Steady State Periodic Response Research Articles

Bifurcation Diagrams of Nonlinear Oscillatory Dynamical Systems: A Brief Review in 1D, 2D and 3D.

We discuss 1D, 2D and 3D bifurcation diagrams of two nonlinear dynamical systems: an electric arc system having both chaotic and periodic steady-state responses and a cytosolic calcium system with both periodic/chaotic and constant steady-state outputs. The diagrams are mostly obtained by using the 0-1 test for chaos, but other types of diagrams are also mentioned; for example, typical 1D diagrams with local maxiumum values of oscillatory responses (periodic and chaotic), the entropy method and the largest Lyapunov exponent approach. Important features and properties of each of the three classes of diagrams with one, two and three varying parameters in the 1D, 2D and 3D cases, respectively, are presented and illustrated via certain diagrams of the K values, -1≤K≤1, from the 0-1 test and the sample entropy values SaEn>0. The K values close to 0 indicate periodic and quasi-periodic responses, while those close to 1 are for chaotic ones. The sample entropy 3D diagrams for an electric arc system are also provided to illustrate the variety of possible bifurcation diagrams available. We also provide a comparative study of the diagrams obtained using different methods with the goal of obtaining diagrams that appear similar (or close to each other) for the same dynamical system. Three examples of such comparisons are provided, each in the 1D, 2D and 3D cases. Additionally, this paper serves as a brief review of the many possible types of diagrams one can employ to identify and classify time-series obtained either as numerical solutions of models of nonlinear dynamical systems or recorded in a laboratory environment when a mathematical model is unknown. In the concluding section, we present a brief overview of the advantages and disadvantages of using the 1D, 2D and 3D diagrams. Several illustrative examples are included.

Harmonic balance for differential constitutive models under oscillatory shear

Harmonic balance (HB) is a popular Fourier–Galerkin method used in the analysis of nonlinear vibration problems where dynamical systems are subjected to periodic forcing. We adapt HB to find the periodic steady-state response of nonlinear differential constitutive models subjected to large-amplitude oscillatory shear flow. By incorporating the alternating-frequency-time scheme into HB, we develop a computer program called FLASH (acronym for Fast Large Amplitude Simulation using Harmonic balance), which makes it convenient to apply HB to any differential constitutive model. We validate FLASH by considering two representative constitutive models, viz., the exponential Phan-Thien–Tanner model and a nonlinear temporary network model. In terms of accuracy and speed, FLASH typically outperforms the conventional approach of solving initial value problems by numerical integration via time-stepping methods often by several orders of magnitude. Exceptions to this rule are sometimes encountered for materials that are strongly shear thinning or described by constitutive models with discontinuous derivatives. We discuss how FLASH can be conveniently extended for other nonlinear constitutive models, which opens up potential applications in model calibration and selection, and stability analysis.

The effect of instrument inertia on the initiation of oscillatory flow in stress controlled rheometry

In a recent paper [Hassager, J. Rheol. 64, 545–550 (2020)], Hassager performed an analysis of the start up of stress-controlled oscillatory flow based on the general theory of linear viscoelasticity. The analysis provided a theoretical basis for exploring the establishment of a steady strain offset that is inherent to stress controlled oscillatory rheometric protocols. However, the analysis neglected the impact of instrument inertia on the establishment of the steady periodic response. The inclusion of the inertia term in the framework is important since it (i) gives rise to inertio-elastic ringing and (ii) introduces an additional phase shift in the periodic part of the response. Herein, we modify the expressions to include an appropriate inertial contribution and demonstrate that the presence of the additional terms can have a substantial impact on the time scale required to attain the steady state periodic response. The analysis is then applied to an aqueous solution of wormlike micelles.

On a principle for mass sensing using self-excited template dynamics of coupled oscillators and root-finding algorithms

In this work, we analyze a mass sensing mechanism that relies on self-excited template dynamics of a simple network of two coupled oscillators, one linear and one nonlinear, to detect changes in the mass of the linear oscillator with tunable sensitivity. The resultant shift in the ratio of oscillation amplitudes is predicted using perturbation analysis and verified against numerical results obtained via parameter continuation. A physical realization of the mass sensing mechanism is proposed, in which the linear oscillator is represented by a periodically excited actuator/microcantilever assembly, the nonlinear oscillator is simulated in silico on a finite interval of time while driven by the periodic steady-state response of the linear oscillator, and a nonlinear root-finding algorithm is used to identify the input to the actuator that reproduces the sought, coupled, self-excited dynamics. The analysis shows that the sensor gain can be adjusted by appropriate tuning of the coupling stiffness without modifications to the physical components. Numerical results for two different actuator models demonstrate rapid convergence of the root-finding algorithm from a trivial initial solution guess over ranges of values of model parameters and added mass ratios.

Transient Analysis and Low Power Consumption of Oscillator using C-mos Technology at 90 nm

The design and analysis of an oscillator in 90 nm technology using the cadence virtuoso tool. This oscillator has less noise and power consumption. There is also evidence of oscillator periodic steady state response. The portable power supplies, such as solar cells or other devices, it is crucial that electronic devices and circuits that can work at low voltages for portable electronic applications.

Influence of Fiber Angle on the Steady State Periodic Response of Annular Sectorial Plates

Structural elements in the form of annular sectorial plates are widely used in aeronautical, biomedical, and marine engineering. When these components are exposed to dynamic load, large-amplitude vibrations occur. The vibration response analysis based on the linear strain-displacement relation typically yields a conservative estimate and can be used as a first approximation to the actual solution for thin structural elements. As a result, geometric nonlinearity must be incorporated for the efficient and fail-proof design of such elements. This paper shows non-linear and linear steady-state forced vibration responses of the annular sectorial plates. The governing equations of motion have been solved in the time domain by using a modified shooting method and an arc-length/pseudo-arc length continuation technique at the bifurcation point to obtain the complete response curve comprised of stable and unstable branches. This work investigates the effect of fibre angle on the non-linear steady-state forced vibration response of annular sectorial plates. The strain/stress fluctuation throughout the thickness of the laminated annular sectorial plate is determined to explain why hardening nonlinearity has increased. The cyclic fluctuation of the non-linear steady-state normal stress during a time period at the centre of the top and bottom surface is also provided in relation to the forcing frequency ratio of peak amplitude in the non-linear response. Due to the change in restoring forces, the frequency spectra reveal much increased harmonic involvement along with the fundamental harmonic for all fibre angles.

Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutions

Highly accurate simulations of complex phenomena governed by partial differential equations (PDEs) typically require intrusive methods and entail expensive computational costs, which might become prohibitive when approximating steady-state solutions of PDEs for multiple combinations of control parameters and initial conditions. Therefore, constructing efficient reduced order models (ROMs) that enable accurate but fast predictions, while retaining the dynamical characteristics of the physical phenomenon as parameters vary, is of paramount importance. In this work, a data-driven, non-intrusive framework which combines ROM construction with reduced dynamics identification, is presented. Starting from a limited amount of full order solutions, the proposed approach leverages autoencoder neural networks with parametric sparse identification of nonlinear dynamics (SINDy) to construct a low-dimensional dynamical model. This model can be queried to efficiently compute full-time solutions at new parameter instances, as well as directly fed to continuation algorithms. These aim at tracking the evolution of periodic steady-state responses as functions of system parameters, avoiding the computation of the transient phase, and allowing to detect instabilities and bifurcations. Featuring an explicit and parametrized modeling of the reduced dynamics, the proposed data-driven framework presents remarkable capabilities to generalize with respect to both time and parameters. Applications to structural mechanics and fluid dynamics problems illustrate the effectiveness and accuracy of the proposed method.

Efficient Hybrid Symbolic-Numeric Computational Method for Piecewise Linear Systems With Coulomb Friction

Abstract A wide range of mechanical systems have gaps, cracks, intermittent contact or other geometrical discontinuities while simultaneously experiencing Coulomb friction. A piecewise linear model with discontinuous force elements is discussed in this paper that has the capability to accurately emulate the behavior of such mechanical assemblies. The mathematical formulation of the model is standardized via a universal differential inclusion and its behavior, in different scenarios, is studied. In addition to the compatibility of the proposed model with numerous industrial systems, the model also bears significant scientific value since it can demonstrate a wide spectrum of motions, ranging from periodic to chaotic. Furthermore, it is demonstrated that this class of models can generate a rare type of motion, called weakly chaotic motion. After their detailed introduction and analysis, an efficient hybrid symbolic-numeric computational method is introduced that can accurately obtain the arbitrary response of this class of nonlinear models. The proposed method is capable of treating high dimensional systems and its proposition omits the need for utilizing model reduction techniques for a wide range of problems. In contrast to the existing literature focused on improving the computational performance when analyzing these systems when there is a periodic response, this method is able to capture transient and nonstationary dynamics and is not restricted to only steady-state periodic responses.

Semi-analytical solution to the steady-state periodic dynamic response of an infinite beam carrying a moving vehicle

Vehicle-induced vibration is among the most important aspects for riding comfort and structural durability. It is well known, however, that the span of structure is much larger than the size of vehicle, resulting in a huge computational cost for dynamic analysis, and meanwhile, the multi-degree-of-freedom (MDOF) vehicle and the nonlinear interaction between beam and vehicle are difficult to be involved in an explicit solution. In the present work, a joint harmonic balance and Fourier transform (HBFT) method is proposed for the steady-state periodic dynamic response of an infinite beam carrying a vehicle taking into account the surface roughness and the nonlinear interaction between beam and vehicle, and the semi-analytical steady-state periodic (SASP) solution is then obtained. First, the vehicle and beam are separated as two subsystems, where the infinite beam is supported by a viscoelastic foundation and the vehicle is described by a MDOF discrete model. The response of the beam is analytically solved from the Fourier transform approach and the response of the vehicle is explicitly formulated considering the Fourier expansion of the nonlinear interaction forces. Next, the Fourier coefficients for such interaction forces are numerically solved from the harmonic balance method based on the compatibility between the subsystems. By substituting the solved interaction forces back into the closed-form analytical solutions, the responses of both the subsystems are evaluated explicitly. This allows for accurate and efficient analysis for the steady-state periodic response of vehicle–beam system considering the infinite beam, the discrete vehicle and the nonlinear interaction between them. After validating the SASP solution by comparing with the assumed modal method, the effect from the vehicle speed on the response of the vehicle–beam system is discussed, indicating the existence of a few resonant speeds, having potential value for the optimal design of vehicles and road pavements. Apart from the present example, the SASP solution can be applied in broader cases and the proposed HBFT method is generally feasible to a variety of dynamic coupling problems consisting of a continuum and a discrete model.

Harmonic balance analysis of magnetically coupled two-degree-of-freedom bistable energy harvesters

Because a magnetically coupled two-degree-of-freedom bistable energy harvester (2-DOF MCBEH) shows the rich, complicated nonlinear behaviors caused by its coupled cubic nonlinearities, understanding the dynamics remains challenging. This paper reports and investigates the important nonlinear dynamical phenomena of the 2-DOF MCBEHs by performing the harmonic balance analysis (HBA). All periodic solution branches are identified in order to study and comprehend the complicated dynamics of the 2-DOF MCBEHs. This end requires care when truncating the harmonic balance solution. For a 1-DOF MCBEH, which is the conventional type, the fundamental harmonic is able to approximately describe the steady-state periodic response. However, high-order harmonics are significant for the 2-DOF MCBEH. This paper demonstrates that the harmonic balance solution should involve the high-order terms instead of using the oversimplified single-harmonic solution. By performing the proposed HBA, important solution branches are reported, and their dynamical behaviors are studied. Moreover, the complete architecture of the frequency response of the 2-DOF MCBEH is disclosed across the entire frequency range. The HBA also reveals the underlying physics of building a bridge between the first and second primary resonant areas under a strong excitation. In the future, the findings in the present report can be utilized in the design process of the 2-DOF MCBEHs.

Influence of geometric imperfections on the nonlinear forced vibration characteristics and stability of laminated angle-ply composite conical shells

The forced vibration characteristic of laminated composite truncated conical shell has been analyzed by including von Karman kinematic non-linearity. The geometric imperfection is included in the analysis by providing an initial doubly sinusoidal transverse displacement. The finite element analysis has been carried out using the displacement field of FSDT and the constrained strain terms are interpolated using field consistent modified shape functions in order to avoid membrane/transverse shear locking. The steady state periodic response has been obtained by solving the governing equations in time domain using modified shooting method and arc length/pseudo-arc length continuation techniques. The main aim of the paper is to explore the effects of initial imperfections, fiber-angle and curvature on the nonlinear periodic response characteristics. Further, the stability of the periodic response has also been investigated and the ensuing bifurcations resulting in loss/gain of stability has been discussed. The parameters affecting the softening/hardening nonlinear behavior of the conical shells have also been assessed. The nonlinear frequency response reveal greater amplitude during inward half cycle due to destabilizing nature of the nonlinear membrane stress resultants during inward motion at large amplitudes. Multiple stress reversals is observed during a periodic cycle which has large ramifications on the fatigue design.

COMPLEX HYBRID SYMBOLIC ANALYSIS OF NONLINEAR ANALOG CIRCUITS DRIVEN BY MULTI-TONE SIGNALS

The paper presents a general symbolic method for generating the complex hybrid matrix necessary for computing the periodic or nonperiodic steady-state response of a nonlinear analog circuit driven by multitone signals. This method is remarkable by its great efficiency and generality, and it is very useful in frequency-domain approach based on harmonic balance and least square approximation. For the general case of the nonperiodic steady-state response there are three basic methods: frequency-domain approach based on Voltera series; time-domain approach and a frequency-domain approach based on harmonic balance and least square approximation. The last one is significantly more efficient when the total number of nonlinear resistors, inductors and capacitors is significantly less than the total number of linear inductors and capacitors in the circuit, as is often the case in practice.

Modal reduction-based finite element method for nonlinear FG piezoelectric energy harvesters

In this study, harvesting energy from a nonlinear functionally graded (FG) piezoelectric cantilever beam under harmonic excitation is investigated. The material properties of the piezoelectric are assumed to be a combination of piezo-ceramic and piezo-polymer materials for high performance and flexibility. The neutral axis is obtained in order to eliminate bending–stretching coupling. The geometrical nonlinearity and electromechanical coupling are incorporated in the coupled nonlinear equations that are derived using the generalized Hamilton’s principle and solved using the combination of modal reduction and finite element methods. The shooting method is employed to obtain steady-state periodic response of an FG nonlinear harvester with appropriate initial conditions. Also it is shown that at least two-mode approximation is required for accurate estimation of nonlinear response and harvested power. Using the method of nonlinear modal reduction, the unstable branches for frequency domain solution are estimated and the computational time is reduced considerably compared to full finite element method. A case study is also accomplished in detail to analyze the effects of base amplitude values and material distribution on harvested power and bandwidth.

Near-resonant dynamics, period doubling and chaos of a 3-DOF vibro-impact system

A mechanical system composed of two weakly coupled oscillators under harmonic excitation is considered. Its main part is a vibro-impact unit composed of a linear oscillator with an internally colliding small block. This block is coupled with the secondary part being a damped linear oscillator. The mathematical model of the system has been presented in a non-dimensional form. The analytical studies are restricted to the case of a periodic steady-state motion with two symmetric impacts per cycle near 1:1 resonance. The multiple scales method combined with the sawtooth-function-based modelling of the non-smooth dynamics is employed. A conception of the stability analysis of the periodic motions suited for this theoretical approach is presented. The frequency–response curves and force–response curves with stable and unstable branches are determined, and the interplay between various model parameters is investigated. The theoretical predictions related to the motion amplitude and the range of stability of the periodic steady-state response are verified via a series of numerical experiments and computation of Lyapunov exponents. Finally, the limitations and extensibility of the approach are discussed.

Design of a broadband piezoelectric energy harvester with piecewise nonlinearity

In this study, a novel broadband piezoelectric vibration energy harvester (VEH) is constructed using piecewise nonlinearity. The VEH consists of a piezoelectric cantilever beam and a cylindrical cam-roller-spring structure that can produce negative stiffness and piecewise nonlinearity. The designed piecewise nonlinear force consists of two sections. The first section keeps the nonlinear force small, so that the VEH generates a high peak power, and the second section makes the nonlinear force increase sharply, so that the VEH has a wide bandwidth. Therefore, the designed VEH has higher peak power and wider bandwidth than those VEHs based on smooth magnetic nonlinearity or spring nonlinearity. Firstly, a dynamic model is developed and the steady-state periodic responses are obtained by the average method. Then, the influences of the structure parameters of the cam-roller-spring on the performance of the VEH are studied. The results show that compared with the linear system, the piecewise nonlinearity can increase the peak voltage from 28.9 to 64.6 V, increasing by 123%; it can increase the peak power from 0.84 to 4.17 mW, increasing by 396%; and it can increase the operating frequency range from 0.99 to 6.48 Hz, increasing by 554%. Finally, the energy harvesting performance of the designed VEH is compared with the widely used magnet-type and spring-type VEHs, in which the nonlinear forces are smooth. The results show that the proposed VEH has a 9.67% increase in peak power, a 79.2% increase in the frequency band, and a 4.2% increase in maximum efficiency than the magnet-type VEH; and has a 23.5% increase in peak power, a 5.01% increase in the frequency band, and an 8.8% increase in maximum efficiency than the spring-type VEH.

Integration of vibration control and energy harvesting for whole-spacecraft: Experiments and theory

Giant magnetostrictive material (GMM) is integrated into a nonlinear energy sink (NES) to construct a NES-GMM device for vibration control and energy harvesting. The NES-GMM device is experimentally embedded in a scaled model of a whole-spacecraft system. LMS Test.Lab software and an oscilloscope are used to collect vibration and energy signals of the whole-spacecraft system with and without the NES-GMM device under sine sweeping frequency excitations. The transmissibility amplitudes, resonant frequencies, and voltage values demonstrate the effeteness of vibration control and energy harvesting of the NES-GMM device under different various conditions. The experimental system is modeled as a two-degrees-of-freedom (2DOF) linear primary system with a nonlinear NES-GMM additional subsystem. The complexification-averaging (CX-A) method verified with numerical verifications is developed to investigate analytically the vibration reduction and energy harvesting of the NES-GMM device for the whole-spacecraft system. The experimental and theoretical results show that the NES-GMM device reduces the vibration and produces the electricity and that the device hardly changes the resonance frequencies of the original whole-spacecraft system. The saddle-node (SN) bifurcation is detected in the steady-state periodic response of the system. The performance of the NES-GMM device is examined for its varying parameters.

Study on dynamic characteristics of bio-inspired vibration isolation platform

Inspired by the body movement of the kangaroo, a multi-degree-of-freedom vibration isolation platform containing three units, that is, a protected object, a nonlinear energy sink, and an X-shaped structure, has been modeled, and the differential equations of the system have been given in the form of uniform relative coordinates. Furthermore, the displacement transmissibility analysis and numerical calculation are supported by the method of harmonic balance and Runge–Kutta algorithm, which shows that (a) there are nonlinear behaviors and resonant phenomenon in the time–frequency response and (b) quasi-periodic motion may be a predictor of periodic steady-state response or strong resonance, and the displacement evolution before quasi-periodic motion may be used to distinguish the two phenomena. In addition, based on the numerical method, the system energy changes in a selected frequency are discussed. Finally, the correctness of the theoretical analysis is verified by simulation data in Adams. Taken together, these results demonstrate that the dynamic characteristics are adjustable and designable of structural parameters in a specific frequency band and can provide a useful way to reduce the amplitude of resonant peaks and improve the vibration isolation performance for practical engineering applications.

Exploring 1:3 internal resonance for broadband piezoelectric energy harvesting

Recent literature has shown that the spectral characteristics of a harvester can be significantly improved by invoking internal resonance between the modes. Taking motivation from this premise, the present work analyzes the nonlinear dynamics and harvesting performance of a 1:3 internally resonant piezoelectric cantilever beam with a lumped mass under harmonic excitation. The governing modal equations are derived using Galerkin’s approach and Hamilton’s principle of least action. The method of multiple scales has been used to study the dynamics of the harvester in the proximity of primary resonances. The steady state periodic responses of the harvester are obtained using a pseudo-arc continuation technique and subsequently, their implications on the harvested power are discussed. The frequency responses of the harvested power are shown to boast significantly high magnitudes across a wide frequency band due to the energy transfer between the modes near the primary resonances. Results presented in the study show that the higher mode invoked through internal resonance plays a significant role in broadband power generation. The transfer of energy between the modes is found to occur only within a certain threshold of excitation level which in turn affects the magnitude of harvested power. Numerical simulations have also been carried out the results of which are compared against the analytical results. The outcomes of this first-hand analysis shall aid in the proposition of an efficient design for a broadband energy harvester.

Model Parameter Identification and Validation for Energy Saving Light Bulbs

Due to electrical energy reduction need, in the recent years, alternative light sources like Compact Fluorescent Lamps (CFL) and Light Emitting Diode lamps (LED) are replacing the classical incandescent light bulbs. Although they consume less power, these lamps have the disadvantages of being a significant source of harmonics. This means that the lightning network should be redesigned taking into account the magnitude of the current harmonic components and of the null current. To this aim, an accurate model easy to build and a fast analysis are very useful. This paper presents three models for different types of energy saving light bulbs, two existing ones and a new implementation, a procedure for model parameters identification and a comparison of the models. Some errors criteria are considered, both qualitative and quantitative. The periodic steady state response is computed using a frequency domain analysis.

Initialization of Electromagnetic Transient Simulation Using Differential Quadrature Method

ABSTRACT The initialization problem of electromagnetic transient simulation is to calculate the steady-state solution of the differential equations describing the electromagnetic transients of power systems. Currently, the widely used initialization method in electromagnetic transient programs (EMTP) is the fixed-point iteration or the so-called EMTP-based approach where the steady-state periodic response is found for a given initial state by simply integrating the system equations until the response becomes periodic. In lightly damped systems, this type of method could extend over many periods making the computation costly. In this paper, the initialization is described as a two-point differential boundary value problem. On this basis, the differential quadrature method (DQM) is used to solve this problem, therefore, a new electromagnetic transient initialization method is derived. The proposed initialization method is a rigorous Newton algorithm and thus has better convergence than the fixed-point iterative method. In order to solve the problem that the Jacobian matrix involved in the DQM solution is very large, a block recursive solution method based on V -transformation is proposed. Case studies conducted on two typical networks have confirmed the effectiveness of the proposed method.