrum, and the evolution of the total intensity of the wave field.

The frequency estimation for non‐conservative nonlinear oscillation

AbstractImmediate frequency estimation of a damped nonlinear oscillator is very much needed in engineering. A developed He's frequency formula is derived to cover the damping nonlinear oscillator in the current work. The frequency due to the odd nonlinear damping force's presence is established immediately after converting the nonlinear oscillator to a linear damping model based on the conservative restoring force. A frequency of the non‐conservative system is performed. The analytical solution is compared with a numerical solution that shows the high accuracy of the estimated frequency.

Uniform global asymptotic stability for oscillators with nonlinear damping and nonlinear restoring terms

Second-order nonlinear differential equations with a time-varying coefficient describe various damped oscillators to explain the actual phenomena observed in numerous research areas. In actual phenomena, it is often important to immediately suppress the vibration of an object. For this purpose, the equilibrium of the oscillator needs to be uniformly globally asymptotically stable. This study examines uniform global asymptotic stability for a damped nonlinear oscillator. A first-order nonlinear differential equation, referred to as the characteristic equation, related to the second-order differential equation plays an important role in uniform global asymptotic stability for the oscillator. To be more precise, the property of particular solutions of this characteristic equation becomes a criterion for determining whether the equilibrium of the nonlinear oscillator is uniformly globally asymptotically stable. Careful mathematical analysis is used for the proof. In addition, a few examples are presented to show that the property of the particular solutions can be examined by drawing solution curves via a simple numerical simulation. Furthermore, detailed explanations are provided for the assumptions used to obtain the results. Our results clarify the appropriate damping coefficient required for the equilibrium to become uniformly globally asymptotically stable and have the advantage of being able to predict the speed at which a vibrating object returns to its original equilibrium state.

Dynamic features of a non-polarized Erbium-doped fiber ring laser subjected to external cavity-loss modulation

An experimental study of an Erbium-doped fiber ring laser subjected to external cavity-loss modulation using a loudspeaker as the modulator has been carried out and analyzed via modulation of the driving amplitude and frequency in the range of 1–20 kHz. The linear and non-linear (chaotic) dynamical behaviors have been observed from the bifurcation, time domain and frequency domain diagrams presented. The bifurcation diagram showed that the loss modulated laser is more sensitive to the acoustic wave generated from the loudspeaker when the driving amplitude is < 200 mV and < 1000 mV (< 16.5 mW and < 82.5 mW optical power). At driving amplitudes of 500 mV – 1000 mV (41.25 mW – 82.5 mW optical power), the ring laser system behaves like a damped non-linear oscillator. Optical bi-stability was also observed in the bifurcation diagrams near or exactly at the fundamental resonance frequency and first super harmonic resonance frequency. The availability of different control parameters during external cavity-loss modulation in an Erbium doped fiber laser makes the study of its non-linear response features easy in a very precise way. The control parameters considered in this paper are the driving amplitude and the modulation frequency.

Influence of dissipation on extreme oscillations of a forced anharmonic oscillator

Dynamics of a periodically forced anharmonic oscillator with cubic nonlinearity, linear damping, and nonlinear damping, is studied. To begin with, the authors examine the dynamics of an anharmonic oscillator with the preservation of parity symmetry. Due to this symmetric nature, the system has two neutrally stable elliptic equilibrium points in positive and negative potential-wells. Hence, the unforced system can exhibit both single-well and double-well periodic oscillations depending on the initial conditions. Next, the authors include position-dependent damping in the form of nonlinear damping (xẋ) into the system. Then, the parity symmetry of the system is broken instantly and the stability of the two elliptic points is altered to result in stable focus and unstable focus in the positive and negative potential-wells, respectively. Consequently, the system is dual-natured and is either non-dissipative or dissipative, depending on location in the phase space. The total energy of the system is used to explain this dual nature of the system. Furthermore, when one includes a periodic external forcing with suitable parameter values into the nonlinearly damped anharmonic oscillator system and starts to increase the damping strength, the parity symmetry of the system is not broken right away, but it occurs after the damping reaches a threshold value. As a result, the system undergoes a transition from double-well chaotic oscillations to single-well chaos mediated through a type of mixed-mode oscillations called extreme events (EEs) in which the small-amplitude single-well chaotic oscillations are interrupted by rare and recurrent large-amplitude (double-well) chaotic bursts. Furthermore, it is found that the large-amplitude oscillations developed in the system are completely eliminated if one incorporates linear damping into the system. Hence, it is believed that a novel means has been identified for controlling the EEs that occur in forced anharmonic oscillator system with nonlinear damping. The numerically calculated results are in good agreement with the theoretically obtained results on the basis of Melnikov’s function. Further, it is demonstrated that when one includes linear damping into the system, this system has a dissipative nature throughout the entire phase space of the system. This is believed to be the key to the elimination of EEs.

Application of variable- and distributed-order fractional operators to the dynamic analysis of nonlinear oscillators

Many physical processes in nature exhibit complex dynamics that result from a combination of multiscale, nonlinear, non-local, and memory effects. Recent experimental measurements conducted in a variety of physical domains have shown that, at the macroscale level, these effects typically result in significant deviations from the behavior predicted by classical models. Notably, the underlying dynamics was often shown to be of non-integer order and possibly better captured by fractional-order models. Fractional operators are intrinsically multiscale; thus, they provide a natural approach to account for non-local and memory effects. In this study, we present the possible application of variable-order (VO) and distributed-order (DO) fractional operators to a few classes of nonlinear lumped parameter models that have great practical relevance in mechanics and dynamics. More specifically, we present a methodology to define VO and DO fractional operators that are capable of capturing various physical transitions characteristic of contact dynamics, nonlinear reversible systems, hysteretic systems, and nonlinear damped oscillator systems. Despite using simplified lumped parameters models to illustrate the application of VO and DO operators to mechanics, we show numerical evidence of their unique modeling capabilities as well as their connection to more complex systems at the continuum scale. Further, for a selected problem involving distributed nonlinear damping, we provide approximate analytical solutions that are helpful to better understand the underlying dynamics and to quantify the accuracy of our numerical models.

Correction to: Vibrational resonance in a higher-order nonlinear damped oscillator with rough potential

In our recently published paper one of the author’s affiliation (O. O. Popoola) was wrongly indicated.