Nonlinear Oscillators Research Articles

Approximate analytical solutions to nonlinear oscillations via semi-analytical method

A modified algebraic method is proposed in this study as an effective semi-analytical method for obtaining analytical solutions for two types of nonlinear differential equations. The trial solution is assumed to be a polynomial, and the algebraic approach is used to find the unknown coefficients. To enhance the solution, first, we apply the Laplace transformation on the series solution and then create the Padé approximations of the resultant equation. Finally, for the nonlinear problems under discussion, the inverse Laplace transformation is used to obtain periodic solutions.

Study on the ultrasonic cavitation damage to early atherosclerotic plaque

Ultrasonic cavitation can damage surrounding material and be used for destruction of the target tissue. In this paper, we investigated the interaction between atherosclerotic plaque (AP) and cavitation bubbles to determine whether the mechanical effect of cavitation damage could be potentially useful in therapy for treating atherosclerotic plaques. A two-bubble–fluid–solid model was established to study the dynamic behavior of bubbles near the AP and the AP damage by ultrasound-induced cavitation. A low-intensity focused ultrasound (LIFU) transducer was used for testing cavitation-based AP damage. We found that the nonlinear oscillation of bubbles causes the relative positions of the bubbles to shift, either toward or away from one another, these phenomena lead to changes in the bond failure rate between the fiber bundles, and the value of BRF exhibits an upward trend, this is the reason why the fibers suffered from reversible stretching and compressing. However, the AP damage is irreversible and diminishes as the number of cycles in the ultrasonic burst. It appears that the bigger the radii, regardless of whether the bubble (3 − i)’s and bubble i's radii are equal, the greater the AP damage. Ultrasonic cavitation therapy may not be appropriate for advanced AP patients, and the calcified tissue has a greater impact on the stability of the plaque. The damage area should be strictly selected. Additionally, the tissue damage phenomenon was found in experimental results. This work shows that the severity of AP damage is correlated with acoustic parameters and the surrounding environment from both simulation and experimental perspectives. The results show that ultrasonic cavitation may provide a new choice for the treatment of AP.

Elementary examples of renormalization from mechanics and integral calculus

Two classical mechanical systems—block-and-tackle machines and nonlinear oscillators—exhibit renormalization of numerical parameters. A simple one-dimensional integral illustrates several other facets of renormalization (regularization, renormalization group, resummation). These examples provide insight for students and non-specialists into the workings of a subtle, complicated and historically misunderstood subject.

Nonlinear oscillation of microscale fiber-reinforced composite laminated beams under a thermal loading

The work analyzes the nonlinear dynamic response of fiber-reinforced composite laminated microbeams in thermal environment, accounting for the dependence of the material properties on temperature. The governing equations of the nonlinear partial differential equations are based on the Euler-Bernoulli beam theory, and the modified couple stress theory involving the von Kármán geometrical nonlinearity. The governing nonlinear equations are reduced to a single equation by neglecting the axial inertia, which is discretized in the form of nonlinear ordinary differential equation, according to the Galerkin method, and it is solved analytically using the multiple time scales (MTS) method. A systematic investigation checks for the effect of the material length scale parameter, temperature increase, and damping parameter of the microscale beam on its time history and phase plane trajectory. Different stacking sequences are also considered within the parametric investigation. Based on results, it is noticed that the system is in a damped period-doubling bifurcation for four cycle oscillation states. It is also noticed that the stacking sequence affects significantly the nonlinear dynamic behavior of the system. It is found that the microscale composite laminated beam with [0/0/0] lay-ups has the highest dynamical deflection and the highest nonlinear frequency, followed by [0/90/0], [30/-30/30] (angle ply), [90/0/90], [45/-45/45] (angle ply) and [60/-60/60] (angle ply). However, the microscale composite laminated beam with [90/90/90] lay-ups has the lowest dynamical deflection and the lowest nonlinear frequency.

Haque’s approach with mickens’ iteration method to find a modified analytical solution of nonlinear jerk oscillator containing displacement time velocity and time acceleration

AbstractHaque’s approach with Mickens’ iteration method has been used to obtain the modified analytical solutions of the nonlinear jerk oscillator, including displacement time velocity and acceleration. The jerk oscillator represents the features of chaotic behavior in numerous nonlinear phenomena, cosmological analysis, kinematical physics, pendulum analysis, etc., such as electrical circuits, laser physics, mechanical oscillators, damped harmonic oscillators, and biological systems. In this paper, we have used different harmonic terms for different iterative stages using the truncated Fourier series. A comparison is made between the iteration method, the improved harmonic balance method, and the homotopy perturbation method. After comparison, the suggested approach has been shown to be more precise, efficient, simple, and easy to use. Furthermore, there was remarkable accuracy in the comparison between the numerical results and the generated analytical solutions. For the third approximate period, the maximum percentage error is 0.014.

Two-dimensional Coupled Asymmetric van der Pol Oscillator

A reliable algorithm based on an adaptation of the standard differential transform method (DTM) is presented, which is the multi-step differential transform method (Ms-DTM) since it may be difficult to directly apply differential transform method (DTM) to obtain the series solutions for the present two- dimensional coupled asymmetric van der Pol oscillator. The solutions of a two dimensional coupled asymmetric van der Pol oscillator were obtained by Ms-DTM. Figurative comparisons between the Ms-DTM and the classical fourth order Runge-Kutta method (RK4) are given. The obtained results reveal that the proposed technique is a promising tool to solve the considered van der Pol oscillator and yield same information on the phase portrait confirming the stability of the system, effectively. It can be said that the considered approach can be easily extended to other nonlinear van der Pol oscillator systems and therefore is widely applicable in engineering and other sciences.

Phase control of escapes in the fractional damped Helmholtz oscillator

We analyze the nonlinear Helmholtz oscillator in the presence of fractional damping, a characteristic feature in several physical situations. In our specific scenario, as well as in the non-fractional case, for large enough excitation amplitudes, all initial conditions are escaping from the potential well. To address this, we incorporate the phase control technique into a parametric term, a feature commonly encountered in real-world situations. In the non-fractional case it has been shown that, a phase difference of ϕOPT≈π, is the optimal value to avoid the escapes of the particles from the potential well. Here, our investigation focuses on understanding when particles escape, considering both the phase difference ϕ and the fractional parameter α as control parameters. Our findings unveil the robustness of phase control, as evidenced by the consistent oscillation of the optimal ϕ value around its non-fractional counterpart when varying the fractional parameter. Additionally, our results underscore the pivotal role of the fractional parameter in governing the proportion of bounded particles, even when utilizing the optimal phase.

High-power and high-speed all-optical all-fiber wavelength-swept source at 2 μm based on a femtosecond laser and time stretch technique

We present an all-fiber all-optical high-speed wavelength-swept laser source at 2 μm with a 3-dB bandwidth of 33 nm and repetition rate of 18.4 MHz, by using optical time-stretch technology, which to our knowledge is the first report of a 2 μm wavelength-swept source without any bulk, free-space optical elements. The light source is built upon a nonlinear polarization rotation mode-locked femtosecond oscillator. A chirped fiber Bragg grating (CFBG) is employed to realize the all-optical wavelength sweeping. In comparison with the commonly used dispersive fibers, the CFBG possesses outstanding power efficiency and temporal stability, especially for wavelength sweeping in the 2.0 μm wavelength regime. A two-stage fiber amplifier system is utilized to boost the output power up to 1.01 W. This all-fiberized mode-locked swept source with high speed and high stability shows great potential as a valuable source for various spectroscopy applications and swept source optical coherence tomography, particularly for highly scattering medium with low water content.

Cluster formation due to repulsive spanning trees in attractively coupled networks.

Ensembles of coupled nonlinear oscillators are a popular paradigm and an ideal benchmark for analyzing complex collective behaviors. The onset of cluster synchronization is found to be at the core of various technological and biological processes. The current literature has investigated cluster synchronization by focusing mostly on the case of attractive coupling among the oscillators. However, the case of two coexisting competing interactions is of practical interest due to their relevance in diverse natural settings, including neuronal networks consisting of excitatory and inhibitory neurons, the coevolving social model with voters of opposite opinions, and ecological plant communities with both facilitation and competition, to name a few. In the present article, we investigate the impact of repulsive spanning trees on cluster formation within a connected network of attractively coupled limit-cycle oscillators. We successfully predict which nodes belong to each cluster and the emergent frustration of the connected networks independent of the particular local dynamics at the network nodes. We also determine local asymptotic stability of the cluster states using an approach based on the formulation of a master stability function. We additionally validate the emergence of solitary states and antisynchronization for some specific choices of spanning trees and networks.

Transient gap resonance between two closely-spaced boxes triggered by nonlinear focused wave groups

A two-dimensional (2D) viscous numerical wave flume is developed based on OpenFOAM® to investigate the transient gap resonance problem triggered by nonlinear focused wave groups. The narrow gap is formed by two closely-spaced boxes, where the weather-side box is allowed to heave freely under wave actions, but the rear one is firmly fixed. The overset mesh method is introduced in the numerical model to capture the weather-side box's large-amplitude motion accurately. This work mainly focuses on the coupled effects of free-surface nonlinearity and heave-freedom of the weather-side box on the fluid oscillations in the gap and heave motions of the weather-side box. The heave-freedom of the weather-side box is demonstrated to play an important effect on the period in which the maximum value of non-dimensional amplitude of fluid oscillation in the gap occurs. It is revealed that the free-surface nonlinearity significantly affects the nonlinear fluid oscillations in the gap but has a limited effect on the heave motions of the weather-side box. The four-phase combination method is further employed to conduct the higher harmonic component analysis to clarify the effects of higher harmonic components on the nonlinear response of fluid and box.

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.

A Note on a Simple Equation for Solving Nonlinear Undamped Oscillations

A Note on a Simple Equation for Solving Nonlinear Undamped Oscillations

A study of controlling the transverse vibration of a beam-plate system by utilizing a nonlinear coupling oscillator

In the engineering field, prolonged exposure of complex structures to high-strength vibration environments can lead to various engineering challenges. This underscores the importance of effectively managing the vibration of complex structures. Given the extensive utilization of beams and plates in the construction of intricate structures, this study incorporates a nonlinear coupling oscillator into the beam-plate system and formulates a vibration analysis model for this system. The governing equations of the system are derived theoretically and solved numerically using the Galerkin truncation method. This study focuses on the operational modes of the nonlinear coupling oscillator based on accurate numerical results. Meanwhile, the influence of parameters belonging to the nonlinear coupling oscillator on the transverse vibration responses of the beam-plate system is systemically studied. Upon meticulous examination of the numerical results, the operational states of the nonlinear coupling oscillator encompass the normal vibration suppression mode and the quasi-periodic vibration suppression mode. The alteration in the linear elastic coupling stiffness of the beam-plate system impacts the effectiveness of vibration suppression in the nonlinear coupling oscillator. This influence stems from the fact that changes in the linear elastic coupling stiffness directly affect the nonlinear forces exerted on the beam-plate system within a nonlinear coupling oscillator. Within a feasible range, the nonlinear stiffness and viscous damping can be chosen as the adjustable parameters for the nonlinear coupling oscillator to regulate the vibration of the beam-plate system by adjusting the motion mass of the nonlinear coupling oscillator, which presents a challenge. The appropriate utilization of the nonlinear coupling oscillator demonstrates favorable control effectiveness in managing the transverse vibration of the beam-plate system. The study presented in this work offers a method to simultaneously control the vibration of each component within beam-plate coupling systems.

Stochastic dynamics of mechanical systems with impacts via the Step Matrix multiplication based Path Integration method

AbstractIn this work we propose the Step Matrix Multiplication based Path Integration method (SMM-PI) for nonlinear vibro-impact oscillator systems. This method allows the efficient and accurate deterministic computation of the time-dependent response probability density function by transforming the corresponding Chapman–Kolmogorov equation to a matrix–vector multiplication using high-order numerical time-stepping and interpolation methods. Additionally, the SMM-PI approach yields the computation of the joint probability distribution for response and impact velocity, as well as the time between impacts and other important characteristics. The method is applied to a nonlinear oscillator with a pair of impact barriers, and to a linear oscillator with a single barrier, providing relevant densities and analysing energy accumulation and absorption properties. We validate the results with the help of stochastic Monte-Carlo simulations and show the superior ability of the introduced formulation to compute accurate response statistics.

Nonlinear analysis of the synchronous reference frame phase-locked loop under unbalanced grid voltage

The synchronous reference frame phase-locked loop (SRF-PLL), also called dqPLL, is an electric circuit commonly used in power electronics to estimate the phase angle of a three-phase AC grid. If the voltage is unbalanced, the PLL is modeled as a periodically forced nonlinear oscillator and is known in practice to converge to a steady oscillation. In the existing literature, the oscillation has been studied via linearization assuming a low level of unbalance. Aiming for stronger nonlocal statements, we present nonlinear analysis. We apply the method of autonomous comparison systems and incremental stability to show that the steady oscillation is unique and attractive in a wider neighborhood. Its lock-in domain is estimated using numerical phase portrait analysis. The oscillation is further approximated up to the terms of the second order in the unbalance factor—it yields an estimation of the time average of the PLL’s phase error which is not visible by linearization only. The results provide stability guarantees and can guide the tuning of SRF-PLL.

Terahertz electric-field-driven dynamical multiferroicity in SrTiO3

The emergence of collective order in matter is among the most fundamental and intriguing phenomena in physics. In recent years, the dynamical control and creation of novel ordered states of matter not accessible in thermodynamic equilibrium is receiving much attention1–6. The theoretical concept of dynamical multiferroicity has been introduced to describe the emergence of magnetization due to time-dependent electric polarization in non-ferromagnetic materials7,8. In simple terms, the coherent rotating motion of the ions in a crystal induces a magnetic moment along the axis of rotation. Here we provide experimental evidence of room-temperature magnetization in the archetypal paraelectric perovskite SrTiO3 due to this mechanism. We resonantly drive the infrared-active soft phonon mode with an intense circularly polarized terahertz electric field and detect the time-resolved magneto-optical Kerr effect. A simple model, which includes two coupled nonlinear oscillators whose forces and couplings are derived with ab initio calculations using self-consistent phonon theory at a finite temperature9, reproduces qualitatively our experimental observations. A quantitatively correct magnitude was obtained for the effect by also considering the phonon analogue of the reciprocal of the Einstein–de Haas effect, which is also called the Barnett effect, in which the total angular momentum from the phonon order is transferred to the electronic one. Our findings show a new path for the control of magnetism, for example, for ultrafast magnetic switches, by coherently controlling the lattice vibrations with light.

A review of the frequency-amplitude formula for nonlinear oscillators and its advancements

The frequency-amplitude relationship is pivotal in understanding oscillatory systems, dictating their dynamic behaviors and responses. This paper delves into the intricacies of the frequency amplitude formula, elucidating its foundational role in both linear and nonlinear oscillations. Through comprehensive analysis, we highlight the formula’s significance in predicting system behaviors, especially in environments with varying amplitudes. Our findings underscore the formula’s potential as a robust tool for enhanced system characterization, offering profound implications for diverse applications across scientific and engineering domains. This study delves deep into the formulation of the frequency-amplitude relationship, extending its application beyond the conventional cubic powers of the restoring force. We introduce three novel and equivalent formulations of the generalized frequency amplitude, breaking traditional boundaries by not confining the restoring force to just odd functions. The formula to determine the frequency of the singular oscillator has been set forth. Our approach, characterized by its simplicity, offers a robust tool for analyzing high nonlinearity vibrations, ushering in a richer, multidimensional perspective to vibration analysis.

Exploiting multi-stiffness combination inspired absorbers for simultaneous energy harvesting and vibration mitigation

In scenarios where wide-amplitude shock excitation is encountered, a routine bistable configuration struggles to achieve satisfactory energy transfer, and a sensible combination of multiple nonlinear oscillators with different stiffness mechanisms can be exploited to achieve optimal vibration mitigation performance. In this paper, a nonlinear energy conversion and multi-stiffness combination inspired dynamic vibration absorber (NECMC-DVA) is proposed with a magnetic-spring tunable nonlinear stiffness element as the substructure. Parametric studies are performed for different modes of magnet-spring tunable nonlinear stiffness unit arrays considering transient impulses and harmonic excitation. The NECMC-DVA device, which comprises bistable oscillators, cubic stiffness oscillators, and hybrid stiffness oscillators, is also experimentally demonstrated for its synergistic performance in vibration mitigation and energy harvesting. The performance of the NECMC-DVA is measured under transient impulses and harmonic excitations. It has been found that the NECMC-DVA, which consists of a bistable oscillator, a cubic stiffness oscillator, a hybrid stiffness oscillator, energy-capture elements, and motor sequences, achieves high-efficiency nonlinear energy transfer and significant shock robustness in wide-amplitude impulsive scenarios. The NECMC-DVA with hybrid energy sinks demonstrates more remarkable targeted energy transfer and energy conversion performance than its pure bistable oscillator counterpart. Furthermore, the NECMC-DVA prototype can achieve adaptive vibration control modulation and energy conversion management through motion control boards by monitoring ambient excitation signals. Therefore, the NECMC-DVA enables integrated active and passive mitigation and energy harvesting to simultaneously accommodate multi-source indeterminate vibration and shock environments.

Use of Mathematical Methods in Oncology

Solitons are non-linear oscillations that exist in a number of objects, including those in DNA. They appear there because histone tails oscillate with great frequency. Solitons arise as a result of the rapid fluctuation of histone tails in the DNA of any biological object, be it an animal or a person. Up to a certain age, these fluctuations are useful for a given biological object, but with age they begin to harm it, which can lead, in particular, to cancer.

Experimental characterization of a nonlinear mechanical oscillator with softening behaviour for large displacements

This paper presents an experimental insight into the performance of a mechanical oscillator consisting of an X-shaped-spring configuration. This configuration achieves an overall softening characteristic with quasi-zero stiffness behaviour far away from the static equilibrium point. Such a geometrical nonlinear configuration has attracted significant research attention in the last few years, particularly for its application as a vibration isolator with the possibility to extend the quasi-zero-stiffness region beyond that of the classical three-spring nonlinear isolator. However, previous experimental evidence has been limited to small amplitude vibration excitation only. Furthermore, it has been focused mainly on the isolation region, rather than on the large amplitude response, thus circumventing an insight on the damping effects and its modelling. To address this gap, in this paper, both frequency sweeps and random excitations are applied to a prototype device for experimental characterization. A nonlinear stiffness model is developed based on the geometry of the system and a nonlinear damping model is assumed based on experimental observation. The proposed model accurately describes the dynamic behaviour of the system as shown by comparison of theoretical and experimental data.