Damped Dynamic Systems Research Articles

Strong Convergence of Trajectories via Inertial Dynamics Combining Hessian-Driven Damping and Tikhonov Regularization for General Convex Minimizations

Let be a real Hilbert space, and be a convex twice differentiable function whose solution set is nonempty. We investigate the long time behavior of the trajectories of the vanishing damped dynamical system with Tikhonov regularizing term and Hessian-driven damping where are three positive constants, and the time scale parameter β is a positive nondecreasing function such that . Under some assumptions on the parameter , we will show rapid convergence of values, strong convergence toward the minimum norm element of , and rapid convergence of the gradients toward zero. Note that the Hessian-driven damping significantly reduces the oscillatory aspects, and the time scale parameter β improves the rate of convergences mentioned above. As particular cases of , we set , for , and , for and . The manuscript concludes with two numerical examples and comments on their performance.

Modeling and analysis of a piezoelectric transducer embedded in a nonlinear damped dynamical system

This paper focuses on the dynamical analysis of the motion of a new three-degree-of-freedom (DOF) system consisting of two segments that are attached together. External harmonic forces energize this system. The equations of motion (EOM) are derived utilizing Lagrangian equations, and the approximate solutions up to the third order are investigated using the methodology of multiple scales. A comparison between these solutions and numerical ones is constructed to confirm the validity of the analytic solutions. The modulation equations (ME) are acquired from the investigation of the resonance cases and the solvability conditions. The bifurcation diagrams and spectrums of Lyapunov exponent are presented to reveal the different types of the system’s motion and to represent Poincaré maps. The piezoelectric transducer is connected to the dynamical system to convert the vibrational motion into electricity; it is one of the energy harvesting devices which have various applications in our practical life like environmental and structural monitoring, medical remote sensing, military applications, and aerospace. The influences of excitation amplitude, natural frequency, coupling coefficient, damping coefficient, capacitance, and load resistance on the output voltage and power are performed graphically. The steady-state solutions and stability analysis are discussed through the resonance curves.

Applications of Elzaki decomposition method to fractional relaxation-oscillation and fractional biological population equations

Elzaki decomposition method (EDM) is adopted to deal with the fractional-order relaxation and damped oscillation equation along with time-fractional spatial diffusion biological population model in different suitable habitat situations. In accordance with the graphs for the solutions obtained, the fractional relaxation exhibits super-slow phenomenon due to its extended descent, and fractional damped oscillation is an intermediate process that explains damped oscillation dynamic systems induced by some attenuated oscillations. The biological population model of time-fractional spatial diffusion depicts a rapid rise in population density in an ecosystem in a suitable habitat that is migrating from an unfavourable zone.

Analytical stochastic framework for assessment of base- and mass-excited structures under equal input energy pertinent to multi-hazard engineering

ABSTRACT Analytical framework for probabilistic assessment of dynamical system under base- and mass- excitations with equal input energy is presented with reference to multi-hazard engineering. Closed-form solutions for equivalent undamped and damped dynamical systems under harmonic sinusoidal base- and mass- excitations are obtained. Uncertainties are considered in the excitation frequency and peak amplitude for sinusoidal base excitations, whereas for the mass excitations, the input forcing functions have variations in excitation frequency, mean speed, and fluctuating wind speeds. Dynamic response quantities (acceleration and displacement) are obtained under the suite of stochastic base and mass excitation forces. Structural failure under the independent excitations is determined as joint probability density functions and fragility surfaces. Under equal input energy from such separate dynamic excitations, dissimilar vulnerability is exhibited by the structure when exposed to two independent time-dependent excitations, base-induced earthquake and mass-induced wind forces. Moreover, the uncertainty in the input excitation parameters has significant influence on the probability of failure in the design life of structure under the non-simultaneous excitations.

The dynamical functional particle method for multi-term linear matrix equations

Recent years have seen a renewal of interest in multi-term linear matrix equations, as these have come to play a role in a number of important applications. Here, we consider the solution of such equations by means of the dynamical functional particle method, an iterative technique that relies on the numerical integration of a damped second order dynamical system. We develop a new algorithm for the solution of a large class of these equations, a class that includes, among others, all linear matrix equations with Hermitian positive definite or negative definite coefficients. In numerical experiments, our MATLAB implementation outperforms existing methods for the solution of multi-term Sylvester equations. For the Sylvester equation AX+XB=C, in particular, it can be faster and more accurate than the built-in implementation of the Bartels–Stewart algorithm, when A and B are well conditioned and have very different size.

Numerical simulation of generalised Maxwell-type viscous dampers with an efficient iterative algorithm

In this work, an extended numerical model for viscous dampers, which can be deemed as a generalised Newtonian fluid model, is proposed based on the conventional power-law fluid viscosity. With a flexible definition of damping coefficient, a wide range of damper response can be simulated using the new damper model. An efficient direct integration algorithm with an iterative scheme is proposed for the state determination of the corresponding Maxwell model that employs the proposed dashpot and spring with either elastic or elastoplastic response. Compared to an ODE solver based approach, which has limitations in terms of applicability/accuracy, compatibility/consistency and efficiency/robustness, the proposed algorithm has a higher computational performance and is less sensitive to numerical instability issues. The proposed viscous damper model and the state determination algorithm can be readily used in simulations of various damped dynamic systems.

Finite-time stability results for fractional damped dynamical systems with time delays

This paper is explored with the stability procedure for linear nonautonomous multiterm fractional damped systems involving time delay. Finite-time stability (FTS) criteria have been developed based on the extended form of Gronwall inequality. Also, the result is deduced to a linear autonomous case. Two examples of applications of stability analysis in numerical formulation are described showing the expertise of theoretical prediction.

Scalability of resonant motor-driven flapping wing propulsion systems

This work aims to develop an integrated conceptual design process to assess the scalability and performance of propulsion systems of resonant motor-driven flapping wing vehicles. The developed process allows designers to explore the interaction between electrical, mechanical and aerodynamic domains in a single transparent design environment. Wings are modelled based on a quasi-steady treatment that evaluates aerodynamics from geometry and kinematic information. System mechanics is modelled as a damped second-order dynamic system operating at resonance with nonlinear aerodynamic damping. Motors are modelled using standard equations that relate operational parameters and AC voltage input. Design scaling laws are developed using available data based on current levels of technology. The design method provides insights into the effects of changing core design variables such as the actuator size, actuator mass fraction and pitching kinematics on the overall design solution. It is shown that system efficiency achieves peak values of 30–36% at motor masses of 0.5–1 g when a constant angle of attack kinematics is employed. While sinusoidal angle of attack kinematics demands more aerodynamic and electric powers compared with the constant angle of attack case, sinusoidal angle of attack kinematics can lead to a maximum difference of around 15% in peak system efficiency.

Controllability of Higher Order Fractional Damped Delay Dynamical Systems with Time Varying Multiple Delays in Control

This paper is concerned with the controllability of higher order fractional damped delay dynamical systems with time varying multiple delays in control, which involved Caputo derivatives of any different orders. A necessary and sufficient condition for the controllability of linear fractional damped delay dynamical system is obtained by using the Grammian matrix. Sufficient conditions for controllability of the corresponding nonlinear damped delay dynamical system has established by the successive approximation technique. Examples have provided to verify the results.

Dynamical representations of constrained multicomponent nonlinear Schrödinger equations in arbitrary dimensions

We present new approaches for solving constrained multicomponent nonlinear Schrödinger equations in arbitrary dimensions. The idea is to introduce an artificial time and solve an extended damped second order dynamic system whose stationary solution is the solution to the time-independent nonlinear Schrödinger equation. Constraints are often considered by projection onto the constraint set, here we include them explicitly into the dynamical system. We show the applicability and efficiency of the methods on examples of relevance in modern physics applications.

The role of surrogate models in the development of digital twins of dynamic systems

Digital twin technology has significant promise, relevance and potential of widespread applicability in various industrial sectors such as aerospace, infrastructure and automotive. However, the adoption of this technology has been slower due to the lack of clarity for specific applications. A discrete damped dynamic system is used in this paper to explore the concept of a digital twin. As digital twins are also expected to exploit data and computational methods, there is a compelling case for the use of surrogate models in this context. Motivated by this synergy, we have explored the possibility of using surrogate models within the digital twin technology. In particular, the use of Gaussian process (GP) emulator within the digital twin technology is explored. GP has the inherent capability of addressing noisy and sparse data and hence, makes a compelling case to be used within the digital twin framework. Cases involving stiffness variation and mass variation are considered, individually and jointly, along with different levels of noise and sparsity in data. Our numerical simulation results clearly demonstrate that surrogate models, such as GP emulators, have the potential to be an effective tool for the development of digital twins. Aspects related to data quality and sampling rate are analysed. Key concepts introduced in this paper are summarised and ideas for urgent future research needs are proposed.

A Random Attractor Family of the High Order Beam Equations with White Noise

In this paper, we studied a class of damped high order Beam equation stochas-tic dynamical systems with white noise. First, the Ornstein-Uhlenbeck process is used to transform the equation into a noiseless random equation with random variables as parameters. Secondly, by estimating the solution of the equation, we can obtain the bounded random absorption set. Finally, the isomorphism mapping method and compact embedding theorem are used to obtain the system. It is progressively compact, then we can prove the existence of ran-dom attractors.

Taylor Approximation of Second Degree for Complex Mode in Asymmetrical Damped System

Algorithms for the first- and second-order sensitivities of the complex mode for an asymmetrical damped linear discrete dynamic system with respect to design parameters are presented. The first- and second-order sensitivities are used to construct the gradient matrix and the Hessian matrix of the complex mode and to establish its Taylor approximation of the second degree. The symmetrical dynamic system possesses a set of biorthonormal right modes so that the two-sided normalization method is engaged. In contrast, in an asymmetrical dynamic system, the two-sided normalization technique is not effective. In our presentation, the results are derived in terms of a complex mode so that the uniqueness of the complex mode and its sensitivities are kept. The algorithms make the calculations more precise and easier to perform. The usefulness and stability of the derived expressions are demonstrated by considering a half-car model of 7 degrees and a whirling beam of 10 degrees.

The digital twin of discrete dynamic systems: Initial approaches and future challenges

This paper employs a discrete damped dynamic system to investigate the emerging concept of a digital twin. Dynamic systems are well understood across engineering and science domains, and represent a familiar and convenient platform for exploring the various aspects of a digital twin design. The aim is to create a framework accessible to engineering sciences related to the aerospace, electrical, mechanical and computational area. The virtual model of the physical system is expressed as a differential equation in two-time scales, with the concept of a slow time being used to separate the evolution of the system properties from the instantaneous time. Cases involving stiffness variation and mass variation are considered, individually and together. It is assumed that the damped natural frequency and the time response are measured through sensors placed on the physical system. Issues of errors and reduced sampling rate in sensor measurements on the digital twin are investigated. The digital twin is expressed as an analytical solution through closed-form expressions and the effect of sensor errors is brought out through the simulations. Several key concepts introduced in this paper are summarized and ideas for urgent future research needs are proposed. The current work breaks free from the qualitative description of digital twins pervading the literature and can be used as benchmark solutions to validate digital twin of experimental dynamic systems and their implementation using sensors, the internet of things and deployment on the cloud computing systems.

Simulating longitudinal vibrations of coupled oscillator using the fourth-order Runge–Kutta method by programming spreadsheet

This paper presents a simple way of simulating the motion of two oscillators coupled via springs. A linear model using Hooke’s law describes the motion of a coupled oscillator through a set of two coupled second-order linear differential equations. A spreadsheet is programmed to obtain the numerical solutions of coupled differential equations using the fourth-order Runge–Kutta method. The main thrust behind programming the spreadsheet is to solve the differential equations for a physical system under different initial condition and to understand the motion of coupled masses graphically. The programmed spreadsheet can also be modified to include damping in the system and visualize a damped dynamical system. The algorithm developed for programming presents the use of inbuilt functions of the spreadsheet and is user-friendly, especially for those who are not familiar with the spreadsheets. Simulations of the coupled oscillator would help students to visualize the effect of variation of different parameters, such as mass, initial velocity, spring constant, and damping factor, on the motion of coupled oscillators.

Simplified conditions of initial observability for infinite-dimensional second-order damped dynamical systems

In the paper, the initial observability of abstract second-order infinite-dimensional dynamical systems is considered. Two cases of observation operator are explored. First, the observation operator is assumed to be bounded. Then results are extended to more general class of observations related to the observation operators that are assumed to be bounded with respect to square root of elasticity operator which covers the case of some unboundedness in observation. Using the frequency-domain method it is proved, that initial observability of second-order system under discussion can be verified by the initial observability conditions for the corresponding simplified first-order system. General results are then applied for initial observability investigation of infinite dimensional systems described by partial differential equations modelling flexible mechanical structures. Some special cases are also discussed and practical comments and remarks are given. The paper extends earlier results on initial observability for a class of infinite-dimensional second-order abstract dynamical systems.

Controllability of fractional order damped dynamical systems with distributed delays

This paper deals with the controllability criteria for fractional-order damped dynamical systems with distributed delays using Caputo derivatives for both linear and nonlinear cases. Controllability results are established by utilizing the Mittag-Leffler function (MLF) and Schauder’s fixed point theorem. Finally, two numerical examples are provided to show applicability of the proposed results.

Controllability of higher order fractional damped delay dynamical systems

The article isconcerned the controllability results for higher order fractional damped delay dynamical systems of any different orders of Caputo derivatives. To establish a necessary and sufficient conditions for the controllability criteria of linear fractional damped delay dynamical systems by Grammian matrix. Sufficient conditions for controllability of the corresponding nonlinear damped delay dynamical systems areobtainedusing successive approximation technique. Examples are includedto verify theoretical results.

Strategies for Reduced-Order Models for Predicting the Statistical Responses and Uncertainty Quantification in Complex Turbulent Dynamical Systems

Turbulent dynamical systems characterized by both a high-dimensional phase space and a large number of instabilities are ubiquitous among many complex systems in science and engineering. The existence of a strange attractor in the turbulent systems containing a large number of positive Lyapunov exponents results in a rapid growth of small uncertainties, requiring naturally a probabilistic characterization for the evolution of the turbulent system. Uncertainty quantification in turbulent dynamical systems is a grand challenge where the goal is to obtain statistical estimates such as the change in mean and variance for key physical quantities in their nonlinear responses to changes in external forcing parameters or uncertain initial data. One central issue in contemporary research is the development of a systematic methodology that can recover the crucial features of the natural system in statistical equilibrium (model fidelity) and improve the imperfect model prediction skill in response to various external perturbations (model sensitivity). A general mathematical framework to construct statistically accurate reduced-order models that have skill in capturing the statistical variability in the principal directions with largest energy of a general class of damped and forced complex turbulent dynamical systems is discussed here. The methods are developed under a universal class of turbulent dynamical systems with quadratic nonlinearity that is representative in many applications in applied mathematics and engineering. The validity of general framework of reduced-order models is demonstrated on instructive stochastic triad models. Recent applications to two-layer baroclinic turbulence in the atmosphere and ocean with combinations of turbulent jets and vortices are also surveyed.

The controllability of nonlinear fractional damped dynamical systems with control delay

The controllability of nonlinear fractional damped dynamical systems with control delay